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0.15). In the case of the double increase of damping the zone will not occur for J.L ~ 0.3. The magnitude and position of the instability zones for PI are not sensitive to the changes of the damping coefficient. In the case of J.LH I = 10s- 1 the proper zone of the frequency PI exists for J.L > 0.034. When J.LHI = 20s- 1 damping is doubled, and the lower border of the occurrence of the zone is shifted to the value of J.L = 0.07. The parametric instability zones for P3 are presented in Fig. 2.12. The magnitude of the zones depends on the initial conditions of the system motion. The diagrams have been prepared on the assumption that a3 = b3 = 0.01 m, where a3 = Y3(0), b3 = Y3(0). The calculations, in the case of the resonance coordinate Y:3, have been performed with with a precision of up to second order, hence the inclination of the unstability zones in the direction of the growing values of the parameter ..x~ has appeared. For the first approximation, the zones remain symmetrical in relation to the straight line ..x~ = 1. As in the cases considered above, the increase of the unbalance considerably expands the instability zone. The changes of the value of the quotient 0:/ {3 and damping have a negligible influence on the magnitude of the zone. Figure 2.13 presents the parametric instability zones for various
n?
w?
76
2. Discrete Systems
a)
0.7 0
b)
0.1
0.2
0.3
1.3 - r - - - - - - - - - - - - - - - - - - - - - , H1=0, /-LP=21·1Q-4 m .x 21 HI =0, /-Lp=7·1Q-4 m 1.2 HI=P=O
1.1 1.0
0.9 0.8 0.7+------.....--------,r---------l
o
c)
0.1
0.2
0.3
1.3,...--------------------,
.x~ 1.2
0.8 0.7+-------.--------,.--------l
o
0.1
Fig. 2.10a-c. Instability zones for PI
0.2
0.3
2.6 Parametric and Self-Excited Oscillation
a)
1.02 rHl=O, ttp=7·1Q-4 m
,\~
rHl=O, ttP=3.5·1Q-4 m
1.01
r
HI - P - O
c.
1.00
'-P:
O. ttHI 0.05s- 1
0.99 I
0.98
b)
o
0://J=0.5m2s- 2
1
0.2
0.1
J-L
0.3
1.02
,\~
rHl=O, ttP=14.1Q-4 m
1.01
rHl=O, ttP=7·10-4 m
r Hl =P=O ~
1.00
""'C.p 0, ttHI 0.05 S-1
0.99
10://J= 0.65 m 2s- 2 1 0.98
c)
o
0.2
0.1
J-L
0.3
1.02 ,\2
rHl=O, ttP=21.1Q-4 m
2
1.01
rHl=O, ttP=7·1Q-4 m , H1
1.00
p 0 ~ '-p 0, ttH} 0.05s- 1
0.99
0.98
2 2 I 0://J= 1 m s- ]
o
0.1
Fig. 2.11a-c. Instability zones for P2
0.2
J-L
0.3
77
78
2. Discrete Systems
2.0.......----------------~
\2 "'3
ttP=ttQ=21.1Q-4 m ttP=ttQ =7.10- 4 m
1.5
l.0t========:::::::::::=;;;;~
0.1
0.2
J-L
0.3
Fig. 2.12. Instability zones for 2 2 P3 (0./(3 = 0.5m s- , 0./(3 = 0.65m2 s- 2 , 0./(3 = 1 m 2 s- 2 )
values of the parameters {}2, {}? and w? For the zones denoted by 1 we get {}2 = 14400s- 2 , {}? = 1920s- 2 , w? = 19200s- 2 ; for the zones denoted by 2 we have {}2 = 3600s- 2 , {}? = 480s- 2 , w? = 4800s- 2 , and for the zones denoted by 3: {}2 = 900s- 2 , {}? = 120s- 2 ,"w? = 1200s- 2 . In all the cases the magnitudes of the other parameters are as follows: J-LHI = lOs-I, e = 0.2, o/f3 = 0.5m2 s- 2 , VQ = 0.5ms- 1 , J-LP = 0.0015m. As shown in Fig. 2.13a,b the growth of the squares of frequencies {}2, {}?, w? (resulting from the increase in rigidity of the elastic elements in the system, or from the decrease in the values of the masses) causes the instability zones for PI and P2 to expand. For example, when the parameters {}2, {}?, and w? increase by four times, it brings about an approximately doubled expansion of the zones. The unstability zone for pa is not influenced by the frequency changes in the system (Fig. 2.13c). Figure 2.14a, b presents the influence of the velocity changes of the belt VQ on the magnitude of the instability zones for PI and P2. Calculations have been performed for the data denoted by 1, except for the velocity VQ, whose value has been changed. In each case the increase in the belt velocity causes the expansion of the instability zones. For VQ < 0.3 JIlS- 1 these changes are less evident. The influence of the velocity changes VQ on the parametric instability zone for Pa is practically negligible. We can summarize the obtained results as follows. (1) The method of seeking a solution as a power series of the two perturbation parameters J-L and e used in the considerations makes it possible to
2.6 Parametric and Self-Excited Oscillation
a)
79
1.5 1
2
c
1.0
3
0.5J------..---------,..----------l o 0.1 0.2 {L 0.3
b)
1.5
\2 ""2
1 2
0.5
c)
3C
C
1.0
o
0.1
0.2
0.3
1.5
1.0
0.5+--------..---------,..-------~
o
0.1
0.2
{L
0.3
Fig. 2.13. Influence of parameter changes on the instability zones for: (a) Pl; (b) P2; (c) P3
80
2. Discrete Systems
investigate the single resonances of any order for the systems with weak nonlinearity and weakly modulated systems (J.L « 1). When we perform calculations with a p,recision of up to second order, it turns out that the limits of instability zones incline in the direction of the growing values of the parameter >.~ (Fig. 1.7). For the first approximation, the limits remain symmetrical in relation to the straight line >.~ = 1. (2) The parametric instability zones for Pll and P2 expand with the increase in the rotor unbalance. Depending on the value of the quotient alp, this tendency has different intensity. In the case of a/ (3 = 0.5 m 2 s-2, the double increase in the unbalance has brought about a considerable expansion of the unstability zones, for PI as well as for P2. For a/(3 = 1 m 2 s-2 the unbalance, causes a rather small expansion of the instability zones for PI, while for P2 the expansion is still almost doubled. In the case of a lack
a)
1.5
r
tb=O.4ms- I
; ; '\)=0.35ms- 1 I Vo 0.25ms-
r
"- tb-0.15ms-I
1.0
! \
0.5
b)
0
0.1
1.5
\2
A
2
c
1.0
0.5
o
c:
0.1
0.2
0.3
,;;=r
'\)=0.4 ms-
/r
vo=0.25ms- I
~
I
tb= 0.35 ms- I
tb°.15ms-I
0.2
Fig. 2.14. Influence of the belt velocity changes PI; (b) 112
J.L Vo
0.3
on the insta.bility zones for: (a)
2.7 Modified Poincare Method
81
of rotor unbalance the changes of the quotient a/{3 do not influence the magnitude of the parametric instability zones. The influence of damping on the magnitude of the instability zones corresponding to PI and P2 is also very different. The minimum damping (J.LHI = 0.05s- 1 ) causes a considerable shift of the zone for P2 in the direction of the growing values of the modulat,ion depth (J.LHI = 0.15s- 1 ). The magnitude and position of the instability zones for PI are not so sensitive to the damping coefficient changes. The regularities indicated here are the more clear, the greater the difference between the values of the frequency PI and P2 (i.e. for PI » P2) is. The increase in the unbalance also produces a considerable expansion of the instability zone for P3; however the changes of the parameter and of damping have no essential influence on the magnitude of the zone. The growth of the frequency squares n2 , n? and w~ causes the expansion of the instability zones for PI and P2. The parametric instability zone for the frequency P3 is not sensitive to the frequency changes in the system. In the case of the belt velocity increase (vo), the instability zones for PI and P2 are expanded. This property is noticeable within the range of great velocities (vo > 0.4ms- 1 ). The influence of the velocity changes Vo on the parametric instability zone for P3 is practically negligible. (3) For the frequencies PI and P2, the position of the instability zone limits does not depend in the first approximation on the initial conditions of the system motion. The magnitude of the instability zone limits for P3 depends on these conditions.
2.7 Modified Poincare Method 2.7.1 One-Degree-of-Freedom System Consider a one-degree-of-freedom nonlinear system governed by the equation [33, 117]
ddt + 2
y
2
W
2
Y = eQ
(d y, dt ,e
Y )
.
(2.7.1)
The above e > 0 is a small perturbation parameter and the function Q is analytical with regard to its arguments y, dy/dt and e for 0 < e < co. We are going to find a periodic solution to (2.7.1) depending on the perturbation parameter e. The system under consideration is autonomous, therefore we can arbitrarily take
dY(O) = 0 (2.7.2) dt ' because starting with the initial conditions for to, (2.7.1) will not change with a shift of time.
82
2. Discrete Systems
For c = 0 we have d2 y _ + w2 Yo - 0, Yo(O) = 0, dt and a solution to (2.7.3) is given by -2
(2.7.3)
yo(t) = A o cos wt,
(2.7.4)
where the amplitude A o is not yet defined. However, contrary to the previous investigations and following Proskuriakov [65d] , we are going to find an invariant orbit depending also on the second parameter b = b(c), where b(O) = O. Therefore, we have
y(O) = A o + b(c).
(2.7.5)
On the basis of the assumption of analycity of F, this function will be analytical also with respect to A o + b. We are focused on finding a periodic solution
y(t) = y(t + T), y(t) = y(t + T),
(2.7.6) (2.7.7)
and the period T is defined as T = To
+ a(c),
(2.7.8)
where
To = 21rw- 1 , a(O) = O.
(2.7.9) (2.7.10)
From (2.7.6) and (2.7.7) we get
y(To + a, A o + b, c) = y(O, A o + b, c) = A o + b, y(To + a, A o + b, c) = y(O, A o + b, c) = O.
(2.7.11) (2.7.12)
A general solution form of (2.7.1) is a function of the two parameters b and c: K
y(t, Ao + b, c) = (A o + b) coswt
+ LYk(t, A o + b)c k,
(2.7.13)
k=l
and K
y(t, Ao + b, c) = -w(A o + b) sin wt + LYk(t, A o + b)c k .
(2.7.14)
k=l
From (2.7.13) for t
Yk(O, Ao + b) = 0, Yk(O, Ao + b) = O.
=
0 and taking into account (2.7.2) and (2.7.5) we get (2.7.15) (2.7.16)
2.7 Modified Poincare Method
83
We develop the right-hand side of (2.7.1) into a power series of e in the neighbourhood of e = 0 putting fJ = 0 into series (2.7.13) and (2.7.14). We obtain (8 Q dy 8Q dy ad + F de e=O=O y e e=o=O y e=o=O 2 y + 8Q ) + e2(~ 8 Q (d )2 8e e=o=O 2 8 y2 de e=O=O 2 2Q +! 8 Q2 Y) + ~ (8 2 ) 2 8y de e=o=O 2 8e e=O=O 2 2 8 Q dy dy 8 Q dy 8 2Q dy +---+ --+ --8y8y de de e=o=O 8y8e de e=O=O 8i/8e de e=o=O 2 2 y 8Q d 8Q d y ) + O( e + - -2 +-2 8y de e=o=O 8y de e=O=O
. {. eQ(y,y,e) = e Q(yo,yo,O)
+e
(d
3)}
= e{ Q{yo,Vo)o +e (( ~~). YI + (~~)o VI + (~~)J
+e
2
[~ (~:~) °y? + ~ (~:~) °V? + ~ (~:~)
2 82Q ) . ( 8 Q) + ( 8y8i/ Yl Yl + 8y8e
2Q ( 8 ) 0 Yl + 8fj8e
+ (~~) °Yl + (~~) °V2] + 0{e
3 )}.
0
0
. Yl (2.7.17)
Now we demonstrate that using the periodicity conditions we are able to solve the problem, Le. to find A o, o:(e), and then fJ(e). To show this, let us begin with (2.7.12). From (2.7.14) and (2.7.2) for T = To + 0: we obtain y(To +
0:,
A o + fJ,e) = -w(A o + fJ) sinw(To +
0:)
(2.7.18)
K
+ LYk(To + 0:, Ao + fJ)e k = 0, k=l
which leads to the equation !
T/c) the real terms of the eigenvalues become positive. The Hopf conditions ensure that the intersection of the imaginary axis with the complex conjugate eigenvalues occurs with non-zero velocity. A method for searching for bifurcation solutions was presented in [91,98]. Nevertheless, it is troublesome to use because of the time-consuming calculations for equations of dimension bigger than two. As the Hopf bifurcation occurs in nonlinear systems, it would be desirable to use for their analysis the analytical methods widely known in the field of nonlinear vibrations. The most popular and effective are the methods of harmonic balancing and the perturbation method. The main defect of the harmonic balancing method, which makes it useless for Hopf bifurcation analysis, is the necessity of a priori knowledge about the solution. An adv..antage of the perturbation method lies in constructing the solution by the subsequent solving of the perturbation equations of the linear differential equations, when it is only necessary to know the solutions of the undisturbed differential equation system. The combination of the two methods makes it possible to solve the Hopf problem (the method of harmonic balancing is used to solve each of the perturbation equations of the linear, differential equations). We consider the system of differential equations, whose characteristic equation is of the form (u - ut)(u - (2)P(U) = 0,
(2.8.1)
where
Ul.2 = e(T/) ± iW(T/),
w(T/c)
i= 0,
e(T/c) = 0,
°
a~c) i= 0,
and the roots of the polynomial P (u) = have negative real parts. It results from the centre manifold theorem [88] that the critical subsystem is mainly responsible for the bifurcation and bifurcated solution, and for the qualitative assesment of the bifurcated solution it is possible to limit oneself only to the solution of the two-dimensional critical differential equation. Here this solution serves as the initial approximate solution of the full nonlinear differential equation system, and the "detailed'· solution is determined by
2.8 Hopf Bifurcation
95
the method of slllccessive approximations. The latter also makes it possible to solve the problem where there are nonanalytical nonlinearities. Let us consider the differential equation system having the form
d dt (x) = F(l1, x),
n
x E IR
(2.8.2)
,
where T/ is the parameter vector and F (T/, x) is a nonlinear function, analytical in the state variables T/ and x. For the purpose of the further analysis it has been assumed that T/ is a one-dimensional bifurcation parameter. Let XQ fulfil the equation F(T/, XQ) =
o.
(2.8.3)
Examination of the stability of the equilibrium pathxQ is known to be limited to the determination of the eigenvalues of the Jacobian Fx(T/, xQ), where Fx(T/,xQ)
(~:~)
=
]
, (i,j = 1, ... ,n).
(2.8.4)
x=xo
Let the equilibrium path XQ for T/ < T/e (the critical value of the parameter) be the stable solution of system (2.8.2). On the other hand, when T/ = T/e the two complex conjugate eigenvalues cross the imaginary axis with non zero velocity, Le. let 0"1
= e(T/) + iW(T/),
w(T/e) = We
i= 0,
(2.8.5)
For T/ > T/e, the real parts 0"1, 0"2 become positive. A family of periodic solutions is created at the critical point. Let us assume that equation system (2.8.2) can be presented in the form .
Ku(T/)u
+ Kv(T/)v + K(T/,u,v),
U
=
iJ
= Su(T/)U + Sv(T/)V + S(T/,u,v),
2
IR , v E IRn - 2 , U
E
(2.8.6)
where x = colon(u,v), and the matrices KC*)(T/), and SC*)(T/) are the linear parts of the expansion of F(T/, x) into the Taylor series in the equilibrium path XQ. Let the characteristic equation (2.8.6) have the form of (2.8.1), while p ((j) = 0 has roots with negative real parts, and 0"1 and 0"2 are the eigenvalues of the two-dimensional matrix Ku(T/). The matrix K u, known also as the critical matrix, decides about the Hopf bifurcation. From the centre manifold theorem it follows that in the neighbourhood of the equilibrium path XQ = 0 there exists a function v = f(u) which in a sufficiently close neighbourhood XQ = 0 has the property 8f/8u = O. This allows us to assume to a first approximation that
v = f(u) =
o.
(2.8.7)
96
2. Discrete Systems
Taking into account (2.8.7) in (2.8.6), we obtain the following .
U
= Ku(Tf)u
+ K(Tf,u,O),
U
E
2
lR .
(2.8.8)
Later we shall assume that K(Tf, u,O) = K(Tf, u). Let us develop the matrix K(Tf) into a Taylor series in the neighbourhood of the critical point Ku(Tf)
= Ku(Tfc) +
1 2 K U17 (Tf - Tfc) + "2KUT1T1(Tf - Tfc) + ... ,
(2.8.9)
where etc. Let u(Tf) = 0 be the solution of (2.8.8) for 7J < Tfc, and for Tf = Tfc the periodic solution u(t; e) = u(t + T; e) of the period T bifurcates, which is dependent on one formally assumed small parameter e connected with the amplitude. After the transformations this parameter can be arbitrarily assumed to be e = 1. In order to obtain the period T = 211", we shall introduce the dimensionless time r = wt and, as a result, we will obtain the following expression from (2.8.8): d W(e) dt u(r;e) = K(Tf)u(r;e) + K(Tf, u(r;e)).
(2.8.10)
The periodic solution u( r; e) will be sou~ht in the form of a certain Fourier series, where the amplitudes and the freqttencies depend on the parameter e: K
ui(rj e) = 2:)Pik(e) cos kr
+ Tik(e) sin kr).
(2.8.11)
k=O
Because the system (2.8.10) is autonomous, then Tl1(e) = O. Moreover, pik(e), Tik(e), Tf(e) and W(e) are developed into a power series of the parameter e of the following form .()_ ' +1,,2 P,k e - Piko +Pike "2Pike
.()_ T,k e - Tiko Tf (e") -_ Tfc
W(e) =
We
+ ... ,
+' 1"2 + ... , Tik e + "2Tike
+' Tf e + "21 Tf " e 2 +
(2.8.12)
... ,
1 + W'e + "2w"e2 + ... ,
where Piko = Tiko = 0, because ui(rj 0) = 0 at the critical point. The solution of ui(rj e) is also sought in the form of a power series ui(rje) =
u~(r)e) + ~u~'(r)e2 + ... ,
(2.8.13)
2.8 Hopf Bifurcation
97
where K
up (r) = 2:)p~~ cos kr + T~J sin kr).
(2.8.14)
k=O
We can now proceed in two ways. We can either introduce relations (2.8.11)-(2.8.14) into (2.8.10) and by comparing terms in the same power c obtain the perturbation equations of the linear differential equations, or obtain these equations by means of successive differentiation of (2.8.10) with respect to c. As an example, let us consider the mechanical system with 1 ~ degrees of freedom, presented in Fig. 2.15. The vibration equations of the system have the form
fil Ultt = -k l Ul - k(Ul - ua)a + (au~ CUat = -kaua - k(ua - ut}a.
1J)Ult,
(2.8.15)
c Fig. 2.15. Mechanical system with 1 ~ degrees of freedom
The bifurcation parameter 1J is related to damping, and the other coefficients in (2.8.15) are positive. After applying the first step of the perturbations, we obtain
1 1 a 1 -3- aul + -1JUl! fil fil fil = -k l Ul - k(Ul - ua)a, ka k a = --Ua - -(ua - ut} . c c
WUl T = -U2 WU2T WUaT
(2.8.16)
98
2. Discrete Systems
The roots of the characteristic equations (2.8.16) are 0"12 ,
~2 (..2L.. ± (..2L..) 2 _ ml ml
=
4k l ), ml
k3
(2.8.17)
= --.
0"3
c
= TIc = 0 we have
For TI
±iJ
l (2.8.18) k . • ml The first two equations of (2.8.16), after the assumption of U3 = 0, have the form 1 1 a 3 WUl r = -TlUl + - U 2 - -3-Ul' ml ml ml (2.8.19) WU2r = -k l Ul - ku~, 0 branch. In the second case the phase difference increases monotonically, running through the sequential values nrr /2 (n = 0, 1, ... ) at the extremal times. These two types of oscillatory regime with oscillating and monotonically increasing phase differences will respectively be called modulations of the first and second type. The solution of (2.10.19) has the form
± Irol J( ~
1
Fl2 (e) - F22 (e)) -1/2 de
= T2 -
T20,
eo = e(T20).
(2.10.22)
~o
Suppose el,'" ,e4 are the roots of the fourth-degree polynomial Fr(e)arranged in increasing order, with 6 and 6 lying inside the domain
Fi (e),
112
2. Discrete Systems
bounded by the parabolas ±Fl(e). The modulation semiperiod (for the oscillating phase case) corresponds to e varying over the interval (6, e3), and so the modulation period is equal to {3
T· = IFJ _ 2Frll/2
f ((e - 6)(e - 6)(e - 6)(e -
e4))-l/2 de· (2.10.23)
{2
For "non-coarse" systems one must consider singular cases for the position of the F 2 (e) curve (Figs 2.19b and c): the passage of F 2 (e) through the points e = 0 or e = 1 (Fig. 2.19b), and "external" touching of the parabolas F l (e) and F 2 (e) (corresponding to lines 1 and 2 in Fig. 2.19c). In these cases two of the roots ej coincide: in the first case 6 = 6 = 0 or 6 = e4 = 1, and in cases 2 and 3 6 = 6. Because the improper integral in (2.10.23) diverges when two of the roots ej coincide, the modulation period tends to infinity as these regimes are approached. These are "boundary" regimes separating the modulations of the two types distinguished above (lines 1 and 3) and associated with separatrices in the plane (e, 1), or regimes of stationary oscillations without modulation (curve 2). We remark that the" aperiodic" oscillations described in [ld] correspond to these boundary regimes. Consider the possible AP-portraits in the plane (e,1) which are given by integral (2.10.18) and which graphically describe the oscillatory modes of the system. Stationary points corresponding to oscillations with no modulation are found using (2.10.15) from the system of equations
.
I \
eo - e) sin 21 = 0, eFl
+ F2 + F2(1/2 -
e) cos 21 = 0,
which can have the solutions 2F2 = 0, cos 21 = - - , Fo
e
e= 1
1,
cos 21 =
= ± ~1r (n =
(2.10.25)
2(F~ + F2) , Fo
0, 1,2, ...),
(2.10.24)
e= e: = ±~/~ ~oF2.
(2.10.26)
(2.10.27)
These solutions exist when the following conditions are satisfied: (1) 12F21 < IFol, (3) 0 ~ e-:- < 1,
(2) 21Fl + F21 < IFol, (4) 0 < e: ~ 1.
(2.10.28)
Using the periodicity with respect to 1, we confine ourselves to the plane rectangle (0 < e < 1; 0 < 1 < 1r). The stationary points (2.10.25)-(2.10.27) can be positioned on the boundary lines of this rectangle and on the mid-line 1 = 1r /2 (with not more than one point on aline). It is easiest to investigate the nature of a stationary point with the help of (2.10.18), considering the
2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom
113
form of the integral curves in the neighbourhood of the stationary point. The stationary points at ( = 0 and = 1 are saddle points and therefore unstable. From this it follows that the presence of a second degree of freedom makes oscillations along the first generalized coordinate unstable if the stationary points (2.10.25) or (2.10.26) exist. In the neighbourhoods of the stationary points on the lines 1 = ±n1l" /2 the trajectories can be of either elliptic or hyperbolic type, and consequently these stationary points can be stable or unstable. The stability conditions for odd and even n, respectively, have the forms
e
(5) Fo(Fo + FI) > 0,
(6) Fo(Fo - FI) < O.
(2.10.29)
On the characteristic graph (Fig. 2.19c) the "externally" touching hyperbolas (curve 1) correspond to the stable stationary points and the "internally" touching ones (curve 2) to the unstable points. The stationary points on the lines 1 = ±n1l" /2 correspond to synchronous single-frequency modes, Le. normal oscillations of the nonlinear system. It follows from (2.10.3) and (2.10.8) that the points on the lines 1 = 0 and 1 = 11" correspond in the (U1, U2) configuration space to the two straight lines U2 = ±hU1' where
h
=
/ a2 = (1_e;)1/2 = (Fo/2 - F1 - F2)1 2 a1 Fo/2 + F2
e;
The stationary points on the lines 1 = 11"/2 and 311" /2 correspond to the ellipses
u2
1
et
+
u2 2 1- et
= Eg2,
e
e
and the points on the lines = 0 and = 1 correspond to straight lines along the axes OU2 and Ou 1. The separatrices pass through the possible unstable stationary points. For separatrices passing through the "left" points (2.10.25) one should put C = 0 in (2.10.18). We obtain equations for two branches: (2) cos 21 = -
eF1 + 2F2 no(1- ),
e
(2.10.30)
which exist when condition 1 of (2.10.28) is satisfied. The "right" separatrix, passing through the stationary points (2.10.26), exists when condition 2 is satisfied. The equations of the branches of this separatrix are obtained from (2.10.18) with C = F 1 + 2F2 (1)
e=
1,
(2) cos 21 =
(e + 1~~ + 2F2 .
(2.10.31)
The central separatrix (CS) passing through the stationary points (2.10.27) for odd (or even) n exists when condition 3 (condition 4) is satisfied and condition 5 (condition 6) is violated. Substituting the coordinates of points
114
2. Discrete Systems
(2.10.27) into (2.10.18), we obtain C = (-F2 ± Fo/2)2 /C~Fo - Fd and the equations for the branches of the central separatrix
e= B± VB2_D, Fo cos 21' + F2 - Fo cos 21' - Fl'
B -
D -_
(F2 =F Fo/2)2 . (Fl ± Fo)(Fl - Fo cos 21')
(2.10.32)
The stationary points and separatrices possess the following properties. 1. If the "left" stationary points (2.10.25) exist (i.e. condition 1 is satisfied), then in the rectangle (0 < ~ < 1, 0 ~ l' < 'IT) there is at least one "intermediate" stationary point (2.10.27) on the line l' = 1r /2 or l' = 0, and this point
is stable. Indeed, when condition 1 is satisfied the sign of the numerators in condition 3, 4 is given by the sign of their first term, and for their moduli we have I ± Fo/2 - F21 < IFol. If the signs of Fl and Fo are the same, then the sign of the denominator in condition 3 is the same as the sign of the numerator, and because we then have IFl + Fol > IFol, condition 3 is satisfied and, clearly, condition 5. In the case of opposite signs for Fl and Fo, the signs of the numerator and denominator in condition 4 are the same and IFl - Fal > IFo!' so that conditions 4 and 6 are satisfied. A similar assertion holds for the "right" stationary points (2.10.26).
2. If one stationary unstable point (2.10.27) exists on the line l' = 1r /2 (or l' = 0), then a stable stationary point exists on the line l' = 0 (or l' = 1r /2); and there are no separatrices (2.10.30) ati.d (2.10.31). Suppose condition 3 is satisfied and condition 5 is not satisfied (Le. the stationary point at l' = 1r /2 is unstable). Then Fo and Fl have opposite signs and IFll > IFol. It follows from condition 3 (because the sign of the denominator is governed by the sign of F 1 and is opposite to the sign of Fo) that the signs of Fo and F2 are the same and IF21 > IFl l/2. Condition 1 is therefore violated. Considering the case F l > 0 and F l < 0 separately, and taking into account that the sign of F 1 is opposite to the signs of Fo and F2 and that IFll > IFol, we find that in both cases condition 2 is violated, and the (right) inequality in condition 4 is also violated, which proves the assertion. These properties enable us to describe the various possible AP-portraits in the 'plane (e,1'). Each side separatrix (SS) joins two unstable stationary points at = 0 or ~ = 1. The branches of these separatrices surround a single stable stationary point at l' = 1r /2 or l' = 0 (0 < e< 1). One can verify that if, for example, between the "left" separatrices there is a point on the line l' = 0, then the abscissa of the point of intersection of the separatrix with the line l' = 0 is twice the abscissa of the stationary point it is obvious that ~ < 1/2. A similar property is satisfied by the right separatrix: here it is necessary for the stationary point surrounded by its branches to be in
e
e;
2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom
115
e
the right half of the rectangle. The separatrix originating from = 0 cannot intersect the line ~ = 1, and conversely. The branches of the CS join the two unstable stationary points (2.10.27), corresponding to even or odd values of n, and surrounding the stable stationary points. The CS cannot intersect the lines = 0 or = 1. Inside the domains surrounded by the SS or CS a modulation regime of the first type exists, and outside these domains, there is a regime of the second type. Thus, four qualitatively different types of the AP-portrait, governed by conditions 1-6, are possible and they are shown in Fig. 2.20
e
1 0
a)
b)
e
1 0
c)
1
d)
Fig. 2.20a-d. Four qualitatively different types of the AP-portrait
1. Conditions 1 and 2 are satisfied. There are stable stationary points at "y = n1l" /2 (2.10.27) for even and odd n, in the left section (e < 1/2) and right section (e > 1/2) of the rectangle, Le. three stable normal modes exist (and two trivial unstable ones Uk = 0, k = 1,2). Each of the stationary points is surrounded by the corresponding SS; there are no CS (Fig. 2.20a). 2. Only one of conditions 1 and 2 is satisfied. There is a stable stationary point (2.10.27) only for an odd or even n, and only one SS (on the left if condition 1 is satisfied, and on the right if condition 2 is satisfied); there are no CS (Fig. 2.20b). All the normal modes, apart from the single Uk = 0, k = 1 or k = 2 mode, exist and stable modes also exist that are either rectilinear (if condition 4 is satisfied), or elliptic (when condition 3 is satisfied). 3. Neither condition 1 nor 2 is satisfied, but condition 3 is satisfied. The stable and unstable stationary points (2.10.27) alternate (with the point for odd n being stable if condition 5 is satisfied). There is a CS, but no SS (Fig. 2.20c). Three normal modes exist, where either the rectilinear one is stable (when conditior.. 6 is satisfied), or the elliptic one is stable (when condition 5 holds). 4. Conditions 1-3 are not satisfied. There are no stationary points (normal modes) or separatrices. All oscillatory modes are of modulation type 2, with the modulation being relatively small if compared with cases 1-3 (Fig. 2.20d).
116
2. Discrete Systems
In cases 1-3 one can distinguish subcases. In. case 1 there are two subcases distinguished by the position of the left stationary point: on the line '"Y = 0 or on '"Y = 11"/2. Similarly, in case 3 the stationary point can be stable at '"Y = 0 or at '"Y = 11"/2. Four subcases are possible for case 2: a left or right separatrix, and a stationary point at '"Y = 0 or '"Y = 11"/2. The corresponding AP-portraits can be obtained from those shown in Fig. 2.20. We introduce the parameters
°
bu
b22 40' 02 = b12' 0' = b 2E ' 1 Then, conditions 1-6 can be represented in the form 01
= b12'
(1) (2) (3) (4) (5) (6)
302 - 3 < 0'0 - 301 + 1 < 0'0 - 301 + 1 < 0'0 30 2 - 1 < 0'0 302 - 3 < 0'0 - 301 + 3 < 0'0 01 + 02 > ~, 01+02 ~, 02 < 3' 02 < 2, 02 > 2,
(2.10.33)
(2.10.34)
Unlike 0' and E, the dimensionless frequency detuning parameter 0'0 does not depend on the choice of c and can be written in the following form 0'
°
-
40'. --"....------,,,..--
- bI2(U~(0)
+ u~(O))'
(0'.
= c 20' = w22 -
2)
WI .
(2.10.35)
As can be seen from (2.10.34), the ty~e of AP-portrait is determined by the relative positions of the points Cl = 302 - 3, C2 = 302 - 1, d1 = -301 + 1, d2 = -301 + 3
(2.10.36)
and the quantity 0'0. Four possible positions of the intervals (Cl' C2) and (dl,d 2) are shown in Fig. 2.21 (C2 < d 1,Cl < d 1 < C2,Cl < d 2 < c2,d2 < C2)' The type of the AP-portrait (easily determined from (2.10.34)) is shown above the intervals. In case (a) the interval (C2' dd contains the stable stationary point at '"Y = 0 (11"), Le. the rectilinear normal mode is stable, and the unstable one is at '"Y = 11"/2 (311"/2) (i.e. elliptic). In case (d) these points (and normal oscillations) "exchange" stability. Figure 2.21 graphically demonstrates the influence of the parameter 0'0 on the'system behaviour. If 0'0 lies in the interval
01
we expa~d the logarithm in (2.11.50) to identify all terms proportional to e1wt and e 21wt • Such terms oscillate at the frequency wand thus give rise to the seculiar behavior in Yl . The coefficient of e iwt is -2iwA'(T)
+ (1
- w 2 )A(T) In(2IAI 2 )
1- w ~ 1 4 + 2 A(7") 2k + 1 2
6
k [
2
_
1 -2 w A(T)
2k + 1 ] k+1 .
~ ~4-1 [
2; ]
(2.11.51)
2.11 Nontraditional Asymptotic Approaches
127
Evaluating the sums in (2.11.51) gives
-2iwA'(r)
w 2 )A(r) In(2IAI 2 )
+ (1 -
- (1 - w 2 )A(r) In 2 + (1 - w2 )A(r).
(2.11.52)
Thus, the condition that there is no secular behaviour in Y1(t, r) is that the expression in (2.11.52) (as well as its complex conjugate) vanishes:
- 2iwA'(r)
+ (1 -
w 2 )A(r)[1 -In(IA2 1)]
= O.
(2.11.53)
To solve (2.11.53) we let
A( r) = R( r )ei9 ('T"),
(2.11.54)
substitute (2.11.54) into (2.11.53), and decompose the result into its real and imaginary parts: R'(r) = 0, (J' ( r)
=
w 2 -1
2w
(1
+ 2ln R).
(2.11.55)
Hence, R(r) is a constant R(r) =
and
(J ( r)
Flo,
(2.11.56)
is a linear function of r,
(J(r) =
w 2 -1
2w
(1
+ 2ln Flo)r + (Jo.
(2.11.57)
The initial conditions in (2.11.46) imply that Flo our final result for To(t, r) is
¥Ott, r)
= cos [wt + r
w2 2W
= 1/2 and
1 (1- 21n 2)] .
(Jo
= 0,
thus
(2.11.58)
Finaliy, we eliminate r in favour of 8t to obtain the MSA result TMSA =
w - 6w
2: ~1r, (21n 2 - 1)
(2.11.59)
:1
which we expand to order 6:
TMSA=~ [1+6w~~1(2ln2_1)]
+0(6 2 ).
(2.11.60)
To our surprise, (2.11.60) agrees exactly with the order-6 result we obtained in (2.11.44) using the 6-perturbation method at a quarter-period. It is a long but routine calculation to carry out the 6-perturbation series to order 62 . Using the quarter-period method we find that at 6 = 1,
T =
~ [2'-+ 0.5238 w~~ 1 + 0.6041 (w~~ 1)'] .
(2.11.61)
128
2. Discrete Systems
Table 2.1. Comparison of the exact value of the period of the anharmonic oscillator with the period calculated from the order-8 quarter-period method (same 88 MSA) and the order-8 2 quarter-period method f:
1 3 8
W
J2 2 3
T( exact) 4.76802 3.52114 2.41289
T( order 82 ) 4.73488 3.50794 2.40871
T(order 8) 4.87195 3.59669 2.45397
In Table 2.1 we compare three results: the exact numerical calculation of the period T; the order-8 quarter-period calculation, which is the same as the order-8 MSA result in (2.11.60); and the order-8 2 quarter-period calculation in (2.11.61). We set 8 = 1 and look at three values of c = w 2 -1. As expected, the MSA and order-8 results are excellent, having an accuracy of about 2%. The order-8 2 results are even better, having a relative error of less than 0.5%.
2.11.3 Asymptotic Solutions for Nonlinear Systems with High Degrees of Nonlinearity As an example, one can consider the equation
x.. + xn= 0,
n = 2k
+ 1,
k = 1,2, ... ,
for which we will seek a single-parameter family of periodic solutions which are skew-symmetric with respect to the origin of coordinates in the limit as n
--+ 00.
e
.
Let us introduce the function = xj!A (A is the amplitude) for which the inequality 0 < lei < 1 holds. Note that the function e is continuous and periodic. The initial equation can then be represented as follows:
e.. + An-len = O.
(2.11.62)
en
We will expand the function in a series in 1I n as n do this, we first transform the function
_{en,
1
0
using the Laplace transformation [152] 1. If we suppose 1/J2 in the form 1/J2 = 1/J(x), where x = X N + 1 j(N + 1), we have the following equation for the function t/;(x):
1fi;tx + Nx- N - 1 if;;t + Ex- 2N if; - if; = O.
(2.11.81)
After expanding the function x- 2N and x- N - 1 in a series of the power 1j(2N + 1) and 1j(N + 2) as described above, one can obtain 00
x- 2N =
I)-1)i O(i)(1 - 1jx)(2N + 1)-1 ... (2N
+ 1 + i)-I,
(2.11.82)
i=O 00
X- N - 1
=
I)-1)i O(i)(1-1jx)(N+2)-I ... (N+1+i)-I.
(2.11.83)
i=O
Substituting expressions (2.11.82), (2.11.83) into (2.11.81) and splitting it into 1 j N, we have a recurrent sequence pf equations, whose solution gives us the possibility formulating the boundar~ conditions for 1/J~i). In conclusion, we may say that the approach proposed above is the natural asymptotic method for solving the differential equations which contain the term x 1+ e5 for 0 ---. 00. A similar asymptotical approach for the case of small o was proposed in [30d]. Matching solutions for 0 ---. 0 and 0 ---. 00 by means of a two-point Parle approximant one can obtain a solution for any value of
O.
2.12 Pade Approximants 2.12.1 .One-Point Pade Approximants: General Definitions and Properties
The principal shortcoming of the perturbation methods is the local nature of solutions based on them. As the technique of asymptotic integration is well developed and widely used, such problems as elimination of the locality of expansion, evaluation of the convergence domain, construction of uniformly suitable solutions, are very urgent.
2.12 Pade Approximants
133
There exist a lot of approaches to these problems [94, 154]. The method of analytic continuation (for example, the Euler transformation t = c(1 +e)-I) requires a knowledge of the positions of the singularities' of the sought function of the parameter e [154, 70d]. It is useful to apply those methods in cases when a great number of expansion components is known. It is then possible, using, for example, the Domb-Sikes diagram [40, 70d] , to determine the positions of the singularities' and to perform analytic continuation. A significant number of expansion components is also necessary to apply the methods of generalized summation. Not diminishing the merits of these techniques, let us, however, note that in practice only a few of the first components of the perturbation theory are usually known. Lately, the situation has indeed changed a little due to the application of computers. However, up till now, there are usually 3-5 components available of the perturbation series, and exactly from this segment of the series we have to extract all available information. For this purpose the method of Pade approximants (PA) may be very useful [4, 5, 123, 144, 64d, 67d]. Let us consider PAs which allow us to perform the most natural, to some extent, continuation of the power series. Let us formulate the definition. Let 00
F(e) =
L Cie i , i=O
",m
Fmn (e) =
L...i=O ",n
L...i=O
i
aie b. i '
Ie
where the coefficients ai, bi are determined from the following condition: the first (m + n) components of the expansion of the rational function Fmn(e) into the Maclaurin series coincide with the first (m + n + 1) components of the series for F(e). Then Fmn is called the [min] Pade approximant. The set of Fmn functions for different m and n forms the Pade table. The diagonal PAs (m = n) are the mostly widely used in practice. Let us notice that the PAis unique when m and n are specified. To construct the PAs, it is necessary to solve a system of linear algebraic equations (for optimal methods for the determination of PA coefficients see [42, 43, 99]). The PAs have found wide utilization in a series of branches of mathematics and physics, and particulary for enlarging the domain of applicability of series of perturbation methods. The PA performs meromorphic continuation of the function given in the form of power series, and for this reason it allow us to achieve success in the cases where analytic continuatioin cannot be applied. If the PA sequence converges to a given function, then the roots of its denominators tend to singular points. It allows us to determine the singularities with a sufficiently great number of series components, and then to perform the analytic continuation. The data concerning convergence of the PA could have applications in practice only as options which would enhance the reliability of the results. Indeed, in practice it is possible to construct only a limited number of PAs, while all convergence theorems require information about an infinite number of them.
134
2. Discrete Systems
Gonchar's theorem [39d] states that if none of the diagonal PAs ([n/n]) has any pole in a circle of radius R, then the s:equence [n/n] converges uniformly in the circle to the initial function f. Futher, the lack of poles in the sequence [n/n] in the circle of radius R implies the convergence of an initial Taylor series in the circle. As the diagonal PAs are invariant with respect to the fractional-linear transformations z ---. z/( CtZ + (3), then the theorem is valid only for the open circle containing the expansion point and for any domain being a union of such circles. The theorem has one important consequence for continuous fractions, namely: the holomorphity of all suitable fractions of an initial continuous fraction inside a domain n implies uniform convergence of the fraction inside
n.
An essential disadvantage in practice is: the necessity of verifying all diagonal PAs. The point is that if inside a circle of radius R only some subsequence of the diagonal sequence PA has no pole, then its uniform convergence to the initial holomorphic function, in the given circle, is guaranteed only for r < ro, where 0.583R < ro < 0.584R. There exists a counterexample showing that in general r < 0.8R. Since in practice only a finite number of components of the series of the perturbation theory is known and there are no estimations of the convergence rate, then the above theorems could only increase the likelihood of the results obtained. This likelihood is also augmented by known "experimental results" since the practice of PA application shows that the convergence of PA series is usually wider than the convergence do~ain of the initial series. Let us note that widely applied continued fractions form a particular case of PAs. In fact, the suitable fractions, representing the sequence of approximations of the continued fraction, coincide with the following PA sequence: [0/0]' [1/0]' [1/1], [2/1], [2/2]' _.. _Therefore we shall not separate the case of the application of continued fractions. The following circumst8J}.ces are essential. In the perturbation theory asymptotic series, divergent for all values of the parameter c =f:. 0, are very often obtained. This does not permit us to evaluate the value of the sought function with arbitrary precision for any c. At the same time a transformation with a PA (or into a continuous fraction) gives an expression suitable over in a wide range of problems. The approach is strictly mathematically proved for those series where (-l)nCn (Cn is the n-th coefficient of the series) is the n-th moment of some mass distribution, but numerous applications of similar approaches also show their applicability to more general cases. 2.12.2 Using One-Point Parle Approximants in Dynamics
We shall consider the Duffing equation which can be studied with different methods, which allows us to compare their efficiency. We shall apply the perturbation method combined with a PA to the problem.
2.12 Pade Approximants
135
The equation is written in the form
it. + u + u 3 = O.
(2.12.1)
The vibration frequency w has its exact value W=
1rv'1 + A2 2V2K(B) ,
(2.12.2)
where B
7r/2
jA2
= arctg V2+"A"2'
K(B) =
J VI -
o
d1/J
,
A2(2 + A2)-1 sin 1/J 2
K (B) is an elliptic integral of type I. The asymptotic expansion of w in terms of small A 2 (where A is the amplitude of vibrations) has the form w = 1+
3
2
SA -
21 4 256 A
81
6
6549
+ 2048 A - 262144 A
37737 A 10 _ 9636183 A 12 67108864
+ 2094152
+ ....
8
(2.12.3)
First, we shall restrict ourselves to the first three components of the series (2.12.3) and we shall construct the PA [2/2] 32 + 19A2 W2 = 37 + 7 A2 .
(2.12.4)
Taking into consideration the components""' A6 of the frequency expansion, we have W4
1 + 1.13A2 = 1 + 0.756A2
+ 0.261A4 + 0.0599A4·
(2.12.5)
Continuing the process we obtain a sequence of diagonal PAs in the form W2n =
2n
~ (tiC
2i
( 2n
~ Pic
2i
)-1
Along with the diagonal PAs we shall study an element of the Pade table of order [2/4] 1 + 0.513A2 w = 1 + 0.138A2 + 0.030A4'
constructed with the first three components of the series (2.12.3). The result of the frequency calculations according to (2.12.3)-(2.12.5) are graphically presented in Fig. 2.22. The curves 1, 2, 3 correspond to the sums of three, seven and eleven components of the series (2.12.3). Curves 4, 5, 6, correspond to the Pade approximants of order [2/3], [3/3], [4/4]. The exact
136
2. Discrete Systems
s....----r--------,-----,-----,----,
-
w 4
3
4 2 I----J--~~---+----_+--~
.----1~-
I
------+----1
3 2 1 20
40 A 2 50
30
Fig. 2.22. Frequency of amplitude dependence for the Duffing equation constructed with the perturbation method and Pade approximants
solution is represented by the dashed curve. The nondiagonal approximant is represented by curve 7. It can be seen that the best approximation is achieved with the diagonal PAs. Lately solitons and solutions close to them have been widely used in mechanics. These are essentially nonlinear soh1tions which cannot be constructed using the quasi-linear approach when any nttmber of components is conserved. It is still more interesting to note that the PA allows the construction of solutions of that type, beginning from local (quasi-linear) expansions. Moreover, the term "padeon" has appeared. A model example is presented by the boundary problem
y" - y + 2y 3 = 0,
y(O) ,- 1,
y(oo) = 0,
which has the exact solution ("soliton") y = cosh-1(x).
(2.12.6)
A solution in the form of the Dirichlet series y =·Ce- x [1 - 0.25C 2 e- 2x
+ 0.0625C4 e- 4x + ... J,
C = const,
after rearrangement into the PA and determination of C becomes the exact solution (2.12.6). PAs often give a good result even for a small' number of components of the perturbation series. Obviously, however, the efficiency of the PA increases when the number of approximations increases. So, in [4, 5] many components of the expansion series of the amplitude e 2 of the period of the Van-der-Pol equation have been constructed by PAs which has led to the discovery of the
2.12 Parle Approximants
137
singularities of the ,sought period as a function of e 2 and then, using analytic continuation, the construct of a solution applicable throughout the range of e 2 . At present there is the a possibility of obtaining the approximations of a higher-order with computers. It can be imagined that in the case where a complicated problem of the construction of the approximation of a higherorder in the perturbation methods is solved, then it is desirable to try to apply PAs and other methods of convergence acceleration. At the same time it must be noticed that iterative methods are essentially simpler to realize by means of computer technology. PAs can be used to improve these methods. Let the iterative process have the form
T(uo) = 0,
Un = Tl(Un-d,
n = 1,2, ....
We introduce the function Sn(e):
Sn(e) = Uo + CUt - Uo)e + (U2 - ude 2
+ ... + (un
- Un_den.
(2.12.7)
For e = 0 we have Sn(e) ~ Uo, for e = 1 Sn(e) ~ Un' Then, we rearrange the series (2.12.7) with the PA and suppose e = 1: U
~
Sn
E:l Oi , m + p = n. = Uo +""p 1 + LJj=l {3j
(
2.12.8
)
Let us consider, as an example, the problem of big deflections of round isotropic plate of radius R, with a free opening of radius Ro and a rigidly restrained external outline on which a superficial pressure of constant intensity is acting. The problem solution was found in [50d] using the method of finite central differences for the Young modulus E = 62.4 kg/m3 and the Poisson coefficient v = 0.335, RoR- 1 = 0.1. The method of succesive approximations applied for the solution of the system of nonlinear algebraic equations, for comparatively big loads, converges for some 150-200 iterations, and the convergence to the solution has an osbillating nature. Table 2.3. Radial forces T in a round isotropic plate - iteration procedure Approxim. number
T
o
5.27286 1.09640 4.81246 1.45039 4.55120 1.67086 4.37191 1.82867 4.23735 1.94992
1 2 3 4 5 6 7 8 9
Approxim. number 10 145 146 147 148 149 150 151 152
T
4.13072 3.02320 3.11416 3.02603 3.11236 3.02680 3.11063 3.02849 3.10890
138
2. Discrete Systems
Table 2.3 gives the result of computations of the dimensionless radial force T = N r R 2 D-l for p = R- 1 , where r is the polar coordinate; q* = O.5qR4 (Dh)-1 = 35 is the intensity of the external load. Applying the method of generalized summing the situation can be improved (Table 2.4). Let us present the proposed method. The PA (2.12.8), taking into account four approximations, will have the form 5.319 - 284.883e - 27.606e 2 T = 1 _ 52.762e _ 47.992e2 (2.12.9)
Table 2.4. Radial forces T in a round isotropic plate - using of Pade approximants T
Approxim. number
Approxim. number
T
6
3.0656 3.0760 3.0791 3.0789 3.0789 3.0789
o 2.6955 3.0140 3.0941 3.0656
1
3 4
5
7 8 9
Solution [5Od)
When e = 1, the formula (2.12.9) gives T = 3.079. The boundary problem considered above demonstrates the high efficiency of Parle approximants to accelerate the convergence of iterative processes. PAs can be used for a heuristic evaluation of the domain of applicability of the perturbation theory series. The e Values, up to which the difference between calculations according to the segment of the perturbation series and its diagonal PA does not exceed a given value (e.g. 5%), can be considered as approximative values for the domain of applicability of the initial series. A transformation to a rational functional allows us to describe nontrivial behaviour at infinity and to take into consideration the singular points of the solutions. We shall consider,' as an example, the problem of the flow around a thin elliptical airfoil (Ixl < 1, Iyl ~ e, e « 1) by a plane stream of perfect liquid incoming with velocity v. The first few components of the asymptotic expansion of the relative stream velocity q* on the airfoil surface are: 1
q* =
2_ V -
1
+e
_
~ e2 x 2
2
1-x 2
_
~ e3 x 2
1-x2 + ...
The written solution diverges for x (2.12.10) by the PA, the singularity for x 2
q* =
(1 - x )(1 + c) 1 - x 2 + 0.5e2x2
+ O(e 4 ).
2
(2.12.10)
= 1. After replacing expansion = 1 disappears: (
2.12.11
)
Fig. 2.23 presents for e = 0.5: 1 - the exact solution, dashed line - the solution (2.12.10), 2 - PA (2.12.11), and the point line - the solution according to the
2.12 Pade Approximants
139
1.6 ....------oy----...,..---~---.,
q.
o
o
1.2
0.8
I 0.4
L-~
----
-----+----~
--
I 0.8
0.4
0.6
x 0.2
Fig. 2.23. Compansion of the PA approach and Lighthill method
Lighthill method [108, 154]' which gives in this case worse results than the PA. 2.12.3 Matching Limit Expansions From the physical point of view, every nontrivial asymptotic usually has an inverse. In other words, if an asymptotic for e --+ 0 (e --+ 00) exists, the asymptotic for e --+ 00 (e --+ 0) can be constructed. Then there appears one of the principal sharpest problems for the asymptotic approach - namely the construction of solutions appropriate for 0 « e « 00. This may be solved both on the level of solutions and on the level of equations. In particular, one can try to synthesize the limit equations with the purpose of obtaining a "complex" relationship allowing for a smooth transition from e --+ 0 to e --+ 00. For a synthesis of solutions one can utilize two-point PAs (TPPAs) [72, 8.1, 85, 87, 117, 127, 128]. The definition of TPPAs is given below. Let 00
F(e) ~
L aiei
for e
--+
OJ
(2.12.12)
i=O 00
F(e) ~
L
bie i for e
--+ 00.
(2.12.13)
i=O
The following function will be called the TPPA
where the coefficients Ok, (3k are defined so that the first p coefficients of the proper part of the Laurent expansion of n.
There are n possible proper forms of vibrations: kS1r
Yk = As sin - - cos(wst + 4>s), s = 1,2, ... ,n, n+l and the appropriate frequencies of free vibrations are given by
w. = 2ft; sin 2(:: 1)'
(2.12.15)
(2.12.16)
Let us construct the asymptotic expansions of the frequency W s in the vicinities of the points s = 0 and s = 2(n + 1). We substitute variables in the expression for w s putting
x=
x(0.51r -
X)-l,
X
= s1r[2(n
+ l)t 1 .
In the same way, instead of the segment [0, 2(n + 1)] for s, we obtain the semi-interval x E [0,00). Enumerating the expansions for x --+ ~ and x --+ 00, we obtain sin
2(1"~ x)
=;
[x - x2+ (1 - ~~)'x3
- (1 sin
1rX
2(1
+ x)
1-
=
2 1r
~2) x4 + ...J, x --+ 0, x- 2 +
8'
(1 _
- (1 - ":) x- 4
1r
2 )
12
+ ... ,
(2.12.17)
x- 3
x --+ 00.
(2.12.18)
A solution, appropriate for 0 ::; x ~ 00, can be obtained with the TPPA method 2 W -= p 1.57x + 0.81x J s 1 + 1.57x + 0.81x2 . (2.12.19)
ra [
y:;;;
The results of frequency calculations according to (2.12.16)-(2.12.19) are presented in Fig. 2.24. The exact solution (2.12.16) is designated by 1, the expansions (2.12.17) and (2.12.18) by 2 and 3. The rearranged Parle solution coincides very well with the exact solution over the considered interval. An analysis of the diagrams shows that the TPPA has enabled us to construct an approximative solution appropriate for any frequency of vibrations.
2.12 Pade Approximants
2.0
141
~---..,-----,----"""",--------,
w 1.6 --+------+.-------
1.2
-----------I
0.8 0.4
o -. -D.4l.....-.....---ll..--...L...----..J------'-------' 4 2 6 8 o
Fig. 2.24. Two-point PA in the theory of the oscillations of chain
An important TPPA application may be the inverse Laplace transform. Indeed, having a given transform, it is possible to investigate the inverse transform behaviour for t ---+ 0 and t ---+ 00. The problem is the inverse transform description for 0 « t « 00 [30d, 68d]. It is proposed in the monograph [68d] that only those components which give an asymptotic for p ---+ 0 and p ---+ 00 (whare p is the transform parameter) and also the principal singularities of the expression should be left under the integral sign of Mellin's integral. If the simplified integral can be calculated, then an approximate analytic solution is obtained. In spite of the unquestionable utility of this approach, convincingly proved in th~ above monograph, the problem on the whole remains unsolved. The PAs have been applied to the inverse Laplace transform to widen the domain of applicability of power expansions. One of the TPPA application methods for the solution of the inverse Laplace transform is the F(p) transform expansion in Taylor and Laurent series in the vicinities of the points p = 0 and p = 00, which is then followed by the replacement of F(p) by a rational function according to this scheme. Then the transition to the inverse transform is realized ac~ording to well-known rules. But the aim is achieved more quickly by applying the TPPA directly to the asymptotics of the inverse transform f(t). Here is an example:
F(p) = Ko(p)e- P , where K o is the McDonald function,
f(t) =
1 v 2t
rn; -
f(t) = t- 1
-
v7In + ...
4y 2
t- 2
+ ...
for for
t
t
---+
OJ
---+ 00.
142
2. Discrete Systems
The exact value of f(t) is f(t) = [t(t + 2)r 1/ 2 ,
t > O.
(2.12.20)
The TPPA has the form
f(t) ~ (t
+ V2t)-1.
(2.12.21)
Figure 2.25 presents the solutions (2.12.20) and (2.12.21) (curves 1 and 2 as appropriate). If the asymptotics are not of the power form, the difficulties are also surmountable. Sometimes the asymptotic for p --+ 00 may be represented as a sum of exponential functions or of sines and power series [85]. In other cases, it is necessary to introduce nonpower functions into the fractional-rational expressions and expand the latter into power series for t --+ 0 [11]. 0,5
f 0,4 0,3 0,2 0,1 0 0
4
2
6
t
8
:R.ig. 2.25. Matching limiting solution in the theory of the Laplace transform
Another interesting example is the Van der Pol equation. We give some necessary preliminary information according to [94]. The Van der Pol oscillator is governed by the equation
x
+ kx(X 2 - 1) + X = O.
The solution tends in time to an oscillation with a particular amplitude which does not depend on the initial conditions. The period of this limit oscillation T is of interest and is plotted in Fig. 2.26 as a function of the strength of the nonlinear friction, k. The continuous line gives the numerical results obtained by means of the Runge-Kutta method. The dashed curves give the second-order perturbation approximations
T = 21r(1
1
+ _k 2 ) + O(k 4 )
16 T = k(3 - 2ln 2)
as k ~ 0
+ 7.0143k- 1/ 3 + O(k-1ln k)
(2.12.22) as k
--+ 00.
(2.12.23)
The TPPA formula uses two terms of the expansion (2.12.22) and the first term of the expansion (2.12.23):
2.12 Pade Approximants
143
15 .----------,-----.----r---,----,---.,--------:-;]
T 12 - --
- - ---- --I----I----:o,.!.j'~--.._'I:~=-----=--!'~-___i
9
t---------t-----------
6 I -1-----
3
- ----
---l---------------+--- ---
OL--_ _L - -_ _L - - _ - - - - l_ _----L_ _ 2 3 4 1 5 o
---l._ _----'-_ _- - - '
6
k
7
Fig. 2.26. Various solution for the period of the Van der Pol equation
6.2832 + 1.5294k + O.3927k 2 T = ---I-+-0.-24-3-3-k--and shows good agreement with the numerical results for all values of k (curve 1 in Fig. 2.26). 2.12.4 Matching Local Expansions in Nonlinear Dynamics! Interesting results were obtained by the use of two- point Parle approximants in the theory of normal vibrations in nonlinear finite-dimensional systems [118, 56d].
Consider a conservative system
II = all ) + II = 0, Xi. = dx dt ' az' i = 1,2, ... , n, (2.12.24 II = II(x) is the potential energy, assumed to be a positive definite
'..
miXi
i
Z
Xi
where function; and X = (Xl,X2, ... ,xn )T. The power series expansion for II(x) begins with terms having a power of at least 2. Without reducing the degree of generalization, assume that mi = 1, since this can always be ensured by dilatation of the coordinates. The energy integralJor system (2.12.24) is n
~ LX~ + II(Xl,X2, ... ,xn ) =
h,
(2.12.25)
k=l
h being the system energy. Assume that within configuration space, bounded by the closed maximum equipotential surface II = h, the only equilibrium position is Xi = 0 (i = 1,2, ... , n). 1
By courtesy of Yu.V. Mikhlin
144
2. Discrete Systems
In order to determine the trajectories of normal vibrations, the following relationships can be used [135]: - II 12 + Xi1( - II) -- - II Xi (i = 1,2,3, ... ,nj X = Xi ).( 2.1 2 .26) 2x"i 1 h~n + L.Jk=2 x k These are obtained either as Euler equations for the variational principle in Jacobi form or by elimination of time from the equations of motion (2.12.24) with consideration for the energy integral (2.12.25). An analytical extension of the trajectories on the maximum isoenergy surface II = h is possible if the boundary conditions, Le. the conditions of orthogonality of a trajectory to the surface, are satisfied [135]: x~ [-IIx(X,
X2(X), ... ,xn(X)] = -IIXi (X, X2(X), ... ,Xn (X)), (2.12.27)
(X,X2(X), ...,xn(X)) being the trajectory return points lying on the II = h surface where all velocities are equal to zero. If the trajectory Xi (X) is defined, the law of motion with respect to time can be found using
x + IIx (XI,X2(X), ... ,xn(x)) =
0,
for which the periodic solution x(t) is obtained by inversion of the integral. Now consider the problem of normal vibrational behaviour in certain nonlinear systems when the amplitude (or energy) of the vibrations is varied from zero to an extremely large value. Assume that in the system
z + II
Zi
(Zl'
Z2, ... ,zn) = 0
(2.12.28)
the potential energy II(ZI' Z2, ..:, zn) is:a positive definite polynomial of Zl, "" Zn having a minimum power of 2 ~nd a maximum power of 2m. On choosing a coordinate, say Zl, substitute Zi = CXi where c = ZI(O). Obviously, XI(O) = 1. Furthermore, without loss of generality, assume XI(O) = O. Then
(2.12.29) where V = E~:OI ckV(k+~) (Xl, X2, ... ,xn ), v(r+l) contains terms of the power (r + 1) of the variables in the potential
V(c, Xl, X2, ... ,Xn ) = II(ZI (xd, Z2(X2), ... , Zn(X n )). It is assumed below that the amplitude of vibration c = z(O) is the independent parameter. At small amplitudes a homogenous linear system with a potential energy V(2) is selected as the initial one while, at large amplitudes, a homogenous nonlinear system with a potential energy V(2m) is selected. Both linear and nonlinear homogenous systems allow normal vibrations of the type Xi = kiXI, where the constants k i are determined from the algebraic equations
ki Vx(;) (1, k 2, ... ,kn ) = V:J;) (1, k 2, ... ,kn ).
2.12 Pade Approximants
145
A number of vibrations of this type can be greater than the number of degrees of freedom in the nonlinear case. In the vicinity of a linear system at small values of c, trajectories of the normal vibrations X~I)(X) can be determined as a power series of x and c (assuming that Xl = x), while in the vicinity of a homogenous nonlinear system (at large values of c), x~2)(x) can be determined as a power series of X and c- l . The construction of the series is described in [55d]. The amplitude values (at ± = Xi = 0) define the normal vibration mode completely. Therefore, for the sake of simplicity, only the expansions of p?) = xP)(I) and p~2) = xi(l) in terms of the powers of c wil be discussed below: 00
p~l) =
00
La;i)d,
(2) _ '"'" a(i) - j
Pi
-
LJlJj
C
.
(2.12.30)
j=O
j=O
In order to join together the local expansions (2.12.30) and to investigate the behaviour of the normal vibration trajectories at arbitrary values of c, fractional rational TPPAs are used: (i)
P8
=
~8 (i) ~1 L.Jj=O a j f..,-
E
8
. 0 J=
b(i)' J'
cJ
(2.12.31)
or ~8 (i) ~1-8 p(j) = L.Jj=O a j f..,8
~8
L.Jj=o
b(i) j
'-8'
(2.12.32)
cJ
Compare expressions (2.12.31) and (2.12.32) with the expansions (2.12.30). By preserving only the terms with the order of cr ( -8 < r ~ 8) and equating the coefficients at equal powers of c, n - 1 systems of 2(8 + 1) linear. algebraic equations will be obtained for the determination of a;i), b;i) (j = 0,1,2, ... ). Since the determinants of these systems L1~i) are generally not equal to zero, the systems of algebraic equations have a single exact solution, a;i) = b~i) = O. J Select a TPPA corresponding to the preserved terms in (2.12.30) having the nonzero coefficients. ay), bY). Assume that b~i) 1= 0, for otherwise as c ---+ 0 X~l) ---+ 00. Without loss of generality, it can also be assumed that bg) = 1. Now the systems of algebraic equations for the determination of a;i), by) become overdetermined. All the unknown coefficients a~l), ... , a~l),
bp), ... , b~l)
(i = 2,3, ... , n) are determined from (28 + 1) equations while the
"error" of this approximate solution can be obtained by a substitution of all coefficients in the remaining equation. Obviously, the "error" is determined
146
2. Discrete Systems
by the value of L1~i) , since at L1~i) = 0 nonzero solutions and, consequently, the exact Pade approximants will be obtained in the given approximation in terms of c. Hence, the following is the necessary condition for convergence of a succession of the TPPA (2.12.31), at s ---+ 00, to fractional rational functions: p(i)
=
E~ a~i)d
~) i b. ci =0 1
1=0
,,~ LJ 1
(b~i) = 1),
(2.12.33)
namely, lim
L1(i)
8-00
=0
(i
= 2,3, ... , n).
(2.12.34)
8
Indeed, if conditions (2.12.34) are not satisfied, nonzero values of the coefficients A~i), b~i) in (2.12.33) will obviously not be obtained. Conditions (2.12.34) are necessary but not sufficient for the convergence of the approximants (2.12.31) to the functions (2.12.33); nevertheless, the role of conditions (2.12.34) is determined by the following consideration. Since in the general case there is more than one quasilinear local expansion and essentially nonlinear local expansions are alike, the Dl.J,mbers of expansions of the respective type being not necessarily equal, it is the convergence conditions (2.12.34) that allow one to establish a relation between the quasilinear and essentially nonlinear expansions, that is, to decide which of them corresponds to the same solution and which to different ones. For a concrete analysis based on the above technique, consider a conservative system with two degrees of freedom, , whose potential energy contains the terms of the 2nd and 4th powers or the variables ZI, Z2. Substituting ZI = CX, Z2 = C1J, where c = ZI(O), (x(O) = 1), one obtains 2
V = c
X2
(
d l 2"
y2
)
+ d2"2 + d3xy + c4
x2y2
(x4
'1'1"4
+ '1'2 x3 y
4) =c2V(2) + c4V(4).
+ '1'3-2- + '1'4x y3 ,+ '1'5 ~
The equation for determining the trajectory y(x) is of the form
2y"(h - V)
+ (1 + y'2)(-y'Vx + Vy ) =
0,
(2.12.35)
while the boundary conditions (2.12.27) can be written
(~y'Vx + Vy)lh=v = O. For definiteness, let d 1 = d2 = 1 + '1'; d3 = -'1'; "II = 1; '1'2 = 0; '1'3 = 3; '1'4 = 0.2091; '1'5 = '1'. Write the equations of motion for such a system: Ii + x + '1'(x - y) + c2(X 3 + 3xy2 + 0.2091 y3) = 0, ii + Y + '1'(y - x) + c2(2 y 3 + 3x 2y + 0.6273y2x ) = O. (2.12.36) In the linear limiting case (c = 0) two rectilinear normal modes of vibrations y = kox, k~l) = 1; k~2) = -1 are obtained, while a nonlinear system
2.12 Pade Approximants
147
(where the equations of motion contain only the third power terms with respect to x, y) admits four such modes: k~3) = 1.496; k~4) = 0; k~5) = -1.279; k o(6) -- -5 . In order to determine nearly rectilinear curvilinear trajectories of normal vibrations, (2.12.35) is used along with the boundary conditions. By matching the local expansions the following Pade approximants are obtained I -IV II-V -1-1.11c2-O.275c4 1+1.202 p = 1+1.61c2+O.72C4 p = 1+1.00c2+O.215C4 "y
"y
= 0.5
1+1.062 p = 1+2.06c2+3.20c4
-1-2.76c2-1.36c4 P = 1+2.31c2+l.04c4
= 0.2
1+1.70c2 P = 1+3. 96c 2+ 13.29c4
-1-6.41c2-9.03c4 P = ~1:-+--=5:-:.3:-::::0--;c2""+--=7:-:.0:-=2---:c4'--·
(2.12.37)
The two additional modes of vibration exist only in a nonlinear system; as v increases (the amplitude c decreases), they vanish at a certain limiting point. For the analysis of these vibration modes, assume a new variable u = (p - 1.496)/(p - 5). By using the variable u, two expansions in terms of positive and negative powers were obtained; therefore, fractional rational representations can be introduced as above. By comparing these expansions, the following TPPAs are obtained III - VI 8.874u+1.126u 2 v = 1+4.300u+2.836u 2+O.549u 3 "f =
"'V
,
0.5
= 0.2
35.497 U +5.108u 2 v = 1+3.021u-0.794u 2+O.622u3
(2.12.38)
88.986u+1.470u 2 - 1-0.143u+3.747u 2+O.072u3 ·
v -
-:--~~--=-",",:,",-:-----;;:~~~
Now proceed to the determination of the limiting point. Obviously, it can be found from
8v =0 8u . From (2.12.38): at "Y = 2 the limiting point is v ~ 1.21, c ~ 0.91; at "Y = 0.5 the limiting point is v ~ 11.10, c ~ 0.30; at "Y = 0.2 the limiting point is v ~ 23.93, c ~ 0.20. Hence, as "Y -+ 0 the limiting point is characterized by the amplitude c -+ O. Therefore, the two additional vibration modes in a nonlinear system can exist at rather small amplitudes of vibrations. Note that the quasilinear analysis does not allow one to find these solutions even at small amplitudes.
148
2. Discrete Systems
In the limit, when I = 0, a linear system decomposes into two independent oscillators having identical frequencies and admits any rectilineal modes of normal vibrations. Obviously, the full system (2.12.36) at I = 0 admits four modes of vibrations (in the nonlinear case) Y2 = kY1, k = {1.496, 0, -1.279, -5}. Thus, fractional rational Pade approximants allow us to estimate the nonlocal behaviour of normal vibrations in nonlinear finite-dimensional systems. For system (2.12.36) the evolution of the modes of normal vibrations is shown in Fig. 2.27 using parameters ( = In(l + c2 h 2 ) and r.p = arctg p (the picture shows periodicity in (1:") ~nt
=
/.l
~n
+ 21rMi'' na(~) t = /.l
~n
-
21rMi .
Then, one can rewrite the system in the form Pmn - 'Ya:nK
(a~ + Jl2,6~)(a~ + Jl2,6~ + A);
en = (+) _
eni - -
'Ya'!nK
.
][~ K
]'
[a~ + Jl2(a~~))2] [a~ + Jl2(a~~))2 + A]'
(:-) = _ nt
[a~ + Jl2(a~~))2
a~ + Jl2(a~~))2 + A
i
~ 1, 2, ....
Substituting expressions for en, e~t), e~~) (i = 1,2, ...) into (3.2.14), one can obtain K as a function of A: K = Pmn S(1
+ 'Ya~S + S(+) + S(-))-I,
(3.2.15)
where
5 = [(a~ S(+) =
+ Jl2,6~)(a~ + Jl2,6~ + A)J -1 ;
f: [a~ + Jl2(a~~))2] f: [a~ + Jl2(a~~))2]
-1
[a~ + Jl2(a~~))2 + A] -1 ;
-1
[a~ + Jl2(a~~))2 + A] -1 .
j+l
S(-) =
j+l
Taking into accoun~ formulae (3.2.13)-(3.2.15) one obtains (3.2.12) as a transcendental equation (with respect to the unknown A) that may be solved routinely by numerical methods. Then we will obtain K (using formula (3.2.15)) and the amplitudes en, e~t),e~~) (i=I,2, ... ). For the numerical investigation we choose a square plate loaded by the lateral load Q
=
Q10 sin (":'1 ) ~ (O.~:X2) .
162
Xl
3. Continuous Systems
We also suppose v = 0.2, EcIM/(Da2) = 200; = 0.5 a l, X2 = a 2lx 2, M2 = (Dh/a2)M2. The numerical results are plotted in Fig. 3.5.
qlO
ol----+--A---+--+--~~-___r-_r_:::;;;;;;;~__,
o
0.5
o
0.5
o
0.5
1
l:i 1
x2
1
Fig. 3.5. Bending moments in the Stringer plate in the perpendicular direction to ilieri~ .
3.2.3 Perforated Membrane
Consider the problem of transverse oscillations of a rectangular membrane weakened by a double periodic system of regularly spaced identical circular holes of radius a. The ratio c of the period of perforations to the characteristic size of the region n is a small quantity. The outer contour 8n of the membrane is rigidly clamped, while the edges or'the apertures 8ni are free. Let us begin from the linear case. In mathematical language we have the boundary value problem
2 2 2 (8 U 8 U) _ 8 u c 8x + 8 y 2
2
2
-
iJt2
in
n,
(3.2.16)
3.2 Homogenization Procedure U
= 0
(3.2.17)
an,
on
163
au an
(3.2.18)
is the transverse displacement of the points of the membrane, c2 = pi P, p is the tension in the membrane, and p is the density. We take a solution for the characteristic oscillations of the membrane in the form u(x, y, t) = u(x, y)eiwt , where A = w 2/c 2 and w is the circular frequency. Then, instead of (3.2.16) we obtain where
.
U
a2 u a2 u
ax 2 + a y 2 + AU = O.
(3.2.19)
We have presented the solution of the problem (3.2.19), (3.2.17), (3.2.18) just posed as an asymptotic series in the powers of a small parameter
u = uo(x, y) + e: (UlO(X, y) + Ul (x, y, c;, 1])) +e: 2 (U20(X, y) + U2(X, y, C;, 1])) + "', where c; = xle: and 1] = yle: are the "fast" variables. The functions Uo, UlO, U20, .. · depend only on the "slow" variables, and the other variables ui(i = 1,2... ) are periodic together with their derivatives with respect to the "fast" variables and have a period equal to that of the structure. Similarly, we expand the frequency A = Ao
+ e:AI + e: 2A2 + ....
After separating e: we obtain from (3.2.19) an infinite system e:- l
a2Ul
a2Ul
+ 87]2
0,
(3.2.20)
a2uo a2Ul a2Ul + 8 y 2 + 2 8x8c; + 2 8y81] a2U2 a2U2 + ac;2 + a1]2 + AoUo = 0,
(3.2.21)
ac;2
=
a2uo 8x2
a2Ul
8x 2
a2U2 a y 2 + 2 8x8c;
a2Ul
+
a2UlO
+ ax 2
a2U2 + 2 8y81]
+
a2U3 ac;2
+
a2U3 87]2
a2UlO
+ a y 2 + Ao(Ul + UlO) + AlUO =
O.
(3.2.22)
The corresponding boundary conditions (3.2.18) assume the form aUl e:o
an +
auo an = 0;
e: l
aU2
+ aU1 + aUlO
where
an
an
on ani -
an -
0
,
(3.2.23)
on an.
.
81 an is the derivative with respect to the
"fast" variables.
164
3. Continuous Systems
The solution of the boundary problem for a complex multiply connected region now breaks up into three stages. The first stage is the solution of the "cell problem" (3.2.20), (3.2.23). On the opposite sides of the "cell" the function Ul must satisfy the periodicity conditions
uIIE=b = uIIE=-b, 8Ul
_ 8Ul
8(, E=b -
ull71=b = ull71=-b, 8Ul _ 8 Ul
8(, E=-b' 81] 71=b -
(3.2.24)
81] 71=-b'
Using Galerkin's variational method to solve problems (3.2.20), (3.2.23) and (3.2.24), we represent Ul as Ul
~ ~ (
. m1r(, n1r1] m1r(, . n1r1]) Almnsm-b-cos-b- +A2mnCOS-b-sm-b- .
= 2of=:'o
After performing the necessary operations, we obtain A Imn
=a
8uoA* 8x mn'
A
2mn
* = a OO° 8y A mn,
A:n
where n is a constant determined from the vanishing of the variation of Galerkin functional. The second stage of the solution of the problem is the construction of the averaged relations. Applying the averaging operator to (3.2.21) i>(x,y) =
I~;I
JJ
q»(x,y,(,,1])d(,d1]\
{li
n;
where is a "cell" without holes, we obtain the averaged equation with a boundary condition on the outer contour of the membrane: 8 2uo 82uo 8x 2 + 8 2 + BAouo = 0 in n*, Uo = 0 on 8n, y where n* is a membrane without perforations, B =
(1 _1r (1 _1r 2 4 b2
a )
2 4 b2
a _
1r a 2 ~ ~ A* 2 b2
20 f=:'o
mn
J
1(%1rvm + n 2))-1 Vm+ n 2
2
2
'
and J 1 , is the Bessel function of order 1. At the third stage we find the first correction to the frequency AI. To do this we must determine the function U2 as a solution of the boundary value problem 82u2 8(,2 8U2
+
an +
a2U2 82uo 82uo 82ul 82ul 81]2 = - &:2 - 8y2 - 2 8x8(, - 2 8y81] -
8Ul 8n
+
OO lO
8n = 0
on
8ni ,
AOUO
in
ni,
3.2 Homogenization Procedure
165
with the periodicity conditions similar to (3.2.24). Proceeding as in the determination of the function Ul, we can obtain 00 00 ( . m1rE n1r1] U2 = ~~ Clmnsm-b-cos-b-
m1rE . n1r1]) + C 2mn cos -bsm -b- + cp(E, 1]), where
G1mn
= Al mn (~o
.,. a.;~o),
C 2mn
= A 2mn ( : ;
.,.
a.;~o),
and cp( E, 1]) is a function satisfying the condition cp( -E, -1]) = cp( E, 1]). The form of the function c.p(E, 1]) is unimportant, since it makes no contribution to the averaged equation and, consequently, is not used in the determination of AI. After averaging (3.2.22) we obtain
82ulO 82ulO 8x 2 + 8 2 + B(AOUlO + Al uo) = y
o.
To determine Al we multiply the equation just obtained termwise by Uo and integrate over the region r}* [120, 122]. If UlO = ill = 0 on 8n, then the differential operator
82ulO L(UlO) = 8x 2
+
82ulO 8y 2
is self-adjoint. Then Al = 0 and the expansion of the characteristic frequencies begins with A2 - a term of order c 2 . In that case, if ill does not satisfy the boundary condition on the contour of the membrane and consequently UlO ;;J 0 on 8n, we obtain a nonzero first correction to the characteristic frequt:ncy. Now let us investigate the nonlinear but quasilinear (the terms with derivatives in the governing equation are linear) case - a membrane on nonlinear support. The governing equation is 2
2
2
(8 U 8 U) 2 3 8 u C 8x 2 + 8 y 2 +ClU = 8t 2 ' 2
and the boundary conditions are given by (3.2.17), (3.2.18). Here c? is the rigidity of the nonlinear support. After splitting into c, one obtains the cell problem in the form (3.2.20), (3.2.23). Then its solution coincides with the solution of the linear problem, and the homogenized nonlinear equation may be written in the form 2 ( 82UO 82uo) 2 3 82uo C 8x 2 + 8 y 2 + BclUo = B &t 2 in 8n·,
Uo = 0 on 8n.
166
3. Continuous Systems
3.2.4 Perforated Plate
We use the averaging method for the computation of densely perforated plates. As has been mentioned above, having the solution of the static local problem, this approach gives the possibility to obtain, without basic difficulties, the solution of dynamical and quasilinear problems. We consider the problem of the bending of a rectangular plate, weakened by a doubly periodic system of holes. Let n be the domain occupied by the pl~te, let the exterior contour be 8n and let 8ni be the boundary of the hole. The periodic e of the structure is the same in both directions and small in comparison with the characteristic dimension of the domain n (e « 1). The boundaries 8ni of the holes are free, and the exterior contour 8n of the domain is fastened in a definite manner. We have the boundary value problem 8 4u 8 4u 8 4u . (3.2.25) ax4 + 2 8x 28 y 2 + a y 4 = fInn· M r = 0,
~ =
°
(3.2.26)
on 8ni ,
where 2 82u 2 8 u M r = vLlu + (1 - v) ( cos a 8x 2 + sin a 8 y 2 2
~
8 2u ) + sin 2a 8x8y ;
a
8
= cos a axLlu + sina 8yLlu 2
+(1- v)~ [cos2a 8 u as
8x8y
+ ~sik2a (8
2
2
U_ 8 U)] '. a y2 ax 2
2
u is the normal deflection, and a is the angle between the exterior normal n to the contour and the x axis. We represent the solution of the problem in the form of a series of powers of a small parameter e: _ U - Uo
' + eUI + e2 U2 + .. "
(3.2.27)
where Ui = Ui(X, y, E, TJ) (i = 0,1,2, ...), variables. Taking into account the relations
e
x/e,TJ
y/e are the "fast"
88188818
-'=-+--' ---+-ax ax e 8e' ay - ay e 8r} the initial equation and each of the boundary conditions splits with respect to e into an infinite system of equations
84uo 8e4
84uo
+ 28e28TJ2 +
8 4uo 8TJ4
=
0,
in n i ;
3.2 Homogenization Procedure
167
(3.2.28)
[( I - v )
sin
20 cos 0
2
] B3UQ
+coso - -3+
ae
+ sin a] a;;;.0 + [(1 - v) ( + sin a] :;2~
[(1- v) (
3 . ] a uQ +smo aea1]2
= 0,
a4Ul
B4Ul
[(
cos2asina -
cos 2a cos a -
on
ani;
0,
in
a4Ul
ae4 + 2 ae 2a1]2 + a1]4
=
)
ni ;
sin 20 cos 0 2
I-v - - - -
~ sin2a cos a )
~ sin2asina)
(3.2.29)
(3.2.30)
(3.2.31 )
168
3. Continuous Systems
(3.2.32)
(3.2.33)
(3.2.34)
(3.2.35)
3.2 Homogenization Procedure
169
where {}i is a characteristic cell of the structure. Thus, the solution of the formulated problem (3.2.25), (3.2.26) for a composite multiply connected domain splits into a series of steps in domains with a simpler geometry, from which we can distinguish two fundamental problems: a local problem ("a problem on the cellU ) which consists in the solving of the biharmonic equation in the domain (}i with the given boundary conditions on the contour of the hole and the periodic continuation conditions on the opposite sides of the "cells" ull€=b = ull€=-b 8Ul
8~
8Ul
I€=b
= 8~
82ul
8~2
82ul I€=b =
83ul
8~3
I€=-b
ull 11 =b = ulI 11 =-b 8Ul 8Ul 81] 111=b = 8T] 111=-b
8~2
I€=-b
82ul 82ul 81]2 111=b = 81]2 111=-b
I€=-b
83ul 83ul 11=b 81]3 1 = 81]3 111=-b;
aJUl I€=b =
8~3
(3.2.37)
and a global problem which consists in the solving of an averaged equation of the form (3.2.36) in the domain {}* without perforations and with the initial boundary conditions on the contour of the plate. As follows from relations (3.2.28), (3.2.37) and (3.2.29), (3.2.37), the functions UQ, Ul do not depend on the fast variables, Le., UQ
= uQ(x, y);
Ul
= Ul(X, y).
(3.2.38)
Consequently, the solution of the problem is represented in the form of a sum of a certain smooth function and a small fast oscillating correction; moreover, the expansion starts with the U2 O(e 2 ) term. After the successive solving of the "cell problems" (3.2.30)-(3.2.32), (3.2.37) and (3.2.33)-(3.2.35), (3.2.37) and the determination, of the functions U2, U3, we determine the principal part of the solution, Le., the function UQ. Applying to (3.2.36) the averaging operator (...) f'.I
(.
~.) = I~:I
JJ(...)d~
d1],
{li
we obtain the averaged equation in the form
170
3. Continuous Systems
84uo ( 8x 4
84uo
+ 2 8x 28 y 2 +
If ( + In; I
84uo 8y4 -
4 8 u3 8x8€3
1
D.
8 4u 2 + 8y 28€2
I
)
Ini I In;1
(3.2.39) 4u
84u3 84u3 (J4u3 3 8 2 + 8y8€2817 + 8x8e817 2 + 8y8173 + 8x 28€2
3 84u2 + 8y28172
+
84u2 8x28172
+
4
4 8 u2 ) de d - 0 8x8y8€817 c., 17 .
We obtain the solution of the "cell problems" (3.2.30)-(3.2.32), (3.2.37) and (3.2.33)-(3.2.35), (3.2.37) for a plate with a square net of perforation holes of radius a by making use of the Bubnov-Galerkin method, modified for the case of natural boundary conditions. We consider them successively. We represent
~ ~(
U2 = ~o
f='o
. m1r€ . n1r17 A 1mn sm -b- sm -b-
. m1r€ n1r17 + A 3mn sm -b- cos -b-
m1r€
n1r17
+ A 2mn cos -b- cos -bm1r€ . n1r )
+ A4mn cos -b- sm - b17 - '
(3.2.40)
where A 1mn , A 2mn , A 3mn , A4mn are constants, defined from the conditions of the vanishing of the variation of the Galerkin functional. The selection of the function U2 in the form (3.2.40) allows us to satisfy the periodic continuation conditions (3.2.37); then the variation of the Galerkin functional becomes
where by M r - I, ~ we have denoted expressions (3.2.31), (3.2.32), respectively. As one can see from (3.2.41), by virtue of the symmetry of the considered domain the constants, A 3mn , A4mn are equal to zero. The unknowns A 1mn , A 2mn are determined after carrying out the standard procedure of the Galerkin method: 282uo2 82uo= 2 82uo * 1mn 2 A*2mn + b - 8 A = b 8x8y A 1mn ; A 2mn = b - X8 y 2 A*2mn,(3.2.42) . where Aimn' A*2mn, A*2mn are numerical coefficients. By means of the same scheme, after transforming the right-hand sides of the equations and the boundary conditions, we obtain the solution of problem (3.2.33)-(3.2.35), (3.2.37). From similar considerations, we represent
~~( . m1re n1r17 U3 = ~~ B1mnsm -b- cos-b +U2(UO
-+
ud;
m1re . n1r17 ) + B 2mncos-b-sm -b-
(3.2.43)
3.3 Averaging Procedure
171
and with the aid of the Galerkin method we find 3 (-
b
B 1mn =
B2mn
B"'lmn
eJ3uo 8x 3
eJ3uo B"'2mn 8 3 y -
3 (-
=b
-
u
eJ3 o ) + =B"'lmn 8x8 y2
;
3
=
8 uo ) + B"'2mn 8x 2 8y ;
being numerical coefficients. This approach to solving local problems with the aid of the modified Bl1bnov-Galerkin method turns out to be especially efficient for the determination of the global characteristics, displacements and averaged coefficients, since for the determination of the latter one Can use integral representations. As an example we consider a plate for which alb = 1/3. If in expansions (3.2.40), (3.2.43) we restrict ourselves to one-term approximations, then for the coefficients we obtain B"'lmn, B"'lmn, B"'2mn, B"'2mn
Aill = 0.0102; A"'210 B"'110 B"'lll
A "'210 = A "'201 = -0.0156;
= A"'201 = -0.0090; = B"'201 = 0.0074; = B"'211 = 0.0042;
B"'110 B"'lll
= =
B"'201 B"'211
= 0.0014; = 0.0055.
Then, after some transformations of (3.2.39), we obtain the averaged equation in the form
- 84uo A 8x4 where
A, 2B
A=
- 84uo
- &4uo 8 y 4 = f,
+ 2B 8x 2 8 y 2 + A
are the averaged coefficients
0.860;
2B = 1.690.
In the case of one-term approximations, comparison with the known values [40d] shows the satisfactory accuracy of the results.
3.3 Averaging Procedure in the Nonlinear Dynamics of Thin-Walled Structures 3.3.1 Berger and Berger-Like Equations for Plates and Shells In 1955 Berger proposed approximate nonlinear equations for the deformation of rectangular and circular plates, neglecting the second invariant of the strain tensor in the potential energy expression (the "Berger hypothesis" [51]). Berger's equations have become widely used due to their simplicity and visualization. Later Berger's results were generalized for shallow shell and sandwich plate problems.
172
3. Continuous Systems
Similar equations were applied to dynamic problems. The adequacy and applicability of the "Berger hypothesis" were frequently and widely discussed in scientific papers. It has been shown that the "Berger hypothesis" leads to insufficient results when applied to orthotropic plates; there is no obvious pattern of generalization to shallow shell equations (for example the direct application to dynamic equations of a shallow shell was shown in [57] to be erroneous) . Various approaches were proposed to verify the "Berger hypothesis", including extravagant ones (propositions to regard the (1 - v) term as a small parameter, and to neglect the second invariant of the stress tensor instead of the strain tensor in the potential energy terms). Here we describe a noncontradictory derivation procedure of Berger-type equations in the application to rectangular and circular isotropic plates, and isotropic and sandwich shallow shells. It is shown that the second invariant of the strain tensor is small in a random way and this takes place only for isotropic single layered and transversally-isotropic three-layered plates; logically sequential procedures for the composition of Berger-type simplified theories require us to apply the homogenization approach. First of all, let us consider several intuitive considerations. The applicability of the "Berger hypothesis" to isotropic rectangular plates was justified by considerable amount of numerical analysis and appears to be beyond doubt. In other words, the contribution of the second invariant J 2 of the strain tensor to the potential is undoubtedly smaller than that of the first invariant J 1 . Taking into account '. J1
= Cl + C2;
Cl = U x
+ O.5w;;
J2
= CIC2 C2 =
O.25c~2;
v y + O.5w;;
C12 =
uy + Vx
+ wxw y
the corresponding inequality for a rectangular plate 0 < x < a, 0 < Y < b may be written as a
a
!
!(A+B 1 +C)dXd Y »(I-v) j j(A-B 2 )dXd Y
o
0
b
b
(3.3.1)
0 0
A = 2ux v y + w;u x + w;v y ,
+ B 12 , B 2 = O.5B ll + B 22 , B ll = u; + u~, B 12 = uxw; + VyW~, B 22 = (uy + vx)wxwy, C = O.25(w; + W~)2. B 1 ,= B ll
The main difference between the left and right hand parts of (3.3.1) is connected with the C-term. Let us consider the eigenvalue problem assuming that the displacements and bending moments are equal to zero along the plate boundary. Applying Galerkin's procedure for the one-term approximation (u, v, w) = Ai(t) x sin(m1rx/a) . sin(n1ry/b) one can see that the (A + Bd
3.3 Averaging Procedure
173
and (A - B 2 ) terms contribute equally (at least, by order of magnitude) to the potential energy, except for the special case a = b, m = n. Hence, the Cterm contribution to the potential energy must prevail, as for the m, n ~ 1 case. Then, due to differentiations, the magnitude of the C-term becomes significant. Moreover, the C-term contains a slowly varying part instead of the rapidly varying B 12 , B 22 terms, and the integrals of the former ones become small. These considerations have led us to the decision to use the homogenization method (the nonlinear WKB-method [160]), based on the high variability of the solution along spatial coordinates, for the purpose of composition of Berger-type equations. The nondimensional equations of motion of a rectangular plate may be written as (12(1 - V2))-I£\72\72w + £(FEEWT]T] - 2FET]WET] + FT]T]WEE) + WTT = 0, \72\72F + £( WEEWT]T] - W~T]) = 0, FT]T] = (1 - V2)-I(11.E + 0.5£w~ + V(VT] + 0.5w~)), FEE = (1 - V2)-I(YT]
+ 0.5Ew~ + v(11.E + 0.5w~)), FET] = -0.5(1 + V)-I(UT] + vE + £wT]wT]). where £ = h a; (E,1]) = (x/y)a; P = P/ Eha; (11., V, w) = (11" r = dt p(1- n11, 2)/E; \72 = 8 2/8E 2 + 8 2/81]2.
v, w)/h;
The most natural way of introducting "rapid variability" into the nonlinear system requires one to include the "rapid" variable £OO(E, 1]), regarding it as an independent variable. The value of a would be specified during the limiting (£ -+ 0) system derivation process. Now, following the multiple scale method, we obtain (the notation E, 1] describes the "slow" variables, as before)
8 8 00 8 8~ = 8E + £ T] 80 ;
8 81]
8
= 81] + £
00 8 T] BO .
We suggest that the functions P, w, 11., v are sums of "slow" (Le. depending upon the "slow" variables only) and "rapid" periodic components of the unknown period Oo(E, 1]) [12, 3d, 4d]: F = pO(E, 1]) + £{3t p1 (E, 1], £00),
+ £{32 W1 (E, 1], £00), 11. = 11,O(E, 1]) + £{33U1(E, 1], £00), v = vO(E, 1]) +£,B4V1(E,1],£OO).
W = wO(E, 1])
The following relations are to be used, too: pO ,..., £/'I WO ;
w0,..., £/'2;
11,0,..., £/'3;
V°,..., £/'4;
8/8r( ... ),..., £6( ... ).
There are asymptotic integration parameters ,Bi, /'1, /'2, /'3, /'4, fJ describing the relative orders of magnitude of the "slow" and "rapid" components: pO and wO, 11,0, vO and £. The noncontradictory choice of its values, being routine
3. Continuous Systems
174
work, has to be managed while satisfying the conditions of the noncontradictive character of the limiting (e ---+ 0) systems. The nontrivial limiting systems may be obtained from (3.3.1), assuming
a
= -0.5,
11 = 1,
(31
12
= 0,
= 0,
(32
< 0,
/33, /34
13, 14
> 0,
8
> -0.5,
=0
and may be written as (12(1 - v2))-IWJooo(lJ~
+ 0~)2 +
+(F~€O~ - 2F€110€011 + F~110VWJO + W~T = 0, FJooo(O~ + O~)
=
0,
(3.3.2) (3.3.3)
e- 1 FJoO~
+ F~l1 = 0.5(1 - v2)-I(WJ)2(0~ + vO~), e- 1 FJoO~ + F~€ = 0.5(1 - v2)-I(WJ)2(O~ + vOV, e- 1 FJOO€Ol1 + F~l1 = -0.5(1 + V)-I(WJ)20 xi0l1 ·
(3.3.4) (3.3.5) (3.3.6)
The underlined term in (3.3.2) may be derived using a O-averaging pro-
o
cedure, (... ) = 00 1 J( ... )dO --
0
1
-0
Foo = 0,
U
r..o
-0
0
0
0
(F€€, .l'ryl1' F€l1) = (F€€, F1J1J , F€l1) ,
= 0.5(1 - v2)-I(WJ)2(0~
+ O~).
Using the previously introduced variables we get a
b
:
(wm9~ + 9~) = :b I I (w~ + w~)~ dy + O(E); o 0 equation (3.3.2) becomes the Berger equation D\1 2\1 2w + N\1 2w + phwtt = 0; \12
2
=
8 8x 2
fj2
+ 8 y2;
II o
B =
Eh 1 _ v2'
(3.3.7)
b
a
- B N2ab
2
D' Eh B = 12;
2 2 (wx+wy)dxdy.
0
The strain compability equation becomes linear:
\14F = O.
(3.3.8)
One could easily obtain Berger's equations fOT the viscoelastic plate,
h2
12 r\12\12w - r N\1 2w T
where
+ phwtt
= 0
r'l/J = 'l/J J R(t - rI)drl, and R is the relaxation kernel. o
3.3 Averaging Procedure
175
Circural plates are to be considered separately for centre-holed and continuous plates. In the first case, Cartesian coordinates may be used, regarding (3.3.7), (3.3.8). In the second (ro = 0) case, r- 1 varying coefficients are to be taken into account. Finally, one obtains (3.3.7), where
82
1 fJ 'l = 8r2 + ;: 80 2
+
1 82 r 2 8e2 '
2n R
N =
2:R2 !J(w; + w;) dr de.
o0 Let us consider a shallow shell, the curvatures of which are k 1 , k 2 , and the in-plane dimensions are a and b. Assuming a ""' b, ki ""' 1 and following the procedure described herein, we obtain Berger's equations
4 + h\1kF + N\1 2w + ~ (W xx
D\1 w
a
! !(k + 1
o a
+W••
b
vk2)w
dydx
0
b
! !(k + 2
Vk1)WdYd Y)
+ phwtt
= 0,
o 0
'l4p
+ E'lkw = 82
'l k
0,
(3.3.9)
82
= k 1 8x 2 + k 2 8 y 2.
Two points of special value may be outlined. Firstly, equations (3.3.9) allow all the possible natural limiting passags: a Berger plate; a Kirchhoff nonlinear bar, a shallow arc and, finally (the problem which the Berger hypothesis approach failed to overcome), linear shallow shell equations. Secondly, the "Berger's hypothesis u applied to (3.3.9) appears to be invalid (the second invariant of the strain tensor energy term is not smaller in order of magnitude compared with the first invariant term). Let us look for approximate equations of transversally-isotropic sandwich shells. Introducing Do = Deo;
where the parameters e, jL depend on the bending stiffness of the load-carrying layers, and on the sandwich shear-resistance (e, eo, /3), can be calculated using the formulas given in [2] (Pk' h k denote the density and thickness of the k-th layer).
176
3. Continuous Systems
1, J.L ""' hI R, 0 ""' 1, approximate equations can written as Do(1 - OJ.LR 2V'2)V' 2V'2 X + hV' kP + NV'2 w
Assuming 00
""'
a b
+ ~ (W xx /
/
o
+ IIk 2)w dydx
0
b
a
+W yy / o
(k,
/
(k 2 + Vk1)WdYdX)
+ PIWtt =
0,
(3.3.10)
0
V' 4p + EV'kW = 0, 0.5(1 - v)J.LR 2V'21/J = 1/J } W = (1 - J.LR 2V'2)X .
(3.3.11)
Considering the case 0 < 1, the underlined term in (3.3.10) must be omitted as well. Limiting systems, governing static and dynamic behaviour of nonlinear bars, and linear plates are of no interest to us and are omitted. This investigation can be concluded as follows: 1. "Berger's hypothesis" in its initial formulation appears to be true for
isotropic single-layered and transversally isotropic multi-layered plates only. 2. As a matter of fact, Berger's equations represent the first approximation a of homogenization procedure (the nonlinear WKB method) when the rapid variability of the solution witH respect to spatial coordinates is assumed.
3.3.2 "Method of Freezing" in the Nonlinear Theory of Viscoelasticity The classical averaging method (in the form of the "method of freezing" [37d]) is a very usefull approach for solving the integro-differential equations of nonlinear dynamics. Let us consider for example the equation of the nonlinear oscillation of the viscoelastic rectangular plate (0 ~ x ~ a, 0 < Y ~ b) rV'4 w - 12rh- 2JV'2 w + phD-1Wtt = O. (3.3.12)
r are the polar coordinates: r E [0,1], 4> E [0,0], R is the sector radius, w(r, 4>, t) is the deflection function as a fraction of r, and (J = R2(ph/ D)I/2. For the sake of argument, we shall assume elastic-restraint conditions on the contour of the plate: 2 8w 8 W] (3.4.44) wlrl,r2 = [ u r8
2
1
2
+ 24>1;
-2.... ( aU 2 + aUI) + ~4>I4>2; 2R
OXI
U2
= - R aX2 - R'
OX2
1
4>
(8u 2
2
aUI)
= 2R aXI + OX2 .
Equations (3.5.29)-(3.5.30) are the nonlinear quasimembrane shell motion equations (only bending moments in the circumferential direction are considered) without tangential inertia. These equations may be obtained as a result of the reduction of general relations if it is assumed that the following relations are satisfied e22
= 0,
eI2
= O.
These relations denote physical conditions of extension in the circumferential direction and the absence of shear in the middle shell surface.
202
3. Continuous Systems
For ok > 0 it is possible to omit the term in brackets in relations (3.5.27)(3.5.30). The equations of membrane and quasimembrane vibrations are of fourth order with respect to the axial coordinate Xl, and they can be satisfied by two boundary conditions on every shell edge only while integrating the corresponding limiting systems. From the point of view of singular perturba.tion theory we deal with outer solutions and we must construct inner solutions (boundary layers) [83, 120, 122, 29d]. The boundary layer solution has a large variability index in the Xl direction, and its variability in the circumferential direction and in time is the same as with the inner solution. Now let us present all the stress-strain state components U as follows =
U
U(O)
+ U(k) ,
(3.5.31)
where the indexes (0) and (k) indicate the components of the outer solution and the boundary layer respectively. It is necessary also to introduce the parameter v characterizing the order of w(O) with respect to w(k): w(k) '" crw(O).
The value of the parameter v and the boundary layer variability in the Xl direction depend on the boundary conditions and are defined in an asymptotic, splitting process. As an illustration of the method used we consider the boundary conditions for the variant G~~. First of all, we write asymptotic orders of the components of the boundary conditions for the inner solutions u(O) '" 1
c- 1/
2W(0)'
aw(O)
1/2-2a;w~)
' a Xl '" c 1
'" 1
3/2-a;
N(O)
;
12
'" C 1
(0). W,
(3.5.32) U(k) '" 1
N(k) 12
c 1/ 2W(k)
'" c 1/ 2 + v W(0).
1 c- 1/
'" '" 1
1
2- a ;
W
(k)
aw(k)
'aXl
1/2-a;+v (0) '" c 1 W;
-1/2 (k) '" c1 w
W
(k)
'"
-1/2+v c1
(0). W,
v (0) '" c 1 W .
We choose the value of the parameter v from the condition of the absence of a contradiction in the limiting boundary value problems (in other words, the number of the boundary conditions for the limiting system must coincide with the order of the differential equation with respect to xd. In the case under consideration the unique possible value of v is v = 1 - 20 2 > O.
(3.5.33)
Let us emphasize that the boundary layer nonlinearity order estimation results immediately from (3.5.33) w(k) '"
cl R .
3.5 Regular and Singular Asymptotics
203
Estimating (3.5.32) and taking into account (3.5.33), we find w(O) '" c~(1-2a;)w(k) (0)
Ul
»
w(k),
-(1-2ai) (k) --..
'" cl
u l
8w(0) '" 8w(k) 8Xl
(k)
~ u l
,
N(k)
N(O) 12 '"
8Xl'
12'
Then, the splitting of the boundary conditions may be represented in the following form W (O) 12:1=0,l --
u(O) 1 12:1=0,l --
8W(k) 8Xl
o·,
8w(0) 12:1=O,l
=
8Xl
Izl=o,l;
(3.5.34) (k) 1 _ (0) I N 12 zl=O,l - - N12 Zl =O,l·
(3.5.35)
Therefore, for the outer and inner solutions, boundary conditions (3.5.34) and (3.5.35) must be, respectively, given. In the same way other boundary conditions are split. The results are presented in Table 3.3. To obtain the boundary layer equations, let us take (3.5.31) into the initial equations and take into account the outer solution equations (3.5.33)(3.5.34). Table 3.3. Splitting of the boundary conditions for the half-membrane state and boundary effect
c
Splitting boundary conditions w(O)
12
C34
,
8u~k)
U(l)
w(k)
'Xl'
+ ...l.. (8UJ(k») 2 = L(k) 2R 8X2 1 w(O) w(O) L (k) M(O) + M(k) , Xl' l' 11 11 W(O) N(O) L(k) w(O) + w(k) , 11' l' Xl Xl W(O) N(O) L(k) M(O) + M(k) , 11' 1'11 11 W(O) u(O) w(k) N(O) + N(k) , l' Xl' 12 12 W(O) N(O) w(O) + w(k) N(O) + N(k) , 11' Xl xl' 12 12 w(O) w(O) N(O) + N(k) M(O) + M(k) , Xl' 12 12' 11 W(O) N(O) N(O) + N(k) M(k) + M(k) , 11' 12 12' 11 11 w(O) u(O) Q(k) w(k) + w(O) , l' l' Xl Xl w(O) N(O) w(O) + w(k) Q(k) , 11' Xl xl' 1 w(O) w(O) M(O) + M(k) Q(k) , Xl' 11 11' 1 w(O) N(O) M(O) + M(k) Q(k) , 11' 11 11' 1 u(O) N(O) w(O) + w(k) Q(k) l' 12' Xl Xl' 1 N(O) N(O) w(O) + w(k) Q(k) 11' 12' Xl Xl' 1 u(O) N(O) M(O) + M(k) Q(k) l' 12' 11 11' 1 N(O) N(O) M(O) + M(k) Q(k) 11' 12' 11 11' 1 _
w(k)
8X2
C 38 12 13 C46 13 C68 23 C 45 34 C 56 23 C 58 35 C68 f2
C 47 14 C67 12 C 78 16 C 78 24 C 57 45 C 67 25 C 78 56 C 78
1~
204
3. Continuous Systems As a result, we get the following limiting systems *
02
*
1 = 2'
0 1
1 = 0 ,v * = 1 ,05* = 2'
0 3
*
06
= 1-
ok = Ok,
(k = 1 - 6), v* = v for WS and SS; = Ok, (k = 2,3,4,6), v* = v - 1/2 for RS. The equations of motion have the form Here
ok
*
1 < 2'
N(k) 8 _..;;..;11 ..... 8Xl
+
8 2 M~~) 2 8 Xl
8N(k) 12 = 8X2
+
0 '
8N(k) 12 8Xl
2 (0) 8 w(k) N ll 8 2 =
(k) RN22 -
0,
0;
(3.5.36)
N~~) = B2lC~~) + B22c~~), M~~) = Dll~~~) + Knc~~) j
N~~) = B33C~~) ,
1
8u(k) 1 (8 2W(k)) 2 + _ 1 8w(0) 8w(k) ; l_+_ R 8Xl 2R 8xr R2 8Xl 8Xl
1
(k)
1 8w(0)
w(k)
c22 = -
R + R2
aw(k) .
8X2
8X2'
(k 1 (8U~k) 8U~k)) c )-- --+-12 -
oi = 20 1, Os = 20 5,
Xl
o = Bl1c~~) + B12C~~) + Kll~~~) c(k) 11 -
8N(k) 22 = 8X2
+
*
02'
2R
8X2
8Xl
1 +_
(8W(0) 8w(k)
R2
8w(k) 8w(k)
+ 8w(0) 8w(k)) I .
8X2
8Xl
1
+ ---R2 8Xl 8X2 8X2
8i l
'
(k) _ 1 8 2 w(k) ~ll - - R2 8x~ .
We add equations (3.5.36) to conditions (3.5.35) and get a well-posed approximate boundary value problem to satisfy the boundary conditions. In (3.5.36) ihe variable coefficients, which are obtained by the outer solution term, can be "frozen" with respect to Xl on the shell edges. This is valid because the outer solution variability index in the axial direction is much smaller than the boundary layer index, and in the zone localized near the shell edges, the inner solution may be assumed constant with respect to the X 1 variables. We can limit the equations for the boundary layers, and thus give the possibility of satisfying the boundary conditions, which differ from (3.5.36) by the presence of the inertial term in the third equation of motion. The above derived sets of equations are sufficiently accurate to describe the bending state in a shell. Let us formulate simplified WS boundary value problems: 01
=
1
02
= "2'
03
= 0,
04
= 1,
05
= 02
=
1
"2'
3.5 Regular and Singular Asymptotics
205
In this case the limiting equations are equal to well-known equations for the shallow shell theory. One has to extract the parameter cs for these shells of this class from geometrical-rigidity parameters, which are the tangent rigidity and the extension-compression ratio. If reinforcement is strong enough, these values are small. Let us introduce the parameters of asymptotic integration Qk (k = 1- 6): a
-01
-
cs
f'V
a aX2
;
aXl W
R
Q4
f'V
cs
.
U6
Cs W,
f'V
a
j
-
-03 cs ;
f'V
at
05
til
;
-02
Cs
f'V
06
f'V
Cs
w.
As a result of the asymptotic procedure we get the following limiting systems, which do not have an analogue in the isotropic case
(a)
Q1
aN(l) _....::.:ll~ aXl
= Q2,
+
= Q4 = 2Q2,
Q3
aN(l) 12 aX2
a 2 M(l)
_~ll=- + 2 a Xl2
aN(l) 12 aXl
= 0,
a 2 M(l) 12 a Xl a X2
+
D
a
(1) 1~ (1) _ _ cll - R aXl (1) _ c12 -
+2
-.!... (aU~l) 2R
aXl
aN(l) 22 aX2
(3.5.37) a 2 w(1)
( )
D
pR 2 _......,...._ = 0', a t2
-
aU~l)) aX2
(1) 22 K 22
Ng) = B33C~~),
(1). + K 22c22 ,
M(l) 12 -
(a (1)) R +
2 (1) _ (ifJ(l)). 1 ,c22 -
+
= 0,
+ RN221
M(l) 22 -
I
~
= Q6 = Q2;
0 = B22C~~) + K22K~~,
(1). + K llcll
(1) llK ll
+
a 2 M(l) 22 a X2 2
o = Bllcg) + KllKg) , (1) M II -
as
~
~ aX2
_
(1)
W
~ 2
D
(1). 33 K 12 I
2. (ifJ(l)) 2 ,
~A;,(1)A;,(1) + 2 '.i'1 '.i'2 .
The stress-strain state of a WS is approximately described by an equation of three types. U nUke in the case considered above, in this case it is necessary to introduce not one, but two parameters VI and V2 characterizing the ratio of the order of magnitudes of the quantities defining each of the three states W(l), w(2), w(3): w(2)
f'V
c~lw(l),
w(3)
c~2W(1).
f'V
(3.5.38)
The VI and V2 values are defined in every case by the boundary condition splitting process. Moreover, it is possible to have the following VI and V2 values 3 3 VI
= 2j
V2
= 2;
VI
= "2 j 5
= "2;
V2
=
VI V2
2;
= 2;
VI
= 2;
V2
= "2'
5
The corresponding limiting systems are:
(b)
Q1
= Q2 -
!
I
Q3
= 2a 2,
VI
= ~ (VI =
2), Qs
= Q2 - ~,
a6
= a2;
206
3. Continuous Systems
(3.5.39)
Ng) = B 11 c Ng)
= B 33 cg),
"'(121) _
2.. aui2) .
-
L(w,4»
+ PR2~~.
212
3. Continuous Systems
As a result we obtain the following ordinary differential equation with constant coefficients for the time function e(t):
~:; + o{ [ G¥; Y+e~:n + A,O A e' + A 2
3{5
= O.
(3.5.49)
JB
Here tl = l (pR2)-1 t; Al = e~e4+2e~e3e4P-2+e~e2e4P-4+s24(1-e~s~)2; A 2 = 116 + !s~de4 - ~(1- e~s~); A3 = ~s~; a = 332s~. Let us consider a practically important case of steady-state periodical vibrations and use the method of strained coordinates [119, 120, 122] for solving (3.5.49). We change the independent variable tl to a new one r = wt l , where w is an unknown frequency of the periodic solution. Then (3.5.49) must be replaced by . d (3.5.50) dr'
()=
The initial time point may be chosen in any way because of the periodicity of the solution.. Without loss of generality let r =
e= I, e= O.
0,
. (3.5.51)
Let us introduce a formal small parameter and pose
e(r) = e6(r) + e 26(r) + e 36(r) + ... W = Wo + eWl + e2W2 + e3W3 + ...
(3.5.52) (3.5.53)
with the constraint that expansion (3.5.53~ is uniformly asymptotic [119, 120, 122]. Now substituting (3.5.52), (3.5.53) into (3.5.50) and comparing the coefficients of en in the usual way, we find 1
e
2
e 3 e
e
4
2"
wo6 + AIel = 0,
(3.5.54)
2
"
wo6 + A l e2 = 2WO,W16, 2"
(3.5.55)
2 . . . .
wo6 + A le3 = -(WI + 2WOW2)6 - 2WOW16 2 '2 .. 2 3 - awO(6el + 6el) - A 2 l ,
e
(3.5.56)
W5e4 + A le4 = -2(WOW3 + WOW2)el - (w~ + 2WOW2)e2 - 2WOWle3 - a [W5(26ele2 + 26e16 +
+2WOWl(6e~
+
e~e'l)]
-
3A3e~6,
e~e2 + e~e2) (3.5.57)
The initial conditions (3.5.54) give us 6(0) =
I,
ei(O) = 0,
el(O) = 0, ei(O) = 0, i = 1,2, ...
(3.5.58) (3.5.59)
3.5 Regular and Singular Asymptotics
213
The solution of the initial value problem (3.5.57), (3.5.58) is Wo =~;
6
= fCOST.
(3.5.60)
Let us rewrite (3.5.55), taking into account (3.5.60): ..
6 +6
=
2f
Jjf;Wl cos T.
(3.5.61)
From the conditions of the absence of secular terms, it follows that WI = O. Then initial value problem (3.5.59), (3.5.61) has the solution 6 = O. In the same way one obtains W2 = 0.125~(3Cl - 20)f2; C l = A 2A 1l ; 6 = -0.03125Cl(COS T - cos 3T)f3 + 0.06250 (cos T + COS3T)f3; W3
C2
= 0.039062~ h'1(-Y2 - 6CI) - 2')'2h'2 + 80) + 80C2] f4; = A 3 A 1 , 11 = C l - 20, "1'3 = 3Cl - 20.
= 0;
W4
l
As a result, we have approximate expressions for the frequency of the periodic nonlinear vibrations:
w=~n;
n = 1 + 0.125 {')'2 + 0.03125 [')'2(')'1 -
2')'2 - 160)
- 6C l ')'1 + 80C2] f 2c 2 } f 2c 2.
For the clamped edges of the shell the displacement w may be approximated as W(Xl' X2, t) = II (t) sin 2 SlXl cos S2X2 + h(t) sin 2 SlXl, where h = l36s~R-l ff. The coefficients of time equation (3.5.49) in this case are
[2
A 1 ="38 2ClC4 A 2 = 12 1
2 c 4P -2 + gClC2P 3 2 -4 + 2s -4( 1 + elc3 2
2 4] ; s 2 -1) + 0.125clC4S2 +"23 [(c62
21 4 A 3 = 128S2;
0
9 4 = 64 s2'
n
are obtained for the following
= 0.012;
C4 = 0.6; C5 = 0.75; = 500; 7 = f Rh- l ; c = 1;
The numerical results for the frequency values of the parameters Cl C6
c6 s 2)2] 2 ;
= 0.015; = 0.005;
C2 C7
= 0.00016;
= 0;
LR- l
C3
= 2;
Rh- l
m = 1,n = 8.
The typical amplitude-frequency dependencies are represented in Fig. 3.7. From these results it follows that the vibration frequency n decreases with increasing amplitude. Consequently, for the stringer shell we have the weak nonlinearity of the soft type.
3. Continuous Systems
214
1.0...-------r------,-----,
I
\
::: ~.~~~..~~-----l_e\. -+c~~~~---
,
0.4 -- --- -~- ------~0.2
-----~-
----~--
0.5
e--------
1
n
1.5
Fig. 3.1. Frequency-amplitude dependence for the stringer shell
As can be seen from the above simplified boundary value problem, after asymptotic decomposition the computation of the shell can be done in several stages; at each stage all equations not higher than the fourth order in Xl must be considered with the corresponding boundary conditions. Simplified boundary value problems, which are novel in the literature concerning shells, may serve as a basis for calculations of a very wide class of problems for elastic shells. They can also be used as a starting point when seeking further reduced equations under additional simplifying assumptions. We note that the results obtained with the use of this separation are in good agreement with the results of numerical computations carried out without separation of the boundary conditions. '. 3.5.3 Nonlinear Oscillations of a Cylindrical Panel The equations of the nonlinear vibrations of an elastic rectangular cylindrical panel [151] together with the initial and boundary conditions can be written in the dimensionless form L1~w
+ 64 a;w -
k62a~p' = 64 q(x, t)
1 L1~P + 2a62L(w,w) + ak6 2a;w
=
+ 62 L(w, p),
(3.5.62)
0,
[w, atw]t=o = 0, [a;p, aXaypa~W + v6 2a;w, a~w + (2 - v)6 2a;ayw]Y=±1 [w; a;w, 62a;p - va~p + (2 + v)axa~p]Y=±1 = 0, L1 I -- ay ':}.2
(3.5.63) = 0,
(3.5.64) (3.5.65)
+ 62 ax2 ,6 = -a2. al
Here Xl
2
= alx,
c =
X2 = a2X,
pha4I D- I ,
a
=
W = alW, F = Dp, T Eh al2D- I , k = a - lR- 1 ,
= ct, Q = qD a -3 ., l . (3566)
3.5 Regular and Singular Asymptotics
215
It is assumed that the transverse load Q is a function of the longitudinal coordinate Xl and the time 'T. The panel platform occupies the rectangle Ixpl < ap, /3 = 1,2. The boundary conditions (3.5.64) correspond to a free edge, and (3.5.65) to a fixed hinge support. Besides problem (3.5.62)-(3.5.65), the nonlinear integro-differential equation of the vibrations of a circular arch, written below in dimensionless form
2 (1- v )a:w + 8,w -
~(k + &,w)
1
J[~(axW)2
- kW] dx = q
(3.5.67)
-1
[w, atw]t=o = 0,
[w, a;W]X±l = 0
is considered.
A natural small parameter 6 occurs in the system of (3.5.62)-(3.5.65). Therefore, there is a problem of constructing an asymptotic form as 6 --+ O. Asymptotic expansions are constructed in the form
~ 6m [ wm(x, Y,t) W = ~o ~=
l; 6
m
00
[
1+x 1+X] , + um(-6-,y,t) + vm(-6-,y,t) 1+x
(3.5.68)
1+x
~m(x, Y, t) + CPm(-6-' Y, t) + '¢m(-6-' Y, t) ] .
The functions Wo m , ~m are found by using a first iteration process. For this the solution is sought in the form 00
{w,~} = L
6m{wm,~m}.
(3.5.69)
m=O
We substitute (3.5.69) into (3.5.62)-(3.5.65) and collect coefficients of identical powers of 6. Equating the coefficients of 60 and 6 1 to zero, to determine Wo, ~o and WI, ~1 we obtain
a~Wm = OJ
[a~Wm, a;Wm]Y=±l = 0;
[w m , a;Wm]X=±l = 0;
(3.5.70)
m =0,1
a~~m = 0; Seeking W m
, ~m
[a;~m, axay~m]Y=±l = 0;
[a;~m, axa;~m]x=±l = OJ
in the form 3
{Wm,~m} = Lyj{Wm,j,~m,j} j=O
we have from (3.5.70)
Wm = wm,o(x, t)+YWm,l(X, t), [wm,o, a;wm,O]X=±l = 0, ~m = 0.(3.5.71) The function WO,O is still uknown and will be determined below. The function ~o is taken to be equal to zero since it follows from the formulation of
3. Continuous Systems
216
problem (3.5.62)-(3.5.65) that the function g> is determined to the accuracy of the linear components in x and y. Continuing the iteration process it is found that the functions Wm,j, tPm,j vanish for odd values of m and j. Consequently, henceforth in this chapter we speak only about evaluating the function Wm,j, tPm,j for even m and j. Equating the expression for 6 2 to zero and taking (3.5.71) into account, we deduce
W2 = W2,O(X, t) + y2 w2 ,2(X, t); tP = C2(t)y2.
2W2,2 = -v8;wo,o;
(3.5.72)
The function W2 ,0 is still unknown and will be determined below. At this stage of the first iteration process, the conditions on the boundary x = ± 1 are not satisfied. The discrepancies occurring here are later compensated by using boundary layer functions. To determine C 2 (t) we will use the well-known identity connecting the functions tP and W for a fixed reinforcement of the boundary x = ± 1 in the longitudinal direction 1
1
f(a~ 2), needed to close (3.5.77) and (3.5.78) are obtained here simultaneously. We substitute (3.5.68) into (3.5.62)-(3.5.65), we take account of the results of the first iteration process, we make a change of the variables x = -1 + 6e (x = 1 + 6() and we collect coefficients of identical powers of 6. Equating the coefficients for 60 to zero, we find a system of nonlinear equations with zero
218
3. Continuous Systems
right-hand side for Uo, CPo from which we obtain uo coefficients for 61, 6 2, 63 to zero we deduce
+ CPo =
O. Equating the
L1~CP2 = 0, [a~CP2' aeayCP2]y+±1 = 0 (3.5.79) Acp21e=0 = 2VC2(t), Bcp21e=0 = 0, [Acp2' Bcp2]e=l-ooo --+ 0 L1~u4 = ka~CP2, [U4, alU4]e=l-ooo --+ 0 (3.5.80)
Ul = CPl = U2 = U3 = 0,
[a~U4 + ValU4' ~U4
+ (2 - v)a~ayU4]y=±1 = 0
= -w4Ix=-l,
alU4le=0 = -o;w2Ix=-1
u41e=0
(L1 2 = al +a;,
A
= al- va;,
B
2
= al + (2 + v)aea~, 1=6'
We note that the boundary value problems for u m , CPm are linear for m > 1. The functions v m , 1/Jm are found analogously. We will illustrate the calculation of the boundary layer function U4 for the case of a rectangular plate (k = 0). We construct the solution in the form 00
U4 = aoeSOeFo(Y)
+ 2Re
L
ame-Sont; Fm(y).
m=l The Papkovich functions Fm(y) are determined from the boundary value problem (the prime denotes the derivative with respect to y):
+ 2s~F~ + s~Fm = 0 [F~ + vs~Fm, F::: + (2 - v)s~F:n]y=±l = 0
F/nv
(3.5.81)
(so, Sm are, respectively, the real and co~plex roots of the equation !P(s) = (3 + v) sin 2s -- (1 - v)2s = 0). . To calculate am from boundary conditions (3.5.80), the problem is posed of representing the two real functions /I = -W4( -1, y, t) and h = -a;W2( -1, y, t) in the form of the series 00
{/I, f2} =
L {I, s~}amFm(Y).
(3.5.82)
m=l Here the time t plays the role of a parameter. To obtain the initial conditions for t = 0 for the function W2m ,0, we substitute (3.5.68) into (3.5.63), and we collect the coefficients of identical powers of 6 and equate them to zero. In particular, the coefficient for 6° yields the initial conditions written in (3.5.67) for wo,o, The consistency conditions
q(±I,O)
= a;q(±I, 0) = at q(±I, 0) = 0
should be satisfied here. The coefficients of 62 and 64 are reduced, respectively, to the zero-initial conditions for the functions W2,0, atW2,O and W4,0, atW4,0. Analogous consistency conditions on the higher derivatives of q are added to construct the next terms of the expansion.
3.5 Regular and Singular Asymptotics
219
After evaluating the principal terms of expansion (3.5.68), the process of constructing the next terms of the asymptotic form is continued analogously: functions of the first and second iteration processes are determined alternately. The boundary values of the functions of the first iteration process W2m,O, 8;W2m,o are determined simultaneously in the solution of the boundary layer problems. In the case of rigid clamping of the panel edges Xl = al ([w, 8x W]x=±1 = 0) the principal term of the expansion is also determined from the equation of arch vibrations, but with the boundary conditions [w, 8x W]x=±1 = O. In the case of the hinge of supports or rigid clamping of the boundaries X2 = ±a2 there is no passage to the limit from the equations of the vibrations of a cylindrical panel to the equations of the vibrations of an arch.
3.5.4 Stability of Thin Spherical Shells Under Dynamic Loading 1 As shown below, the theory of two-point Pade approximants gives the possibility of obtaining a solution of very complicated problems. The very interesting example of using this technique is matching by TPPA coefficients of limiting equations and constructing on this basis the constituting equation, which may be exploited for any values of the parameters. In the course of solving the problem of the stability of shells under dynamic loads, it becomes necessary to describe the motion of its mid-plane with deflections that are large when compared to the thickness. Characteristic forms from the linear theory are usually chosen as the approximating functions when approximate analytical methods are used to reduce the initial system of partial differential equations to a Cauchy problem for ordinary diferential equations. However, the effectiveness of such approaches is limited to the region of small values of the deflection amplitude. Here an asymptotic method is used to obtain the corresponding differential equation describing the motion of a shell with significant deflections. Since one may use as a small parameter a quantity which is proportional to the ratio of the thickness of the shell to the amplitude of its deflection Wo, the resulting equation will be more accurate, the greater the deflection of the shell. To describe the motion of the structure throughout the entire range of displacements, we obtain an ordinary differential equation whose coefficients are determined by combining the corresponding expansions for large and small deflections. The thus-formulated Cauchy problem is solved numerically by the Runge-Kutta method. The efficiency of the proposed approach is evaluated by comparing the results of calculations with known experimental data. As the initial equations, we will examine the equations of motion of an orthotropic spherical shell written in terms of the stress function ifJ and w for the case of the axisymmetric deformation 1
By courtesy of A.Yu. Evkin
220
3. Continuous Systems
D1l
WIll) -D22 ( -wI - -wI ) = h a (aw aifJ) IV +2----2 3 r r r r 8r ar ar 2 h a ( raifJ) aw +-- +q+-ph-; rR ar 8r 8t 2
(w
u
B
( ifJI
V
(>II 1 )
+ 2-r-
( (>1
-
~ +.!. a 2w aw + ~~ (>1 )
B22 ?'" -
r ar 2 ar
B U B 22 - B~2
r
R ar
(3.5.83)
(r
aw) = ar
o.
After the substitution of variables r2
hifJ F=-
z---
- woR'
w W=-, Wo
Wo
qR2
_
Bu
Bu
q=4
we obtain c: 2[z 2W IV
+ 4zW III + W II (9 -
= 2(zw I FI)I + (zFI{q -
Ao)/4] 2w a C fh2 '
(3.5.84)
c: 2[z2 FIV + 4zF III + F II (9 - a)/4]
= WI (2zW I + WI) + (zWI)I,
(3.5.85)
where
Ao = D 22 , Du w Wo
Bu
a _ B 22 - Bu '
= h'
= wot,
and Wo is the natural frequency of linear vibrations of the shell. In the case of an isotropic sph~re
T
c:
2
C
=
2h woV3 (1 - v 2 ) ,
= pR2
w5V3(1 2E
Ao
= 1,
a
= 1,
v 2)
.
When the amplitudes of the deflection wo are sufficiently great when compared to the thickness of the shell, the parameter c: 2 becomes small and can be used in an asymptotic integration of system (3.5.84)-(3.5.85). In [36d], the corresponding procedure was performed for the case of a static load on an isotropic spherical shell. It was established that the main approximation of the asymptote yields good results when wo/h > 4. In the case of an orthotropic sphere, the parameter c: 2 can also be regarded as small for the corresponding deflection if we exclude from consideration shells in which there is a substantial increase in flexural rigidity in the meridional direction D 11 .
3.5 Regular and Singular Asymptotics
221
When c = 0, (3.5.85) has two solutions. The first, WI = 0, corresponds to the momentless state of the shell. The second solution, WI = -1, corresponds to the mirror reflection of the part of the shell relative to the plane whose intersection with the sphere gives a circle of radius rl = v'woR. Thus, in the case of large deflections, the form of the shell becomes determinated and in the initial variables it is described by the function
In the neighbourhood of r = rl (z = 1), discontinuities in the derivatives are compensated for by rapidly changing functions of the internal boundary layer. We obtain the following equations for them in the main approximation of the asymptote: v IV
+ 2(v I uI)I
- u lI = 0,
u IV
+ vI (1 -
2v I ) = 0,
(3.5.86)
these equations coinciding with the corresponding equations of the static problem [36d]. Here, the functions u and v are differentiated with respect to the variable x = (1 - z)/c, W = c(r)v(x), F = c(r)u(x). The boundary conditions fo.r the sphere which is rigidly fixed at its boundary (at r = ro) take the form vI
= 0,
vI
= 1,
u lI uI
= 0,
= 0,
= -xo; X -+ +00, X
(3.5.87) (3.5.88)
where
xo =
Zo -1 c
r5
Zo = woR'
The relations of boundary value problem (3.5.86)-(3.5.87) were obtained with the assumption that
q «1,
a2w C ar 2 '" 1.
(3.5.89)
In this case, neither the load nor the inertial term enters into the equations or the boundary conditions for the functions u and v describing the stress state of the internal boundary layer. It should be noted that the satisfaction of relations (3.5.89) is necessary only for large deflections. Thus, it is satisfied in all cases of practical importance. In particular, it is possible to study the action of shock loads in which the parameter q can rearch large values. However, due to the instantaneous nature of these loads, the given parameter remains small even when the deflections are substantial. The equation needed to determine the function wo(t) or c(r) can be obtained by the variational method. As in [36d], we obtain the following equation for the total potential energy of the system when a uniformly distributed radial pressure acts on the surface:
222
3. Continuous Systems
where D1 =
321rD ll hYb R '
I [( 00
Jo =
2] dx -_ 0.56 + 0.3/22 . u II) 2+ (II) II Xo
-:to
An expression for Jo(xo) was obtained after the numerical solution of boundary value problem (3.5.86)-(3.5.87) for different values of Xo and a subsequent approximation of the corresponding function in the interval 0, 5 < Xo < 00. The last term is connected with the effect of the edge on the deformation of the shell. One obtains the following expression in the main approximation for the kinetic energy of the system:
K=
1rh4 Rpj 12w~ , 2
where 1 = wo/h. The corresponding equation of motion has the form
j 1 + j2 + w~V1 = 4/Yb (qiT) + qo) .
(3.5.90)
Here, one has isolated the dynamic q(T) and static qo components of the load:
A=
w~pR2Bll
3 2
B ll B 22 - B?2 '
w 2 = 6J ob / * A
0 h/lb3 / 4
-
J 0 = 0.56 + ' (y'2H/h -
,v1)
5a
(I < 2H),
where H is the camber of the shell. One may describe the motion of a shell with small deflections by the Ritz method. Here, we make use of the following approximation of the deflection function:
w(;, t) =
{
f(t)h
o
[1- (:.rr 0 m 2 + n 2k } i 2 < m 2 + n 2k i2
{
2(0.5 - J.L) '"'tim =
4
1
'7r(m 2 -i 2 )
(-1 \Tn
+ ~ sin 21rJ.Lm,
,[{
i'1.
.
for i = m
m JSlll1rJ.L'tcos1rJ.Lm+
- { ';' } sin
"I'm "I'i] COS
for i
#
m,
and E' denotes the summation without the component i = m. Let us compare the frequencies given by this method with the exact values for the limit case (J.L = 0) when both sides y = ±0.5 are completely clamped. For the square plate A = (1.47831r)4. The PA of a segment of series (3.6.4) is
232
3. Continuous Systems
_ ao + ale A(e ) - bo + bl e ' where ao = Ao, bo = 1, al = Al + bIAO, bl = -A2/AI· For e = lone obtains A = (1. 70811r)4; the numerical solution is A (1. 70501r)4. Figure 3.12 presents the diagram of the relation of A to J.L for the cases of the symmetrical and the nonsymmetrical restraint layout. 1.75 . . . . - - - - , - - - - , - - - - , - - - - , - - - - r - - - ,
A/1r 1.70
1.65 _ _ _ _ _ -J:
1.60
y
--. - ~ -1-- .--.-
I . I I I
II I
__ (l.~~L .
O~--tri~
__
L.!=_~~_~
1.55
1.50 -- .---
e(1.8%)
. -.-. - --.Y
----~-t~ _~.--I_-...
1.40 '----'----'----'-----'-----'-----' -0.6 -0.4 -0.2 0 0.2 0.4 J1. 0.6
Fig. 3.12. The relationship be'tween the vibration frequency of the plate partially clamped on two opposite sides and the clamped segment length
Let us consider the application of this approach to the static analysis of the rectangular plate (-O.5a < x < O.5a; -O.5b < fj < O.5b), subjected to a uniform lateral load ij. The plate is simply supported along x = ±a/2 and subjected to mixed boundary conditions ("clamped-hinged"), symmetrical with respect to fj. The governing differential equation may be written as DV 4 W = ij.
(3.6.5)
Let us denote
W W=b'
x
x =-, b
3.6 One-Point Parle Approximants
k
ijb 4
a
= b'
q
= D'
233
(3.6.6)
Taking (3.6.6) into account, the governing equation (3.6.5) may be rewritten as \74W = q.
(3.6.7)
The boundary conditions may be formed as k W=O, Wxx=O, whenx=±'2;
(1 - H(x))Wyy ± H(x)Wy = 0,
W=O,
(3.6.8) 1
when Y = ±-, 2
(3.6.9)
where H(x) = H(x - J.Lk) + H( -x - J.Lk). Introducing the parameter £ into the boundary condition according to the procedure, one obtains 1
W = 0,
W yy = H(x)£(Wyy ± W y) when Y = ±'2'
(3.6.10)
The case £ = 0 gives us a plate which is simply supported along the boundary; the case £ = 1 corresponds to the problem under consideration (3.6.8)-(3.6.9). The intermediate values of £ are related to mixed conditions of a "simply supported-elastic clamping" kind with the elastic support coefficient u =
£/(1 - c). In order to solve the problem, let, us represent the deflection of the plate as
W =W1 +W2, q WI = 8k
(3.6.11) (_1)(m-I)/2 (
L
00
m=I,3,S,...
am
amth am + 2 12h ch 2a m y c am
ham ysh 2a m y ) cos 2a m xj c am 1 00 (_1)(m-I)/2 Am 2 W = -8 2 h (amthamchamy am c am m=I,3,S, ... -2amysh 2a m y) COS 2a m xj
+
(3.6.12)
L
1T'm
am =-·
(3.6.13) (3.6.14)
2k
Expression (3.6.12) describes the deflection of the simply supported plate subjected to the uniform lateral load q. Expression (3.6.13) describes the deflection of the simply supported plate, caused by the edge bending moments, distributed along y = ±0.5: 00
Myly=±o.s=
L m=1.3.S....
Am(-1)(m-I)/2cos2amx.
(3.6.15)
234
3. Continuous Systems
Satisfying boundary condition (3.6.10), one obtains the infinite linear algebraic system for the coefficients Ai as the unknowns:
f
Ai( _1)(i-I)/2 = c
'Yim( _1)(m-I)/2 Am [1 _ _1_ ( 40 m
m=I,3,5, ...
q
+th am )] + +c 8k
~. (m2~i2)
'Yim = {
'Yim
m=I,3,5, ...
.(ch~:m -thom), 2 (0.5 - J.L -
(_I)(m-I)/2
00
L
~m
ch am
o~
i = 1,3,5, ... ,
2;#£ sin 21rmJ.L) ,
(isin1rJ.Licos1rJ.Lm - msin1rJ.Lmcos1rJ.Lm) ,
(3.6.16)
i
= m
i
=1=
m.
Let us apply the perturbation technique to system (3.6.16), representing Ai as the c-expansion 00
Ai
=
L Ai(j)c
j
(3.6.17)
.
j=O
Substituting (3.6.17) into system (3.6.16) and splitting it into the powers of c, one obtains the reccurent formulas for Ai:
(3.6.18)
Ai(o) = 0;
L 00
Ai(l)
=
(_I)(i-I)/2
m=I,3,5, ...
q (_I)(m-I)/2 'Ymi 8k , 4 (am . am
(3.6.19) 00
L
Ai(n) = (_I)(i-l)/2
'Ymi( _1)(m-I)/2 Am(n-l)
m=I,3,5, ...
(3.6.20) The truncated perturbation expansion (with holding three initial non-zero terms) may be PA-transformed:
+ alc)(bo + blc)-l, al = A i (2) + bI A i (I),
A i [I/I](c) = c(ao
= A~(I), bo = 1,
(3.6.21)
bl = -Ai (3)/A i (2). Let us consider the limit cases for (3.6.21). Firstly, J.L = 0.5 corresponds to the simply supported edge y = ±0.5. The W'Yrni = 0, therefore Ai(j) = 0, W 2 = 0, W = WI, i.e., the exact solution for the plate, simply supported along the boundary. Secondly, J.L = 0.0 corresponds to the fully clamped edge y = 0.5. Here ao
I,
'Ymi = { 0 ,
when i when i
=m =1=
m
3.6 One-Point Pade Approximants
235
and the recurrent relation yields (3.6.22)
Ai(o) = 0; q 1 ( ai Ai(l) = 8k4 h2 ai
c
ai
-
thai
)
(3.6.23)
;
q 1 ai 1 A.( ) = - - ( - thai) 1 - - 4 [ , n 8k ch 2 ai 4a i
at
(
ai h2 c ai
n-l
+ thai ) ]
(3.6.24)
For c = 1, the PA for the truncated expansion (3.6.22)-(3.6.24) is
A'[l/l](c ,
ai - thai(aithai
q
= 1) = -2ar . ai -
+ 1)
.
(3.6.25)
0.5
Fig. 3.13. The relationship between the normal displacement of the plate partially clamped on two opposite sides and the damped segment length
th ai(aith ai - 1)
4.5 ,....----,...---.,.----..,.----..,-------, I
---J--~
W 10-3 q 4.0
I
3.5
3.0
2.5
- -
-
0.5
0
x
0.5
2.0
1.5
0
0.1
0.2
0.4 J1
The formula (3.6.11), taking (3.6.25) into account, describes the plate deformation when the x = ±0.5 edge is simply supported and the y = ±0.5 egde is clamped. The analysis, listed below, was carried out for the square plate. Expansion (3.6.21) for Ai was truncated to ten (initial) terms for c = 1. The deflection and bending moments in the centre of the plate are calculated for several given values of the parameter J1 (see Fig. 3.13-3.14).
236
3. Continuous Systems
5.0 .------,.-----.------,-----,-------,
M 10-2 q
4.5 .. -
4.0 .
-i--~ I
"----+----,.".H-
Arl-~~---t-1I1 x =0
y=o
3.5
3.0
I
2.0
---J -"""----"I
1.5 ' - - _ - - - J ._ _----'-_ _---'-_ _--'--_------' o 0.1 0.2 0.3 0.4 J1 0.5
Fig. 3.14. The relationship between the moment M 11 in the plate partially damped on two opposite sides and the c1aInped segment length
The results, obtained by the coupled series method [70d], are shown as dots. The dotted curves display the data computed by the finite element method. The discrepancy of the deflection, as well as the bending moments, does not exceed 5%, which confirms the acceptable accuracy of this method. Figure 3.15 shows the values of the edge moments My along y = ±0.5. Various problems of statics and dynamics of plates and shells, subjected to mixed boundary conditions, may be solved effectively on the basis of the approach presented here. Nonlinear problems can very often be solved by means of the perturbative approach. (
3.7 Two-Point Parle Approximants: A Plate on Nonlinear Support The present section deals with the problem of oscillations of a plate (a beam in the limit case) on a nonlinear elastic support. This problem may be solved by numerical or asymptotic methods. In the latter case quasilinear asymptotics are usually used [66, 67]. Then, one cannot obtain solutions for large amplitudes. Here a new nonquasilinear asymptotic is proposed. Heuristically it may be described as follows. In the long-wave approximation plate bending rigidity may be neglected, and we may investigate oscillations of a rigid body
3.7 Two-Point Pade Approximants
237
16 ...----......---------,--------;---,.-----,
Mq 10- 2
-r ,
14
~=0.3
!
.--1 ~-----~-~ /-£=0.2
12 10 . 8 --
/-£=0 6 I
-l
4
- -_.
2
--~
-L
0 -2
I 0.1
0
0.3
0.2
0.4
X
0.5
Fig. 3.15. The values of the moment My alone the line y = ±0.5 for various
p.
on an elastic nonlinear spring. In the short-wave approximation nonlinearity of the foundation may be neglected. It is typical for asymptotic methods that approximate solutions may be formulated for some limiting values of the parameters. For the intermediate case the analysis is very difficult. It is possible to overcome these difficulties using tw~point Pade approximants. Let us consider free vibrations of a beam on a nonlinear elastic support. The governing equation may be written in the form
+ k(w + '(31h2w3) + J.LttW = O. Jl/4; W = wi hI; x = (Trx)1 Lj
EJw xxxx
(3.7.1)
Here hI = {3 = k 2/k 1; k k 1L 41Tr4j J.L = 4 4 J.LL ITr : k 11 k2 are foundation coefficients. Let (without loss of the general character of the solution) the beam be simply supported:
w
(~Tr
1
t) = w (±2Tr t) = xx
1
O.
(3.7.2)
238
3. Continuous Systems
The initial conditions we assumed are as follows:
W(x,O) = Acosnx;
Wt(x,O) = O.
(3.7.3)
e,
Taking into consideration the new independent variables rewritten as weeee + e(w + aw 3 ) + w 2w rr = O. Here = nx; T = n 2(EJ/J.L)1/2tw; e = k/(EJn 4); a = {3h?
T,
(3.7.1) may be (3.7.4)
e
We assume a "'" 1 (this is the case of the essentially nonlinear foundation) and investigate two limit cases. For n "'" 1 (the long-wave case) e ~ 1, for n ~ 1 (the short-wave case) e «: 1. Let e « 1 (the short-wave case). The displacement W and the "frequency" w may be expressed as the e-expansions
W = Wo + eWI + e2 W2 + ... , W = Wo + eWI + e2W2 +...
(3.7.5)
After substituting (3.7.5) into (3.7.4) and splitting it bye, the following recurrent sequence of equations may be obtained:
w~4)
+ w5worr = (4) 2 _ WI + WoWlrr -
0,
2 3 - WOWIWO rr - Wo - aow ,
Satisfying the boundary and initial conditions (3.7.2), (3.7.3), we obtain , W = A cos ( cos T. (3.7.6)
.
The conditions of the absence of the secular term lead to the expressions Wo = 1;
9
WI = 0.5 + 32aA2;
(3.7.7)
W2 = -0.125 - ~aA2 _ 459 a 2 A 4. 32 ,2048 Now we are going to investigate the long-wave case (e » 1). The beam displacement and frequency square "ansatzes" are W -_ Wo + e-1 WI +-2 e W2 + ... , W _eO. 5 [W(O) + e- 1w(l) + e- 2w(2)
+ ...J.
(3.7.8)
Substituting expressions (3.7.8) into (3.7.4) and performing the e-splitting, one obtains the system of equations which permits us to determine the unknown expansion coefficient 2 w(O) = 1 + aA cos(. (3.7.9) Using tw~point Pade approximants, one may obtain an analytical solution for any value of the paremeter e.
3.7 Two-Point Pade Approximants
239
In our case we have =
W
Wo + (WI + w)cO. 5 + w(O) wc 1. 5 1 +wcO. 5
(3.7.10)
where w = W2/(W(O) - wd. We can obtain the solution for the linear case from formulae (3.7.7), (3.7.9), (3.7.10), assuming f3 = O. There is an exact solution in the linear case, and we can compare it with the approach presented above. The numerical results are plotted in Fig. 3.16, where the curves correspond to: 1 - the exact solution; 2 - the matched spectrum expression (3.7.10) ({3 = 0). The results are consistent with the physics of the problem and confirm the reliability of the approximate solution. Twcrpoint Pade approximants overcome the locality of the asymptotic expansions. Curve 2 coincides satisfactorily with the exact solution everywhere in the interval considered.
3,....-----r------r-------,.---.,....---.,...------r---...."..
I I --1- "-----_.
w 2.5
---------t--~~'"
2---- -
1.5
lL.---~--....l...--
o
_
L.-_
__l__ _~_ _____l.
234
1
5
6
_____J
c
7
Fig. 3.16. Comparison of the exact solution and the two-point Pade approximants formula
Now we investigate the case of a plate. Only final results of the asymptotic analyses are displayed here. The initial partial differential equation may be written as weeee
2 2l weeTJTJ
W
= w/h;
+
l4WTJTJTJTJ
+ c(w +
3 QW )
+ W2WTITI =
o.
= my; y = (tryd/L 1; l = L/L 1; Q = f31h2; = p2(mn)4w ; p2 = L 4/J1(Dtr 4); ko = kd(Dtr 4).
Here 7'1
+
TJ
Let the plate be simply supported:
trn
W
= wee = 0 for l ( €,y{t),t,u{t,e), at ' 8t 2 '
(4.2.9)
where: F= L
{I + TJ2)l2 1r 22 c
!I,
(4.2.10)
['(1,,: 1/)1' y(p)(t) - T}(l,,: 1/) y(P+l)(t)
= ~~a...{
+~T;y(P+2) (t) }, 4> =
['(1,,: 1/) l'
'1'.
The nonlinear functions ¢> and F as well as the solution sought, y, u and TJ, are presented in the form of power series: . _
2
2
¢> - ¢>o + €¢>e + € ¢>u + + T¢>T + T ¢>TT + + T€¢>u + , F = Fo + €Fe + €2 Fu + + TFT + T2Fn + + T€Fu + , _ 2 2 Y - Yo + €y~ + € Yu + + TYT + T Yn + + T€YTe + , (4.2.11) u = Uo + cUe + €2Uu + + TUT + T2UTT + + T€U Te + , TJ -_ TJo + €TJe + € 2 TJee + ... + TTJT + T2 TJn + ... + T€TJTe + ....
270
4. Discrete-Continuous Systems
Having substituted (4.2.11) into (4.2.9), and having equated the expression representing the same powers of the small parameters T and £ as well as the same powers of their products T m £l (m,.l = 1, 2 ... ), the recurrent systems of linear equations are obtained. While solving the subsequent equations of the system, we use the harmonics balance method. Let us assume that we have determined the first system of recurrent equations, where the operator (*) means T or £. Having substituted the solutions (4.2.6) for the nonlinear functions F(*) and ¢>(*) (this time for the equation we assume tl = t and ao = 1) and having developed these functions into a Fourier series, we obtain 00
00
F(*)(t, x) = L L sin n;x [A~*2 cos(kt) p=Ok=O
+ B~~ sin(kt)],
00
¢>(*) (t) = L[Ci*) cos(kt)
+ Di*) sin(kt)],
(4.2.12)
k=O
where:
ff = ff C~*) ~ f ~f
21t'
l
A~*2 = :l
o
0
21t'
l
2 1rl
(*) B nk
F(*)(t, x) sin n;x cos(kt) dt dx,
o
.
n1rX F(*) (t, x) sm l - sin(kt) dt dx,
0
21t'
¢>(*) (t) cos(kt) dt,
=
.,
.
(4.2.13)
o
21t'
Di*) =
¢>(*)(t) sin(kt) dt.
o We seek the solutions of'the system of equations formed by comparison of the expression next to (*) in the form of N K U(*) (t, x) = L L sin n;x [a(*)nk cos(kt) n=l k=O
+ b(*)nk sin(kt)] ,
K
y(*;(t) = L[C(*)k cos(kt) k=O
+ d(*)k sin(kt)].
(4.2.14)
The solution of the first equation of system (4.2.9) is explicitly determined only when
4.2 Simple Perturbation Technique
Qe.)n{a?)8' b?)6' 11e) =
271"
ff l
271
Fe.) (t, x) sin n;x sin{nt) dt dx = O. (4.2.15)
o 0 Conditions (4.2.15) allow us to neglect the resonance terms which exponentially grow with time. Thus we obtain 2N of the equations, whereas the unknowns a?)s' b?)8 and 11e are 2N + 1. In this case, however, dealing with an autonomous system, we may assume that be.)N = o. As an example let us consider the discrete-continuous system described by the equations 2 2 8 u = (30)2 8 u{t., x) 2{t )]8u{t.,x) !:U2 8 2 + £[0003. u. ,X !:U UL.
X
~
UL.
+£8{x - x)y{t.) - £T8{x - x) dd
y
t.
8u{t., x) _ 8 2u{t., X)) _ (8U{t., X)) 3 +T ( at. 8x2 T at. '
(4.2.16)
d2 y dy 2 + 10£-d + 400y{t.) = 10£u{t., x), u{t., 0) = u{t., 1) = 0, d t. t. where for the sake of simplification of the calculations, the delay T and the small parameter £ are in the evident form (4.2.1) and x E [0,1] is the association point of the discrete system with the continuous one. In the discrete system described by the second equation of system (4.2.16) accompanied by the lack of interaction on the side of the continuous system and as a result of damping in the system, oscillations cannot occur. The oscillations are excited in the continuous system because of damping of the Van der Pol type described by the second term on the right-hand side of the equality sign. For T = £ = 0 the period of this solution is equal to To = ~ /15. We seek the periodic solution of system (4.2.16) with period T, insignificantly different from the period To. Accordingly, let us first make use of the independent variable
t. = 1 + ~~£, T) t.
(4.2.17)
Having substituted (4.2.17) into (4.2.16), we obtain 2 8 u{t, x) = 2- (I + )2 B2u{t, x) + (I + 11) [0 003 _ 2{t )] 8u{t, x) at2 ~2 11 8x2 £ 30 . u, x fJt +£
(I +'11)2 1+11 dy 900 8{x - x)y{t) - £T 8{x - x) dt
30
1 + 11 8u{t, x) (I + 11)2 B2u{t, x) +T 3O at - T 900 8x2
-T~ 1+11
2
(8U{t,X))3. fJt
'
d y 1( ) dy 4 2 £ 2dt 2 +£3 1 + 11 dt + 9{1 + 11) y{t) = 90{1 + 11) u{t,x).
(4.2.18)
272
4. Discrete-Continuous Systems
The parameters T and € are treated as independent. Assuming one of them to be equal to zero, the problem is reduced to the classical perturbation method. We assume the starting solution in the form of u(O) = u(O) + u(O) = a(O) sin1rxcost + a(O) sin1rxcost , T E: E: T y(O) =
O.
(4.2.19)
The amplitude sought, a~O), will be determined from the first recurrent equation formed by the compan~ion of expressions that are coefficients of the parameter €, whereas the amplitude a~O} will be determined from the first recurrent equation formed by the comparsion of expressions that are coefficients of the parameter T. From the first equation of system (4.2.18), having equated the expressions that are coefficients of the parameters T, we obtain
8 2u 1 8 2u 277T 82u~0) 1 8u~0) - 2= -2- -2 + + --8t 1r 8x 1r 2 8x 2 30 8t 1
82u~0)
- 900 8x2 - 30
) at" .
(8u~0)
3
(4.2.20)
Having equated the resonance terms to zero, we obtain 77T = 0.0055, a}O) = 0.044.
(4.2.21)
The solution of (4.2.20) is UT
=
•
1~8 (a~0))3 sin 31rxsint - 1~8 (a~0))3 sin1rx sin 3t +a sin 1rX cos t,
(4.2.22)
T
where the amplitude aT will be determined from the subsequent reccurent equation. This equation is of the form 8 2u TT 1 8 2u TT '2 8 2u T 2 82u~0) 8t 2 1r2 8x 2 + 1r 277T 8x2 + 1r 277TT 8x 2 1 8u~0) 1 Bu T 1 8 2u T 2 82u~0) 77T + 30 77T ~ + 30 8t - 900 8x 2 - 900 8x 2 8{u~0))3 8{u(0))2 uT -90 at + 3077T 8t
(4.2.23)
From (4.2.23), having equated the resonance terms to zero, we obtain
-2(0) _ 2_ ~ (0) 2 _ 30 77TaT 30 aT 16 (aT ) aT 2 (0) - 77Ta T - aT 77TT
0,
1 2 2 (0) 2 + 900 aT1r + 90077TaT 1r
135{ (0))5_ 135 {a(0))3=0. aT 8 77T T
+ 206
(4.2.24)
4.2 Simple Perturbation Technique
273
Solving the system of equations (4.2.24) we get: aT
= 0.00002,
17TT = -0.00006.
(4.2.25)
From the second equation of system (4.2.18), we obtain YT = O.
(4.2.26)
Let us now determine the perturbation equations formed owing to the comparison of the expressions that are coefficients of the parameter £. From the first equation of system (4.2.18), we obtain 8 (0) 82 82 (0) 82 e ~ = ~ U [0 003 _ { e(0) )2] ~ (2 2 ) at2 1r2 8x2 + 1r2 217e 8x2 + 30' u fJt' 4.. 7
2-
2-
2-
and having equated the resonance terms to zero, we obtain a system of algebraic equations. Solving it, we have a~O) = 0.12649,
17e = O.
(4.2.28)
The solution of (4.2.27) is Ue
=
ae sin{1rx) cost - 5.10- 7 sin31rxcost.
(4.2.29)
From the second equation of system (4.2.18), we obtain
d y ~ _ 2- (0) dt2+9Ye-90ue (t,x). 2
(4.2.30)
We seek the solution of.{4.2.30) in the form
Ye = be cost
+ Ce sint.
(4.2.31)
Having substituted (4.2.31) into (4.2.30), we find
be = -0.088 sin{1rx) ,
Ce
(4.2.32)
= O.
From the second equation of system (4.2.18), having equated the coefficients of £2, we obtain
d 2 yu dt 2
4
+ gYu
=
1. "3besmt
1
.
_
+ 90ae sm{1rx) cost.
(4.2.33)
We seek the solution of (4.2.33) in the form
Yu = bu cos t + Cu sint. Having substituted (4.2.34) into (4.2.33), bee
= -
8
5~ sin{ 1rx),·
(4.2.34) ~e
calculate:
u = -0.0048 sin 1rX.
(4.2.35)
C
From the first equation of system (4.2.18), having equated the coefficients of £2, we obtain 2 82 u _-:::e_e _ _1 8 u U at 2 1r 2 8x 2
+ _2217u 8
1r 1 (0) 8u~0) - 15 u e ue----at
2 e(0) U 8x 2
+ 0 0001 8 U e _
+ Yeo{x -
.
_
x).
at
1 8 {(0))2 U e e 30 u at (4.2.36)
274
4. Discrete-Continuous Systems
From (4.2.36) we finally calculate: 2 a = 0, TJu = -0.01 sin ?rI.
(4.2.37)
e
By means of analogous calculations it is possible to determine the recurrent equations occurring with the combinations €krl, where k ~ 1 and l > 1. In the case when the characteristic equation (4.1.1) does not have imaginary eigenvalues, the periodic solution is sought in the form [37] K
=
v(t,x)
L
LL€k/LlVkl{x,a(t),tb(t)}, k=ll=O K
y(t) = a(t){aei,p(t)
L
+ a:e-i,p(t)} + L
L
€k /LlYkl{ a(t), tb(t)}.
(4.2.38)
k=ll=O
where K
L
~; = LL€k/LlAkl{a(t)}, k=l l=O
dtjJ
ill =
K
w
L
+ L L €k /Ll Bkl{a(t)}.
(4.2.39)
k=l l=O
a and
a: are determined from the equations
p
p
(Ape-Tpwi - Ewi) a = 0,
L
L
p=O
(ApeTpWi
p=O
+ Ewi) a: = o.
(4.2.40)
: '.
The eigenvectors (3 and ~ of the adjoint set of equations are obtained from p
L
p
(A;e
Tpwi
+ Ewi) (3 = 0,
p=O
L
(A;e- Tpwi - Ewi) ~ = 0,
(4.2.41)
p=O
where A; are the matrices, conjugate to the A p matrices. From the first of equations (4.2.38) we have K
':: =
2 8 v _ fJt2 -
L
{;~E.I"
{a;:, (~;) + a~, (~~)},
2 ~ ~ k l{ 8 Vkl L-L-€ /L 8a
k=l l=O
+ 8Vkl (d2a) 8a
dt 2
+
2
(da)2 dt
dtjJ da + 282Vkl -
B2vkl (dtb )2 8a 2
dt
+
8Vkl (d 2tb ) 8:,p dt 2
From the first of equations (4.1.1) we obtain 2
8 v _ fJt2
L(2m){ (t z v,
x
)}
=
~ ~ k l { B2V kl L.; L.; € /L 8a 2 k=l l=O
(4.2.42)
8a8tb dt dt
(da)2 dt
}
.
4.2 Simple Perturbation Technique
82Vkl dt/J da 8Vkl (d 2 a) 8 2Vkl (d1jJ) 2 +2 8a(Jt/J dt dt + 8a dt 2 + 8a 2 dt Vkl (d 21jJ) 8 + 81jJ dt2 - Lx(2m) {Vkl} } ,
275
(4.2.43)
while from the second one we obtain dy = da (aeh/J(t) dt dt K
+ ae- I1/J(t») + ia(t) dt/J
(ael1/J(t) _ ae-I1/J(t»)
dt
L
' " ' " k l { 8Ykl da + L- L- e J.L 8a dt
+
k=l l=O
8Ykl dt/J } 8t/J dt
(4.2.44)
.
From (4.2.39) it follows that 2
d a _ 2 A dA IO dt 2 - e 10 da
2
+ e J.L
20 +e3 (dA ~ A IO da dt/J dt dt = ewA IO
(dA IO A da 11
+
dA 11 A ) da 10
dA IO A20 ) + O(e k J.L l ; k + l = +~
2
4),
2
+ e (wA20 + A IO B IO ) + eJ.LWA11 + e J.L(wA21 +A ll B IO + A IO B ll ) + eJ.L2A I2 W + e 3 (A30w +A20BIO + A 10 B20) + O(e kJ.Ll ; k + l = 4),
2 d 1jJ _ 2 dB lO A 2 dt 2 - e da 10 + e J.L +e 3
{
dB 10
(4.2.45)
{
dB IO A da 11
dB 20
~A20+ ~AIO
+
(4.2.46)
dB ll A } da 10
} +O(ek J.L l jk+l =4),
(4.2.47)
(4.2.48)
(4.2.49)
t _ ) = ~ .!.- dny(t) (_ )n Y( J.L L- n! dt n J.L, n=O
lI(t - J.L {) = ,
~ .!.- dnll(t, {) (_II.)n L- n! dtn fA"
n=O
(4.2.50)
276
4. Discrete-Continuous Systems
then the function ef and eF can be expanded in a power series of the small parameters /.L and c. Further calculations were carried out for n = 1 (YI -/.L(dy/dt) and VI = -/.L(dv/dt)) and under the assumption that N
•
_ _
VI -
L
~~ k l /.L L- L- e /.L
da 8a dt
{ 8Vkl
k=1 l=O
in
= -I' { _
~~ (",,''''(t) + tie-I'" (t»
~~
k l (8 Y kl
da
8Vkl
+
d'I/J }
81/; dt
I'a(t) { ~; i (o<e'''' (t)
-
8Ykl
(4.2.51)
' - tie-I'" (t»
d1/;)}} .
8a dt + 8'I/J dt
/.L L- L- e /.L k=ll=O
The necessary derivatives of the functions f and F were calculated at the point /.L = e = 0 and Vo = 0, Yo(t) = a(t){aeiw(t) + ae-iw(t)}. The sequence of recurrent linear differential equations obtained are of the form
e:
2
W
B2 VlO81/;2 (x, a, 'I/J)
+ f e,
= L(2m) {lI; } x
10
p
- -iW) , W8YlO(a, &¢ 1/;) = ~ L- A pYIO (a, 'I/J - TpW ) - A 10 (aeiw + ae p=o p
e2
-iaBIQ(aeiW - ae- iW ) - LTpA p{ AlO(aei(w-Tpw) (4.2.52) p=o +ae-i(W-TpW)) - iaBIQ(aei(~-TPW) - ae-i(W-TpW))} + Fe; , 2 w 2 8 V20 (x, a, 'I/J) = L(2m) {l/. } +' f 81/;2 x 20 u, p
'I/J) = """ W8Y20(a, &¢ L- ApY2O(a, 'I/J - TpW) - A 2O (ae'l11' p=o
+ ae- 1'11' )
p
-iaB2O(aei~ - ae- iW ) -
L A p{ TpA20(aei(W-TpW)
(4.2.53)
p=o +ae-i(W-TpW)) + iaB2O(aei(~-Tpw) - ae-i(W-TpW))} + F u 2
W28 Vj, I (x, a,1/;) 81/;2
=
;
L(2m) {V; } + f x 11 ep.,
p
(a, 'I/J) = """ W8Yll&¢ L- ApYll (a, 'I/J - TpW) --:- All (ae1'11' + ae- 1'11' )
p=o p
-iaB ll (aeiW - ae- iW ) -
L
TpA p{ All (aei(W-TpW)
(4.2.54)
p=o
+ae-i(W-TpW)) - iaB u (aei(W-TpW) - ae-i(W-TpW))} + Fep.;
4.2 Simple Perturbation Technique
W
2 82V3C1{X, a, tP)
= L(2m){TT } +
8tP 2
x
"30
t
u
3
277
,
p
W 8Y30{a, 8t/! tP) = '" L- ApY30 (a, tP
- TpW ) -
A 30 (aeiw + o:e - -iW)
p=O p
-iaB30 {aeiW - ae- iW )
- LTpAp{ A 30 {ae i (W-TpW)
(4.2.55)
p=O
+ iaB 12 (aei(W-TpW)
+ae-i(W-TpW))
2
W
- ae-i(W-TpW))}
+ F e3;
82V12{x,a,'I/J) _ L(2m){l!, } +1
8tP 2
-
12
x
e1J.2,
p
" ApYI2{a, 'I/J - TpW) - A 12 {ae1'w + ae- 1'w ) W8YI2(a, 8,p 'I/J) = 'Lp=O p
-iaB I2 {ae iW - ae- iW ) -
LAp {TpA I2 (aei(W-TpW)
(4.2.56)
p=O +ae-i(W-TpW)) + iaB I2 {ae i (w-Tpw) - ae-i(W-TpW))} +
F e1J.2;
p
-iaB21 (aeiW -
ae- iW ) -
LA
p { Tp
A 21 (aei(W-TpW)
(4.2.57)
p=O +ae-i(W-TpW))
+ iaB 21 (aei(W-TpW)
- ae-l(w-Tpw))}
Here
le
3
_ 8 21 2 81 8v2 Vio + 8v V20 +
m
L l=1
8 21 2 8 2 Y(10)l + y,
81 Y(20)l L a l=1 Yl m
+ Fe 2 w
278
4. Discrete-Continuous Systems
4.2 Simple Perturbation Technique l aF { - AIQ(ae'w +2 ~ L- ~
n=1
+ae- 'w
279
I
)
YIn
- -iW) - --w 8yll } -aB 10 (Q,eiw + a:e at/; . To achieve a complete ordering of all the recurrent equations we take the additional condition that €i-I < Ii, where i is a positive integer. After expanding the function f(.) into a Fourier series, one obtains 00
f(.) =
00
~ ~ {b(.)sn(a) cosnt/; + C(.)sn(a) sinnt/; }Xs(x),
(4.2.59)
s=1 n=l
where l
b(.)sn(a) =
211'
JJ 2~l J J
2~l
dx
o
(x, a, t/;)Xs(x) cosntj.J dtj.J,
f(.)
(x, a, t/;)Xs(x) sin ntj.J dt/;.
0 211'
l
C(.),m(a) =
f(.)
dx
o
(4.2.60)
0
The functions F( *) are expanded into a complex Fourier series (4.2.61 ) n=-oo
where
C(.)n(a) =
2~
21t'
J
F(.)e inW dt/;,
n = ±1, ±2, ...
(4.2.62)
o We describe a procedure of solving the recurrent set of ODE's based on (4.2.52). In order to avoid terms ascending unrestrictedly in time in these equations, the following conditions must be satisfied: -{A lO (a)
+ iaB lO (a)} {("', m+ ~ 7". (Ap"" {3)e- T • Wi }
+(C(lo)l(a),f3) :...- O.
(4.2.63)
Here (a, b) denotes the scalar product. By equating to zero the real and imaginary parts of (4.2.61), we obtain two equations to determine the quantities AIQ(a) and BIQ(a). Then we can find VIQ and YIQ, and further successfully solve the recurrent set of equations. Analytical conditions for the existence of a two-parameter family of periodic and quasiperiodic orbits in autonomous and non-autonomous systems Can be found in [32, 35, 36, 38].
280
4. Discrete-Continuous Systems
4.3 Nonlinear Behaviour of Electromechanical Systems 4.3.1 Introduction UsuaHy the dynamics of nonlinear discrete-continuous systems governed by ordinary and partial differential equations (the case considered here) causes some difficulties in nonlinear analysis. It is often brought about by the timeconsuming numerical techniques used to find the solution of the partial differential equations. Additionally, real physical systems possess many parameters which can be changed over wide ranges and in practice direct simulation of the governing equations is costly and tedious. For simple dynamical systems, averaging formulas can be derived without computers. However, in the case of complex systems (such as the example considered in this book) this classical approach leads to serious difficulties. Therefore, the idea of applying the averaging technique has been supported by symbolic computation with the use of the Mathematica package. A program has been written in the it Mathematica language which has yielded the averaged equations. This set of equations has been transformed (using one of the Mathematica options) to Fortran expressions, and further, a numerical analysis has been carried out. Admittedly, in this section, on the one hand, the electromechanical system serves as an example for a systematic strategy of solving many other relative problems, which can be found in nonlinear mechanical, biological or chemical dynamical systems. It consist~ of a few steps: 1. Dynamical equations are derived. 2. The averaging method is proposed and the program for symbolic computation yields the averaged equations (AVE). 3. Further systematic study of the obtained ODE's is developed. The model was first discussed by Rubanik [137], where attention was focused on the averaging procedure only starting with the governing equations. Here, the system under consideration is discussed in some detail, and the symbolic computation is used to obtain the differential amplitude equations based on the application of the Mathematica package. Contrary to the approach in [137], also the numerical analysis of the averaged differential equations found is carried out to show interesting nonlinear phenomena. The averaging method, also based on the symbolic computation, has been proposed by one of the authors earlier [21d, 22d]. Here two parts of the project are clarified. First, more attention is paid to the discussion of the electromechanical nonlinear system including its electrical model. Second, instead of an approximation of the time delay function by a Taylor series, a Galerkin method is applied. As has been pointed out by Elsgolc [33d], the truncation of the Taylor series to one term, as well as the use of more Taylor terms in the function with the time delay approximation, does not lead to an improvement in accuracy of the numerical calculations. For this reason a new approach, based on a Galerkin approximation, which does not possess any
4.3 Nonlinear Behaviour of Electromechanical Systems
281
limitations because of the delay magnitude, is used. Third, the qualitatively new numerical results are discussed and illustrated now in comparison to the author's previous works [34, 24d, 25d]. 4.3.2 Dynamics Equations
A string (a continuous system) is embedded in a magnetic field. For a certain set of parameters the stationary position of the string becomes unstable and the string starts to vibrate. Because the string possesses an inductance L, a resistance R and a capacitance C the movement in the magnetic field causes the occurrence of voltage and current. The amplifier controls the change of the current amplitudes with the time delay as the experiment shows. Figure 4.2 presents the electromechanical model and its electric scheme.
c s L
~u 'X
F, P R
U;up",! N
Uoutput
a)
b)
Fig. 4.2. Scheme of a string embedded in a magnetic field (a) and its electrical model including an amplifier (b)
The magnetic induction B(x) acting along the string generates voltage at the ends of the string according to the following equation
J () l
Uinput () t =
B
X
8u(t,x)
at
dx,
(4.3.1)
o where x is the spatial coordinate, t denotes time, u(t, x) is the amplitude of oscillations of the string at the point (t, x) and 1 is the length of the string. The amplifier gives the output voltage Uoutput(t)
where
hi (i =
=
hIUinput(t) -
h2Uinput(t),
1,2) are constant coefficients.
(4.3.2)
282
4. Discrete-Continuous Systems
The current oscillations including the time delay in the amplifier are governed by the equation
i(t) + 2Ai(t) + kI(t) = Uoutput(t - v)
(4.3.3)
where:
2A = RL- I ,
k = (LC)-I,
Uoutput(t) = hI Uinput(t) - h2Ui~put (t)
hI
= L -1-hI,
h2
(4.3.4)
= L -1-h2 ·
In the above expresion, the "dot" denotes differentiation with respect to t, I(t) denotes the changes of the current and v is the time delay. The changes in time of I(t) and the changes in x of B(x) play the role of the force acting on the string, whose oscillations are governed by the equation
8"':i:; x) _ t?8"~~ x) ~ ~ (2ho8u~ x) _ B(X)I(X)) ,
(4.3.5)
where h o is the external damping coefficient, p is the mass density per unit length, c2 = F / p, F denotes the string's tension and € is a small positive parameter. The frequencies of free oscillations of the string are given by W s = (s = 1,2, ...), and the homogeneous boundary conditions are as follows:
1rcs/l
u(t,O)
= u(t, l) = O.
(4.3.6)
4.3.3 Averaging
..2w ?
2 '
The further analysis is straightforward for the perturbation technique. Because B(x) and I(t) are defined, therefore (4.3.5) can be solved using a modified classical perturbation approach (it is assumed that UI (x, alt a3, (h, (3) is a limited and periodic function). Substituting (4.3.8) for (4.3.4) and taking into account that ai = ai(t) and Oi = Oi(t) (i = 1,3) slowly change in time, the following resonance terms are calculated from the right-hand side of (4.3.5) (further referred to as Ri) l 211'
~e =
:l JJR
sin 1T";X cos !PiO d!PiO
o Ria
= :l
0 l 211'
JJ
o !PiO
(4.3.17)
R sin 1T";X sin!PiO d!PiO
0
= iw + 0i,
(4.3.18)
i = 1,3
where Ric, Ria correspond to the coefficients of cos i!PiO and sin i!PiO, respectively. The comparison of the coefficients of cos i!PiQ and sin i!PiO and generated by the left-hand side of (4.3.5) to those defined by (4.3.17) leads to the following averaged amplitude equations ; I
· ehoal al = p · ehoa3 a3 = -
P
eBl . -2- {(bleMI - blaNI) sm 0 1 {JW eB3 . -2- {(b 3e M 3 - b3a N 3) sm03 {JW
· eB I 01 = - 2 {(bleMI ..... blaNI) cos 01 al{JW · eB3 03 = - 2 {(b 3e M 3 - b3a N I) cos 03 a3{JW
+ (bleNI
-
blaMI) cos 0d ,
+ (b 3e N 3 -
b3a M 3) COS03} ,
+ (bleNI
•
-
blsMI) sm 0d ,( 4.3.19) .
+ (b3e 1V3 - b3a M 3) sm 03} .
The analysed set of equations has some properties which can cause difficulties during numerical analysis. First of all, this is a stiff set of equations (note tbe occurrence of a3 in the denominator ofthe last equation of (4.3.19)). As is assumed by the averaging procedure, the amplitudes ai and the phases Oi change with time very slowly, and a long integration to trace the behaviour of the system is required. For the further analysis of the time dependent solutions we transform (4.3.19) into the amplitude equations. For this aim we assume
4.3 Nonlinear Behaviour of Electromechanical Systems
Uo =
(Yl COSWlt + Y2 sinwlt) sin
285
(~X)
+(Y3 cos w3t + Y. sin w3t) sin
(~x)
.
(4.3.20)
Comparison with (4.3.7) yields the following relations: Y1 (t) = al (t) cos 81 (t) Y2(t) = -al (t) sin 8 1 (t)
(4.3.21)
Y3(t) = a3(t) cos 83(t) Y4(t) = -a3(t) sin83(t).
In what follows, the set of the amplitude differential equations has the form
Yi (t)
= al (t) cos 81 (t) - al (t)B I (t) sin 81 (t),
Y2(t) =
-al (t)
sin 81 (t) -
al
(t)B I (t) cos 81 (t),
(4.3.22)
Y3(t) = a3(t) cos 83(t) - a3(t)B3(t) sin 83(t), Y4(t) = -a3(t) sin 83(t) - a3(t)B3(t) cos 83(t)
where
ai
and 8 1 are given by (4.3.19) and
8,
= arctan ( - ~: ),
al
= (y12 + Yl)1/2,
83 a3
= arctan ( - ~:) ,
(4.3.23)
= (y32 + Yl)I/2.
4.3.4 Numerical Results We consider both time-dependent and time-independent solutions. In order to get the stationary solutions we solve the nonlinear algebraic equations obtained from (4.3.19). For this a Powell hybrid method and variation of Newton's method have been used. It takes a finite-difference approximation to the Jacobian with high precision. Figure 4.3 presents an example of numerical calculations for the following fixed parameters: l = 0.1, w = 30.0, A = 0.01, k = 35.0, h 2 = 5.0, p = 1.0, ho = 0.005, II = 5.0, B I = 5.0, B3 = 1.0. hi has been taken as the control parameter. The increase of hi damps the value of the first harmonic oscillations. However, during the change of hi the amplitude a3 as well as the phases 81 and 83 remain constant. Now (4.3.22) and their time-dependent solutions will be analysed. The system of equations is stiff and the Gear method routine from the IMSL Library is used to solve the problem. Let us consider the following set of parameters: l = 0.1, W = 190.0, A = 0.1, hI = 0.58, h2 = 0.02, II = 0.1, p = 1.0, Bl = 6.3, B 3 = 0.08, € = 0.05. Calculations have been performed with TOL = 10- 8 , which is proportional to the calculation step. As can be seen from Figure 4.3a, the variables YI,2(t) change in an oscillatory manner, whereas Y3,4 (t) decay exponentially. This allows the interpretation that the ampli-
286
a)
4. Discrete-Continuous Systems 0.020
"I ...............
al
a3 0.018--~----~~-~,-~"'~~----.j~f--.-a-' 1---
0.016 ~-0.014
'---1
----_____
-'T.---.-.-----.-.. . .-.-. --~
0.012·--·'-·~-
I------- - - -
-
--
----=9"'"-~;;::::_=1
. - - -~
------~
--
a3
0.010 I-----+------+-------.---~----+_-------____i 0.008 "-
--
---1-.---. ! _.
0.006 0.004
------1--------
-_._.
--.-
0.02
0
--I--.-------\-- - - - ' - " -
0.04
-'
0.06
-~-
"
..
0.08
~--
---'
hi
0.10
b) (2) -- -- - -
(2.5) - --
-~---.--
t I
-
l
(3)
(3.5) (4)
I-- -.- -
-~-'-
-
i
-~--.- -
----I--~
I --.-
L..-_ _---'-I
o
----
~.- --~--
-------- -
(Ji-- __
.----
l-
....1.-_ _- - . 1
-
--.
-
-------
~
.
.-
_
---'-_ _- - - - - '
0.02 0.04 0.06 0.08 Fig. 4.3. Amplitudes (a) and phases (b) versus parameter hI
hi
0.10
tudes of the first and the third modes of the string behave quite differently. Furthermore, it implies that quite different analytical types of solutions can be assumed from the mathematical point of view. The other figures illustrate the change of the mode amplitudes with the incre~e of the coefficient h2 . As can be seen from these figures, the increase in the nonlinear term h2 leads to the extension of transient periodic oscillations. Finally, let us consider the following fixed parameters: l = 0.1, W = 900.0, k = 7250, A = 0.1515, hI = 5.48, h2 = 65.0, P = 1.0, B I = 0.65, ho = 0.00001, B 3 = 0.089, € = 0.009, II = 0.00001. Strong nonlinear behaviour is observed. In the beginning all variables do not exhibit oscillatory behaviour. After the time of about 20000 units a sudden occurrence of strong nonlinear oscillations of YI,2(t) Can be seen, whereas the variables Y3,4(t) do not change in an oscillatory manner again (Figure 4.4a). Increasing the time delay a strange
4.3 Nonlinear Behaviour of Electromechanical Systems
287
~'Y;'YJ'~
a)
0.12 0.10 0.08
.~-+---~-~-~
-----
0.06 0.04 0.02
oL:..=---l-~.--L-=:~J=~:::~..~..~..===l""I!l!I-----1 (0.02) (0.04)0
100000
200000
300000
400000
500000
600000
time
b)
~'Y;'Y:I'Yt 0.15 - - - - + ~.- ------ -
0.10 0.05
"'--Y4-~---------+-·_-~+------- - - -
.........': •
}J:·
.. a.
··~.":':."::.-::oIftolft
3
o '\y
'. 2
". (0.05) --- -~--~-.. (0.10) 0
-.
---->........
........
100000
-~--
200000
300000
400000
500000
600000
time Fig. 4.4. Time evolution of the amplitudes with an increase of coefficient
h2:
(a)
h2 = 0.02; (b) h 2 = 0.04
transitional state is observed: strong nonlinear oscillations Yl,2(t) vanishing in time are shown. The amplitude Y3 decreases linearly, and its derivative remains constant. This means that the first and the third mode amplitudes behave qualitatively differently. To summarize, the analysis is focused on the numerical observation of the averaged differential equations derived from the dynamical examination of the string-type electromechanical generator. Considerable attention is paid to the derivation of the averaged equations through the application of a modified perturbation technique supported by symbolic computations. The obtained set of equations is nonlinear and stiff. The numerical calculations based on solving the initial value problem have been performed to reveal some interesting results, which are here briefly summarized.
288
c)
4. Discrete-Continuous Systems ~,Y;,Y;,~ 0.15 ...-------,-----,----r-----,------,------,
0.1 0.05 0 (0.05) (0.1) (0.15)
d)
0
100000
200000
300000
400000
500000
600000
time
~,Y;,Y;,~ 0.15 ,,
0.1
I
--
,I
0.05 0 (0.05) (0.1) (0.15) 0
e)
1ססoo0
200000
30ססoo
400000
500000
600000
time
~,Y;,Y;,Yt 0.15 .----,------,------r------.------..----,
0.1 0.05
o (0.05) t--\----l~H_+-_H~-\-I-_____+____I__~~\__-+____::_--I (0.1) 1----'\--,f+-t-l----+--\----I-4r.-V-~~~--~--.I (0.15)
L - - _ - - - - L_ _--L-_ _~_ _- ' - -_ _..i...._._
o
100000
200000
300000
400000
500000
_..J
600000
time
Fig. 4.4(continued). Time evolution of the amplitudes with an increase of the coefficient h 2 : (c) h 2 = 0.05; (d) h2 = 0.055; (e) h 2 = 0.0554
4.3 Nonlinear Behaviour of Electromechanical Systems
289
~,Y;,r;,~
a)
2
1.5 1
0.5
-
~
..-.
~ tA~., .....
... -_....-..-
~.-
(1)
- ...
....-•....•
o
10000
5000
15000
"i
~·:·::~..
y
25000
.
30000
35000
time
._
l~
,
!~
I
·.. ···h-···
"', I
1':'
~· . ·. •·.. ~ x..... r--...... .- ··
,'\Y,4
T
"'r;l
'JI'.
.
,.
--
-.. . .
_.
- ~ - -- ...----I----+--.--.-~c----~--+--
(1) (1.5)
20000
•
4
........,
(0.5)
"
f - - - - - +----~-~+--~+-~~+_--+_---_+_-__1
. . R o
0.5
.... /c. ", I} .. ....... r;1 ~
~
f
-
~,~,r;,~ 1.5 ... 1
.....-1"; ..... .........,
""
-- -----t
(0.5)
b)
----
'c-
~
o
(1.5)
~ l'1
-
1------
--t--------j
-~ - --------+---------+~_.-- - -+-----+----+---~---I
L.-_....!.-_-----J._ _....L..-_---l--_ _" - - _ . . . . l . - _ - - - - '
o
10000
20000
30000
40000
50000
Fig. 4,5. Strong nonlinear amplitudes oscillations: (a) v
60000
70000
time
= 0.00001; (b)
v = 0.0001
It has been shown that in the case of stationary solutions the increase in the control parameter hI damps the first harmonic oscillations. The amplitude a3 and the phases remain constant during the change of the control parameter. In the case of time-dependent solutions interesting nonlinear behaviour has been reported in general. Amplitudes of the first two modes behave in a qualitatively different manner. YI,2(t) change in time with oscillations, whereas Y3,4(t) decay exponentially. Furthermore, strange nonlinear phenomenon has been exhibited. After a long nonoscillatory transitional state strong nonlinear oscillations suddenly occur.
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