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(z) = 0 £ IRm . Looking for a minimal representation s e I R / , / = 6n — m, requires z = z(s). Thus, differentiating both the variables z and the constraint w.r.t. time followed by partial differentiation of the latter w.r.t. s yields a representation of z in terms of s, see Table 1. However, due to the time differentiation, the result is a velocity s which needs by no means to be integrable. (s then represents "quasicoordinates", a highly successful slang expression, obviously introduced by Whittaker 3 3 ). But, because the constraints themselves are (up to here) purely holonomic, there exists also an integrable subset {q} G {s} : J qdt —> q. These are chosen for positional description, leading to a (regular linear) combination of the q's for velocity representation. Table 1:
Constraints, Minimal Velocities: The Holonomic Case (g=f)
z(s):
"
(3)i„(2)->:
*-(2)
4:
(SMt
^s=(|j)(|i')q=(|sjq
=
H(q)q€lR»
(4)
Remark. The differentiation rules are not unique in literature, sometimes even pretty mixed up. We use here the following concept: Let s = a T x be a scalar with variable x and constant a. Differentiation w.r.t x then yields (ds/dx) = a T (a row vector while a represents a column). Consequently, one obtains for (/dz) by (d^f/dq) and z by q). Table 3:
(d*^ \*l)
Constraints, Minimal Velocities: The Nonholonomic Case (g=f-k)
= 0: h= i
fd±\
w
(OA) \dsJ
= 0
=>• s =
U) h = = H(q)q e IR
9
\*l)
Remark. We retain the dots in formulating the nonholonomic constraints for the same reason as we did for s: All these axe expressions on the velocity level. The usefulness of this notation lies in the comparability of holonomic/nonholonomic terms and can easily be seen in the above treatment of minimal velocities: The solutions {s} (see Table 1) may be integrable or not, thus the dot represents a total differential or not. Remark. For 4? being nonlinear w.r.t. q one may replace (d^f/dq) by (dif?/dq) in Table 3, see Table 2. However, do nonlinear nonholomic constraints exist? Let us assume that: first, the position z e IR6™ is constrained with * ( z ) = 0 e lR m , leading to minimal coordinates q e IR /==6n ~ m . Then, clearly, the volocity z and the acceleration z (and any higher derivative) is automaticly also constrained (the holonomic case). Second, the velocity may additionally be constrained with * ( q , q) = 0 G IR (not being integrable, of course, otherwise one would have additional holonomic constraints) leading to minimal velocities s e ]R9=6™~m~ . Third, the acceleration may additionally be constrained with 0 ( q , q, q) = 0 G IRr (not being integrable, of course, otherwise one would have additional nonholonomic constraints) leading to minimal accelerations s e IR = 6 n _ m ~ ~ r . However, this third assumption does not make much sense because of the basic axioms (momentums are already linear w.r.t. accelerations). Therefore, operating on the acceleration level for nonlinear nonholonomic constraints is nothing but the same "trick" as for honolomic ones: Differentiation reveals linear terms which can then elementarily be treated. But the way back is always possible (integrability of differentiated constraints is of course trivially assured). Therefore, the frequent statement that nonlinear nonholonomic constraints do not exist because corresponding acceleration constraints would contradict the basic "natural laws" does not hold (consequently, not having the integrability in mind, holonomic constraints would also contradict the axioms). The only statement one can extract here is that "nonlinear nonholonomic constraints do not seem to exist in daily experience" (Hamel) 19 , although for control purpuses velocities and
6
H. Bremer
even accelerations may be computed anyhow with no restriction at all. Next, if we introduce with Hamel 14 a set of "variables"
s T = (s£ P . C d e p .) = f*i = 0 • • • ** = 0 I h • • • sg], ^ f (q, q) € ffi/ (1)
(hence denoted as "Hamel's variables") then the q = (dq/(q, q) = 0 € 1R identically3,2. From the engineering point of view, Eq.(l) consists of / (translational, rotational) velocities, where, in the presence of nonholonomic constraints, some of them are "forbidden". These are, of course, in advance known from the modeling assumptions. Thus, one "natural" choice of velocity variables is given with a coordinate representation in a reference frame taking the "forbidden" and "allowed" velocities directly into account. ( R e m a r k : mathematicians, in general, do not like this kind of argumentation because of loosing, in their opinion, generality. However, we should not forget the sequence of foregoing: The first step is modeling. The second is its mathematical description. So why forget the first step while discussing the second? This kind of "generality" is, from the engineering point of view, shadow-boxing.) Clearly, using Eq.(l) for description, f (q, q) may be used as a whole which enables to cancel the dependent componentents at the end of calculation. However, one has to be cautious: The motion equations one would abtain for Si = 0, i = 1 • • • k are not (at least not directly) fulfilled. The reason is: looking at the corresponding virtual work in the sense [• • -]Ssi = 0, the 5si are arbitrary for the independent part but zero for the dependent part (as direct consequence of Sdep = 0). In view of structurally variant systems where one has to deal with closed and open constraints simultaneously, this situation is not very satisfactory. On the other hand, one may easily introduce Lagrange parameters to obtain arbitrary 5si for the dependent part as well and thus free the dynamics from the corresponding constraints 16 . Generally, one has for the (generalized) constraint forces e.g. (d4?i/ds)T\i where (d^i/dsj indicates its directions. Evidently, using \i> itself for coordinate, then {d^i/d^)T reduces to the i-th unit vector. One has then just to add A^ to the corresponding equation: [• • • — Xi]5si = 0, i = 1 • • • k. R e m a r k . In order not to trivialize Hamel's "Principle of the relaxation of the constraints" one should emphasize that the above denotes a very simple case. What Hamel did in his famous paper was of course much deeper: Starting with the rigid body, formulating the conditions of rigidity along with the correspond-
On the Use of Nonholonomic
Variables in Robotics
7
ing Lagrange parameters, then freeing the system from the constraints where the constraint forces (Lagrange parameters) convert into impressed forces, he obtains immediately the (symmetric) stress-strain relations of continuum mechanics. 2.2
On Virtual Displacements and Variations
Although already mentioned in the context, "virtual" terms have not yet been defined up to here. These have caused a lot of confusion during the centuries. It may be that Lagrange 27 himself initiated these misleadings with his "I have to emphasize ... that I introduced a new characteristic 5. By this, 5Z shall express a differential of Z which is not the same as dZ but which is built by the same rules." .44 years after Lagrange, Poinsot published a paper (reprinted in his textbook 35 ) where he stated that the "virtuals" leave something obscure in one's mind and he therefore replaced the virtual displacements by the actual velocities, calling that a "new principle". This is completely wrong (see Lagrange: same rules, but different things), although contemporary authors call this "a better foundation, where Lagrange's principle appears as a simple corollar" 36 . Other contemporarians share Poinsot's opinion: "virtual displacements ... are the closest thing in dynamics to black magic", they are "ill-defined, nebulous, hence objectionable" (Kane 3 7 , followed by his disciples and colleagues: "too vague for practical use" (Levinson), "esoteric quantities" (Angeles)). This kind of nonunderstandment (not misunderstandment) can easily be rejected by trivial interpretation of Lagrange's concept: A mechanical system the position of which is restricted to * ( z ) = 0 undergoes (generalized) constraint forces which are perpendicular to this (hyper-) plane and force the (considered reference point of the) system to remain in that plane. Thus, any arbitrary but "allowed" motion direction Sz is element from the tangent plane ((d&/dz)Sz = 0 => Sz) except the singular solution (d$>/dz)dz = 0 => dz (see Poinsot's error, see Lagrange: same rules, but different things). The reason is simple: using dz, the scalar product projects motion into the unconstrained minimal space, eliminating all nonworking forces at once, not only the constraint forces (as desired) but also the Coriolis reactions. The question arises, how to calculate "allowed" 5z (or any other S( )). To start with, let us first investigate m = u)u = 0 (mostly used for linear momentum, see (Euler) 6 ), or body-fixed frame: u3JB = UJ1B = u)c (mostly used for momentum of momentum, see (Euler) 7 ) . However, in many applications, e.g. rotor dynamics, the use of a guidance frame ("Fiihrungssystem") R is advantageous, see example 3. The constraint forces and torques are eliminated in Eq.(31) by means of functional matrices. This means, in other words, motion is projected into the unconstrained space which is characterized by s. Eq.(31) shall therefore be called The Projection Equation. With this interpretation, Eq.(31) is the typical representative of the synthetical method the aim of which is to synthesize motion equations by force/torque relations (simultaneously, of course, eliminating constraint forces). It should, therefore, not astonish that the huge amount of "synthetical procedures", mainly elaborated within the last four decades, can eventually and without any exception be interpreted in the sense of Eq.(31). On the other hand, premultiplying once more with M i + T | , 1 M 4 T 4 i + T ^ M 5 T 5 1 G a —» Gi + T 4 1 G 4 T 4 1 + T 5 1 G 5 T 5 i Qi^Qi
(46)
TliQ 4 + TiiQs
since T 4 i , T s i are constant. One is thus left with the independent yt, and one has to chose s. This can be done in the following way: If in Eq.(43) at any stage new possible contraints arise, then the previous velocities are expressed by the new ones, and simultaneously new variables have to enter calculation in order to take the new situation into account. This is the case here for ygC. Denoting the possible constraint herein by vye (that means possibly (hopefully) no front wheel sliding) one obtains
cos 76 sin 76 a sin 76 — sin 76 cos 76 a cos 76 0 0 1
front wheel
(47)
Resolving for y^ c by means of Eq.(42) yields cos 76 — sin 7 6 0 sin 76 cos 76 —a 0 0 1
(48)
Replacing y\° in Eq.(47) with Eq.(48) does not make sense because: if both constraints will never be active - although correct, why should one? If, on the other hand, two constraints may become active (vyi = 0 , ^ 6 = 0), then, inserting the above relation, one takes care of vye = 0 but neglects vyi = 0, or vice versa if one leaves everything as it is. The aim is therefore to introduce new variables such that with these, replacing the old yfc, both constraints may become active simultaneously. That means that both (possibly active) constraints have to enter the term one is looking for. This is easily achieved: resolve the possible constraint in Eq.(48), i.e. (0 1 0)y^ c , for one of the remaining free variables:
On the Use of Nonholonomic Variables in Robotics 27 lLgi. = vx6 sin76 + % 6 . c o s 7 6 - a-ji : either 1 — cos 7 6 a 0 sin 76 0 0 0 sin 76 in (48) =>•
/ vxi ' Uy±
V 7i ,
cos 76 —1 0 sin 76 0 0 sin 76 [ 0 0 sin 76 1
(49)
or
in (48)
0 0
0 1
1 0
-a
Scos76
ssin76
0 1
(50)
— sin 76 COS 76 0 0
4 il cos 76 I sin 76
\VX6 J
\wxi /
Although arbitrary in principle, the first choice contains singularities (for vanishing steering angle 75) while the second does not. T h e minimal velocities are composed of Sj,i = 1,2,3,6. Collecting the variables yields the situation according to E q . ( l ) , [%j.
Uys
«x6
72
73
76
(51)
T h e corresponding time derivative of minimal coordinates q according to Table 1 are obtained the from the fact t h a t the p a t h (x, y, 71) is given within the inertial plane (in robotics referred to as "world coordinates") cos 7i
sin 7i
-sin7j
COS7J
0
0
(52)
Along with t h e remaining variables (72 73 76) and with Eq.(50) one has A^T
E
H ( q ) + ss :
(53)
28
H.
Bremer
'
X
'
y
— sin 71 — cos 7i sin 76 cos 71 cos 76 1 0 0 cos 71 — sin 71 sin 76 sin 71 cos 76 0 1 0 -i QCOS76 ^ sin76 0 0 1
7i 72 73 -76-
«x6
72 73 76
(54)
([ ] + : generalized inverse). The last step for the motion equations according to Eq.(35) is determination of the Jacobians (ctyj/ds). Because the basis (body nr. 1) does not undergo a "guidance motion" its velocity is characterized by y : = y r l leading with Eq.(40) to Yi Y2
Ys
E T21 E T31 T32
Yrl Yr-2 Yr3
E
(55) E yn
*-n,n — 1
-ral
s
E.
. Jrn -
:
where Tnj *-n,p(n) * *-p(n) ,p{p(n)] X • • • 1 s(j),j ( 0 ) successor of j a,nd p(n): predecessor of n (compare Table 4) Because the minimal velocities are compontents of the relative cartesian velocities, yri = FjSj, one obtains the Jacobians
d_
Yi Y2
Fi T2iF!
Ys
T31F1 T32F2 F3
F2 (56)
ds TnlFj LYn yielding for Eq.(35) the general expression
Fn-i •*• n,n—1* n — 1 -^ n
M s + Gs - Q = 0 € M9
(57)
However, proceeding this way directly yields an g x g mass matrix which needs lateron inversion. This can be avoided using the recursive kinematics.
On the Use of Nonholonomic
3.3
Variables in Robotics
29
Recursive Kinetics
In order to avoid a total mass matrix inversion one can use the fact that the Jacobian (56) is upper triangular and therefore directly apply a Gauss elimination procedure. With Eq.(56), the last two (block-)rows in Eq.(35) (i=n, p=n-l) read pT r p
o
TjiTrpT p - 1 ip
c
FJ
MpYP + G p y p - Q p ^ M,y, + G i Y i - Q,
=
(58)
The last row can be resolved for Sj in terms of predecessor velocity and accelereration (see Eq.(40)):
FTM.F, \Mi{Tipyp
FTX
+ n f T i p y p + F ^ ) + Gi(Tipyp + F A ) - Q,} .
(59)
Insertion into the forelast row then yields the same structure as the last row by definition of F j [M p y p + G y p - Q p ] = 0 : M, V
[Ff MiFi]
def
-l•cT^
E-MiiFMT'Fl)
Mp := Mp TUiMiTi. Gp :— Gp + Ty,Jj(Gj — (60) _ Mjli P j)Tt_P , Q p := Q p + T'fpJiiQi - MiFiSi - GiF^) and can therefore be resolved for Sj_i in the same manner. Applying this scheme repeatedly till p = 0 one can solve si with Eq.(59) (where y = 0) and then go ahead for S2 to s n . This means, in total, a recursion in three steps: First: formulate the kinematics for the last body. All the needed quantities are known from initial conditions. Second: Formulate M p , G p , Q p p = n • • • 1 backwards till the basis is reached. Third: Calculate Sj,i = \---n. The benefit is obvious: the number of operations to calculate the total mass matrix is quadratic w.r.t. the number of d.o.f. and its inversion is at least cubic while the maximum dimension of necessary inversions here corresponds to the rank of F j , although the recursion has to be rerunned three times. Hence, for n > 3 the recursion scheme is preferable, for systems with nontrivial kinetic energy. (Note that the Gaussian scheme is not restricted to the single body dynamics as used here. Any row of Eq.(58) may represent an abritrary subsystem the structure of which is the same as the single body equation.)
30
H.
Bremer
E x a m p l e 5: Plane motion of a serial chain and the double pendulum. We define the z-axes connecting the hinge points (distance Li) and the z-axes perpendicular to the plane of motion. The mass center is situated on the x-axis. One has then
cos 7t sin 7J Lp sin 7, — sin 7J cos 7J Lp cos 7, 0 0 1
Yi = 7 i e 3 , T i ;
£7,
(61)
and all the functional matrices Fj reduce to the third unit vector e 3 . The solution scheme is therefore extremely simple. However, the algorithm is obviously advantageous for the use on the computer, not so much for demonstration by hand. We reduce therefore the chain to two contiguous bodies only (the double pendulum), or the non-moving wheeled robot (t>j = 0 , 7 6 = 0) retaining the indices 2 and 3 for upper and fore arm angle of the SCARA, yielding
M 2 := M 2 + T
m3 0 m3 l _ m f £ £
T32,
0 0
G 2 +T 32 m 3
Q2
:—
Q 2 ~^~ ^-32
-m 3 c 3 w 0 2 - m 3
[UJO2
wo2
0
0
0 Q3x ~ m 3 c 3 w o3 73
0
0
J3
J
+ 73J -32,
3
0
(62)
Hence, • Forward step: Kinematics from 2 to 3
y2 = e 3 7 2 , y3
L cos 72 L sin 72 j 72 + e 3 73
• Backward step: Matrices from 3 to 2
(63)
On the Use of Nonholonomic
72 = - [e^Msea]
eJM2e3 e3rG2e3
1
31
e^ {G 2 e 3 72 - Q 2 }
Ji + rn3L2 ( 1 -
m c
j 3 cos2 73
3 = - ^"303^2(72 + 73) sin 73-1713 L22, /( m - j 3-cg - j 72 sin73 cos73
e ^ Q = Q2z + Q3xL2 sin 73 + [Q 3 „ - ( ^ ) -m 3 c 3 L 2 (72 + 73)73 sin73, m
Variables in Robotics
Qsz L2 cos 73
leading to, with Qi =0 for simplicity,
3-^2 f —j^1 J 72 s m 7 3 cos73 + m3L2c3(j2
72
J2 + m3L22fl~
+ j 3 ) 2 sin73 (64)
^iCos273)
Forward step 73
7713C3.L2 cos 73 + J 3 \ ..
73
J
72
m3c3L2J2 cos 73
^
(65)
The result is easily proven with the well known equations of the double pendulum (relative angles): J2 + J3+ m3L\ + 2m3c3Lcosj3 J 3 + m 3 c 3 £ 2 cos 73 - ( 7 3 + 2 727s) 722
J3 + m3c3L2 cos73 J3
72 73
(66)
m2S2isin73 =
Solving the second line for 73 and insertion into the first line yields j 2 . Once 72 known, the second line yields 73. The basic idea of decomposing the mass matrix in the sense of a Gaussian algorithm is not new. Brandl et. al 2 for example eliminated the the contraint forces step by step from the last body to the root coming out with comparable results. First attemptions in this field have already been done by Vereshagin 40 . There is, however, a big advantage in the present procedure: The use of nonholonomic variables leads to the well structurized element matrices (35), starting already at the velocity level.
32
H.
4
Structurally Variant Systems
4-1
Bremer
Freeing from the Constraints
One of the main advantages in using minimal velocities according to Eq.(l) is, that for the constraint case the corresponding velocities may directly be cancelled in Eq.(35). (Recall that this is not allowed for any of the so-called analytical procedures! There, one has still to calculate derivatives w.r.t. s.) However, following Hamel's "concept of relaxation of the constraints", one may also use Eq.(l) for variables but introduce corresponding (constraint) forces instead of eliminating them, yielding for Eq.(57), along with €> = ( 9 $ / 9 s ) s + [d(d$/ds)/dt}s = 0
Ms+g-(— 9s
0, g = G s - Q
-1 f ^£
(67)
9s : AA + b = 0 (see e.g. (Schiehlen) 39 . The first attempt is obviously due to Jacobi 2 3 : " one has twice to differentiate the constraints...". Note that we are consequently using the "Helmholtz auxiliary equation" and therefore not making difference in the nature of (holonomic or nonholonomic) constraints any more). We release then, secondly, the constraints. This means that the A convert into impressed (generalized) forces which "depend mainly on the afore restricted coordinates by forbidden directions" 16 . The (generalized) force directions are by this in any case known, but the A's have then to be treated as work performing forces, i.e. the configuration space is widened up. 4-2
Remark on the Choice of Minimal Velocities
When freeing the system from the constraints, then minimal velocities according to Eq.(l) offer a clear cut advantage concerning the arising (generalized) force direction. This can easily be seen by inspection of Eq.(57): The chosen variables are such that (
|||-> Load
OR Variable resonance type side branch resonator
Mulit DOF type Helmholtz resonator
Compensators
3.2
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
55
Novel devices
1. Common idea for all novel devices Although the concept of mechanical vibration compensation is commonly known in Engineering mechanics, a solution of the equations of motion for the simple case of an un-damped single degree-of-freedom (DOF) mass spring oscillator with mechanical compensation will be presented.
Secondary (compensatory) system
Primary (engineering) system
Excitation x0(t) = XQ cos(fit) Figure 4. Schematics of Engineering and compensatory system
The equations of motion for this system may be written as [14, p. 449] mi X\ = -cx(xi
— x0) + c2(x2 - xi)
m2 x2 = -C2{x2 — xi).
(4a) (4b)
Using the functions x\(t) — X\
cos(f21)
x2(t) = X2 cos(Q t)
(5a) (5b)
for the steady state solution of the equations of motion, the amplitudes
56
J. Mikota
X\ and X 2 of the masses mi and m 2 respectively are given as
*1
X2--
( - m 2 Q2 + c2) ci X0 (6a) —mifi m 2 + mi Q.2 c2 + c\ m 2 fi2 — c\ c2 + c2 m 2 D.2 c2 c\ X0 .(6b) 4 2 -mifi m 2 + mi fi c2 + ci m 2 fi2 — C\ c2 + c2 m 2 Q,2 4
A closer look at the numerator of Eq. (6a) makes clear that a properly tuned secondary system, i. e. the natural frequency
(7)
UJC
of the compensatory system placed at the harmonic of the excitation xo(t) = XQ cos(fit) may be used to effectively cancel the movement of the primary system with respect to the excitation xo(t). In that case the amplitudes Xi and X 2 are given as (8a)
Xi = 0 X2 =
C\
XQ
(8b)
f22 m 2
Furthermore, the mechanical system Fig. 4 may be associated with its dual hydraulic system depicted in Fig. 5.
c,
c, Qo
>Q2
Qi—•
L,
Li Pi
P2
Figure 5. Dual hydraulic system to mechanical system Fig. 4
Compensators
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
57
The equations describing the dynamics of the dual hydraulic system are 1 (9a) P i = ~-(Qo-Qi) P2 = 7 ^ ( Q i - Q 2 )
(9b)
Pi = L\ Qi + V2 V2 = Li Ql-
(9c) (9d)
Taking the derivatives of pi and p 2 (10a) (10b)
Pi = Li Qi +p2 t>2 = L2 Q2 and re-arranging the terms in the equations results in 7 ^ ( Q o - Q i ) = ii• L
x=>Q. Coming back to our original problem of reducing fluid borne noise in a hydraulic circuit, the excitation of the Engineering system xo(t) (in case
58
J. Mikota
of the mechanical system) and Qo(t) (in case of the hydraulic system), may be compensated by a properly tuned oscillator where the natural frequency of the oscillator u>c is placed at the harmonic of the excitation ft. 2. Mass spring oscillators • Single degree-of-freedom mass spring oscillator Since a mass spring resonator for the compensation of fluid borne noise needs to be sealed from the hydraulic circuit and the dynamic behaviour of a single degree-of-freedom (DOF) mass spring system is commonly known, the attention of this paragraph is focused on the influence of damping.
damping between mass m and cylinder wall
I (a) Arrangement in hydraulic circuit
(b) Equivalent system
Figure 6. Single DOF system with damping
The equation of motion and the dimensionless damping ratio £ for a (homogeneous) system depicted in Fig. 6 are given as
0 — mx + dx + cx
(14) (15)
The different responses of a single DOF mass spring system with m = 1 kg, C € {0.01,0.1,1} ^ and c = 1 N/m are depicted in Fig. 7.
Compensators
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
59
i... UJN at a certain base harmonic Q,\ and N — 1 integer multiples of it 0,2 •• • QN, will be presented [8]. The equations governing the dynamics of an N-body oscillator de-
Compensators
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
F>>
Figure 9. Multi DOF oscillator in hydraulic circuit
picted in Fig. 10 may be written as follows:mi xi + c\ xi - c2 (x2 — xi) = 0 (17a) m 2 x2 + c2 (x 2 - xi) - c3 (x3 - x2) = 0 (17b)
mjv-i Xjv-i +
CAT_I (XTV_I
- zw_ 2 ) - cN (xN -
XJV-I)
= 0 (17c)
mpf XM + CN (XN — XN-I) + dx±N = F (17d)
m.
"fr
F(t) Excitation
A/V Figure 10. Structure of a third order system
In vectorial notation, above equations may also be written as Mx+Dx+Cx=F
(18)
where x = [xi.. . XJV] T , M is the mass matrix, D is the damping matrix, C is the stiffness matrix and F is the force vector of the system.
61
62 J. Mikota Assuming an undamped system {d^ = 0), the natural frequencies of the oscillator may be calculated as the roots of the characteristic polynomial l - u / M + q =0.
(19)
A closer look at Eq. 19 reveals that the characteristic matrix (—w2 M + C) has tri-diagonal structure and may be written as / —CJ
mx+ci+C2 Q~
—C2 I.I
0
nin-L/'nJ.Cn
-CN-1 \
\
f*r*
0
—U mjV-1+cjV-l+CJV — CN —CN — w 2 m j v + c j v /
By defining the first natural frequency of the oscillator as Q, and integer multiples of it as
fix = fi ft=22fi
(20a) (20b)
= ATQ
(20c)
QM
the natural frequencies of an (un-damped) chain structure oscillator uj\... WJV may be placed at fii... QN simply by making the masses m\... mjv to m\ = m
(21a)
m 2 = m/2
(21b)
mN = rn/N
(21c)
and making the stiffnesses of the springs c\... CN to C! = N c c2 = (N - 1) c
(22a) (22b)
cN =c = Q.2 m.
(22c)
Compensators
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
63
To make this approach more plausible, the characteristic matrix [13,16] may then be written as / -u>2 + {2N-l)Q.2 -(AT-l)fi2 0 \ -(N-i)n2 -±£+(2N-3)n2 -(N-2)n2 -2fi 2
\
-(Jy_1)+3n2 2
o
-Q
-n2
-£+n2/ (23)
Although a formal proof for arbitrary N cannot yet be presented, the characteristic polynomial has the form
mNl[(—+itf),
(24)
which places the natural frequencies u> of the oscillator exactly at fii...fijv. Due to the fact that the resonance frequencies of a damped multi body oscillator are only marginally different from the natural frequencies in an un-damped case (see Fig. 11), the frequency tuning concept presented in this paragraph is also suitable for lightly damped oscillators (£ < 0.1), where the dimensionless damping ratio £ is defined as C=
dN
r— 2 mjv,
(25)
3. Analogy between mechanical and acoustic systems According to the derivations given in the previous paragraph of this section, the natural frequencies of a mechanical chain-structure oscillator may be placed at fii and N — 1 integer multiples f^ • • • ^N- In this paragraph it will be shown that this concept of frequency tuning is not limited to mechanical systems but may also be applied to acoustic systems. Since the general duality between mechanical, electrical and acoustic systems has been described in a number of excellent books (e.g. [15]), the analogy will be shown on a system with 2 DOFs. The dynamic behaviour of the mechanical system depicted in Fig. 12 is described by m\X\ + c\X\ — C2(x2 — x\) = 0 •m2x2 + c2{x2 -xi) = F,
(26a) (26b)
64
J.
Mikota
2*103
4*103 Frequency [rad/s]
103
8*103
10"
Figure 11. Normalised bode diagram of a 3 r d order system featuring damping ratios £ £ {0,0.1,0.01}
whereas the behaviour of the acoustic system is described by p = L2Q2 + P2
(27a)
P2 = ^r(Q2-Qi)
(27b)
p2 = L1Q1+pi
(27c)
Pi = -J-Qiw
(27d)
By using different state variables, the equations describing the behaviour of the acoustic system may be also be written as u = p = L2Q2+P2 = L2Q2 + 7r(Q2-Qi) ^r(Q2-Qi)=P2
= L1Qi +
^rQi.
(28a) (28b)
Compensators
for the Attenuation
of Fluid Flow Pulsations
in Hydraulic Systems
'/////////////A c,
tk
m,
P2
oJJE
m.
T
(b) Acoustic system
(a) Mechanical system
Figure 12. Mechanical and acoustic systems
In vectorial notation these equations are simply
ical system Xmech =
(*1 Xi X2
^•mech ~
"
/0 / " Xmech ~
1
\
A
Vo
(29a) (29b)
X2)
9i+Sa. 0 -a- \ mi
0
£Z_ mi
0
mi
0 0
Xmech 1
Q _ _£2_ m?
1 0 )
+ ( 0 0 ^ 0 ) T umech
(29c)
65
66
J. Mikota
Acoustic system A
a c o u — (Qi
u
acou
Qi Q2
(30a)
Q2)
(30b)
~ P
l/C 2 + l / d
/Q
acou —
1 0 [0
0 v* \ 0 0 0_IZ£a
0 ^ 0
1
+ (0 0 i 0 )
+
1J2
L.2
T
0 y
, acorn
(30c)
where umech and u a c o u are the inputs to the mechanical and acoustic system respectively. Looking at these equations, it should be plausible that the natural frequencies of a multi DOF Helmholtz resonator may be placed at fii . . . Qpj, where 1 fii = n = -== (31a)
Vie
9.2 = 2 ft
(31b)
nN = Nil
(31c)
by making the hydraulic inductivities L\...
LN to
LX=L L2 = L/2
(32a) (32b)
LN = L/N
(32c)
and making the hydraulic capacities C\... CN to C
^ =
Co =
CM
N-C 1 N-l
= C.
(33a) (33b)
(33c)
Compensators
for the Attenuation
of Fluid Flow Pulsations
I
in Hydraulic Systems
67
I Figure 13. Principle of an oscillator based on a circular plate
4. Compensators based on plate/shell elements In order to discuss the principle of compensators based on plate or shell elements, the simplest possible compensator of this kind (see Fig. 13) will be discussed in this section. This is a plate of homogeneous thickness h, clamped at the circumference and tuned in such a way to place the first natural frequency at the base harmonic of the pulsating flow stream. In addition to that, the stresses in the plate due to the maximum hydraulic pressure psys must not exceed the maximum permissible stress amax of the material. The first natural frequency fin of a circular plate of constant thickness h being clamped at the circumference is given as [11] A n S* 10.216
where
fin
and the flexural rigidity of the plate
(34a)
Eh3 . (34b) 12(l-i/2)
D
The induced bending moments per unit length due to a constant pressure distribution at the bottom surface in radial and tangential direction, Mr and Mv respectively, are given as Mr
p a?[l + v- (3 + !/)a2]
(35a)
^+u~(l+3
(35b)
16 M = 1p a 2
* i&
v a2
)\
68
J. Mikota
where a = r/a. At the circumference r — a, above equations simplify to Mr = --p
a2
and result in maximum stresses at the bottom/top layer of 6 Mr <Jr = —f£-
and
6MV ov = —f^1-
(37)
These stress components ar and av respectively may be combined to an equivalent stress &E according to the "von Mises" hypothesis OE = \]0r
~ °r
