Mech. Mach. Theory Vol. 22, No. 4, pp. 385390, 1987 Printed in Great Britain. All rights reserved
0094114X/87 $3.00 + 0.00 Copyright © 1987 Pergamon Journals Ltd
O P T I M I Z A T I O N OF P A R A M E T E R S OF T O R S I O N A L D A M P I N G SYSTEMSt" V. Z E M A N and Z. H L A V A ( ~ Technical University, Nejedlfho sady 14, 306 14 Pilsen, Czechoslovakia AlaftractThe basis for optimization of mass, damping and stiffness parameters of machine drives is a discrete linear torsional mathematical model in a matrix form. Inertia, damping and stiffness matrices are expressed in dependence on the discrete elements' parameters. The method of optimization is based on minimization of objective functions in the form of linear nonequations.
INTRODUCTION A new machine project, developed mostly with respect to the high efficiency, acceptable price and admissible static stress, need not necessarily result in satisfactory dynamical properties of the work. Because of economic reasons reconstruction from the point of view of dynamics should already be solved when projecting. The method of reconstruction which makes possible a quick and satisfying solution in practice (in the first approximation) is therefore the one supposed. The basis for solution must be the information gained from the drawing of the preliminary project. Our paper will only deal with the method of reconstruction of machine drives that, in the first approximation, can be replaced by a discrete linear torsional dynamical model. Excitation is supposed to be polyharmonic or periodic (harmonic). The mathematical model has then a matrix form
STABLE DYNAMIC RESPONSE Stable dynamic response of the mathematical model (1) is given by a particular solution q(t) = Re
qk(cok,p)exp(icokt) , p = [pj].
(3)
k 1
M ( m ) ~ ( t ) + B(b, bE)el(t) + K(k)q(t) 1
 ~ (fkccoSOj +fk, sincokt),
the disc It, from the position given by nominal static load. Vectors fkc of cosine and f~ sinusoidal components together with excitation frequencies cot, k = 1, 2 , . . . , l represent the excitation. Reconstructing the machine drive aims at reducing the stabilized torsional vibrations [displacements q~(t) of the discs It] or torsional moments Mr(t) transmitted by torsional coupling between discs I t_ t and /~. Reconstruction is always achieved by changing the parameters pjf{I~, k~, b~, b~ } chosen from the set of all parameters of discrete elements.
(l)
k~l
where square symmetric mass, damping and stiffness matrices are the known functions of vectors
Vectors of complex amplitudes in (3) can be expressed[l] either by the dynamic stiffness matrix in the form
(cok,p)  [  M(m)~ + r(k)
(2)
+ i~a(b, bE)]'(f~/f~) (4)
of discrete elements' parameters. The meaning of the symbols in (2) is
or by eigenvalues 2~ and eigenvectors v~ in the form
m = [/~],
k = [kj],
b ffi Ibm], b E  [b~]
~(COk, P) ffi ,l~ icokV'VrAv(fkc  if~), It = moment of inertia of the disc (kg m :) k~ ffi torsion stiffness between discs /,_ ~ and /~ where i is an imaginary unit. (kg m 2 s 2) Eigenvectors v, in (5) satisfy the relations b~ ffi viscous internal damping coefficient between discs/i J and It (kg m 2 s  l ) [22M(m) + 2,B(b, b e) + K0c)lv, = 0, b~ ffi viscous external damping coefficient of the where 2, are roots of a characteristic equation coupling of the disc It with frame (kg m 2 s l ). det [22M(m) + 2B(b, b E) + K(k)] = 0, Element q~(t) of vector q(t)[q~(t)] of the finite dimension n has the meaning of the turn off angle of and are standardized by the formulation
(5)
(6)
(7)
v~r[22,M(m) + K(k)]v~ = 1, v = 1, 2 . . . . . 2n. (S) i" Presented at The Fourth Conference on the Theory of Machines and Mechanisms, Liberec, Czechoslovakia, 11I 3 Complex amplitudes of additional (dynamic) torSeptember 1984. sional moments transmitted by torsional couplings 385
386
V. ZEMANand Z. HLAV.~(~
between discs /,._ ~ and /, are given by the formula
M,,k(tok, p) = (k~ + icokbi) x [q~_1.k(09k, P)  q,. k (tok, P)], (9) where q~.k(tok, P) is the ith element of vector qk(tok, P) and q0.k = 0. RECONSTRUCTION AS AN OPTIMIZATION PROBLEM
~s(p) = ~ ~ #X,.~lq,.k(O~k, P) t i
+ V,.klmi, k(Ok,P)l.
(10)
Complex amplitudes of displacements in (10), and in formula (9), for moments can be calculated according to (4). Nonnegative weight coefficients Pi, k or vi,k correspond by index i to displacement qi or torsional moment Mj and by index k to harmonic components of excitation moments with frequency COk" If the excitation frequencies t~k change during operation (quasistationary excitation) and resonance states cannot be eliminated the objective function is formulated as ~'(P) = X X }] [#xi,J,kIq~,k(/TS'P)I k
j
i
+ Vi,x,klg~,k(/Tj, P)I]
(11)
where, according to (5) and (9), 2n
Vi v
T
q,,k(flJ, P) = X "' v,(fkcifg,) v=l iflj ~v
(12)
2n
p) =
(k, + i p,) X x
l ) i  l,v   Ui v
T
iflj  2~' v, (fk<  ifk~)
From the vector of parameters p = [mr, k x, b T, (bE)r]T, by means of a selection vector, we will choose components that will compose optimizing (changeable) parameters pj, j = 1. . . . . s. Movement of these parameters occurs inside sdimensional parallelepiped in the form
pL<~pj<~py,
Reconstruction of machine drive is understood as a vector calculation p = [pj] of parameters chosen from the set of all parameters of discrete elements, which in the feasible domain p~ ~
k
MINIMIZATION PROCEDURE
(13)
and flj are imaginary components of eigenvalues 2j, j e {1, 2 . . . . . n }. Nonnegative weight coefficients /&Zk or V~j.k correspond by index i to displacement q~, by index j to resonance COg= flj and by index k to harmonic components of excitation moments with frequency cok. Minimizing objective functions in equation (10) can suppress vibrations of chosen discs /,., and dynamic load of shafts (of torsional couplings) between discs I~_ ~ and I~ caused by polyharmonic stationary excitation. By minimization of the objective function in equation (11), it is possible to suppress vibrations of discs I~ and dynamic load of shafts between discs li_ 1 and /l in resonance peaks tOg = flj caused by quasistationary polyharmonic excitation.
j = 1. . . . . s,
(14)
where pL and py are vector components of upper and lower limits of optimization parameters. From the mathematical point of view, the optimization problem is the nonlinear programming problem. It is the problem of finding a minimum of the objective function 0(P), while satisfying condition (14). With regard to condition (14), it is always possible to ensure that the starting point of the optimizing process be inside the sdimensional parallelepiped area defined by constraint (14). Therefore, after ensuring the permanent incidence of vector p with feasible domain, the objective function is modified by the additive member 10,_ k ~ (
~rj~ + ~r,+j, k "~
tmjpy pyp/'
(15)
where trj, k or tr,+s,k, j = 1,2 . . . . . s are weight coefficients and correspond to the  j t h nonequation. There are a number of modifications (index k) to enable the finding of a minimum, even for the case of its being on the bound of the feasible domain. In fact, we solve the sequence of minimization problems where, because of the increasing index k, the free minima (across the whole sdimensional euclidean space) converge to the bound minimum in set (14). Because of the complexity of the modified objective function, a method not demanding the knowledge of derivations was chosen. We used a search method in the socalled conjugated directions in relation to the Hessian matrix of the objective function in starting point[2]. This method precisely minimizes objective quadratic functions in the s direction. For general functions it was taken as an iterative method where the last point of the preceding iteration is taken for the starting point of the following iteration. The method of conjugated directions was modified according to Powell's algorithm[2]. The direction with the least decrease of modified objective function is in the following iteration replaced by given difference between the last and starting point of preceding iteration. Finding the minimum of the modified objective function restricted already to one of conjugated directions (socalled onedimensional search techniques) is performed by the method of parabolic interpolation (extrapolation) of the function. Interpolation is performed only when three points which the parabola interlaces satisfy the socalled convex test. This test demands the "middle" point to be
387
Torsional damping systems "over" the line segment connecting both end points• Only under this supposition does the points function of one variable approach the minimum. The minimization procedure is composed of three, in essence, iterative procedures. First, the minimum of the function restricted to the direction is specified. Iterational process of a higher grade is performed by the method of conjugated directions. Decreasing of the barrier member (15) is the highest iteration procedure (successive modification of the objective function). All iteration procedures are stopped in the case that the relative error is lower than the fixed limit• A F O R T R A N computer program, whose given basis are the minimization procedures modified from[3], worked out the optimization of torsional systems• A development diagram of this program is shown in Fig. 1.
fl = b2/b ~ and ~ = k2/k, of the dynamic damper of vibrations (Fig. 4), connected with the torsional system I~, b_t b , k~ at given damping ratio D I = b E/2 l x/k t I l . Harmonically variable excitation moment M j ( t ) = M, cos o~tt acts on the disc I~. Excitation frequency ¢o~can be slowly changed along the whole frequency range Ogle <0, oo). Objective function (11) was rewritten into a dimensionless form (for k = 1, v~,j,k = 0) kl ~'(P) = ~ [Iqt. l(fl,, P)I + Iql. l (f12, P)I + 0.1 Iq2,1(fll, P)I
+ 0.1 Iq~,~(/~2, ~)l] The calculation was repeated for different values of damping ratio D l and for starting parameters determined by the vector
APPLICATION
#0)= [~t0) fit0) ~,t0)]r=[0.05
The method described in the previous section was applied to the optimization of machine drive parameters, which, after reduction, e.g. for highspeed shaft, has the form of a twosided, rigidlyfixed torsional system (Fig. 2). For k~ = 0 or k,+z = 0 it is onesided, rigidlyfixed; and for kl = k,+l = 0 twosided, isolated chain• Vector elements fkc (or f~) in (1) have the meaning of harmonic cosine (or sinusoidal) components of excitation moments Mi(t). Mass, damping and stiffness matrices in (1) are expressed in the form
1 0.05] r
while satisfying the conditions 0.01~<~<0.1;
0.1~
0.01~<~<0.1.
Values of the objective function before and after optimization, and parameters after optimization, are given in Table 1. Decreasing the upper limit ~u decreases the efficiency of dynamic damper as follows from comparison of the achieved values of objective functions for Dt = 0.01 in Table 2.
M(m) = diag (I~) b2 b 2
b2 b2 + b3
0
0
b 3
.. •
0 ...
B(b, b E) = diag (b~) + .. 0 .. 0
Ikl
2
k2
+k~ k2+k3 K(k) =
0 k 3
•. . 0 • .. 0
If the dynamic response cannot be reduced considerably due to the change of parameters, higher efficiency can be achieved by (a) means of inserting (Fig. 3) flexible coupling, (b) passive dynamic damper of the resonance damper type, (c) rubber damper and (d) viscous damper with parameters la, Ib (or/), b, k. Software (in FORTRAN) enables optimizing an arbitrary set of parameters ptr{I~,k~, bi, b~} by minimization of objective function (10) or (11) in the feasiable domain of parameters. The program was tested on the example of calculation of dimensionless parameters a = I 2 / l l ,
bn_ 1 bn_l +b" 0 b.
0 .. 0 ...  k._ 1
k._ 1 + k.
0
k~
b. b.
k]
k~ +k~+l
CONCLUSION The described method enabled, by means of digital computers, optimization of parameters of discrete elements torsional damping linear systems. The optimization aims at suppressing the steady vibrations of chosen discs and dynamic load of chosen shafts caused by polyharmonic excitation. Objective function in the form of weighted sum of amplitude of displacements and amplitude of torsional moments is after barrier members' modification minimized by means of Powell's method of conjugated directions.
388
V. ZEMAN and Z. HLAV,~
Input parameters describing system J
damping and stiffness matrix systems assembLingl
CaLcuLation and I= quasistotlonary inversion l of complex .. dynamic stiffness mofr x
~tationary

l
CaLcuLation of  rcomplex ampLitudesJI (4] matrix  L__
CaLcuLation of complex amplitudes of torsional moments
I Objective function caLcuLation ,
1 J CaLcuLation of J (5] p  I co~nptex ampLItudesJ  J  matrix J

r. . . . .
I
I
L .
} ....
ICaLcuLotion of eigen numbers and standardized eigenvectors (modal analysis)
~
(9)
.
.
.
I I' . . . .
.
J
n (11) I . . . .
rIL   ~(10] 
1
CalcuLation of complex amplitudes of torsional moments
JObjective function I ~ caLcuLation I
1
Tnput parameters for controLLing optimization
I
Barrier terms objective functlonJ mod[ficot on I
1
Mintmizotion of modified objective function by meons
of PoweLL's method (caLcuLation of a new vector p]
i
eters I
Lower then
d e m ~ m d e d
JreLative'~,,
Fig. 1
bigger then ~aon¢l~l
Torsional damping systems
389
M~(t)
M (t)
M.(t) D
z,
( (
Fig. 2 Ir, b
Z
(a)
(b)
k, b
(d)
(c) Fig. 3
M1(t)=M1 cosc~t
k2, b=
•  .   ~  . ~
~.
Fig. 4
Table 1 D
0.005
0.01
0.02
0.05
0.1
0
~,~(o)) ~,(p) =~t ~8opt 7o~
18.722 10.713 0.0996 3.7074 0.0785
16.395 10.319 0.0996 1.8682 0.0774
23.900 9.620 0.0994 0.9479 0.0746
10.349 7.502 0.0959 0.5786 0.0432
5.511 4.875 0.0854 0.8545 0.0752
19.980 11.130 0.0996 oo 0.0793
Table 2 D =0.01
ocU 0.I
=u = 0.05
a U=0.02
~,(p) =o~ ,Oopt 7~
10.319 0.0996 1.8682 0.0774
14.934 0.0493 0.7199 0.0423
24.396 0.0200 0.2218 0.0176
390
V. ZEMANand Z. HLAV,~
The method of parameters optimization can be effectively used for projecting constructional modifications from the point of view of dynamics. REFERENCES
1. V. Zeman, Tuning and optimization of parameters of
vibration machine constructions (in Czech). Doctoral dissertation, Technical University, Pilsen, Czechoslovakia (1984). 2. D. M. Himmelblau, Applied Nonlinear Programming. McGrawHill, New York (1972). 3. C. H. Suh and C. W. Radcliffe, Kinematics and Mechanisms Design. Wiley, New York (1978).
DIE PARAMETEROPTIMIERUNG DER GED.~MPFTEN TORSIONSYSTEMEN Ztttmmnteaf~mmagDiedynamische Optimierung der Massen, Diimpfungsund Steifigkeitsparameterbei den Antrieben geht yon dem linearen diskreten mathematischen Modell in Matrix form hervor. Die Massen, Dilmpfungs und Steifigkeitsmatrizenh~ngen yon den Zugeh6rigen Parametern der diskreten Elemente ab. Die angefiihrte Methode der nichtlinearen Programmierung geht yon der Minimierung der Zielfunktionen mit den Besch#inkungen in der Parametern in der Form von linearen Ungleichungen hervor. Das Programm f/Jr die Optimierung der Parameter der Kettentorsionssysteme wiirde in Fortransprache ausgearbeitet und wird auf dem Beispiel des dynamischen Diimpfers getested.