Sem.org IMAC XVI 16th Int 160906 on Theory Dynamic Vibration Absorption Its Application Vibration

7/24/2019 Sem.org IMAC XVI 16th Int 160906 on Theory Dynamic Vibration Absorption Its Application Vibration

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THE THEORY OF DYNAMIC VIBRATION ABSORPTION AND ITSAPPLICATION TO A VIBRATION SYSTEM

WANGMING LU

Boston University, Department of Aerospace and Mechanical Engineering110 Cummington Street, Boston, MA 02215, U.S. A.

MINGWUWANG

Shanghai Jiao Tong University, Department of Power Machinery Engineering

1954 Huashan Road, Shanghai. 200030, China

ABSTRACT Based on modem theories of functionalanalysis and optimization theory, a new theory ofdynamic vibration absorption is derived after detailedanalvsis of a linear damped multi-degree-of

freedom MDOF) system. The new theory has advantageover the traditional ones in that it involves only oneindependent variable, i.e., the force exerted by dynamicvibration absorber DVA) to the unit to which it is

attached, regardless of the number of freedom of theprimary system. The closed form for the optimizationresults are given. Finally, as a verification, this methodis applied to a linear damped 3-DOF system with threeDVAs attached to the primary system, to demonstrate itsefficiency and simplicity, and the results are compared to

those given by other published literature.

NOMENCLATURE

M K, C mass, stiffness and damping matrixof the primary system.

mass of the ith dynamic unit,

K; , C stiffness and damping between the {i-1 th andith dynamic unit,

m k c.l f l

DVA.

g t)

mass, stiffness and damping of the ith

force vector determined by DV Asattached to the primary system,

X displacement vector,

displacement of dynamic unit andDVA respectively,

g opt t) optimization result of g t ,f t ) external exciting force vector,

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R Mffi jCm K

R; resultant matrix after replacing the ith

column of matrix R with f + g .~ value of algebraic determinant of

R k .i

i) natural frequency,,-----

j ~ 1

* denote the conjugate of a complex.El total energy of l dynamic units.

m number of DVAs attached in theprimary system.

number of dynamic units for

optimization objective function.

Bequation set.superscript:

coefficient matrix of optimization

r , i real and imaginary part of a complexnumber. respectively.

1. INTRODUCTION

The optimization theory of the optimally tuned anddamped dynamic vibration absorber, derived originally

by BROCK[1] is to be found in books on mechanicalvibrations such as the well-known text by DENHAR.TOG[2]. When there is no damping in the primarysystem. the optimization procedure is simplified by theexistence of the fixed points on the family of amplitudefrequency response curves[2], and hence, the optimumparameters of DV A can be obtained by equal peakmethod. All systems, however, contain some damping,and when optimizing parameters of DV A, one can not


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neglect the influence of damping in the primary system.

Optimization by numerical means was obtained for a

three-element-damping dynamic absorber[3], or a

viscously damped dynamic absorber[4],added to a

viscously damped primary system. Finally, a graphical

solution based on the optimization cycle technique was

obtained for a primary system with hysteric damping and

a viscously damped dynamic absorber[5].

To investigate the effects of a dynamic absorber on a

multi-degree-of -freedom (MDOF) primary system, a

method has been suggested by which the latter is reduced

to a single-degree-of-freedom (SDOF) system processing

an equivalent mass[6]. A representative case for a

viscously damped dynamic absorber has been treated

numerically[?]. KAzuTo and KoUICHI discussed the

optimum parameters of DV A for a n-DOF system in thefrequency range by introducing the concept of equivalent

mass[8]. W. Lu has given the optimum parameters of

DV A attached to a damped MDOF system[9-ll].

In what follows. based on the aforementioned literature, a

new theory of dynamic vibration absorption for a linear ,

viscously damped. MDOF system is presented byauthors. In this method, the total vibration energy of

some dynamic units in the primary system or the overall

primary system, is employed as objective function for

optimization. This function in itself is expressed withregard to the optimum force vector exerted by every DV A

on the dyna mic un it to which this DV A is a ttached . By

using functional analysis theory, the optimum force

vector can be obtained in order that function value be

minimized. As the former is determined by the vibration

performance of DV As attached to the primary system, the

optimum parameters of every DV A can be computed. It

is obvious , from the subsequent discussion, that the

above mentioned method has considerable advantages

over traditional ones in that its employed objective

function depends only one independent variableregardless of the number of DOF of the primary system.

So, substantial computational efforts can be avoided, and

consequently, the optimization process turns to be more

efficient and simple.

For convenience, theoretical formulation of proposed

method is given in section 2, as well as the closed

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solutions for optimum force vector determined by

parameters of DV As. Finally, a linear damped 3-DOF

system is employed to demonstrate the effectiveness and

simplicity of this method.

2. OPTIMIZATION THEORY OF DYNAMIC

VIBRATION ABSORPTIONConsider a linear damped n-DOF system, as shown in

figure 1. For generality, one call a mass-spring-dashpot

subsystem as a dynamic unit, and we assume that it is

possible that only m dynamic units to which DVA can beattached to absorb unwanted vibration, and vibration

absorption objective is to minimize total vibration energy

of l dynamic units in primary system. It is obvious that

m l-< n

The effect of DV A attached to the dynamic unit ofprimary system can be represented by a force. g t) .

exerting on this unit. The equation motion for system

shown in figure l can be written in the matrix form as.

x(t) + Ci(t) + KxU) = (t) + g(t) 1)where,

Ml 0 0 0

0 M2 0 0M=

00 0 Mn

KI +Kz) -Kz 0 0- ~ +K3 -K3 0 0

K==

0 0 -Kn-1 K _t + K. -K.

0 0 - K . K.

C1+C2) -c; 0 0-c; C 2 ~ -C3 0 0

C

0 0 -Cn-1 cn 1 c . -c.0 0 c . c.

2-4)

x(t) = [x 1(t) X2 (t) X , (t) rj t ) = h t), 2 (t) ... , , (t) rg(f) = [0,- 0 g n-m+l) (f), g n-m+2) (f) , , g 1 (t)] T

(5a-c)

Equation (1) can be manipulated into:


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-Mco 2 + jCco + K) X j + g 6) optimum is equivalent to the determination of theoptimum g t ) , as denoted as gopt t) . For convenience.

So, it can be seen that, from the right hand side RHS) of let

the above equation, that installation of m DV As to m

dynamic units of the primary system is equivalent to

addition of a force column vector g t ) , defined in 5c),to RHS of equation of motion. By carefully controlling

g t ) , one can alter system vibration performance, and

thus, reach the optimizat ion goals.

As the coefficient matrix. i. e., - Mco2 + jCco + K, ofequation 6) is nonsingular matrix. for steady state forced

vibration of the system. its roots can be obtainedaccording to Crammer law.

X t -Mw 2 + jCw + K) - 1{ +g) 7)

displacement of the ith unit. X; , can be obtained.

8)

Recall the optimization objective function, that is,

1E; = ; ; 2. M;i;: 9)

... i= l

substitution of 8) into 9) yields,

1 IRIE; = - 2. M;co 2 - ; 2

2 i= l IRI10)

\R jcan be written in the formn

IR I= I..ctk+ gk Aki 11)k=l

By substituting 11) into 10), one can obtain,

1 lcol 2 n 2E; = 2 tM; \Rf)~ f k+ gk Aki 12)

It is obvious that E; is positive definite, or at least,

semi-positive definite function. Consequently, the

optimization problem can be converted into the

minimization of E;. After determining the externalexciting force, one can see that depends on only the

equivalent force column vector, g t) . By this way, the

number of independent variables in the optimizationobjective function is reduced to only one. To find the

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2n

e = 2., Jk +gk)Aki 2::0 13)k=l

(J) 2

m IRI) > 0 14)

consequently, when e approves its minimum value. E;

does too. Let introduce the following variables.

rg k = g k j g k

Ak =A; j A ~

15a)

15b)

15c)

By this manipulation. e can be written in the function of

real variables. i.e.,

e ~ [t, f,+, A.,][t,u:+; A ~ J~ {t,lu:+; A ~ -u +: A ~ ] r 161+{t,lu:+; ~ -u: +: A ~ ] r

At its points of minimum value, the following conditions

must be satisfied,

a e;j r = 0, a e;j i =0.;a gn-o ja gn-o

8 = 0. 1. . m - 2. m - 1 7a-b)

After differentiation with regard to the real part, g andimaginary part, g ~ ,of the jth element of force column

vector, i .e, g i and let the results equal zero, one obtain.n

A; 2.,[J; + ; A ~-A;JJ: + ~ A ~ ]k=l

n

+A;; 2.,[J; + ~ ) A ~-A;JJ: + ~ A ~ ]= 0k = l


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II

- A ; L [ f : + ; A ~-ASJJ: + ~ A ~ ]k=l

n

+A;; L [ f : + ; A ~-A;JJ: + ~ A ~ ]=0k=l

(18-19)

multiply equation (19) with j = and hence, addequation(18) to it yields,

n

A;; L [ f : + g;) A ~ +} A ~ ; +k=l (20)

that is.n

A;;L[Ak; fk + gk)] = 0 (21)k=l

We call (21) as optimization equation set. and as A isnot always zero, hence. there must have.

n

L[Ak; fk + gk ] = 0 . i = 1. 2 . ... z (21)k=l

For this equation set, one can have the followingdiscussions.(1). For the case of l>m, (21) has no roots, i.e., when thenumber of DV A is fewer than that o f dynamic unit to

which a OVA can be attached, it is impossible to obtainthe minimum value for the objective function, but only asmaller value, if any, to some extent.

(2). For the case of l


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so, the following non-dimensional equations can beobtained,

1 2 2 rn 1 f Ai ~ ; = Z; (29a)

0 _ 1 2 h = ziQ. A1 f 1A (29b)where.

\ 1 f 2 ) 2 j 2 2~ - ;= - Ai i X i (29c)

From(29.c), it can be seen that. to determine the dynamic

parameters of a DVA. i.e., m. , k. and c., one can fist1 1 1determine one parameter. usually mass of DV A, whichmay be one tenth of the mass of the unit to which it isattached.

3. APPLICATION EXAMPLE AND DISCUSSIONS.

In the aforementioned sections. the dynamic vibrationabsorption theory for linear damped multi-degree-offreedom system is presented. The closed solutions foroptimum parameters of DV A are given, and the detailedoptimization process is demonstrated. In the comingsection. as a practical application example. parameteroptimization of DV A attached to a linear damped 3-DOFsystem. as shown in figure 3. is investigated to show theefficiency o aforementioned theory.

Consider the 3-DOF system shown m figure 3, thedynamic parameters of every units are as follows.M 1 = 0.25kg. M 2 = O.Skg. M 3 = l.Okg,C1 = 0.001Ns I m,

C1 =0.01Nslm.C 1 =0.1Nslm, K1 = K2 = K3=1500 N/m. It is easy to get natural frequencies of thesystem shown in figure 3 as follows,

ffi 1 = 119.5 rad Is) , ffi2 = 69.5 (rad Is) ,ffi

3= 19.8 (rad I s

Quite often, the DV A is employed to abate the vibrationmode corresponding to the dynamic unit to which thisDV A is attached. So, every vibration mode frequencyshould be treated separately while using the above theoryto tackle this problem. based on the aforementionedtheory, A MATI.AB program is developed to calculate theoptimal results o DV As, which are given as follows,

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m1= 0.03kg,

~ opt= 0.178,

c lop := 1.170 (NS I m

m2= 0.05kg,

~ 2opt= 0.142

c lopr= 0.935 (NS I m)

m 3= O.lkg,

~ 3opt= QJ63,

c Jopr= 0.599 (NS I m)

rot= 109.34 rad I s ) .k opt= 358.68 N I m),

m P 1 65.79 rad I s ) .

k Zopz= 216.45 N I m).

rot= 18.38 rad I s ) .k Jopz= 33.37 (N I m).

Here. masses of DV As are determined at first underspecific engineering conditions.

The optimization results given by KAzuTo, SETO et, al[7], and those obtained by authors are compared infigure 4.

It can be seen that effect of DV A based on the theorypresented in this paper is more efficient than that ofDVA based on the KAzuTo s theory, especially for thesecond and the third peak values in the curves.

4. CONCLUSIONS.

Based on considerable endeavors of many scholars.authors presented an advanced optimization theory forparameters of DV A attached to a linear damped MDOFsystem. This theory. as it employs the optimum forcecolumn vector consisting of component force exerted bym DV As on the dynamic units to which the former areattached, is very efficient to get the optimal parameters ofDV As because the optimization objective functioninvolves only one independent variable which can mirrorthe vibration absorption effectiveness of the specificDVA. So the optimization process is simplified mostconsiderably.

Finally, as an application example, parameters of DV Asattached to a linear damped 3-DOF system is presented toillustrated in details the process based on the theorydeveloped by authors.


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ACKNOWLEDGMENT

The research work reported in this paper is financially

supported by The Chinese Ship Structuring Corporation.

This support is greatly appreciated. Many thanks should

be due to other researchers in the Institute of Vibration,

shock and noise in Shanghai Jiao Tong University, whohelp the first author process experimental data.

REFERENCES

l J. E. BROCK A note on the damped vibration

absorber , Transactions of the American Society of

Mechanical Engineers A-284, 1946.

2) J. P. DEN HARTOG, Mechanical Vibrations, New

York: McGraw-Hill Book Company, fourth

edition 1956.

3) J. I SOLIMAN AND E . ISMAILZADEH Optimization of

unidirectional viscous damped vibration isolation

system, Journal of sound and vibration 36, 527-539.,

1974.

4 A. G. THOMPSON Optimum tuning and damping of a

dynamic vibration absorber applied to a force excited

and damped primary system, Journal of sound andvibration 36, 527-539 , 1981.

5) J. B HUNT Dynamic Vibration Isolation and

absorption. London: Mechanical EngineeringPublications, 1979.

6) G. B. WARBURTON, In Dynamic vibration

Isolation and Absorption. London: MechanicalEngineering Publications, 1982.

7) F. W. LEWIS The extended theory of the viscous

vibration damper, Journal of applied mechanics 22,377-382, 1955.

8) KAzUTO SETO, KOUICHI IW AN AMI AND

YOSHIHIRO TAKITA, Vibration control of multi

degree-of-freedom system by dynamic absorbers,

Bulletin of Japanese of Society of Mechanical

Engineers 50. 1962-1977, 1962.

9) W. Lu AND M. WANG The effects of a viscously

damped dynamic absorber on a multi-degree-offreedom system, Noise and vibration control, 1996.

l 0) W. Lu AND H. WANG The numerical method for

parameter optimization of dynamic vibration absorberattached to a damped vibrating system, Journal ofvibration and shock, 1996

11) W. Lu AND Q. Yu, applying a combi ned substructure

transfer matrix method to the optimization of a

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vibration isolation system, Proceedings Of 5th

International Modal Analysis Conference, Feb., 1997.

c --1

Figure I. model of a linear damped n-DOF vibrationsystem with m DVAs attached

figure 2. detailed analysis for the ith dynamic unit

with a DV A attached

~2

Figure 3. The model of a linear damped 3-DOF system


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g1?=:le=:l

0

. - - - - - - - - - - - - - - - - - -

t.1lu .:z:

t.1luLlJ

'

u n

frequency (Hz)

Figure 4. The comparison o optimization results given byKazuto, et, a . and authors

(dashed line, results by KAzUTO, et, a ; solid line, results

obtained by authors)

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Sem.org IMAC XVI 16th Int 160906 on Theory Dynamic Vibration Absorption Its Application Vibration

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