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Osamu NishiharaAssociate Professor,

Department of Systems Science,

Graduate School of Informatics,

Kyoto University,

Yoshida-Honmachi, Sakyo-ku,

Kyoto 606-8501, Japane-mail: [email protected]

Toshihiko AsamiAssociate Professor,

Department of Mechanical Engineering,

Himeji Institute of Technology,

Shosha, Himeji, Hyogo 671-2201, Japan

e-mail: [email protected]

Closed-Form Solutions to theExact Optimizations of DynamicVibration Absorbers(Minimizations of the MaximumAmplitude Magnification Factors)

A typical design problem for which the fixed-points method was originally developed isthat of minimizing the maximum amplitude magnification factor of a primary system byusing a dynamic vibration absorber. This is an example of usual cases for which theirexact solutions are not obtained by the well-known heuristic approach. In this paper, morenatural formulation of this problem is studied, and algebraic closed-form exact solutionsto both the optimum tuning ratio and the optimum damping coefficient for this classic

problem are derived under assumption of undamped primary system. It is also proven thatthe minimum amplitude magnification factor, resonance and anti-resonance frequenciesare entirely algebraic. DOI: 10.1115/1.1500335

1 Introduction

Well-known design formulas for the tuning of dynamic vibra-tion absorbersDVAs substantially originate from a note by J. E.Brock, published in 1946 1. The design criterion considers theminimization of the resonance amplitude magnification factor.The concept was introduced in a paper by J. Ormondroyd and J. P.Den Hartog 2, in which the existence of optimum damping wasnumerically observed. Hahnkamm studied the optimal tuning con-dition, which minimizes the amplitude magnification factor atfixed points. This is considered to be the origin of the fixed-pointsmethod. J. E. Brock completed these works by calculating theaverage of two optimum damping factors for the fixed points.Though this is an empirical method, the formulas were presentedin Mechanical Vibrations written by J. P. Den Hartog 3. Thus,the results have widely circulated in this field 4.

The present paper discusses a more natural formulation of thedesign problem, which results in the exactsolution to this classicproblem. This algebraic approach gives the closed-form solution,which is obviously quite different from numerical optimizations5 7. The fixed-points method tunes a DVA such that the mag-nification factors of two fixed points are equalized. If the dampingratios that make the resonance curves horizontally pass throughthe fixed pointsP and Q are identical, then the optimal solution isproduced. In this case, however, they are not equal, and the reso-nance points do not coincide simultaneously with the correspond-ing fixed points. The readjustment of the damping ratio yields tworesonance points with equal amplitude magnification factors, butthe amendment is almost meaningless as long as the tuning isdetermined by the conventional method.

One of the authors of the present paper has found the newapproach for exact solution 8. To summarize, the approach isbased on an observation of the trade-off between two resonanceamplitude magnification factors. In light of this, the solution to theoriginal problem is reduced to a quadratic equation that is derivedin terms of the discriminant of a quartic equation. In the case ofundamped primary system, it turns out that the optimum param-eters, minimum amplitude magnification factors, resonance andanti-resonance frequencies, and the sensitivities of the amplitude

magnification factors are entirely algebraic. The numerical exten-sions are also shown, enabling the formulation of efficient solu-tions for the damped primary system, resulting in more directapplications.

We have tested the new formulation in several cases, for ex-ample, mobility and accelerance cases, which are simple varia-tions of the compliance case that is discussed in the present paper9,10. It turned out that the exact solutions are not more complexthan those of the fixed-points methods, and in some cases they arerather simpler. Our formulation is also applicable to the hystereti-cally damped vibration absorbers, and in some cases, algebraicexact solutions are obtained assuming hysteretically damped pri-mary system. The present paper explains this new formulation bytaking the most typical case as an example, and refers to the

sensitivities and the optimality of the solution, which are notclearly discussed in the previous reports.

The algebraic method is applicable to the linearized model ofthe passive gyroscopic damper PGD, and the exact solution hasbeen obtained 11. The PGD means a dynamical model of gyro-scopic stabilizer developed by one of the authors. It consists of asingle gimbal and a set of torsional springs and viscous dampersattached to the gimbal axis. The gimbal is a housing of the rotordriven by an electric motor to maintain constant revolution. Thebasic dynamical characteristics are similar to that of the rotationaltype dynamic vibration absorber. The PGD is essentially a semi-active system, which is more effective under ordinary conditions.The assumption of an undamped primary system leads to the al-gebraic solution of optimum parameters.

2 Undamped System

2.1 Model for Analysis. The model for analysis is a systemwith two-degree-of-freedom shown in Fig. 1, consisting of a pri-mary system and a dynamic vibration absorber as an auxiliarysystem. As external force acts on the primary system, the dynamicvibration absorber is provided to reduce the resonant motion ofthe primary system to within safe limits.

The equations of motion for this system are

m1x1c 1c 2x1k1k2x 1c2x 2k2x 2w,

m2x2c 2x2k2x2c 2x1k2x10, (1)

Contributed by the Technical Committee on Vibration and Sound for publication

in the JOURNAL OFVIBRATION AND ACOUSTICS. Manuscript receivedAugust 2000;

Revised April 2002. Associate Editor: R. L. Clark.

576 Vol. 124, OCTOBER 2002 Copyright 2002 by ASME Transactions of the ASME

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where m1 and m2 are the masses of the primary and auxiliary

systems respectively; x1 and x2 are the displacements of the pri-mary and auxiliary systems respectively; k1 and k2 are the springconstants of the primary and auxiliary system respectively; andc 1and c 2 are the viscous damping coefficients of the primary andauxiliary systems respectively.

The equations of motion 2 are entirely dimensionless.

x12Z2z x 11p2x12zx 2p

2x2w,

x22zx 2p2x22zx 1p

2x10, .

(2)

The dimensionless parameters are defined as m2/m1 , z

c 2/ (2m 1k1), Zc 1/ (2m1k1), p2k2/ (k1) , and w

w/ (k1x0). The displacements x 1 and x 2 are transformed as x1

x 1/x 0 and x

2x2/x0 where x 0 is the arbitrary unit length. Theunit time for the nondimensionalization is determined accordingto the undamped eigenperiod of the primary system t0m1/k1, and thus, the dimensionless time is defined as t/t0 . Note that a dot over the displacement in these equationsmeans the derivative with respect to the dimensionless time. For

example, x 1dx1/d. The transfer function from the excitation

force to the primary system displacement G(s)X1(s)/W( s) isnaturally dimensionless. The exact solution in this section as-sumes an undamped primary system (Z0). Hence, the transferfunction, or compliance, is written as:

Gss22zsp 2

s21 s22zsp 2s22zsp2. (3)

2.2 Problem Formulation. Using the present notation, theobjective function is expressed by G(s), i.e., H norm ofG(s); with decision variables p and z. It is reasonable to assumethat G (s) has two distinct resonance points. These are denoted Aand B, with frequencies A and B (AB). This leads to theequation:

Gsmax GjA, GjB . (4)

The new approach outlined in this paper originates from the ob-servation of a trade-off relation between G(jA) and G(jB).It is well recognized that each fixed point is very close to thecorresponding resonance point, and that the trade-off relation be-

tweenG(jP) and G(jQ) can be postulated. On this assump-tion, it is guaranteed that the optimum design is derived usingequivalent resonance magnification factors, i.e.:

GsGjAGjB. (5)

Therefore, the problem is reduced to determiningpand z such thatthe common resonance amplitude is minimized. This assumptionwill be reviewed in detail after the solution of simplified problem.In section 2.10, local optimality of the solutions is revalidated bynumerical evaluation of the sensitivities of amplitude magnifica-tion factors with respect to p and z.

The polynomials n() and d() are introduced to denote thenumerator and the denominator of G(j)2:

np2224z22, (6)

dp 221p 22424z221122,(7)

where all terms are even-ordered in . For simplicity, 2, n() ,and d() are replaced by , N(), and D(), respectively.These notations lead to the equation

N

D h, (8)

which holds at the resonance points A and B, and where h denotesthe square of the resonance amplitude magnification factor. Simi-

larly, another equation holds:NDND

D2 0, (9)

where denotes /. Equation 9 holds because the tangents atA and B are horizontal. Equations 8 and 9 are simplified to

NhD0 and NhD 0, respectively. Therefore, the equa-

tionF()0 has two double roots; AA2 andBB

2 , where

F()DN/h.

2.3 Simplification. The function F() is a monic quarticpolynomial.

F4b 13b 2

2b3b 4 (10)

By definition, the coefficients are

b121p2p 22z24z222z2, (11)

b24p22p 2p42p 42p 4r28z28z2,

(12)

b 32p4p 4p2r22r2z2, (13)

b 4p4r2, (14)

where h is replaced by 1/(1r2). The new parameter, r

1h1 (h1, 0r1) has been introduced for simplicity.The existence of two double roots leads to the factorized form:

F A2B2. (15)

Vietas theorem gives the alternative expressions for b i (i1, . . . ,4):

b12AB, (16)

b2A24ABB

2 , (17)

b 32ABAB, (18)

b4A2B

2 . (19)

Fig. 1 Dynamic vibration absorber attached to single-degree-of-freedom system

Journal of Vibration and Acoustics OCTOBER 2002, Vol. 124 577


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By eliminating A and B, these four equations are reduced totwo equations:

b1b4b 30, (20)

b12

4 2b4b20. (21)

Substitution of definitions1114into Eqs.20and 21yieldsthe simultaneous equationsf10, f20 with respect to p and

z, where ris the given parameter. The first equation is rearrangedas:

f12 12p2rr2z2p21rr1p 20.

(22)

Equation 22 is then solved with respect to z0:

zp21rr1p 2

2r 12p2r . (23)

Substitution of Eq. 23 into f20 eliminates z in f2 . Hence, f20 is expressed as a polynomial in terms of, p, and r. For

simplicity, f20 is now rewritten using xp2. The result is a

quartic equation in x:

f2xa 0x4a 1x

3a2x

2a3xa40, (24)

where

a 016

1r2

, (25)

a 1414r1r2, (26)

a22123r21r2, (27)

a 321rr3222r2r2r, (28)

a4r41r2. (29)

Given a positive solution x to f2(x)0 for the parameter r, thesubstitution ofpx into Eq. 23 determines the value ofz thatsatisfies f10. Thus, the problem reduces to the minimization ofrunder the positive root condition of Eq. 24.

2.4 Derivation of Discriminant. The number of real rootsto Eq. 24 depends on the parameter r. The coefficients a i (i0, . . . ,4) are polynomials in r. The continuity of the root ofalgebraic equation with respect to the coefficients, guarantees thatf2(x)0 yields the multiple root at the boundary between twosections that correspond to the discrete numbers of real solution.

The discriminantD 4 is a factor of the resultant:

Rf2 ,f2a07D4 . (30)

For simplicity, the equation R(f2 ,f2)0 is used instead of D40. The resultant is defined as the determinant of a matrix, whichis convenient for use with a formula manipulation software suchas Mathematica 12.

Rf2 ,f2

a0 a1 a2 a3 a 4 0 0

0 a0 a1 a2 a 3 a 4 0

0 0 a0 a1 a 2 a 3 a 4

4a0 3a1 2a2 a3 0 0 0

0 4a0 3a1 2a2 a 3 0 0

0 0 4a0 3a1 2a2 a 3 0

0 0 0 4a0 3a1 2a2 a3 (31)2.5 Determination of r. The highest order term in Eq.31

is the 26th power of r. The following quadratic equation is ob-

tained by eliminating the common factor 166(1)18r14(1

r)4(r21)3. Here, r0 and r1 correspond to h1 and h , respectively; then both solutions are apparently inadequate.

6480272r216r6410 (32)

The roots of Eq. 32 are

r1r2

8433/2

6480272 , (33)

where r2 is excluded because r10r2 .

2.6 Derivation ofp. Substitutingrr1 into Eq.24resultsin an eighth-order equation with even-ordered terms. The equationcan now be factorized as follows:

f21r216

r2p 212r2p 22p22p22p2

2p222, (34)

r1143

)1, (35)

2r1432

)1, (36)

where is the unique positive solution to Eq. 34. The fact that zis positive can be confirmed algebraically by setting rr1 and

p.

2.7 Optimum Parameters. It is proven algebraically thatthe determination of p and z results in two double roots for Eq.10. The details are omitted here to avoid prolixity. Equations16and 19show that A andB satisfy the quadratic equation

2b1

2b40. (37)

Therefore, the optimum value ofris

roptr18 433/2

6480272 . (38)

The characteristics of the compliance function in proximity to theoptimal solution are described as follows. If rropt , then a

double root occurs to Eq. 24. For the case rropt , it corre-sponds to two real roots. In this case, two equal height resonancepoints are yielded, which are slightly higher than that of the opti-mum case. Note that two distinctive values of p exist for theequivalent height. The remaining case rropt , results in a pair ofconjugate complex solutions. This simply means that such heightequivalence is not achievable.

The optimum parameter values rropt , pp opt , and zzoptyield equal height resonance points at A and B. The optimumparameter values are obtained by step-by-step substitutions as fol-lows:

popt

ropt143

)1, (39)

zoptp opt21roptropt1popt

2

2ropt 12p opt

2ropt

(40)

The optimum resonance magnification factor is

GjAGjB1

1ropt2

. (41)

Hence, maxG(jA),G(jB) is minimized.

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2.8 Resonance Frequencies. The application of the opti-mum parameters given by Eq. 3840 results in the real roots

A2A and B

2B for the quadratic equation 37.

AB

1211popt

2212zopt

2

1411p opt

2212zopt

2 2popt2

ropt

(42)

2.9 Anti-resonance Frequency. The resonance and anti-resonance frequencies correspond to the solutions of the equation

NDND 0, which is defined as the numerator of Eq. 9,because the tangents at the resonance and anti-resonance pointsare horizontal. The frequencies are the real roots of the monicquintic equation:

514p 2p 28z24z222z244p 26p4

4p 48z216p2z224p2z282p2z216z4

32z4162z436p 4p44p65p 6

2p616p2z212p2z28p 4z216p 4z282p 4z2

16z416z42p 44p 22p2p42p42p4

8z28z21p80. (43)

Exploiting the fact that the resonance frequencies satisfy quadraticequation 37, the following cubic equation is derived by dividingthe left-hand side of Eq. 43 by that of Eq. 37:

36z23p22p23p4p 4p2r2z24p2z2

6p2z222p2z24z48z442z4p 22p 4

p4p 6p 6p 2r2p 4rp4r2z28p 2z2

42

p2

z2

2p4

z2

8p4

z2

102

p4

z2

43

p4

z2

4p2rz 24p2rz 222p2rz 28z482z412p 2z4

40p2z4482p 2z4243p2z444p2z48z6

32z6482z6323z684z60 (44)

Here, r, p, and z are the optimal parameters given by Eqs. 3840. There is one real root and a pair of complex conjugate roots.Cubic equation 44 is reducible and its real root is given byCardanos formula. Thus, the algebraic solutions for the resonanceand anti-resonance frequencies of the exact optimization are de-rived.

2.10 Sensitivity Analysis. Next, the sensitivities of the am-

plitude magnification factors of the resonance points are evaluatedwith respect to p and z. Sensitivities are important because of theparametric errors of practical DVAs and the effect on the optimal-

ity of the preceding results. For simplicity, G(j)2 is abbrevi-ated to h. The sensitivities are evaluated by the general expres-sions:

dh

dp

h

p

h

p , (45)

dh

dz

h

z

h

z . (46)

Since h/0 holds at resonance points A and B, the sensitivi-ties now reduce to

dh

d p

h

p, (47)

or

dh

dz

h

z. (48)

In addition, the square amplitude magnification factor h is arational polynomial defined by Eqs. 6 and 7. The derivativewith respect to p is

dh

dp

npdpnpdp

dp2 . (49)

Note that n (p) and d(p) are the numerator and denominator ofh,and that denotes /p in Eq. 49.

Analogously, the derivative with respect to z is written as:

dh

dz

nz dznz dz

dz2 . (50)

These expressions permit precise evaluation of sensitivities. Bynumerical calculations,

dhA

d p0,

dhB

d p0,

dhA

dz0,

dhB

dz0, (51)

can be readily verified for various values of the mass ratio ,where hA and hB indicates h at the resonance points A and B,respectively. Initially, the discussion depended on the assumptionthat any optimal solution yields the same resonance amplitudehAhB , but now the existence of a trade-off between hA and hBhas been confirmed. Now, the proof for the local optimality of thesolution is intuitive.

3 Fixed-Points Method

Using the notations provided in the previous section, the con-ventional results of the fixed-points method are readily expressedas

pfp1

1, (52)

z fp 3813, (53)where pppfand zzpf1,8.

The amplitude magnification factor of the fixed points is:

GjPGjQ1 2. (54)The general equation for the fixed-points frequencies is:

1

221p 2p2p 20. (55)

Substitution of p fp and z fp given in Eqs. 52, 53 results in thesimplified expression:

P

Q 1

11

2 . (56)

The resonance points of the fixed-points method A and B arelocated very close to the fixed points P and Q, respectively. Theseresonance frequencies satisfy Eq. 43, because it is derived from

NDND 0. Similarly, Eq. 43 is simplified by substitutingthe definitions ofp fp and z fp given in Eqs. 52 and 53:



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520201023

413 4

40122

414 3

40452

415 2

5

16

1

170.

(57)

Since the factorization of this equation is unknown, numericalmethods are required to obtain a solution. It is empirically evidentthat the fixed-points method yields two resonance points, andhence has three real roots and a pair of complex conjugate solu-tions. These numerical results enable the precise calculation of themaximum amplitude magnification factor of the fixed-pointsmethod.

4 Numerical Examples

4.1 Compliance Curves. Figure 2 shows the compliancecurves for 0.05. Two curves, corresponding to the exact opti-mization and fixed-points methods, overlap each other and aredifficult to distinguish. The regions proximal to the resonancepoints A and B are shown in detail in Fig. 3. The amplitude mag-nification factors of the left and right resonance points with exactoptimization are G(jA) G(jB) 6.407921. Conversely,G(jP)G(jQ) 6.403124, G(jA)6.407492, andG(jB) 6.408443 for the fixed-points method. The fixed-points method results in a lower left resonance point A and a

higher right resonance point B , while the average is nearly equalto that of the exact optimization. The tuning ratio p fp of the fixed-points theory is highly accurate. The deterioration is considered tooriginate mainly from errors in the damping coefficient z fp of thefixed-points method. Figure 4 shows that the variation betweenthe exact optimization and fixed-points method increases in prox-imity of the antiresonance point.

4.2 Optimum Parameter Values. Table 1a shows thetuning ratios p derived using the fixed-points method and the ex-act optimization for various values of the mass ratio , and bshows the dimensionless damping coefficientz calculated by thesetwo methods. The variations ofp and z are relatively trifling whenthe mass ratio is less than 0.01. The variation of z tends toward0.5% at 0.1, while the variation ofp remains very small.

4.3 Resonance Amplitude Magnification Factors. Table 2shows the resonance amplitude magnification factors calculated

by the exact optimization and the fixed-points methods. The fixed-points method is highly precise. For example, the ratio is only0.023% for a relatively large mass ratio 0.1. The ratios remainbetween 0.5% and 2.3% for larger mass ratios 1 to 10, although

Fig. 2 General view of compliance curves 0.05, Opti-mum parameter values for undamped primary system ,Fixed-points method , Primary system z -"-

Fig. 3 Close-ups of point A and B 0.05, Optimum pa-rameter values for undamped primary system , Fixed-points method , Primary system z -"-

Fig. 4 Close-up of point C 0.05, Optimum parametervalues for undamped primary system , Fixed-pointsmethod



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such high mass ratios are impractical for usual applications. Thefixed-points method is considered highly accurate as a designmethod for the DVA.

5 Damped Primary System

The next task is to extend the preceding method to a dampedprimary system 13,14. The assumption that the optimum solu-tion satisfies the condition G(jA) G(jB) is inherited fromthe undamped case. In this case, the equations originate from thecoincidence of the resonance amplitude magnification factor f10 and f20 include the primary system damping coefficient Z.An eighth order algebraic equation corresponding to the quarticequation 24 is generated by a similar procedure. However, theresultant is more complicated compared to Eq.31. The algebraic

approach applied to the undamped primary system is consideredimpractical in the damped case. Therefore, a numerical approachis used in preference to an algebraic method.

The equation f1(p ,z,r)0 is rewritten as zg 1(p,r). Substi-tuting this into f2(p ,z,r)0 results in f2(p,g1(p,r), r)0. Thecondition of a double root f2(p,r)0 with respect to p is writtenas f2/p0. The expression is evaluated by implicit differentia-tion and rearranged as:

f3f1

p

f2

z

f1

z

f2

p0. (58)

Note that f2(p,g1(p, r) ,r)0 is equivalent to the simultaneousequations f1(p,z,r)0, f2(p,z,r)0, providing a set of threeequations including Eq.58. Equation 58 forces the singularityof the Jacobian matrix off1(p ,z), f2(p,z).

The equations f10, f20 are rather simple. Conventionalformula manipulation systems such as Mathematicareadily com-pute the partial derivative off1 and f2 with respect topand z 12.Then, Eq. 58 is readily derived as a polynomial.

f12 12p2rr2z24p 2rZ zp 21rr1p2

2p 2rp 2Z2 (59)

f21r2412z2414z48z212z21Z

42131z21Z216zZ34Z42r1

413z281zZ41Z2p 2 (60)

The Newton-Raphson method is applicable in solving Eqs.59,60and 58simultaneously. The algebraic exact solution tothe undamped case is a valid initial value for the iteration. If theiteration does not converge, then the continuation method is re-quired, in which Newton-Raphson method is repeatedly appliedwith the gradual increment ofZfrom zero.

Figure 5 shows the compliance curves with various values ofthe primary system damping coefficient. In each case, the numeri-cal optimization is compared to the diversion of the exact solutionto the undamped case. For example, by numerical methods taking

Fig. 5 Reduction of maximum amplitude magnification factorsby numerical optimizations 0.05. Iterative solutions thattake accounts of the primary system damping are displayed forseveral values of the primary system damping ratio . Theadditional curves are provided only for comparison purposes,where the optimum parameter values for undamped primarysystem are diverted to the damped cases .

Table 1 Optimum frequency ratio and damping coefficient

Table 2 Maximum amplitude magnification factor



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accounts of a damping coefficient Z0.01, the maximum ampli-tude decreases by 8.5%. Importantly, the improvement in accu-racy is much greater than the difference between the fixed-pointsmethod and the exact algebraic optimization of the undampedcase.

6 Concluding Remarks

It has been shown that the algebraic exact solution exists for theminimization of the resonance amplitude magnification factor bythe dynamic vibration absorber attached to the undamped primarysystem. The exact algebraic expressions for the resonance and

antiresonance frequencies have been obtained. Initially, a trade-offrelation between two resonance points was assumed, but a sensi-tivity analysis numerically identified the trade-off relation. Thelocal optimality of the solution then became apparent. This ap-proach was extended to the damped primary case, but comple-mentary numerical solution was still required. The fixed-pointsmethod was shown to be highly accurate, especially for smallmass ratios of less than or around unity. This method is applicableto the linearized model of the passive gyroscopic damper whosedynamical characteristics are similar to that the rotational dynamicvibration absorber. Their algebraic solution of optimum param-eters has been obtained under the assumption of undamped pri-mary system. The algebraic approach is also applicable to an airdamped dynamic vibration absorber, which is modeled by a three-element type system. New expressions for the optimum param-

eters have been derived 15 by the extension of the method de-scribed in the present paper.

References

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