Probabilistic Engineering Mechanics 24 (2009) 251–254

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Probabilistic Engineering Mechanics

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Optimal damped vibration absorber for narrow band random excitations:A mixed H2/H∞ optimizationAlok Sinha ∗Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802, United States

a r t i c l e i n f o

Article history:Received 23 May 2007Received in revised form22 May 2008Accepted 12 June 2008Available online 21 June 2008

Keywords:Damped vibration absorberNarrow band randomH2/H∞Optimization

a b s t r a c t

This paper deals with the design of an optimal damped vibration absorber in the presence of a narrowband random excitation. This process is classified as a mixed H2/H∞ optimization. Results from thisoptimization process are compared to those from the classical H∞ (Den Hartog) and the H2 solutions.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The topic of a damped vibration absorber is well-establishedin the vibration literature [1]. Den Hartog [1] provided thealgorithm for selecting optimal absorber parameters such that themaximum amplitude of vibration is minimized over frequenciesof deterministic sinusoidal excitations, which is essentially anH∞ optimization. Crandall and Mark [2] studied the statistics ofresponse of a system with damped vibration absorber subjectedto white noise excitation. Fujiwara and Murotsu [3] numericallycomputed the optimal absorber parameters for white noiseexcitation, and narrow band random excitation with a specifiedfrequency band, which are essentially H2 optimization problems.Asami et al. [4] provided theoretical analysis for the design ofoptimal vibration absorbers subjected to white noise excitation.Using H2 and H∞ optimization techniques [5], Asami et al. [6]and Asami and Nishihara [7] have nicely summarized theexisting literature, and analytically determined optimal absorberparameters for undamped and damped main systems as well.Nishihara and Asami [8] have focused on the generation of optimalclosed-form solutions for theH∞ criterion; i.e., theminimization ofthe maximum amplitude magnification factor for a deterministicsinusoidal excitation.

∗ Corresponding address: The Pennsylvania State University, 330 Reber Building,University Park, PA 16802, United States. Tel.: +1 814 863 3079; fax: +1 814 8634848.E-mail address: axs22@psu.edu.

0266-8920/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.probengmech.2008.06.005

In spite of a vast literature in the damped vibration absorberarea, the optimization of absorber parameters for a narrow bandexcitationwith a variable frequency band has not been considered.The solution of this problem would require the mean-squareresponse, which is equivalent to the area under the frequencyresponse (H2 criterion [5]) for a selected frequency band. Then,the frequency band has to be varied to determine the peakmean-square response (H∞ criterion [5]). Lastly, optimal absorberparameterswill be determined tominimize this peakmean-squareresponse, which can be described as a mixed H2/H∞ criterion foroptimization. This mixed H2/H∞ optimization is different fromthat used by Haddad and Razavi [9], in which the entropy wasminimized with a constraint on the peak frequency response.This paper deals with the optimal design of a damped vibration

absorber with this H2/H∞ criterion for optimization. Unlikeprevious papers [2–4,6–8], themean-square response is computedin this paper by the state-space approach, which leads to theLyapunov equation [5]. Optimal design parameters from thisH2/H∞ optimization are numerically obtained and compared tothe classical Den Hartog’s solution (H∞ criterion) and the recentlydeveloped H2 solution.

2. Formulation

Consider the main system having the stiffness k1, the dampingconstant c1 and the mass m1 subjected to external excitation f (t).The vibration absorber with mass m2, stiffness k2 and dampingconstant c2, is attached to the main system as shown in Fig. 1. The

252 A. Sinha / Probabilistic Engineering Mechanics 24 (2009) 251–254

Fig. 1. Damped vibration absorber.

differential equations of motion arem1x1 + (c1 + c2)x1 − c2x2 + (k1 + k2)x1 − k2x2 = f (t) (1)m2x2 − c2x1 + c2x2 − k2x1 + k2x2 = 0. (2)It is assumed that the external excitation f (t) is random in nature,and is the output of the following filter:

f + 2ξFωF f + ω2F f (t) = ω2Fw(t) (3)

where ξF and ωF are the filter damping ratio, and the filter naturalfrequency, respectively. And,w(t) is the zeromeanwhite noise [5]with intensityw0:E[w(t)w(t + τ)] = w0δ(τ ) (4)where δ(τ ) is delta-dirac function, and E[.] is the expected value. Itshould be noted that f (t)will be a narrow band random excitationaround the frequency ωF for a small value of the filter dampingratio ξF . The width of frequency band increases as ξF increases.Define the non-dimensional time tnd as follows:tnd = ωF t. (5)Using (5), Eqs. (1)–(3) are expressed as follows:ω2F x′′

1 + 2(ξ1 + ξ2)ω11µωFx′

1 − 2ξ2ω11µωFx′

2

+ (ω211 + µω222)x1 − µω

222x2 =

ω211

k1f (tnd) (6)

ω2F x′′

2 − 2ξ2ω11ωFx′

1 + 2ξ2ω11ωFx′

2 − ω222x1 + ω

222x2 = 0 (7)

ω2F f′′+ 2ξFωFωF f ′ + ω2F f (tnd) = ω

2Fw(tnd) (8)

where

x′ =dxdtnd

, x′′ =d2xdt2nd

, f ′ =dfdtnd

, f ′′ =d2fdt2nd

(9)

and

ω11 =

√k1m1; ω22 =

√k2m2; µ =

m2m1;

ξ1 =c1

2m2ω11; ξ2 =

c22m2ω11

. (10)

Dividing (6) and (7) by ω211x0, and (8) by f0,

g2z ′′1 + 2(ξ1 + ξ2)gµz′

1 − 2ξ2gµz′

2

+ (1+ µλ2)z1 − µλ2z2 = h(tnd) (11)

g2z ′′2 − 2ξ2gz′

1 + 2ξ2gz′

2 − λ2z1 + λ2z2 = 0 (12)

h′′ + 2ξFh′ + h = v(tnd) (13)

where

λ =ω22

ω11, g =

ωF

ω11, z1 =

x1x0, z2 =

x2x0

(14a)

h(tnd) =f (tnd)f0

, v(tnd) =w(tnd)f0

, and x0 =f0k1. (14b)

Note that f0 is a nominal force, which can be defined as follows:

f0 = ω211√w0. (15)

Define a state vector y as follows:

y =[z1 z2 z ′1 z ′2 h h′

]T. (16)

Then, Eqs. (11)–(13) can be written as

y′ = Ay(tnd)+ bv(tnd) (17)

where

A =

0 0 1 0 0 00 0 0 1 0 0

−(1+ µλ2)g2

µλ2

g2−2(ξ1 + ξ2)µ

g2ξ2µg

1g2

0

λ2

g2−λ2

g22ξ2g

−2ξ2g

0 0

0 0 0 0 0 10 0 0 0 −1 −2ξF

(18)

and

bT =[0 0 0 0 0 1

]. (19)

In steady state,

E[y(tnd)] = 0. (20)

Let the steady state covariance matrix be

P = E[yyT]. (21)

Then, it is well-known that the matrix P is governed by theLyapunov equation [5]:

AP + PAT + bbT = 0 (22)

where E[vvT] has been assumed to be unity without any loss ofgenerality. The optimal absorber parameters (ξ2 andλ) are selectedby minimizing the following objective function:

I = maxg P(1, 1) = max

g E[z21 ]. (23)

3. Numerical results

The Lyapunov equation (22) is solved via the Matlab [10]routine ‘lyap’, whereas the optimization is performed via theMatlab routine ‘fminsearch’. The value of the objective function isshown in Fig. 2 as a function of g for optimal ξ2 and λ. Similar tothe results for a deterministic sinusoidal excitation [1], both peaksare at the same heights for narrow band random excitation withoptimal parameters.Next, optimal parameters (ξ2 and λ) are obtained as a function

of the mass ratio (µ)for the undamped main system, ξ1 = 0 andfor three values of the filter damping ratio, ξF . Results are shownin Figs. 3 and 4. In these figures, H∞or Den Hartog’s [1] and H2 [4]solutions, which are given below for ξ1 = 0, are also plotted:

λH∞ =1

1+ µ(24a)

A. Sinha / Probabilistic Engineering Mechanics 24 (2009) 251–254 253

Fig. 2. Mean square response as a function of the narrow band filter frequency foroptimal parameters.

Fig. 3. Optimal absorber damping ratio for non-tuned case (ξ1 = 0).

Fig. 4. Optimal frequency ratio for non-tuned case (ξ1 = 0).

and

ξ2,H∞ =

√3µ

8(1+ µ)3(24b)

Fig. 5. Variations in the objective function.

λH2 =1

1+ µ

√2+ µ2

(25a)

and

ξ2,H2 =1

4(1+ µ)

√µ(4+ 3µ)(1+ µ)

. (25b)

It is interesting to note that ξ2,H∞ is close to the optimal ξ2 fornarrow-band random excitation with ξF = 0.01. As ξF → 0.04,or equivalently the frequency band of excitation is increased, theoptimal ξ2 approaches ξ2,H2 . The relation between the optimalfrequency ratio and the mass ratio is almost invariant with respectto ξF , and is indistinguishable from that for the deterministicsinusoidal excitation (Den Hartog’s Solution), Fig. 4. However,this relationship for the narrow band excitation is significantlydifferent from that for the white noise (H2 case) excitation.Lastly, objective functions are computed for the narrow band

excitation with optimal ξ2 and λ obtained from H∞ and H2 cases,Eqs. (24) and (25). Results in Fig. 5 show that optimal ξ2 andλ from the H2 case lead to sub-optimal performance for thenarrowband random excitation (ξF = 0.01) where as those fortheH∞ or deterministic sinusoidal excitation yields almost optimalperformance. Note that the objective functions for each curve inFig. 5 is scaled as follows:

Is =I

IH∞@µ=0.1(26)

where IH∞ is the value of I , Eq. (23), with ξ2 and λ given by Eq. (24).It is also interesting to note that optimal values of the objectivefunctions are almost invariant with respect to inevitable presenceof a small amount of damping (ξ1) in the main system.

4. Conclusions

For a narrow band random excitation, optimal parameters fora damped vibration absorber are almost same as those for adeterministic sinusoidal excitation. At the optimal conditions, bothpeaks in the filter frequency (mean-square) response are at thesame heights, which is exactly the criterion used by Den Hartogfor a deterministic sinusoidal excitation.As the bandwidth of the narrow band excitation is increased,

the optimal absorber damping ratio approaches that for the whitenoise excitation, whereas the optimal frequency ratio remainsalmost unchanged.

254 A. Sinha / Probabilistic Engineering Mechanics 24 (2009) 251–254

The value of the minimum objective function (peak mean-square response) is almost optimal for the Den Hartog’s absorber,in spite of the narrow band excitation being random. Furthermore,the value of the minimum peak mean-square response is almostinvariant with respect to the presence of a small amount ofdamping in the main system. However, the value of this minimumobjective function is sub-optimal for the absorber designed on thebasis of H2 criterion.

References

[1] Den Hartog JP. Mechanical vibrations. Dover Publications; 1985.[2] Crandall SH, Mark WD. Random vibration in mechanical systems. AcademicPress; 1963.

[3] Fujiwara N, Murotsu Y. Optimum design of vibration isolators for randomexcitations. Bulletin of the JSME 1974;17(103):68–72.

[4] Asami T,Wakasono T, Kameoka K, HasegawaM, Sekiguchi H. Optimum designof dynamic absorbers for a system subjected to random excitations. JSMEInternational Journal, Series III 1991;34(2):218–26.

[5] Sinha A. Linear systems: optimal and robust control. CRC/Taylor & FrancisPress; 2007.

[6] Asami T, Nishihara O, Baz AM. Analytical solutions to H∞ and H2 optimizationof dynamic vibration absorbers attached to damped linear systems. ASMEJournal of Vibration and Acoustics 2002;124(April):284–95.

[7] Asami T, Nishihara O. H2 optimization of the three-element type dy-namic vibration absorbers. ASME Journal of Vibration and Acoustics 2002;124(October):583–92.

[8] Nishihara O, Asami T. Closed-form solutions to the exact optimizations ofdynamic vibration absorbers (minimizations of the maximum amplitudemagnification factors). ASME Journal of Vibration and Acoustics 2002;124(October):576–82.

[9] Haddad WM, Razavi A. H2 , mixed H2/H∞ , and H2/L1 optimally tuned passiveisolators and absorbers. ASME Journal of Dynamic Systems, Measurement andControl 1998;120(June):282–7.

[10] MATLAB, The MathWorks, Inc., Natick, MA, Version 7.