Passive vibration suppression of plate using multiple optimal dynamic vibration absorbers

  • M. Ari
  • R. T. FaalEmail author


In the present paper, the optimization problem of the dynamic vibration absorbers (DVAs) for suppressing vibrations in thin plates within the wide frequency band is investigated. It is considered that the plate has simply supported edges and is subjected to a concentrated harmonic force. The vibration suppression is accomplished by the implementation of multiple mass–spring absorbers in order to minimize the plate deflection at the natural frequencies of the plate without absorbers. The governing equations of the plate equipped with DVAs for both isotropic and FG plates are derived and solved numerically and analytically. The formulation of the problem is capable of optimizing the \(L_{2}\) norm of the plate deflection at the wide frequency band with respect to mass, stiffness and position of each absorber attachment point. In this study, the possibility of simultaneous absorption of one or multiple natural frequencies of the plate without any absorbers is also studied. Some numerical results are also presented.


Dynamic vibration absorber \(L_{2}\) norm Optimization Absorption frequency 

List of symbols


Dimensional and dimensionless amplitudes of excitation forcing, respectively

\(\varOmega ,\alpha \)

Dimensional and dimensionless excitation frequency, respectively

\(t,\tau \)

Dimensional and dimensionless time, respectively

\(\left( X_{{0}},Y_{{0}} \right) ,\left( x_{{0}},y_{{0}} \right) \)

Dimensional and dimensionless coordinates of applying point of the force, respectively

\(\left( X_{j},Y_{j} \right) ,\left( x_{j},y_{j} \right) \)

Dimensional and dimensionless coordinates of jth absorber attachment point, respectively

\(\left( X,Y \right) ,\left( x,y \right) \)

Dimensional and dimensionless coordinates of an arbitrary point of the plate, respectively


Dimensional and dimensionless masses of jth absorber, respectively


Dimensional and dimensionless stiffnesses of jth absorber, respectively


Dimensional and dimensionless mass displacement of jth absorber with respect to a fixed reference point, respectively


Amplitude of \(q_{j}\)


Length, width, and thickness of the plate, respectively


Number of dynamic absorbers

\(\bar{W}\left( X,Y,t \right) ,W\left( x,y,t \right) \)

Dimensional and dimensionless deflection of plate, respectively

\(w\left( x,y \right) \)

Amplitude of the dimensionless deflection of plate


Coefficients of the plate mode shapes or components of \(\vec {a}\)


Elasticity modulus of isotropic and FG plates, respectively

\(\nu \)

Poisson’s ration


Flexural or bending rigidity of the plate

\(\rho ,\rho (z)\)

Density of the isotropic and FG plates, respectively

\(\delta \left( . \right) \)

Delta Dirac function


Wave velocity in the plate

\(\beta \)

Aspect ratio (ratio of the plate length to its width)

\(\mu _{j}\left( \alpha \right) , \lambda _{j}\left( \alpha \right) , \tau _{lj}\left( \alpha \right) , \rho _{j}(\alpha )\)

Predefined parameters

\(\alpha _{mn}\)

Dimensionless natural frequencies of the bare plate (plate without absorber)

\(f_{mn}\left( x,y \right) , g_{mn}\left( x,y \right) , \psi \left( x,y,z,v,\alpha \right) ,\theta \left( x,y,z,v,\alpha _{rs} \right) P_{jmnpq}\left( x,y \right) ,Q_{jmnpq}\left( x,y \right) , R_{jmnpq}\left( x,y \right) ,S_{jmnpq}\left( x,y \right) \)

Predefined functions

\(A_{mnpq}\left( \alpha \right) ,B_{mnpq}\left( \alpha \right) ,B_{imnpq}\left( \alpha \right) \) ,\(C_{imnpq}\left( \alpha \right) ,D_{imnpq}\left( \alpha \right) \)

Entries of matrices \({\varvec{A}}\left( \alpha \right) ,{\varvec{B}}\left( \alpha \right) ,{\varvec{B}}_{i}\left( \alpha \right) ,{\varvec{C}}_{i}\left( \alpha \right) ,{\varvec{D}}_{i}\left( \alpha \right) \)

\(\gamma _{pq}\left( \alpha \right) \)

Components of the vector \(\vec {d}\)

\(\delta _{mp}\)

Kronecker delta


Materials constants defined for FG plate


Dimensionless parameters defined in terms of materials constants of FG plate

\(\left\| w \right\| \)

\(L_{{2}}\) norm of the plate deflection

\(\vec {e}\)

A predefined vector with components \(f_{pq}\left( x_{{0}},y_{{0}} \right) \)


Number of the natural frequencies of the bare plate


Numbers of indexes chosen for r and s in frequency \(\alpha _{rs}\)

\({\varvec{J}}_{{4}N\times 4N}\)

Jacobian matrix

\(A_{j},B_{j},\theta _{{11}},\theta _{12},\theta _{22},\theta _{{01}},\theta _{02}\)

Predefined constants



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringUniversity of ZanjanZanjanIran
  2. 2.Composites Research NetworkThe University of British ColumbiaOkanagan NodeCanada

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