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Passive vibration suppression of plate using multiple optimal dynamic vibration absorbers

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Abstract

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.

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Abbreviations

\(F_{{0}},F_{0}^{*}\) :

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

\(M_{j},M_{j}^{*}\) :

Dimensional and dimensionless masses of jth absorber, respectively

\(k_{j},k_{j}^{*}\) :

Dimensional and dimensionless stiffnesses of jth absorber, respectively

\(u_{j},q_{j}\) :

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

\(Q_{j}\) :

Amplitude of \(q_{j}\)

a, b, h :

Length, width, and thickness of the plate, respectively

N :

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

\(a_{mn}\) :

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

E, E(z):

Elasticity modulus of isotropic and FG plates, respectively

\(\nu \) :

Poisson’s ration

D :

Flexural or bending rigidity of the plate

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

Density of the isotropic and FG plates, respectively

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

Delta Dirac function

c :

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

\(A_{11},B_{11},D_{11},A_{12},B_{12},D_{12},A_{33},B_{33},D_{33},I_{0},I_{1},I_{2}\) :

Materials constants defined for FG plate

\(D^{*},I_{1}^{*},I_{2}^{*}\) :

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 \(f_{pq}\left( x_{{0}},y_{{0}} \right) \)

\(N_\mathrm{f}\) :

Number of the natural frequencies of the bare plate

\(N_{{1}},N_{{2}}\) :

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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Authors and Affiliations

  1. Faculty of Engineering, University of Zanjan, P. O. Box 45195-313, Zanjan, Iran

    M. Ari & R. T. Faal

  2. Composites Research Network, The University of British Columbia, Okanagan Node, BC, Canada

    R. T. Faal

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  1. M. Ari
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Correspondence to R. T. Faal.

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Appendices

Appendix A

Using the first equation of (19) for \(N=1\) we have

$$\begin{aligned} a_{pq}(\alpha= & {} \alpha _{rs})=4F_{0}^{*}\left. \Bigg \{f_{pq}\left( x_{0},y_{0} \right) +\mu _{1}\left( \alpha _{rs} \right) \left[ \theta _{01}f_{pq}\left( x_{1},y_{1} \right) -\theta _{11}f_{pq}\left( x_{0},y_{0} \right) \right] \right. \nonumber \\&+\,\left. \frac{f_{rs}\left( x_{0},y_{0} \right) f_{rs}\left( x_{1},y_{1} \right) f_{pq}\left( x_{1},y_{1} \right) -f_{rs}\left( x_{1},y_{1} \right) f_{rs}\left( x_{1},y_{1} \right) f_{pq}\left( x_{0},y_{0} \right) }{\epsilon }\right\} \Bigg / \nonumber \\&{(\alpha }_{pq}^{4}-\alpha _{rs}^{4})\left\{ 1-\mu _{1}\left( \alpha _{rs} \right) \left[ \theta _{11}+\frac{f_{rs}\left( x_{1},y_{1} \right) f_{rs}\left( x_{1},y_{1} \right) }{\epsilon }\right] \right\} ,\alpha _{mn}\rightarrow \alpha _{rs} \end{aligned}$$
(A.1)

where \(\theta _{{11}}=\theta \left( x_{{1}},y_{{1}},x_{{1}},y_{{1}},\alpha _{rs} \right) ,\theta _{{01}}=\theta \left( x_{{0}},y_{{0}},x_{{1}},y_{{1}},\alpha _{rs} \right) \) and \(\epsilon =\mathop {\text {lim}}\limits _{\alpha {mn}\rightarrow \alpha _{rs}}\left( \alpha _{mn}^{{4}}-\alpha _{rs}^{{4}} \right) .\) Therefore for \(p=r\) and \(q=s\), the numerator is \(4F_{0}^{*}\left\{ f_{rs}\left( x_{0},y_{0} \right) +\mu _{1}\left( \alpha _{rs} \right) \left[ \theta _{01}f_{rs}\left( x_{1},y_{1} \right) -\theta _{11}f_{rs}\left( x_{0},y_{0} \right) \right] \right\} \) and the dominator \(-\mu _{1}\left( \alpha _{rs} \right) \left[ f_{rs}\left( x_{1},y_{1} \right) \right] ^{2}\) and thus we deduce that

$$\begin{aligned} a_{rs}\left( \alpha =\alpha _{rs} \right) =\frac{4F_{0}^{*}}{\left[ f_{rs}\left( x_{1},y_{1} \right) \right] ^{2}}\left[ \theta _{11}f_{rs}\left( x_{0},y_{0} \right) -\theta _{01}f_{rs}\left( x_{1},y_{1} \right) -\frac{f_{rs}\left( x_{0},y_{0} \right) }{\mu _{1}\left( \alpha _{rs} \right) }\right] \end{aligned}$$
(A.2)

The above coefficient is certainly bounded. Viewing Eq. (18), for \(N=2\) we also have

$$\begin{aligned}&\tau _{12}\left( \alpha \right) \tau _{21}\left( \alpha \right) -\tau _{11}\left( \alpha \right) \tau _{22}\left( \alpha \right) =\mu _{1}\mu _{2}[\psi ^{2}\left( x_{1},y_{1},x_{2},y_{2},\alpha \right) -\psi \left( x_{1},y_{1},x_{1},y_{1},\alpha \right) \psi \left( x_{2},y_{2},x_{2},y_{2},\alpha \right) ] \nonumber \\&\qquad +\,\mu _{1}\psi \left( x_{1},y_{1},x_{1},y_{1},\alpha \right) +\mu _{2}\psi \left( x_{2},y_{2},x_{2},y_{2},\alpha \right) -1 \nonumber \\&\rho _{2}\left( \alpha \right) \tau _{21}\left( \alpha \right) -\rho _{1}\left( \alpha \right) \tau _{22}\left( \alpha \right) =-\mu _{1}\psi \left( x_{0},y_{0},x_{1},y_{1},\alpha \right) +\mu _{1}\mu _{2} \nonumber \\&\qquad \times [\psi \left( x_{0},y_{0},x_{1},y_{1},\alpha \right) \psi \left( x_{2},y_{2},x_{2},y_{2},\alpha \right) -\psi \left( x_{1},y_{1},x_{2},y_{2},\alpha \right) \psi \left( x_{0},y_{0},x_{2},y_{2},\alpha \right) ] \nonumber \\&\rho _{1}\left( \alpha \right) \tau _{12}\left( \alpha \right) -\rho _{2}\left( \alpha \right) \tau _{11}\left( \alpha \right) =-\mu _{2}\psi \left( x_{0},y_{0},x_{2},y_{2},\alpha \right) \nonumber \\&\qquad +\,\mu _{1}\mu _{2}[\psi \left( x_{0},y_{0},x_{2},y_{2},\alpha \right) \psi \left( x_{1},y_{1},x_{1},y_{1},\alpha \right) -\psi \left( x_{1},y_{1},x_{2},y_{2},\alpha \right) \psi \left( x_{0},y_{0},x_{1},y_{1},\alpha \right) ] \end{aligned}$$
(A.3)

By splitting the singular and regular terms of the predefined function \(\psi \left( x,y,z,v,\alpha _{rs} \right) \) as \(\psi \left( x,y,z,v,\alpha _{rs} \right) =\theta \left( x,y,z,v,\alpha _{rs} \right) +\frac{f_{rs}\left( x,y \right) f_{rs}\left( z,v \right) }{\epsilon },\) the above equalities at \(\alpha =\alpha _{rs}\) are rewritten as follows

$$\begin{aligned}&\tau _{12}( \alpha _{rs} )\tau _{21}( \alpha _{rs} )-\tau _{11}( \alpha _{rs} )\tau _{22}( \alpha _{rs} )=\mu _{1}\theta _{11}+\mu _{2}\theta _{22}+\mu _{1}\mu _{2}[( \theta _{12} )^{2}-\theta _{11}\theta _{22}]-1 \nonumber \\&\quad +\,\{\mu _{1}\mu _{2}\{2\theta _{12}f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )-\theta _{11}{[f_{rs}( x_{2},y_{2} )]}^{2}-\theta _{22}[ f_{rs}( x_{1},y_{1} ) ]^{2}\} \nonumber \\&\quad +\,\mu _{1}[ f_{rs}( x_{1},y_{1} ) ]^{2}+\mu _{2}{[f_{rs}( x_{2},y_{2} )]}^{2}\}/\epsilon \nonumber \\&\rho _{2}( \alpha _{rs} )\tau _{21}( \alpha _{rs} )-\rho _{1}( \alpha _{rs} )\tau _{22}( \alpha _{rs} )=-\mu _{1}\theta _{01}+\mu _{1}\mu _{2}(\theta _{01}\theta _{22}-\theta _{12}\theta _{02}) \nonumber \\&\quad +\,\{\mu _{1}\mu _{2}\{\theta _{01}{[f_{rs}( x_{2},y_{2} )]}^{2}+\theta _{22}f_{rs}( x_{0},y_{0} )f_{rs}( x_{1},y_{1} ) \nonumber \\&\quad -\,\theta _{12}f_{rs}( x_{0},y_{0} )f_{rs}( x_{2},y_{2} )-\theta _{02}f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )\}-\mu _{1}f_{rs}( x_{0},y_{0} )f_{rs}( x_{1},y_{1} )\} / \epsilon \nonumber \\&\rho _{1}( \alpha _{rs} )\tau _{12}( \alpha _{rs} )-\rho _{2}( \alpha _{rs} )\tau _{11}( \alpha _{rs} )=-\mu _{2}\theta _{02}+\mu _{1}\mu _{2}(\theta _{02}\theta _{11}-\theta _{12} \theta _{01}) \nonumber \\&\quad +\,\{\mu _{1}\mu _{2}\{\theta _{02}[ f_{rs}( x_{1},y_{1} ) ]^{2}+\theta _{11}f_{rs}( x_{0},y_{0} )f_{rs}( x_{2},y_{2} )-\theta _{12}f_{rs}( x_{0},y_{0} )f_{rs}( x_{1},y_{1} )\nonumber \\&\quad -\,\theta _{01}f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )\}-\mu _{2}f_{rs}( x_{0},y_{0} )f_{rs}( x_{2},y_{2} )\}/\epsilon \end{aligned}$$
(A.4)

where

$$\begin{aligned} \theta _{22}=\theta \left( x_{2},y_{2},x_{2},y_{2},\alpha _{rs} \right) ,\theta _{12}=\theta \left( x_{1},y_{1},x_{2},y_{2},\alpha _{rs} \right) ,\theta _{02}=\theta \left( x_{0},y_{0},x_{2},y_{2},\alpha _{rs} \right) \end{aligned}$$
(A.5)

Substituting the above relations into the second equation of (19) results in

$$\begin{aligned}&a_{pq}(\alpha _{rs})=\{4F_{0}^{*}\mu _{1}[\theta _{11}f_{pq}( x_{0},y_{0} )-\theta _{01}f_{pq}( x_{1},y_{1} )]+\mu _{2}[\theta _{22}f_{pq}( x_{0},y_{0} )-\theta _{02}f_{pq}( x_{2},y_{2} )] \nonumber \\&\quad +\,\mu _{1}\mu _{2}\{( \theta _{01}\theta _{22}-\theta _{12}\theta _{02} )f_{pq}( x_{1},y_{1} )+( \theta _{02}\theta _{11}-\theta _{12}\theta _{01} )f_{pq}( x_{2},y_{2} ) \nonumber \\&\quad +\,[ ( \theta _{12} )^{2}-\theta _{11}\theta _{22} ]f_{pq}( x_{0},y_{0} )-f_{pq}( x_{0},y_{0} ) \nonumber \\&\quad +\,\{\mu _{1}\mu _{2}\{\theta _{12}\{2f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )f_{pq}( x_{0},y_{0} ) \nonumber \\&\quad -\,f_{rs}( x_{0},y_{0} )[ f_{rs}( x_{2},y_{2} )f_{pq}( x_{1},y_{1} )+f_{rs}( x_{1},y_{1} )f_{pq}( x_{2},y_{2} ) ]\} \nonumber \\&\quad +\,\theta _{11}f_{rs}( x_{2},y_{2} )[f_{rs}( x_{0},y_{0} )f_{pq}( x_{2},y_{2} )-f_{rs}( x_{2},y_{2} )f_{pq}( x_{0},y_{0} )] \nonumber \\&\quad +\,\theta _{22}f_{rs}( x_{1},y_{1} )[f_{rs}( x_{0},y_{0} )f_{pq}( x_{1},y_{1} )-f_{rs}( x_{1},y_{1} )f_{pq}( x_{0},y_{0} )] \nonumber \\&\quad +\,[\theta _{01}f_{rs}( x_{2},y_{2} )-\theta _{02}f_{rs}( x_{1},y_{1} )][f_{rs}( x_{2},y_{2} )f_{pq}( x_{1},y_{1} )-f_{rs}( x_{1},y_{1} )f_{pq}( x_{2},y_{2} )]\} \nonumber \\&\quad +\,\mu _{1}f_{rs}( x_{1},y_{1} )[f_{pq}( x_{0},y_{0} )f_{rs}( x_{1},y_{1} )-f_{pq}( x_{1},y_{1} )f_{rs}( x_{0},y_{0} )] \nonumber \\&\quad +\,\mu _{2}f_{rs}( x_{2},y_{2} )[f_{pq}( x_{0},y_{0} )f_{rs}( x_{2},y_{2} )-f_{rs}( x_{0},y_{0} )f_{pq}( x_{2},y_{2} )]\}/\epsilon \}/ \nonumber \\&\quad ({\alpha }_{pq}^{4}-\alpha ^{4})\{\mu _{1}\theta _{11}+\mu _{2}\theta _{22}+\mu _{1}\mu _{2}[( \theta _{12} )^{2}-\theta _{11}\theta _{22}]-1 \nonumber \\&\quad +\,\{\mu _{1}\mu _{2}\{2\theta _{12}f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )-\theta _{11}{[f_{rs}( x_{2},y_{2} )]}^{2}-\theta _{22}[ f_{rs}( x_{1},y_{1} ) ]^{2}\} \nonumber \\&\quad +\,\mu _{1}[ f_{rs}( x_{1},y_{1} ) ]^{2}+\mu _{2}{[f_{rs}( x_{2},y_{2} )]}^{2}\}/\epsilon \} ,\, for\, N=2 \end{aligned}$$
(A.6)

And finally, for \(m\rightarrow r\) and \(n\rightarrow s\) we conclude that

$$\begin{aligned} a_{rs}(\alpha _{rs})= & {} 4F_{0}^{*}\{\mu _{1}[\theta _{11}f_{rs}( x_{0},y_{0} )-\theta _{01}f_{rs}( x_{1},y_{1} )]+\mu _{2}[\theta _{22}f_{rs}( x_{0},y_{0} )-\theta _{02}f_{rs}( x_{2},y_{2} )] \nonumber \\&+\,\mu _{1}\mu _{2}\{[( \theta _{12} )^{2}-\theta _{11}\theta _{22}]f_{rs}( x_{0},y_{0} )+( \theta _{01}\theta _{22}-\theta _{12}\theta _{02} )f_{rs}( x_{1},y_{1} ) \nonumber \\&+\,( \theta _{02}\theta _{11}-\theta _{12}\theta _{01} )f_{rs}( x_{2},y_{2} )\}-f_{rs}( x_{0},y_{0} )\}/\{\mu _{1}[ f_{rs}( x_{1},y_{1} ) ]^{2}+\mu _{2}[ f_{rs}( x_{2},y_{2} ) ]^{2} \nonumber \\&+\,\mu _{1}\mu _{2}\{2\theta _{12}f_{rs}( x_{1},y_{1} )f_{rs}( x_{2},y_{2} )-\theta _{11}[ f_{rs}( x_{2},y_{2} ) ]^{2}-\theta _{22}[ f_{rs}( x_{1},y_{1} ) ]^{2}\}\},\, for\, N=2\nonumber \\ \end{aligned}$$
(A.7)

Again, it can be seen that the above coefficient is bounded.

Appendix B

The entries of the Jacobian matrix \({\varvec{J}}_{{4}N{\times 4}N}\)

$$\begin{aligned} J_{p,j}= & {} \frac{\partial }{\partial k_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) {\varvec{B}}\left( \alpha _{rs} \right) {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] , \quad r=1,\ldots , N_{1}, s=1,\ldots , N_{2}, p=1,\ldots , N_\mathrm{f} \nonumber \\ J_{iN_\mathrm{f}+p,j}= & {} \frac{\partial }{\partial k_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{B}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(N+i)N_\mathrm{f}+p,j}= & {} \frac{\partial }{\partial k_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{C}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,j}= & {} \frac{\partial }{\partial k_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{D}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{p,N+j}= & {} \frac{\partial }{\partial M_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) {\varvec{B}}\left( \alpha _{rs} \right) {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{iN_\mathrm{f}+p,N+j}= & {} \frac{\partial }{\partial M_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{B}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(N+i)N_\mathrm{f}+p,N+j}= & {} \frac{\partial }{\partial M_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{C}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,N+j}= & {} \frac{\partial }{\partial M_{j}^{*}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{D}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{p,2N+j}= & {} \frac{\partial }{\partial x_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) {\varvec{B}}\left( \alpha _{rs} \right) {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{iN_\mathrm{f}+p,2N+j}= & {} \frac{\partial }{\partial x_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{B}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(N+i)N_\mathrm{f}+p,2N+j}= & {} \frac{\partial }{\partial x_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{C}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,2N+j}= & {} \frac{\partial }{\partial x_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{D}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{p,3N+j}= & {} \frac{\partial }{\partial y_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) {\varvec{B}}\left( \alpha _{rs} \right) {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{iN_\mathrm{f}+p,3N+j}= & {} \frac{\partial }{\partial y_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{B}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(N+i)N_\mathrm{f}+p,3N+j}= & {} \frac{\partial }{\partial y_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{C}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,3N+j}= & {} \frac{\partial }{\partial y_{j}}\left[ \vec {e}^{T}{\varvec{A}}^{-1}\left( \alpha _{rs} \right) \left\{ {\varvec{D}}_{j}\left( \alpha _{rs} \right) \right\} {\varvec{A}}^{-2}\left( \alpha _{rs} \right) \vec {e} \right] \end{aligned}$$
(B.1)

Appendix C

The simplified entries of the Jacobian matrix \({\varvec{J}}_{{4}N\times 4N}\)

$$\begin{aligned} J_{p,j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} ) \left\{ -\frac{8{(\alpha _{rs})}^{4} k_{j}^{*}}{k_{j}^{*}-{(\alpha _{rs})}^{4}M_{j}^{*}} {\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1} ( \alpha _{rs} )\right. \nonumber \\&+\,4( \alpha _{rs} )^{8}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha ){\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[ {\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ) ]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{p,N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{4( k_{j}^{*} )^{2}( k_{j}^{*}+( \alpha _{rs} )^{4}M_{j}^{*} )}{{(M_{j}^{*})}^{2}(k_{j}^{*}-( \alpha _{rs} )^{4}M_{j}^{*})}{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \right. \nonumber \\&-\,4( \alpha _{rs} )^{4}( k_{j}^{*} /M_{j}^{*} )^{2}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{p,2N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{4\pi }{( \alpha _{rs} )^{4}(k_{j}^{*}-( \alpha _{rs} )^{4}M_{j}^{*})}{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&-\,4\pi \{{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{p,3N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{4\pi }{( \alpha _{rs} )^{4}(k_{j}^{*}-( \alpha _{rs} )^{4}M_{j}^{*})}{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&-\,4\pi \{{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{iN_\mathrm{f}+p,j}= & {} \vec {e}^{T}\left\{ -\frac{2\delta _{ij}}{(k_{j}^{*}-( \alpha _{rs} )^{4}M_{j}^{*}}{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-2}( \alpha _{rs} ) \right. \nonumber \\&+\,4( \alpha _{rs} )^{8}{\varvec{A}}^{-1}( \alpha _{rs} )\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}_{i}( \alpha _{rs} )[ {\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-2}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ) ]\}{\varvec{A}}^{-1}( \alpha _{rs} ) \right\} \vec {e} \nonumber \\ J_{iN_\mathrm{f}+p,N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{2k_{j}^{*}\delta _{ij}}{{M_{j}^{*}(k}_{j}^{*}-\alpha _{rs}^{4}M_{j}^{*})}{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \right. \nonumber \\&-\,4( \alpha _{rs} )^{4}( k_{j}^{*} /M_{j}^{*} )^{2}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs})]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{iN_\mathrm{f}+p,2N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \pi \frac{M_{j}^{*}\delta _{ij}}{\alpha _{rs}^{4}k_{j}^{*}(k_{j}^{*}-\alpha _{rs}^{4}M_{j}^{*})}{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \right. \nonumber \\&-\,4\pi \{{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{iN_\mathrm{f}+p,3N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\pi \left\{ \frac{M_{j}^{*}\delta _{ij}}{\alpha _{rs}^{4}k_{j}^{*}(k_{j}^{*}-\alpha _{rs}^{4}M_{j}^{*})}{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \right. \nonumber \\&-\,4\pi \{{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{B}}_{i}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(N+i)N_\mathrm{f}+p,j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{{-\alpha _{rs}}^{4}M_{j}^{*}\delta _{ij}}{k_{j}^{*}(k_{j}^{*}-\alpha _{rs}^{4}M_{j}^{*})}{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&+\,4( \alpha _{rs} )^{8}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha ){\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[ {\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ) ]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(N+i)N_\mathrm{f}+p,N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{k_{j}^{*}\delta _{ij}}{M_{j}^{*}(k_{j}^{*}-\alpha _{rs}^{4}M_{j}^{*})}{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&-\,4( \alpha _{rs} )^{4}( k_{j}^{*} / M_{j}^{*} )^{2}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(N+i)N_\mathrm{f}+p,2N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\{\pi \delta _{ij}\mu _{j}(\alpha _{rs}){\varvec{P}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&-\,4\pi \{{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&+\,{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} )]\}\}{\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(N+i)N_\mathrm{f}+p,3N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\{\pi \delta _{ij}\mu _{j}(\alpha _{rs}){\varvec{Q}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&-\,4\pi \{{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&+\,{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} )]\}\}{\varvec{A}}^{-1}( \alpha _{rs} )\vec {e}\nonumber \\ J_{(2N+i)N_\mathrm{f}+p,j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{{-\alpha }^{4}M_{j}^{*}\delta _{ij}}{k_{j}^{*}(k_{j}^{*}-\alpha ^{4}M_{j}^{*})}{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&+\,4( \alpha _{rs} )^{8}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha ){\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[ {\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} ) ]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\left\{ \frac{k_{j}^{*}\delta _{ij}}{M_{j}^{*}(k_{j}^{*}-\alpha ^{4}M_{j}^{*})}{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )\right. \nonumber \\&-\,4( \alpha _{rs} )^{4}( k_{j}^{*} / M_{j}^{*} )^{2}\{{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&\left. +\,{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{B}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{B}}_{j}( \alpha _{rs} )]\} \right\} {\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,2N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\{\pi \delta _{ij}\mu _{j}(\alpha _{rs}){\varvec{S}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&-\,4\pi \{{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&+\,{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{C}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{C}}_{j}( \alpha _{rs} )]\}\}{\varvec{A}}^{-1}( \alpha _{rs} )\vec {e} \nonumber \\ J_{(2N+i)N_\mathrm{f}+p,3N+j}= & {} \vec {e}^{T}{\varvec{A}}^{-1}( \alpha _{rs} )\{\pi \delta _{ij}\mu _{j}(\alpha _{rs}){\varvec{R}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&-\,4\pi \{{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} ) \nonumber \\&+\,{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )[{\varvec{D}}_{j}( \alpha _{rs} ){\varvec{A}}^{-1}( \alpha _{rs} )+{\varvec{A}}^{-1}( \alpha _{rs} ){\varvec{D}}_{j}( \alpha _{rs} )]\} \}{\varvec{A}}^{-1}( \alpha _{rs} )\vec {e}\nonumber \\ \end{aligned}$$
(C.1)

Appendix D: Flowchart specifying the methodology of computing the optimal parameters of the absorbers

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Ari, M., Faal, R.T. Passive vibration suppression of plate using multiple optimal dynamic vibration absorbers. Arch Appl Mech 90, 235–274 (2020). https://doi.org/10.1007/s00419-019-01607-z

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