Layout optimization of viscoelastic damping for noise control of mid-frequency vibro-acoustic systems

This paper presents an integrated, new, and generic framework for layout optimization of viscoelastic damping for noise control of mid-frequency vibro-acoustic systems. The method is developed based on the concept of power balance among different modal energies between coupled structural and acoustic subsystems and is formulated within the framework of a statistical modal energy distribution analysis (SmEdA). In the novel optimization formulation, the total energy of the acoustic subsystem is chosen as the objective function for minimizing the internal acoustic response in the vibro-acoustic system; and the relative material volume densities for viscoelastic element groups are selected as design variables using a volume-preserving Heaviside function. A new sensitivity analysis formulation is developed in a semi-analytical form via a SmEdA for solving the vibro-acoustic optimization problem. Two numerical examples are presented to demonstrate the efficiency and effectiveness of the present method. The present numerical results reveal two important findings: (a) the total acoustic energy of the chosen vibro-acoustic system can be significantly reduced; and (b) the optimum viscoelastic material layouts not only decrease the peak values of the modal coupling strengths between structural and acoustic subsystems but also create relatively more uniform acoustic modal energy distribution.

a_p(t):

Modal amplitude corresponding to the p^th displacement mode

b_q(t):

Modal amplitude corresponding to the q^th acoustic mode

c _air :

Sound speed of the air in acoustic cavity

c_q(t):

Integrated modal amplitude corresponding to the q^th acoustic mode

C ₁₁ :

Modal coupling factor matrix of structural subsystem

C ₁₂ :

Modal coupling factor matrix between structural subsystem and acoustic subsystem

C ₂₁ :

Modal coupling factor matrix between acoustic subsystem and structural subsystem

C ₂₂ :

Modal coupling factor matrix of acoustic subsystem

E(dB):

Energy units measured by decibels

E(J):

Energy units measured by joules

E _i :

Surrogate Young’s modulus of the design material in the i^th design domain

E _min :

A very small modulus assigned to void regions and is set to be 0.001E₀ in this study

E_p (E_q):

The p (q)^th modal energy of structural (acoustic) subsystem

E _ref :

Reference value when transforming from joules to decibels

E _steel :

Young’s modulus of steel plate

E_str (E_aco):

Total energy of structural (acoustic) subsystem

\( {E}_{visc}^{\ast } \) :

Complex Young’s modulus of the viscoelastic damping material

E ₀ :

Real Young’s modulus of the design material

E ₁ :

Modal energy vector of structural subsystem

E ₂ :

Modal energy vector of acoustic subsystem

f :

Prescribed volume fraction

F_p(t) (F_q(t)):

The p (q)^th generalized modal force acting on structural (acoustic) subsystem

k :

Penalty factor used in modified SIMP model and is designated as 3 in this paper

K_ba (M_ba):

Stiffness (mass) matrix of the base structure

\( {K}_{visc}^{\ast }\ \left({M}_{visc}^{\ast}\right) \) :

Complex stiffness (mass) matrix of the viscoelastic damping material

L_pq(t) (L_qp(t)):

Interaction forces between structural and acoustic subsystems

M :

An arbitrary point in structural domain

M ^′ :

An arbitrary point in acoustic domain

M _e :

Excitation point on the surface of structural subsystem

M_p (M_q):

The p (q)^th modal mass of structural (acoustic) subsystem

N :

Number of design domains

N₁(N₂):

Number of resonant modes of structural (acoustic) subsystem

p :

Modal order of structural subsystem

P(M^′, t):

Sound pressure of acoustic domain at M^′

\( {\tilde{P}}_q\left({M}^{\prime}\right) \) :

The q^th acoustic mode of acoustic subsystem

q :

Modal order of acoustic subsystem

S _coupling :

Fluid-structure coupling surface

S _FF :

Power spectral density of the generalized external force F

\( {S}_{F_p} \) :

Power spectral density of the generalized modal force F_p(t)

U(M, t):

Displacement of structural domain at M

\( {\tilde{U}}_p(M) \) :

The p^th displacement mode of structural subsystem

\( {\tilde{U}}_p\left({M}_e\right) \) :

Modal displacement at excitation point M_e with respect to the p^th displacement mode of structural subsystem

V _i :

Volume of the the i^th design domain

W :

Total weight of the design material

W _pq :

Modal coupling work between the p^th structural mode and the q^th acoustic mode

W ₀ :

Allowable maximum material weight

\( {\tilde{x}}_i \) :

Surrogate material density, i.e., relative material volume density of the i^th design domain

x _i :

The i^th design variable

x :

Vector of design variables

x _min :

Vector denoting the low bounds of the design variables

α, θ :

Two control parameters in the volume-preserving Heaviside function

β _pq :

Modal coupling factor between the p^th structural mode and q^th acoustic mode

\( {\lambda}_p^{\ast } \) :

The p^th complex eigenvalue evaluated from the structural subsystem treated with viscoelastic damping material

η _air :

Modal damping loss factors of the air in the acoustic cavity

η _mat :

Damping loss factor of the viscoelastic damping material

η_p(η_q):

The p (q)^th modal damping loss factor of structural (acoustic) subsystem

ν _steel :

Poisson’s ratio of the steel plate

ν _visc :

Poisson’s ratio of the viscoelastic material

\( {\varPi}_p^{diss} \) :

Time-averaged power dissipated by internal damping

\( {\varPi}_p^{inj} \) :

Time-averaged power injected into the p^th mode (modal input power)

Π _pq :

Time-averaged power flow transmitted from the p^th structural mode to the q^th acoustic mode

Π ₁ :

Vector of modal input powers of structural subsystem

Π ₂ :

Vector of modal input powers of acoustic subsystem

ρ _air :

Mass density of the air in the acoustic cavity

ρ _i :

Surrogate material mass density of the design material at the i^th design domain

ρ _steel :

Mass density of the steel plate

ρ _visc :

Mass density of the viscoelastic material

ρ ₀ :

Real mass density of the design material

ω_p(ω_q):

The p (q)^th angular frequency of structural (acoustic) subsystem

The research project is supported by the National Natural Science Foundation of China (U1508209, 11072049), Liaoning BaiQianWan Talents Program and Dalian Science and Technology Innovation Fund (2018J11CY003). The authors would like to acknowledge the support of these funds. The authors are also grateful to Krister Svanberg of KTH in Stockholm for providing the MMA optimization subroutines.

Authors and Affiliations

1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

   Yang Yu & Guozhong Zhao

2. School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

   Liyong Tong

Corresponding author

Correspondence to Guozhong Zhao.

Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

The corresponding codes can be obtained in the supplementary material. Only the codes for numerical examples in Section 4.1 are given. The source codes includes the mph. files built in COMSOL and the m. files written in MATLAB. The readers should install the COMSOL Multiphysics with MATLAB and replicate the resluts about the case of Plate-cavity system in Section 4.1

Responsible Editor: Emilio Carlos Nelli Silva

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