Sensitivity analysis of a two-plate coupled system in the statistical energy analysis (SEA) framework

Abstract

The main objective of this paper is to perform design sensitivity analysis of two right angle coupled plates, connected by various joint connections, to determine an optimum coupling loss factor (CLF) using optimization in the statistical energy analysis (SEA) framework. The theoretical analysis of obtaining such optimum CLFs can be used during the design stage of a dynamic system to specify the right type of joint. First- and second-order sensitivity analysis using theoretical equations of CLFs of two right angle plates in the SEA framework, coupled to form welded, riveted, and bolted connections, is presented in the audible frequency range. These sensitivities were determined using both analytic (direct differentiation) and finite difference methods; the sensitivities computed by the analytic method agree well with the finite difference method, indicating that the direct differentiation method can be directly used to predict variation in the CLF and the corresponding response of coupled plates joined by various joint connections. The sensitivity analysis also gives a feasible region in terms of frequency range for the determination of optimum values of CLF by selecting the right type of joint at the design stage based on its stiffness. The study presented in this paper will be very useful at the drawing board stage of designing vibro-acoustic systems to reduce vibration or noise of such systems by giving a definite direction for the modification of design parameters, thus eliminating expensive experimentation; it will be helpful in arriving at the optimum values of CLF that would eventually reduce the vibro-acoustic response of dynamic systems with a large number of subsystems connected by welded, riveted, and bolted joints.

Appendix

The mathematical model for determination of theoretical CLF of right angle coupled plates connected by various joint connections (as shown in Fig. 1) is presented by authors Patil and Manik (2015).

The bending wave transmission efficiency (τ₁₂) used in (12) is obtained from the developed mathematical model as

$$ {\tau}_{12}= X\psi {\left|{t}_B\right|}^2 $$

(34)

where

$$ {t}_B=\frac{2\left[{Z}_1+{\beta}_1+{Z}_2+{\beta}_2\right]-2j\left[1-\left({Z}_1+{\beta}_1\right)\left({Z}_2+{\beta}_2\right)\right]}{D} $$

(35)

$$ {\displaystyle \begin{array}{l}D=X\left\{-1+j-2\left({Z}_1+{\beta}_1\right)-\left({Z}_2+{\beta}_2\right)-\left(1+j\right)\left({Z}_1+{\beta}_1\right)\left({Z}_2+{\beta}_2\right)\right\}\\ {}+\psi \left\{-1+j-\left({Z}_1+{\beta}_1\right)-2\left({Z}_2+{\beta}_2\right)-\left(1+j\right)\left({Z}_1+{\beta}_1\right)\left({Z}_2+{\beta}_2\right)\right\}\\ {}+\psi {Z}_3\left\{-\left({Z}_1+{\beta}_1\right)\left({Z}_2+{\beta}_2\right)+2j-\left(1-j\right)\left({Z}_1+{\beta}_1\right)-\left(1-j\right)\left({Z}_2+{\beta}_2\right)\right\}.\end{array}} $$

(36)

in (34) to (36)

X = k_b2/k_b1; $\psi =\frac{k_{b2}^2{B}_{s2}}{k_{b1}^2{B}_{s1}}$; β₁ = (m₂C_B2)/(m₁C_L1); β₂ = (m₁C_B1)/(m₂C_L2);where

m₁, m₂ are the mass of two plates, k_b1, k_b2 are the wave numbers in two plates, B_s1, B_s2 are the bending stiffness of two plates, C_B1, C_B2 are the bending wave speeds in two plates, and C_L1, C_L2 are the longitudinal wave speeds in two plates (Cremer et al. (1973)).

Here,

$$ {\displaystyle \begin{array}{l}{Z}_1={m}_2{C}_{B2}/\left(\left({K}_1/ j\omega \right)+{c}_1\right)\\ {}{Z}_2={m}_1{C}_{B1}/\left(\left({K}_2/ j\omega \right)+{c}_2\right)\\ {}{Z}_3={B}_{s1}/{C}_{B1}\left(\left({K}_3/ j\omega \right)+{c}_3\right)\end{array}} $$

(37)

In (37), K₁, K₂ are the linear stiffness values and K₃ is a rotational stiffness. Similarly, c₁, c₂ are the linear damping values and c₃ is a rotational damping.