Modal analysis of rotating pre-twisted viscoelastic sandwich beams

Abstract

This paper presents an analysis of the vibrational behavior of a rotating viscoelastic sandwich pre-twisted beams with a setting angle and with various viscoelastic stiffness laws. The governing equations of motion are derived using the Lagrange formulation and the assumed modes method. The obtained nonlinear eigenvalue problems are solved by using an iterative nonlinear eigensolver leading to complex eigensolutions composed of damped frequencies and loss factors with high accuracy. Further, the effects of the rotating speed, pre-twist angle, thickness ratio of core layer on the dynamic characteristics are investigated with taking into account the dependence of Young modulus with respect to frequency. Different numerical tests on rotating pre-twisted beams are performed for both isotropic and sandwich materials with a constant then a variable core modulus and the obtained results coincide very well with those provided in literature.

Appendices

Expressions of energies’ partial derivatives

Equations of motion Eqs. (47–50) are obtained by means of the Lagrange’s formulation Eq. (46). All necessary partial derivatives of U and T with respect to the generalized coordinates $q_{ui}$, $q_{vi}$, $q_{wi}$ and $q_{\beta i}$ are expressed hereafter.

1.1 Potential energy’s partial derivatives, $\Big (\frac{\partial U}{\partial q_{i}}\Big )$

$$\begin{aligned}&\displaystyle \frac{\partial U}{\partial q_{ui}}= \int _{0}^L (2E_fA_f + E_c(\omega )A_c) \sum _{j=1}^{m} \varPhi '_{1i} \varPhi '_{uj} q_{uj} dx \end{aligned}$$

(A.1)

$$\begin{aligned}&\displaystyle \frac{\partial U}{\partial q_{vi}}= \int _{0}^L\Big [ \left( 2E_fI_{fzz}+E_c(\omega )I_{czz}+E_fA_f \frac{d^2\sin ^2\theta }{2}\right) \nonumber \\&\quad \sum _{j=1}^{m} \varPhi ''_{vi} \varPhi ''_{vj} q_{2j}\nonumber \\&\qquad +\bigg (2E_fI_{fyz}-E_fA_f\frac{dh_f\sin 2\theta }{4} \bigg )\sum _{j=1}^{m} \varPhi ''_{vi} \varPhi ''_{wj} q_{3j}\nonumber \\&\qquad \bigg ( E_c(\omega )I_{cyz}-E_fA_f\frac{dh_c\sin 2\theta }{4}\bigg )\sum _{j=1}^{m} \varPhi ''_{vi} \varPhi '_{\beta j} q_{\beta j}\nonumber \\&\qquad + N_s \sum _{j=1}^{m} \varPhi '_{vi} \varPhi '_{vj} q_{vj} \bigg ]dx \end{aligned}$$

(A.2)

$$\begin{aligned}&\displaystyle \frac{\partial U}{\partial q_{wi}}= \int _{0}^L\bigg [ \bigg (2E_fI_{fyz}-E_fA_f \frac{dh_f\sin 2\theta }{4}\bigg )\nonumber \\&\quad \sum _{j=1}^{m} \varPhi ''_{wi} \varPhi ''_{vj} q_{vj} +\bigg (2E_fI_{fyy}\nonumber \\&\qquad +E_fA_f\frac{h_f^2\cos ^2\theta }{2} \bigg )\sum _{j=1}^{m} \varPhi ''_{wi} \varPhi ''_{wj} q_{wj}\nonumber \\&\qquad +E_fA_f\frac{h_fh_c\cos ^2\theta }{2}\sum _{j=1}^{m} \varPhi ''_{wi} \varPhi '_{\beta j} q_{\beta j}\nonumber \\&\qquad + \bigg (G_c(\omega )A_c + N_s\bigg ) \sum _{j=1}^{m} \varPhi '_{wi} \varPhi '_{wj} q_{wj} - G_c(\omega )A_c\nonumber \\&\quad \sum _{j=1}^{m} \varPhi '_{wi} \varPhi _{\beta j} q_{\beta j} \bigg ]dx \end{aligned}$$

(A.3)

$$\begin{aligned}&\displaystyle \frac{\partial U}{\partial q_{\beta i}}= \int _{0}^L\bigg [ \bigg (E_c(\omega )I_{cyz}-E_fA_f \frac{dh_c\sin 2\theta }{4}\bigg )\nonumber \\&\quad \sum _{j=1}^{m} \varPhi '_{\beta i} \varPhi ''_{vj} q_{vj} +E_fA_f\frac{h_fh_c\cos ^2\theta }{2}\sum _{j=1}^{m} \varPhi '_{\beta i} \varPhi ''_{wj} q_{wj}\nonumber \\&\qquad +\bigg (E_c(\omega )I_{cyy}+E_fA_f\frac{h_c^2\cos ^2\theta }{2}\bigg )\sum _{j=1}^{m} \varPhi '_{\beta i} \varPhi '_{\beta j} q_{\beta j}\nonumber \\&\qquad - G_c(\omega )A_c \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi '_{wj} q_{wj} \nonumber \\&\qquad + G_c(\omega )A_c \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi _{\beta j} q_{\beta j} \bigg ]dx \end{aligned}$$

(A.4)

1.2 Kinetic energy’s partial derivatives, $\bigg (\frac{\partial T}{\partial q_{i}}$ and $\frac{\mathrm {d}}{\mathrm {d}t}\frac{\partial T}{\partial \dot{q}_{i}}\bigg )$

$$\begin{aligned}&\displaystyle \frac{\partial T}{\partial q_{uj}}= \rho A \varOmega ^2 \int _{0}^L\nonumber \\&\quad \bigg [ \sum _{j=1}^{m} \varPhi _{ui} \varPhi _{uj} q_{uj} + \sum _{j=1}^{m} (r_h+x)\varPhi _{uj}\bigg ]dx \end{aligned}$$

(A.5)

$$\begin{aligned}&\displaystyle \frac{\partial T}{\partial q_{vi}}= \int _{0}^L\bigg [ \rho A \varOmega ^2\sum _{j=1}^{m} \varPhi _{vi} \varPhi _{vj} q_{vj} + \rho _fA_f \varOmega ^2\nonumber \\&\quad \bigg (\frac{d^2\sin ^2\theta }{2}\sum _{j=1}^{m} \varPhi '_{vi} \varPhi '_{vj} q_{vj}\nonumber \\&\quad - \frac{dh_f\sin 2\theta }{4} \sum _{j=1}^{m} \varPhi '_{vi} \varPhi '_{wj} q_{wj}- \frac{dh_c\sin 2\theta }{4}\nonumber \\&\quad \sum _{j=1}^{m} \varPhi '_{vi} \varPhi _{\beta j} q_{\beta j}\bigg ) \bigg ]dx \end{aligned}$$

(A.6)

$$\begin{aligned}&\displaystyle \frac{\partial T}{\partial q_{wi}}= \rho _fA_f \varOmega ^2\int _{0}^L\bigg [ -\frac{dh_f\sin 2\theta }{4}\nonumber \\&\quad \sum _{j=1}^{m} \varPhi '_{wi} \varPhi '_{vj} q_{vj} + \frac{h_f^2\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi '_{wi} \varPhi '_{wj} q_{wj}\nonumber \\&\quad + \frac{h_fh_c\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi '_{wi} \varPhi _{\beta j} q_{\beta j} \bigg ]dx \end{aligned}$$

(A.7)

$$\begin{aligned}&\displaystyle \frac{\partial T}{\partial q_{\beta i}}= \rho _fA_f \varOmega ^2\int _{0}^L\bigg [ -\frac{dh_c\sin 2\theta }{4}\nonumber \\&\quad \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi '_{vj} q_{vj} + \frac{h_fh_c\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi '_{wj} q_{wj}\nonumber \\&\quad + \frac{h_c^2\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi _{\beta j} q_{\beta j} \bigg ]dx \end{aligned}$$

(A.8)

To derive the equations of motion, the derivatives with respect to time t of the partial derivatives of T with respect to the generalized velocities, i.e. $\frac{\mathrm {d}}{\mathrm {d}t}\bigg (\frac{\partial T}{\partial \dot{q}_{i}}\bigg )$ are needed. One obtains

$$\begin{aligned}&\displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\bigg (\frac{\partial T}{\partial \dot{q}_{ui}}\bigg )= \rho A \int _{0}^L \sum _{j=1}^{m} \varPhi _{ui} \varPhi _{uj} \ddot{q}_{1j} dx \end{aligned}$$

(A.9)

$$\begin{aligned}&\displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\bigg (\frac{\partial T}{\partial \dot{q}_{vi}}\bigg )= \int _{0}^L \bigg [\rho A \sum _{j=1}^{m} \varPhi _{vi} \varPhi _{vj} \ddot{q}_{vj} + \rho _fA_f\nonumber \\&\qquad \bigg (\frac{d^2\sin ^2\theta }{2}\sum _{j=1}^{m} \varPhi '_{vi} \varPhi '_{vj}\ddot{q}_{vj} \nonumber \\&\qquad - \frac{dh_f\sin 2\theta }{4} \sum _{j=1}^{m} \varPhi '_{vi} \varPhi '_{wj}\ddot{q}_{wj}\nonumber \\&\qquad - \frac{dh_c\sin 2\theta }{4} \sum _{j=1}^{m} \varPhi '_{vi} \varPhi _{\beta j}\ddot{q}_{\beta j}\bigg ) \bigg ]dx \end{aligned}$$

(A.10)

$$\begin{aligned}&\displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\bigg (\frac{\partial T}{\partial \dot{q}_{wi}}\bigg )= \int _{0}^L \bigg [\rho A \sum _{j=1}^{m} \varPhi _{wi} \varPhi _{wj} \ddot{q}_{wj}\nonumber \\&\qquad + \rho _fA_f \bigg (\frac{h_f^2\cos ^2\theta }{2}\sum _{j=1}^{m} \varPhi '_{wi} \varPhi '_{wj}\ddot{q}_{wj} \nonumber \\&\qquad - \frac{dh_f\sin 2\theta }{4} \sum _{j=1}^{m} \varPhi '_{wi} \varPhi '_{vj}\ddot{q}_{vj}+ \frac{h_fh_c\cos ^2\theta }{2}\nonumber \\&\qquad \sum _{j=1}^{m} \varPhi '_{wi} \varPhi _{\beta j}\ddot{q}_{\beta j}\bigg ) \bigg ]dx \end{aligned}$$

(A.11)

$$\begin{aligned}&\displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\bigg (\frac{\partial T}{\partial \dot{q}_{\beta i}}\bigg )= \rho _fA_f\int _{0}^L \bigg [ -\frac{dh_c\sin 2\theta }{4}\sum _{j=1}^{m} \varPhi _{\beta i} \varPhi '_{vj}\ddot{q}_{vj}\nonumber \\&\qquad + \frac{h_ch_f\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi '_{wj}\ddot{q}_{wj}\nonumber \\&\qquad + \frac{h_c^2\cos ^2\theta }{2} \sum _{j=1}^{m} \varPhi _{\beta i} \varPhi _{\beta j}\ddot{q}_{\beta j} \bigg ]dx \end{aligned}$$

(A.12)

Expressions of matrices: $M_{ij}^{\alpha \beta }$, $K_{ij}^{\alpha \beta }$ and $P_{ij}^{\alpha }$

$$\begin{aligned} M_{ij}^{11}= & {} \displaystyle \int _{0}^L\rho A \varPhi _{ui} \varPhi _{uj} dx,\\ M_{ij}^{22}= & {} \displaystyle \int _{0}^L\rho A \varPhi _{vi} \varPhi _{vj} dx \\ M_{ij}^{33}= & {} \displaystyle \int _{0}^L\rho A \varPhi _{wi} \varPhi _{wj} dx,\\ K_{ij}^{11}= & {} \displaystyle \int _{0}^L 2E_fA_f \varPhi '_{ui} \varPhi '_{uj} dx \\ K_{ij}^{t1}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{d^2}{2}\sin ^2\theta \varPhi '_{vi} \varPhi '_{vj} dx,\\ K_{ij}^{t2}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{dh_f}{4}\sin 2\theta \varPhi '_{vi} \varPhi '_{wj} dx \\ K_{ij}^{t3}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{dh_c}{4}\sin 2\theta \varPhi '_{vi} \varPhi '_{wj} dx\qquad \\ K_{ij}^{t4}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{h_ch_f}{2}\cos ^2\theta \varPhi '_{wi} \varPhi '_{wj} dx \nonumber \\ K_{ij}^{t5}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{h_c^2}{2}\cos ^2\theta \varPhi '_{wi} \varPhi '_{wj} dx,\\ K_{ij}^{t6}= & {} \displaystyle \int _{0}^L\rho _fA_f\frac{h_f^2}{2}\cos ^2\theta \varPhi '_{wi} \varPhi '_{wj} dx \\ K_{ij}^{S1}= & {} \displaystyle \int _{0}^LE_fA_f\frac{d^2}{2}\sin ^2\theta \varPhi ''_{vi} \varPhi ''_{vj} dx,\\ K_{ij}^{S2}= & {} \displaystyle \int _{0}^LE_fA_f\frac{dh_f}{4}\sin 2\theta \varPhi ''_{vi} \varPhi ''_{wj} dx \\ K_{ij}^{S3}= & {} \displaystyle \int _{0}^LE_fA_f\frac{dh_c}{4}\sin 2\theta \varPhi ''_{vi} \varPhi ''_{wj} dx,\\ K_{ij}^{S4}= & {} \displaystyle \int _{0}^LE_fA_f\frac{h_f^2}{2}\cos ^2\theta \varPhi ''_{wi} \varPhi ''_{wj} dx \end{aligned}$$

$$\begin{aligned} K_{ij}^{S5}= & {} \displaystyle \int _{0}^LE_fA_f\frac{h_fh_c}{2}\cos ^2\theta \varPhi ''_{wi} \varPhi ''_{wj} dx,\\ K_{ij}^{S6}= & {} \displaystyle \int _{0}^LE_fA_f\frac{h_c^2}{2}\cos ^2\theta \varPhi ''_{wi} \varPhi ''_{wj} dx \\ K_{ij}^{I2}= & {} \displaystyle \int _{0}^L2E_fI_{fyy}\varPhi ''_{wi} \varPhi ''_{wj} dx,\\ K_{ij}^{I3}= & {} \displaystyle \int _{0}^L2E_fI_{fzz}\varPhi ''_{vi} \varPhi ''_{vj} dx \\ K_{ij}^{R2}= & {} \displaystyle \int _{0}^L\rho A\bigg [\frac{1}{2}(L^2-x^2) + r_h(L-x)\bigg ]\varPhi '_{vi} \varPhi '_{vj} dx,\\ K_{ij}^{I4}= & {} \displaystyle \int _{0}^L2E_fI_{fyz}\varPhi ''_{vi} \varPhi ''_{wj} dx \\ K_{ij}^{R3}= & {} \displaystyle \int _{0}^L\rho A\bigg [\frac{1}{2}(L^2-x^2) + r_h(L-x)\bigg ]\varPhi '_{wi} \varPhi '_{wj} dx,\\ P_{1j}^1= & {} \displaystyle \int _{0}^L\bigg (r_h+x \bigg )\varPhi _{uj} dx \\ K_{ij}^{V1}= & {} \displaystyle \int _{0}^L A_c \varPhi '_{ui} \varPhi '_{uj} dx,\\ K_{ij}^{V2}= & {} \displaystyle \int _{0}^LI_{cyy}\varPhi ''_{wi} \varPhi ''_{wj} dx \\ K_{ij}^{V3}= & {} \displaystyle \int _{0}^LI_{czz}\varPhi ''_{vi} \varPhi ''_{vj} dx,\\ K_{ij}^{V4}= & {} \displaystyle \int _{0}^LI_{cyz}\varPhi ''_{vi} \varPhi ''_{wj} dx \\ K_{ij}^{V5}= & {} \displaystyle \int _{0}^L\frac{A_c}{2(1+\nu _c)}\varPhi '_{vi} \varPhi '_{vj} dx,\\ K_{ij}^{V6}= & {} \displaystyle \int _{0}^L\frac{A_c}{2(1+\nu _c)}\varPhi '_{vi} \varPhi '_{wj} dx \\ K_{ij}^{V7}= & {} \displaystyle \int _{0}^L\frac{A_c}{2(1+\nu _c)}\varPhi '_{wi} \varPhi '_{wj} dx \end{aligned}$$