Vibration of nonlinear bolted lap-jointed beams using Timoshenko theory

Abstract

This paper investigates the vibrational behavior of a system which consists of two free–free Timoshenko beams interconnected by a nonlinear joint. To model the bolted lap joint interface, a combination of the linear translational spring, linear and nonlinear torsional springs, and a linear torsional damper is used. The governing equations of motion are derived using the Euler–Lagrange equations. The reduced-order model equations are obtained based on Galerkin method. The set of coupled nonlinear equations are then analytically solved using the harmonic balance approach and numerical simulation. A parametric study is carried out to reveal the influence of different parameters such as linear and nonlinear torsional spring, linear translational spring, and linear torsional damper on the vibration and stability of the bolted lap joint structure. It is shown that the effect of the nonlinear torsional spring on the response of the system is significant. Interestingly, it is observed that in the presence of the nonlinear spring the softening behavior could be changed to hardening behavior. In addition, the effects of the different engineering beam theories on the modeling of the substructures are studied and it is observed that considering the effect of the rotary inertia and shear deformations is significant. In addition, it is observed that neglecting each of them can yield completely wrong interpretations of the system behavior and incorrect results.

Appendices

Appendix A

$$\begin{aligned} N_1= & {} \int _0^{L_{b1} } \left( {\varphi _1 \left( s \right) } \right) ^{2}\mathrm{d}s \end{aligned}$$

(A.1)

$$\begin{aligned} N_2= & {} \int _0^{L_{b1} } \left( {\varphi _2 \left( s \right) } \right) ^{2}\mathrm{d}s \end{aligned}$$

(A.2)

$$\begin{aligned} N_3= & {} \int _{\frac{L_t }{2}}^{L_t } \left( {\varphi _3 \left( s \right) } \right) ^{2}\mathrm{d}s \end{aligned}$$

(A.3)

$$\begin{aligned} N_4= & {} \int \nolimits _{\frac{L_t }{2}}^{L_t } \left( {\varphi _4 \left( s \right) } \right) ^{2}\mathrm{d}s \end{aligned}$$

(A.4)

$$\begin{aligned} N_5= & {} \varphi _3 \left( {L_t } \right) \end{aligned}$$

(A.5)

$$\begin{aligned} N_6= & {} \varphi _3 ^{{\prime }}\left( {L_t } \right) \end{aligned}$$

(A.6)

$$\begin{aligned} N_7= & {} \int _0^{L_{b1} } (\varphi _1 ^{{\prime }}\left( s \right) )^{2}\mathrm{d}s \end{aligned}$$

(A.7)

$$\begin{aligned} N_8= & {} \frac{1}{2}E_{b1} I_{b1} \int \nolimits _0^{L_{b1} } (\varphi _1 ^{{\prime }}\left( s \right) )^{2}\mathrm{d}s \end{aligned}$$

(A.8)

$$\begin{aligned} N_9= & {} \int _{\frac{L_t }{2}}^{L_t } (\varphi _3 ^{{\prime }}\left( s \right) )^{2}\mathrm{d}s \end{aligned}$$

(A.9)

$$\begin{aligned} N_{10}= & {} \frac{1}{2}E_{b2} I_{b2} \int \nolimits _{\frac{L_t }{2}}^{L_t } (\varphi _3 ^{{\prime }}\left( s \right) )^{2}\mathrm{d}s \end{aligned}$$

(A.10)

$$\begin{aligned} N_{11}= & {} \frac{1}{3}\int \nolimits _{\frac{L_t }{2}}^{L_t } (\varphi _3 ^{{\prime }}\left( s \right) )^{4}\mathrm{d}s \end{aligned}$$

(A.11)

$$\begin{aligned} N_{12}= & {} \int \nolimits _{\frac{L_t }{2}}^{L_t } \varphi _3 \left( s \right) \mathrm{d}s \end{aligned}$$

(A.12)

$$\begin{aligned} N_{13}= & {} \int \nolimits _0^{L_{b1} } \varphi _1 ^{{\prime }}\left( s \right) \varphi _2 \left( s \right) \mathrm{d}s \end{aligned}$$

(A.13)

$$\begin{aligned} N_{14}= & {} \int \nolimits _0^{L_{b1} } \varphi _1 \left( s \right) \mathrm{d}s \end{aligned}$$

(A.14)

$$\begin{aligned} N_{15}= & {} \varphi _3 ^{{\prime }}\left( S \right) \end{aligned}$$

(A.15)

$$\begin{aligned} N_{16}= & {} \varphi _1 ^{{\prime }}\left( S \right) \end{aligned}$$

(A.16)

$$\begin{aligned} N_{17}= & {} \varphi _1 \left( S \right) \end{aligned}$$

(A.17)

$$\begin{aligned} N_{18}= & {} \varphi _3 \left( S \right) \end{aligned}$$

(A.18)

$$\begin{aligned} N_{19}= & {} \frac{1}{2}E_{b1} A_{b1} \int _0^{L_{b1} } (\varphi _1 ^{{\prime }}\left( s \right) )^{4}\mathrm{d}s \end{aligned}$$

(A.19)

Appendix B

$$\begin{aligned} A_1= & {} \rho _{b1} A_{b1} N_1 \end{aligned}$$

(B.1)

$$\begin{aligned} A_2= & {} cN_{15} N_{16} \end{aligned}$$

(B.2)

$$\begin{aligned} A_3= & {} cN_{16} ^{2} \end{aligned}$$

(B.3)

$$\begin{aligned} A_4= & {} -k_{\theta NL} N_{16} ^{4}+N_{19} \end{aligned}$$

(B.4)

$$\begin{aligned} A_5= & {} 3k_{\theta NL} N_{15} N_{16} ^{3} \end{aligned}$$

(B.5)

$$\begin{aligned} A_6= & {} k_{\theta NL} N_{16} N_{15} ^{3} \end{aligned}$$

(B.6)

$$\begin{aligned} A_7= & {} 3k_{\theta NL} N_{16} ^{2}N_{15} ^{2} \end{aligned}$$

(B.7)

$$\begin{aligned} A_8= & {} k_{aL} N_{18} N_{17} +k_{\theta L} N_{15} N_{16} \end{aligned}$$

(B.8)

$$\begin{aligned} A_9= & {} k_1 ^{{\prime }}G_1 A_{b1} N_{13} \end{aligned}$$

(B.9)

$$\begin{aligned} A_{10}= & {} k_1 ^{{\prime }}G_1 A_{b1} N_7 +k_{aL} N_{17} ^{2}+k_{\theta L} N_{16} ^{2} \end{aligned}$$

(B.10)

$$\begin{aligned} A_{11}= & {} \rho _{b1} gN_{14} \end{aligned}$$

(B.11)

$$\begin{aligned} A_{12}= & {} J_1 N_2 \end{aligned}$$

(B.12)

$$\begin{aligned} A_{13}= & {} k_1 ^{{\prime }}G_1 A_{b1} N_2 +2N_8 \end{aligned}$$

(B.13)

$$\begin{aligned} A_{14}= & {} M_t \left( {N_5 +\frac{h_1 }{2}N_6 } \right) ^{2}+I_M N_6 ^{2} \end{aligned}$$

(B.14)

$$\begin{aligned} A_{15}= & {} cN_{15} ^{2} \end{aligned}$$

(B.15)

$$\begin{aligned} A_{16}= & {} -k_{\theta NL} N_{15} ^{4}+N_{16} \end{aligned}$$

(B.16)

$$\begin{aligned} A_{17}= & {} 3k_{\theta NL} N_{16} N_{15} ^{3} \end{aligned}$$

(B.17)

$$\begin{aligned} A_{18}= & {} k_{\theta NL} N_{15} N_{16} ^{3} \end{aligned}$$

(B.18)

$$\begin{aligned} A_{19}= & {} k_2 ^{{\prime }}G_2 A_{b2} N_{19} \end{aligned}$$

(B.19)

$$\begin{aligned} A_{20}= & {} k_2 ^{{\prime }}G_2 A_{b2} N_9 +k_{aL} N_{17} ^{2}+k_{\theta L} N_{16} ^{2} \end{aligned}$$

(B.20)

$$\begin{aligned} A_{21}= & {} \rho _{b2} gN_{12} \end{aligned}$$

(B.21)

$$\begin{aligned} A_{22}= & {} N_{19} \end{aligned}$$

(B.22)

$$\begin{aligned} A_{23}= & {} J_2 N_4 \end{aligned}$$

(B.23)

$$\begin{aligned} A_{24}= & {} k_2 ^{{\prime }}G_2 A_{b2} N_4 +2N_{10} \end{aligned}$$

(B.24)

Appendix C

$$\begin{aligned} a_1= & {} \frac{A_4 }{A_1 } \end{aligned}$$

(C.1)

$$\begin{aligned} a_2= & {} \frac{A_5 }{A_1 } \end{aligned}$$

(C.2)

$$\begin{aligned} a_3= & {} \frac{A_6 }{A_1 } \end{aligned}$$

(C.3)

$$\begin{aligned} a_4= & {} \frac{A_7 }{A_1 } \end{aligned}$$

(C.4)

$$\begin{aligned} a_5= & {} \frac{A_8 }{A_1 } \end{aligned}$$

(C.5)

$$\begin{aligned} a_6= & {} \frac{A_9 }{A_1 } \end{aligned}$$

(C.6)

$$\begin{aligned} a_7= & {} \frac{A_{10} }{A_1 } \end{aligned}$$

(C.7)

$$\begin{aligned} a_8= & {} \frac{A_{11} }{A_1 } \end{aligned}$$

(C.8)

$$\begin{aligned} a_9= & {} \frac{A_{13} }{A_{12} } \end{aligned}$$

(C.9)

$$\begin{aligned} a_{10}= & {} \frac{A_9 }{A_{12} } \end{aligned}$$

(C.10)

$$\begin{aligned} a_{11}= & {} \frac{A_{16} }{A_{14} } \end{aligned}$$

(C.11)

$$\begin{aligned} a_{12}= & {} \frac{A_{17} }{A_{14} } \end{aligned}$$

(C.12)

$$\begin{aligned} a_{13}= & {} \frac{A_{18} }{A_{14} } \end{aligned}$$

(C.13)

$$\begin{aligned} a_{14}= & {} \frac{A_7 }{A_{14} } \end{aligned}$$

(C.14)

$$\begin{aligned} a_{15}= & {} \frac{A_8 }{A_{14} } \end{aligned}$$

(C.15)

$$\begin{aligned} a_{16}= & {} \frac{A_{19} }{A_{14} } \end{aligned}$$

(C.16)

$$\begin{aligned} a_{17}= & {} \frac{A_{20} }{A_{14} } \end{aligned}$$

(C.17)

$$\begin{aligned} a_{18}= & {} \frac{A_{21} }{A_{14} } \end{aligned}$$

(C.18)

$$\begin{aligned} a_{19}= & {} \frac{A_{22} }{A_{14} } \end{aligned}$$

(C.19)

$$\begin{aligned} a_{20}= & {} \frac{A_{24} }{A_{23} } \end{aligned}$$

(C.20)

$$\begin{aligned} a_{21}= & {} \frac{A_{19} }{A_{23} } \end{aligned}$$

(C.21)