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Predicting dynamic response of large amplitude free vibrations of cantilever tapered beams on a nonlinear elastic foundation

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Abstract

The purpose of this paper is to investigate the nonlinear vibration behavior of single and double tapered cantilever beams on the nonlinear foundation and present an alternative analytical method to solve the issue. Unlike other literatures, the admissible lateral displacement function satisfying the geometric boundary conditions of a single or double tapered cantilever beam is derived by using Rayleigh-Ritz method. Analytical approximate solutions in closed and explicit form are obtained by combining the Lagrange method with the Galerkin and Newton linearization method. Compared with the exact (numeric) results, the accuracy of these approximate solutions is presented. The effects of different parameters such as vibration amplitude, and foundation stiff parameters on nonlinear frequencies and displacement response of the tapered beams are easily analyzed, with the brief expressions of the present analytical approximate solutions.

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Abbreviations

x :

The coordinate of the beam neutral axis

y :

The coordinate of the vertical axis

u :

The axial shortening along the beam neutral axis

v :

The transverse deflection v along the vertical axis y

E :

Young’s modulus of the beam

\(\rho \) :

Tensity of the beam

I :

\(={bh^{3}}/{12}\) second moment of the beam cross-section

\(I_0 ,\,I_1 \) :

Moment of inertia at the small end and the large end, respectively

b :

Beam width

h :

Beam thickness

\(L_1 \) :

Beam length

\(b_1 ,\,b_0 \) :

The width of beam at the large end and the small end, respectively

\(h_1 ,\,h_0 \) :

The thickness of beam at the large end and the small end, respectively

\(A_1 \) :

\(=b_1 h_1 ,\) cross-section area of beam at the large end

\(A_0 \) :

\(=b_0 h_0 \) cross-section area of beam at the small end

\(\theta \) :

The angle formed by the tangent from neutral axis to the x axis

\(\tilde{\varphi }\left( s \right) \) :

Single deflection mode

\(q\left( t \right) \) :

An unknown time modulation of the assumed deflection mode

V :

The potential energy of the system

T :

The kinetic energy

\(\kappa \) :

Curvature of the beam neutral axis

L :

The Lagrangian function of the beam

\(\xi \) :

\(=s/{L_1 }\)

\(\varphi \left( \xi \right) \) :

The non-dimensional deflection mode

\(C_1 -C_4 \) :

Arbitrary constants to be determined by Rayleigh-Ritz method

\(\alpha \) :

Taper ratio of the beam

\(\tilde{F}_f \) :

The reaction of the elastic foundation on the beam

\(\tilde{k}_L ,\, \tilde{k}_{\mathrm{NL}} \) :

Linear and nonlinear elastic foundation coefficients, respectively

\(\tilde{k}_s \) :

The coefficient of shear stiffness of the elastic foundation

\(\tilde{\beta }_i ,i=0,1,2,3,4,5\) :

The coefficients of the Lagrangian function of the beam

\(\beta _i ,i=0,1,2,3,4,5\) :

The dimensionless coefficients in Eq. (19)

\(\Theta ,k_s , k_L ,k_{\textit{NL}} \) :

The dimensionless coefficients in Eq. (27)

\(\varepsilon _i ,i=1,2,3,4,5\) :

the dimensionless coefficients in Eq. (31)

\(\tau \) :

\(=t\sqrt{\Omega \Theta }\)

\(\omega \) :

\(=\sqrt{\Omega },\) the dimensionless frequency of nonlinear oscillation

\(\varpi _{\mathrm{La}} , \varpi _L \) :

The approximate and exact fundamental frequency, respectively

\(\omega _{\mathrm{La}} , \omega _L \) :

The dimensionless approximate and exact frequency, respectively

a :

The amplitude of the motion

\(q_i , \omega _i , \Omega _i\), i=1,2 :

The \(i\hbox {th}\) analytical approximation to q,\(\omega \),\(\Omega \)

\(\Delta q_1 ,\, \Delta \Omega _1 \) :

The corrections to \(q_1 (\tau )\) and \(\Omega _1 \left( a \right) \), respectively

\(z_1 \) :

Coefficient to be determined in method of harmonic balance

\(\omega _{\mathrm{e}}\) :

The exact frequency obtained by the improved shooting method

\(\omega _N \) :

The nonlinear vibration frequency of the beam

\(\omega _{\mathrm{LGA}}, \omega _{\mathrm{LGE}}, \omega _{\mathrm{LRR}}\) :

Experimental results from Georgian [29], and the results obtained by Rao and Rao [31], respectively

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Acknowledgements

The work described in this paper is supported by the Science and Technology Developing Plan Project of Jilin Province (Grant No. 20160520021JH), the National Natural Science Foundation of China (Grant No. 11402095), and Innovative Project of Scientific Forefront and Interdisciplinary of Jilin University (Grant No. 2013ZY14).

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Authors and Affiliations

  1. College of Construction Engineering, Jilin University, Changchun, 130021, People’s Republic of China

    Yongping Yu, Hongzhi Zhang & Youhong Sun

  2. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Weipeng Sun

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  1. Yongping Yu
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Correspondence to Weipeng Sun.

Appendix

Appendix

The coefficients of \(B_i \) and \(D_i \left( {i=0,1,2} \right) \) are given by

$$\begin{aligned} B_0= & {} 16a\varepsilon _1 +12\varepsilon _3 a^{3}+10\varepsilon _5 a^{5}-\Omega _1 \left( {16a+8\varepsilon _2 a^{3}+6\varepsilon _4 a^{5}} \right) \\ B_1= & {} -\left( {16a+8\varepsilon _2 a^{3}+6\varepsilon _4 a^{5}} \right) \\ B_2= & {} 16\varepsilon _1 +24\varepsilon _3 a^{2}+25\varepsilon _5 a^{4}+\Omega _1 \left( {-16+5\varepsilon _4 a^{4}} \right) \\ D_0= & {} 4\varepsilon _3 a^{3}+5\varepsilon _5 a^{5}-\Omega _1 \left( {8\varepsilon _2 a^{3}+7\varepsilon _4 a^{5}} \right) \\ D_1= & {} -\left( {8\varepsilon _2 a^{3}+7\varepsilon _4 a^{5}} \right) \\ D_2= & {} -16\varepsilon _1 -12\varepsilon _3 a^{2}-5\varepsilon _5 a^{4}+\Omega _1 \left( {144+56\varepsilon _2 a^{2}+31\varepsilon _4 a^{4}} \right) \end{aligned}$$

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Yu, Y., Zhang, H., Sun, Y. et al. Predicting dynamic response of large amplitude free vibrations of cantilever tapered beams on a nonlinear elastic foundation. Arch Appl Mech 87, 751–765 (2017). https://doi.org/10.1007/s00419-016-1221-x

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