Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams with Spectral Chebyshev Technique and Harmonic Balance Method

Abstract

In this paper, nonlinear forced vibrations of uniform and functionally graded Euler-Bernoulli beams with large deformation are studied. Spectral and temporal boundary value problems of beam vibrations do not always have closed-form analytical solutions. As a result, many approximate methods are used to obtain the solution by discretizing the spatial problem. Spectral Chebyshev technique (SCT) utilizes the Chebyshev polynomials for spatial discretization and applies Galerkin’s method to obtain boundary conditions and spatially discretized equations of motions. Boundary conditions are imposed using basis recombination into the problem, and as a result of this, the solution can be obtained to any linear boundary condition without the need for re-derivation. System matrices are generated with the SCT, and natural frequencies and mode shapes are obtained by eigenvalue problem solution. Harmonic balance method (HBM) is used to solve nonlinear equation of motion in frequency domain, with large deformation nonlinearity. As a result, a generic method is constructed to solve nonlinear vibrations of uniform and functionally graded beams in frequency domain, subjected to different boundary conditions.

Appendix

18.1.1 Calculation of Inner Product Matrix

Values of any two functions f(x) and g(x) at N Gauss-Lobatto points are written as f_N and g_N. Product of interpolated functions has order of 2 N.

$$ {\mathbf{f}}_{2N}={\mathbf{S}}_2{\mathbf{f}}_N\kern8.75em $$

(A.1)

S2 is constructed as follows:

$$ {\mathbf{S}}_{2\mathrm{N}}={\boldsymbol{\Gamma}}_{{\mathrm{B}}_{2\mathrm{N}}}\left[{\mathbf{I}}_{\mathrm{N}};{\mathbf{O}}_{\mathrm{N}}\right]\ {\boldsymbol{\Gamma}}_{{\mathrm{F}}_{\mathrm{N}}}\kern6.25em $$

(A.2)

Here${\boldsymbol{\Gamma}}_{{\mathrm{B}}_{2\mathrm{N}}}\ i$s the 2 N x 2 N backward transformation matrix. I_N and O_N are the N x N dimensional identity and zero matrices. The inner product of f(x) and g(x) can be written as follows:

$$ {\int}_{l_1}^{l_2}\mathrm{f}(x)\mathrm{g}(x) dx={\mathbf{f}}^T\mathbf{V}\ \mathbf{g}={\mathbf{f}}_{2N}^{\boldsymbol{T}}{\mathbf{v}}_{d,2N}\ {\boldsymbol{g}}_{2N}\kern2.75em $$

(A.3)

Here v_{d, 2N}is a matrix whose diagonal has the elements of multiplication ${\mathbf{v}}_{2\mathrm{N}}^{\boldsymbol{T}}{\boldsymbol{\Gamma}}_{{\mathrm{F}}_{2\mathrm{N}}}$. Then the inner product matrix is written as follows:

$$ \mathbf{V}={\mathbf{S}}_{2N}^{\boldsymbol{T}}{\boldsymbol{v}}_{d,2N}\ {\mathbf{S}}_2\kern3.5em $$

(A.4)

When the differential equation has variable coefficients, a weighted inner product is defined with respect to a weighting function γ(x). In the problem given in case study, γ(x) is the variation of the Young modulus distribution, E(x). Since there is a weighting function, the inner product has order of 3 N. Consequently, the inner product and inner product matrix can be described as follows:

$$ {\int}_{l_1}^{l_2}f(x)\mathrm{g}(x)E(x) dx={\mathbf{f}}_{3N}^{\boldsymbol{T}}{\mathbf{V}}_{\mathbf{E}}\ {\mathbf{g}}_{\boldsymbol{N}}\kern0.75em \vspace*{-1pc}$$

(A.5)

$$ {\mathbf{V}}_{\mathbf{E}}={\mathbf{S}}_{3N}^{\boldsymbol{T}}{\mathbf{v}}_{d,3N}\ {\mathbf{E}}_{d,3N}{\mathbf{S}}_3 $$

(A.6)

where v_d,3n and E_d,3N are 3 N x 3 N matrices whose diagonals have the values of f_3N and E_3N

The first and second derivative (with respect to x) of E(x) are E′(x) and E″(x). While finding the inner product matrix as described above, if E′(x) is used, then V_E′(x) is obtained. Similarly, if E″(x) is used, then V_E″(x) is obtained.