A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load

Abstract

A differential quadrature procedure is proposed to study the steady state linear and nonlinear vibrations of an infinite beam resting on an elastic Winkler foundation and subjected to a moving point load. The governing nonlinear partial differential equation of motion of the beam is first expressed with respect to a moving coordinate system. This step reduces the governing nonlinear partial differential equation of motion of the beam to an ordinary nonlinear differential equation. This equation is then converted to a set of nonlinear algebraic equations by application of the differential quadrature method. The Newton–Raphson method is used to solve the resultant system of nonlinear algebraic equations. Issues related to implementation of infinite boundary conditions and modeling the point load are addressed. To accurately predict the dynamic behavior of the beam at high speeds of the moving point load, an efficient and robust absorbing boundary condition is also introduced. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with available analytical solutions in the literature is performed. Numerical results reveal that the proposed procedure can be used as an effective tool for handling nonlinear moving load problems on infinite domains.

Appendices

Appendix 1: Discretization of the Dirac-delta function using the coupled DQM/IQM approach

Consider n grid points with coordinates η ₁, η ₂,…,η _n in the η-direction. Let

$$ F(\eta ) = \delta (\eta - \eta_{0} ) $$

(30)

$$ \eta_{k - 1} \le \eta_{0} \le \eta_{k} , \, 2 \le k \le n $$

(31)

The Dirac-delta function has the following properties

$$ \delta \left( {\eta - \eta_{0} } \right) = 0,{\text{ for all }}\eta \ne \eta_{0} $$

(32)

$$ \int\limits_{ - \infty }^{ + \infty } {\delta (\eta - \eta_{0} )} d\eta = 1 $$

(33)

It is possible to represent the source point (30) by two statically equivalent source points at the grid points located near the source acting point. Thus, Eq. (30) can be rewritten as [37, 38]

$$ F(\eta ) = \delta (\eta - \eta_{0} ) = \frac{{\eta_{R} }}{{\eta_{R} + \eta_{L} }}\delta (\eta - \eta_{k - 1} ) + \frac{{\eta_{L} }}{{\eta_{R} + \eta_{L} }}\delta (\eta - \eta_{k} ), \eta_{L} = \eta_{0} - \eta_{k - 1} ,\eta_{R} = \eta_{k} - \eta_{0} $$

(34)

In view of Eq. (32), Eq. (34) can be written as

$$ F\left( {\eta_{i} } \right) = 0, \quad i = 1, 2, \ldots , k - 2, \quad k + 1, k + 2, \ldots , n $$

(35)

Integrating Eq. (34) numerically and using Eq. (33) we obtain

$$ F(\eta_{k - 1} ) = \frac{1}{{W_{k - 1} }}\frac{{\eta_{R} }}{{\eta_{R} + \eta_{L} }}, \quad F(\eta_{k} ) = \frac{1}{{W_{k} }}\frac{{\eta_{L} }}{{\eta_{R} + \eta_{L} }} $$

(36)

where W _k-1 and W _k are the integral quadrature weights correspond to the grid points η _k-1 and η _k , respectively. When the iterated trapezoidal rule is employed to integrate Eq. (34) numerically, these weights are obtained as

$$ W_{k - 1} = \frac{1}{2}(\eta_{k} - \eta_{k - 2} ) \quad W_{k} = \frac{1}{2}(\eta_{k + 1} - \eta_{k - 1} ) $$

(37)

Appendix 2: Implementation of boundary conditions of the infinite beam

As pointed out earlier, the approach proposed in Ref. [41] is used herein to implement the boundary conditions of the infinite beam. In this approach (say, MWCM approach), the derivative boundary conditions are applied by Modification of the DQM Weighting Coefficient Matrices. The derivative boundary conditions of the infinite beam are

$$ \frac{dw(\eta )}{d\eta } = 0 \quad {\text{at }}\eta = - L_{\infty } {\text{ and }}\eta = + L_{\infty } $$

(38)

$$ \frac{{d^{2} w(\eta )}}{{d\eta^{2} }} = 0 \quad {\text{at }}\eta = - L_{\infty } {\text{ and }}\eta = + L_{\infty } $$

(39)

$$ \frac{{d^{3} w(\eta )}}{{d\eta^{3} }} = 0 \quad {\text{at }}\eta = - L_{\infty } {\text{ and }}\eta = + L_{\infty } $$

(40)

To implement the derivative boundary condition (38), we define

$$ [\tilde{A}]^{(1)} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & \ldots & 0 \\ {A_{21}^{(1)} } & {A_{22}^{(1)} } & {A_{23}^{(1)} } & \ldots & {A_{2n}^{(1)} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {A_{n - 1,1}^{(1)} } & {A_{n - 1,2}^{(1)} } & {A_{n - 1,3}^{(1)} } & \ldots & {A_{n - 1,n}^{(1)} } \\ 0 & 0 & 0 & \ldots & 0 \\ \end{array} } \right] $$

(41)

The second-order DQM weighting coefficient matrix is then obtained as

$$ [\hat{A}]^{(2)} = [A]^{(1)} [\tilde{A}]^{(1)} $$

(42)

The derivative boundary condition (39) can be implemented if we define

$$ [\tilde{A}]^{(2)} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & \ldots & 0 \\ {\hat{A}_{21}^{(2)} } & {\hat{A}_{22}^{(2)} } & {\hat{A}_{23}^{(2)} } & \ldots & {\hat{A}_{2n}^{(2)} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\hat{A}_{n - 1,1}^{(2)} } & {\hat{A}_{n - 1,2}^{(2)} } & {\hat{A}_{n - 1,3}^{(2)} } & \ldots & {\hat{A}_{n - 1,n}^{(2)} } \\ 0 & 0 & 0 & \ldots & 0 \\ \end{array} } \right] $$

(43)

The DQM weighting coefficient matrix of the third-order is obtained as

$$ [\hat{A}]^{(3)} = [A]^{(1)} [\tilde{A}]^{(2)} $$

(44)

Similarly, the derivative boundary condition (40) is implemented as

$$ [\tilde{A}]^{(3)} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & \ldots & 0 \\ {\hat{A}_{21}^{(3)} } & {\hat{A}_{22}^{(3)} } & {\hat{A}_{23}^{(3)} } & \ldots & {\hat{A}_{2n}^{(3)} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\hat{A}_{n - 1,1}^{(3)} } & {\hat{A}_{n - 1,2}^{(3)} } & {\hat{A}_{n - 1,3}^{(3)} } & \ldots & {\hat{A}_{n - 1,n}^{(3)} } \\ 0 & 0 & 0 & \ldots & 0 \\ \end{array} } \right] $$

(45)

Finally, the fourth-order DQM weighting coefficient matrix is obtained as

$$ [\hat{A}]^{(4)} = [A]^{(1)} [\tilde{A}]^{(3)} $$

(46)

As it can be seen, the implementation of derivative boundary conditions in the MWCM approach is very simple and straightforward.