Deterministic and Random Response Evaluation of a Straight Beam with Nonlinear Boundary Conditions

Abstract

Purpose

Deterministic and random responses of one-dimensional continuous structures with nonlinear boundary conditions are evaluated in this article. Dynamic behaviors (i.e., the time-domain response for the deterministic case and mean-square response for the stochastic case) of a straight beam with a cubic nonlinear elastic boundary is investigated numerically.

Methods

The nonlinear vibration problem is discretized by a new global spatial discretization method, and its responses are compared with those from the finite element method and assumed mode method.

Results

The effects of selected various types of linear homogeneous boundary conditions and the number of expansion modes on the accuracy of the responses under harmonic excitation are discussed. Detailed discussions on the influence of nonlinear boundary conditions on dynamic behaviors uncover a counter-intuitive phenomenon: with the increase of the nonlinear spring stiffness, the stationary mean-square displacement of the mid-point descends first and then ascends slightly.

Conclusion

Numerical results show that the present procedure possesses high precision for evaluating harmonic responses for both low-frequency and high-frequency excitations. Furthermore, it can be directly generalized to two-dimensional continuous systems and various types of nonlinear boundary conditions.

Appendices

Appendix 1: Nondimensionalization of the Beam Equation

Considering the straight beam, the equation of motion and boundary and initial conditions are given by

$$ \begin{aligned}& \rho U_{TT} + C_{1} U_{T} + C_{2} U_{XXXXT} + EIU_{XXXX} = F(X,\;T)\quad \\&0 \le X \le L,\;\;T \ge 0 \end{aligned} $$

(44)

$$ \left\{ \begin{aligned}& U(0,\;T) = 0,\quad U_{X} (0,\;T) = 0, \hfill \\& U_{XX} (L,\;T) = 0,\;\;U_{XXX} (L,\;T) = K_{l} U(L,\;T) + K_{\text{nl}} U(L,\;T)^{3} \hfill \\ \end{aligned} \right. $$

(45)

$$ U(X,\;0) = G(X),\;\;U_{T} (X,\;0) = H(X), $$

(46)

where $U(X,\;T)$ denotes the displacement at the reference position X and time instant T; $\rho ,\;E,\;I$ denote the linear density, Young’s moduli and moment of inertia, respectively; C₁, C₂ denote external and internal damping coefficients, respectively, $K_{\text{l}} ,\;K_{\text{nl}}$ denote linear and nonlinear stiffness coefficients, respectively; and $G(X),\;H(X)$ denote the initial displacement and velocity, respectively.

By introducing the dimensionless quantities as follows:

$$ x = \frac{X}{L},\;\;u = \frac{U}{L},\;\;t = \frac{T}{\Delta T},\;\;c_{1} = \frac{{C_{1} \Delta T}}{\rho },\;\;c_{2} = \frac{{C_{2} \Delta T}}{{\rho L^{4} }},\;\;f = \frac{{F\Delta T^{2} }}{\rho L} $$

$$ \Delta T = \sqrt {\frac{{\rho L^{4} }}{EI}} ,\;\;k_{1} = \frac{{K_{1} L^{3} }}{EI},\;\;k_{\text{nl}} = \frac{{K_{\text{nl}} L^{5} }}{EI},\;\;h = \frac{H}{L},\,\;g = \frac{G}{L}, $$

(47)

we obtain the dimensionless expression as shown in Eqs. (12)–(14).

Appendix 2: Exact Solutions of the Clamped-Free Beam

Some exact solutions to the response of the clamped-free beam under deterministic and random excitations are summarized in this appendix. The governing equation and boundary conditions of the clamped-free beam are described by Eqs. (12)–(14) with linear and nonlinear spring stiffness vanishing. To obtain response under deterministic excitations, exact eigenfunction expansion is used [8]:

$$ u(x,\;t) = \sum\limits_{i = 1}^{N} {\varphi_{i} } (x)q_{i} (t), $$

(48)

where $\varphi_{i} (x)$ are eigenfunctions of the clamped-free beam, and $q_{i} (t)$ are the time-dependent modal coordinates. Note that the expression in Eq. (48) is the same as that used in the AMM. Substituting the expression into the governing equation and applying Galerkin’s method, one obtains N decoupled second-order ordinary differential equations as follows:

$$ \ddot{q}_{i} + 2\xi_{i} \omega_{i} \dot{q}_{i} + \omega_{i}^{2} q_{i} = f_{i} ,\quad i = 1, \ldots N, $$

(49)

where $\xi_{i} = \frac{1}{2}\left( {\frac{{c_{1} }}{{\omega_{i} }} + c_{2} \omega_{i} } \right)$ is the ith equivalent damping coefficient, $\omega_{i}$ is the ith natural frequency, and f_i is defined as in Eq. (34).

When the clamped-free beam is subject to free vibration, i.e., $f(x,\;t) = 0$, the solution is

$$ u(x,\;t) = \sum\limits_{i = 1}^{N} {A_{i} } \varphi_{i} (x)\sin (2\pi v_{i} t + \phi_{i} ), $$

(50)

where $\nu_{i}$ is the natural frequency given in Table 1, and $A_{i} ,\;\phi_{i}$ are the amplitude and initial phase determined by initial conditions, respectively.

When the clamped-free beam is subject to Gaussian white noise $W(t)$ with the spectra density K, the exact stationary probability density function of Eq. (49) is given as follows [4]:

$$ |p(q_{i} ) = C\exp \left\{ { - \frac{{\xi_{i} \omega_{i}^{3} }}{{\pi K\varphi_{i} (x_{\text{e}} )^{2} }}q_{i}^{2} } \right\}, $$

(51)

where C denotes the normalization constant. Thus the probability density of the corresponding mode $u(x,\;t) = \sum\nolimits_{i = 1}^{N} {\varphi_{i} (x)q_{i} (t)}$ is given by

$$ p(u_{i} ) = C\exp \left\{ { - \frac{{\xi_{i} \omega_{i}^{3} }}{{\pi K\varphi_{i} (x_{\text{s}} )^{2} \varphi_{i} (x)^{2} }}u_{i}^{2} } \right\}. $$

(52)

According to the basic probability theory that the probability density function of the sum of two random variables is the convolution of the probability density functions of those two random variables, e.g., when three modes are expanded $N = 3$:

$$ p(u) = C\exp \left\{ { - \frac{{s_{1} s_{2} s_{3} }}{{s_{1} s_{2} + s_{1} s_{3} + s_{2} s_{3} }}u^{2} } \right\}, $$

(53)

where

$$ s_{i} = \frac{{\xi_{i} \omega_{i}^{3} }}{{\pi K\varphi_{i} (x_{\text{e}} )^{2} \varphi_{i} (x)^{2} }},\quad i = 1,\;2,\;3. $$

(54)