Nonlinear dynamic characteristics of a vibrator–ground system considering surface topography

Abstract

Nonlinear interaction between the vibrator and ground is important for the vibrator–ground system which is related to structural design, diagnosis and prognosis of the vibrator. Thus, it is necessary to model the vibrator–ground interaction system reasonably and study its dynamic response. In this paper, a nonlinear dynamic model of the vibrator–ground system with consideration of surface topographies is developed and the correlation of surface roughness to the soil–baseplate interaction characteristics is revealed. The developed model is verified by comparing simulation results with field tests. Based on this model, some useful nonlinear dynamic characteristics of the vibrator are analyzed: (a) the baseplate vibration behaviors; (b) the contact uniformity; (c) the ground force. Moreover, a parametric analysis is conducted and the influences of several system parameters are discussed in details. The results indicate that the surface topography has great influence on the vibrator–ground interaction and further, affects the dynamic response of the vibrator. This research can provide references for the operation and optimization of the vibrator.

Appendices

Appendix A: Solution of the nonlinear damping

The internal damping at the interfaces is considered as a constant value $c_{i}$, and the value of the damping ratio $\zeta_{i} = {{c_{i} } \mathord{\left/ {\vphantom {{c_{i} } {2\sqrt {m_{2} n_{1} k_{1} z_{s}^{{n_{1} - 1}} } }}} \right. \kern-0pt} {2\sqrt {m_{2} n_{1} k_{1} z_{s}^{{n_{1} - 1}} } }}$ is in the range of 0.01 to 0.1. For circular contact area, the radiation damping can be expressed as $c = {{3.4r^{2} \sqrt {{{E\rho } \mathord{\left/ {\vphantom {{E\rho } {2(1 + \nu )}}} \right. \kern-0pt} {2(1 + \nu )}}} } \mathord{\left/ {\vphantom {{3.4r^{2} \sqrt {{{E\rho } \mathord{\left/ {\vphantom {{E\rho } {2(1 + \nu )}}} \right. \kern-0pt} {2(1 + \nu )}}} } {(1 - \nu )}}} \right. \kern-0pt} {(1 - \nu )}}$, where $r$ is the radius of the circular loading area, E and $\nu$ are, respectively, the elastic modulus and Poisson’s ratio of the soil, $\rho$ is the mass density of the soil. In this study, the contact domain is discretized by square elements, the equivalent radius of the load area can be obtained by $r = 0.55\sqrt {s_{b} }$, where $s_{b} { = 2}a \times 2b$ is the area of a square element. Hence, the radiation damping of the whole contact domain can be obtained by

$$ c_{r} = \int_{\Omega } {\frac{{0.85s_{b} \sqrt {{{E\rho } \mathord{\left/ {\vphantom {{E\rho } {2\left( {1 + \nu } \right)}}} \right. \kern-0pt} {2\left( {1 + \nu } \right)}}} }}{{\left( {1 - \nu } \right)}}} dn_{k} $$

(A.1)

where n_k is the contact point.

Appendix B: Equations of the baseplate motion

The kinetic equation of the baseplate on the rough earth under harmonic excitation is given by

$$ m_{2} \ddot{z}_{2} + \left( {c_{2} \left( {z_{s} + z_{2} } \right)^{{n_{2} }} { + }c_{i} } \right)\dot{z}_{2} + k\left( {z_{s} + z_{2} } \right)^{{n_{1} }} - F_{t} = A\sin \left( {\omega t} \right) $$

(B.1)

where $\omega$ is the frequency.

It needs to mention that Eq. (B.1) is only valid when the two contact surfaces are in contact with each other, i.e., when $z_{2} \ge z_{s}$. By using the following non-dimensional variables, the equation can be transformed into normalized form, which are given by

$$ x_{2} = \frac{{z_{2} }}{{z_{s} }}, \, \Omega = \frac{\omega }{{\omega_{s} }}, \, \tau = \frac{{\omega_{s} }}{t}, \, \zeta_{r} = \frac{{c_{2} z_{s}^{{n_{2} }} }}{{2m_{2} \omega_{s} }}, \, Y = {F \mathord{\left/ {\vphantom {F {F_{t} }}} \right. \kern-0pt} {F_{t} }} $$

(B.2)

where the static deflection given by $z_{s} = \left( {{{F_{t} } \mathord{\left/ {\vphantom {{F_{t} } k}} \right. \kern-0pt} k}} \right)^{{{1 \mathord{\left/ {\vphantom {1 {n_{1} }}} \right. \kern-0pt} {n_{1} }}}}$, $\omega_{s} = \sqrt {{{kn_{1} z_{s}^{{n_{1} - 1}} } \mathord{\left/ {\vphantom {{kn_{1} z_{s}^{{n_{1} - 1}} } {m_{2} }}} \right. \kern-0pt} {m_{2} }}}$ is the undamped natural frequency at static equilibrium position. The dimensionless equation is subsequently obtained by

$$ \ddot{x}_{2} + 2\left[ {\zeta_{r} \left( {x_{2} + 1} \right)^{{n_{2} }} + \zeta_{i} } \right]\dot{x}_{2} + \frac{1}{{n_{1} }}\left[ {\left( {x_{2} + 1} \right)^{{n_{1} }} - 1} \right] = \frac{Y}{{n_{1} }}\sin \left( {\Omega \tau } \right) $$

(B.3)

The solution of Eq. (B.1) is difficult to resolved directly, so the appropriate solution can be obtained when the expressions $\left( {x_{2} + 1} \right)^{{n_{1} }}$ and $\left( {x_{2} + 1} \right)^{{n_{2} }}$ are expanded into a third-order Taylor series expansion, and then, the equation is reduced as

$$ \ddot{x}_{2} + 2\left[ {\zeta_{r} \left( {1 + n_{2} x_{2} + a_{1} x_{2}^{2} + a_{3} x_{2}^{3} } \right) + \zeta_{i} } \right]\dot{x}_{2} + x_{2} + b_{1} x_{2}^{2} + b_{2} x_{2}^{3} = \frac{Y}{{n_{1} }}\sin \left( {\Omega \tau } \right) $$

(B.4)

where $a_{1} = \frac{{n_{2} \left( {n_{2} - 1} \right)}}{2}$, $a_{2} = \frac{{n_{2} \left( {n_{2} - 1} \right)\left( {n{}_{2} - 2} \right)}}{6}$, $b_{1} = \frac{{n_{1} - 1}}{2}$, $b_{2} = \frac{{\left( {n_{1} - 1} \right)\left( {n_{2} - 2} \right)}}{6}$.

To obtain the undamped free vibration of the baseplate, the damping terms in Eq. (B.3) are set as null. The approximate analytical expression for natural frequency of the baseplate-ground system is solved using multiple scales method. For the initial condition of $x_{2} (0) = x_{20}$, $\dot{x}_{2} (0) = 0$, the nondimensional natural frequency $\Omega_{n}$ is written as

$$ \Omega_{n} = 1 + \left( {\frac{{9b_{2} - 10b_{1}^{2} }}{24}} \right)\left( {1 + \frac{{b_{1} }}{3}x_{20} } \right)^{2} x_{20}^{2} $$

(B.5)