考虑有限深度土体运动的Winkler地基梁1/2次谐波共振响应分析

1/2 sub-harmonic resonance of beams on Winkler foundation with consideration of finite-depth soil motion

土-结构相互作用系统动力响应的基本特征之一是有限范围内弹性地基与其支承结构共同运动,将土体运动引入系统的动力学方程可体现其对系统动力学特性的影响。基于考虑有限深度土体运动影响的Winkler地基上有限长梁的非线性运动方程,利用Galerkin法和多尺度法,求得弹性地基梁1/2次谐波共振的幅频响应方程和位移的二阶近似解。进而通过数值计算,得到了梁1/2次谐波共振的幅频响应曲线,研究了地基深度、质量、弹性模量、Winkler参数和阻尼等对弹性地基梁1/2次谐波共振响应的影响。研究结果表明:有限深度土体运动对Winkler地基梁1/2次谐波共振响应影响显著。运动方程中引入土体运动的影响后,梁1/2次谐波共振区间明显减小。随地基深度、质量和弹性模量改变,弹性地基梁1/2次谐波共振的幅频响应曲线偏转程度、共振区间和响应幅值等均发生定量改变。当弹性地基刚度增大到一定程度,Winkler地基参数变化对系统1/2次谐波共振响应的影响明显减弱。阻尼对系统动力响应起抑制作用,当参数η增大到一定值后将不会出现1/2次谐波共振响应的非平凡解。

One of the essential characteristics of the dynamic response of the soil-structure interaction system is that the elastic foundation and its supporting structure move together in a limited range.The dynamic equations that introduce soil motion into the system can reflect its effect on the system’s dynamic characteristics.The nonlinear equation of motion of the finite-length beam on Winkler foundation were established,with the consideration of finite-depth soil mass motion.The Galerkin method and the multiple scales method were employed to obtain the frequency response equation and the second-order approximation solution of the 1/2 order sub-harmonic resonance of the beam.Then,through numerical calculation,the frequency-response curves of the 1/2 order sub-harmonic resonance of the beam were obtained.Meanwhile,the effects of foundation depth,soil mass,elastic modulus,Winkler parameter,and damping on the 1/2 order sub-harmonic resonance response of beams on elastic foundation were investigated.The results show that the effect of the finite-depth soil mass motion on the 1/2 order sub-harmonic resonance of the beam on Winkler foundation was significant.When the influence of soil mass motion is introduced into the equation of motion of the beam,the range of 1/2 order sub-harmonic resonance of the system is significantly reduced.With the change of the depth,mass,and elastic modulus of foundation,there are some quantitatively changes.Those changes are observed in the frequency-response curves of the 1/2 order sub-harmonic resonance of beam on elastic foundation,such as the degree of deflection,resonance range,and response amplitude.When the stiffness of the elastic foundation increase to a specific value,the effect of changes in the Winkler foundation parameters on the system’s 1/2 order sub-harmonic resonance response is significantly weakened.Because the damping inhibites the dynamic response of the system,when the parameterηincreases to a specific value,the non-trivial solution of the 1/2 order sub-harmonic resonance will not appear.