矩形薄板瞬态响应的卷积型DQ半解析法

Convolution-Type Semi-Analytic DQ Approach for Transient Response of Rectangular Plates

卷积型的Gurtin变分原理是目前在数学上唯一能和动力学初值问题完全等价的变分原理,它完全反映了有关初值问题的全部特征.通过卷积将矩形薄板原始控制方程构造成包含初始条件的新的具有完整初值问题特征的控制方程.对新的控制方程在时间域取解析函数,在空间域采用离散的DQ(differential quadrature)法,从而构造了卷积型DQ半解析法.该方法既可以达到和Gurtin变分原理相同的效果,又避开了Gurtin泛函的繁复.经对矩形薄板的动力响应问题的计算表明,该方法是一种精度好效率高的求解动力响应问题的计算方法.

The convolution-type Gurtin variational principle is known as the only variational principle, that is, from mathematical point of view, totally equivalent to the initial value problem system. The equation of motion of rectangular thin plates was first transformed to a new governing equation containing initial conditions by using convolution method. A convolution-type semi-analytical DQ approach, which involves differential quadrature （DQ） approximation in space domain and an analytical series expansion in time domain, was proposed to obtain the transient response solution. This approach offers the same advantages as Gurtin variational principle and at the same time, is much simpler in the calculation. Numerical results show that it is very accurate, yet computationally efficient for the dynamic response of plates.