Abstract

The longitudinal vibrations of an elastic rod controlled by a distributed force, which is applied to individual sections of the rod, are studied. It is assumed that the force varies in space in a piecewise constant manner. Such a mechanical system can be implemented using piezoactuators attached along the rod. The dynamics of the system is determined from the solution of the variational problem following the method of integrodifferential relations. The variational problem is solved analytically. To do this, traveling waves of the d’Alembert type are introduced on the space-time mesh, which determine continuous displacements and a dynamic potential. The latter relates the momentum density and stresses. A control problem is posed under the condition of the weighted minimization of the vibrational energy stored by the rod at the terminal time instant, and the mean potential energy generated by the control actions. The extremal motion and the corresponding control law are found explicitly by solving the Euler–Lagrange equations. As an example, the control capabilities for certain configurations of piezoelectric elements are studied.

中文翻译：

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使用周期性分布压电力优化弹性杆的纵向运动

摘要

研究了由施加到杆的各个部分的分布力控制的弹性杆的纵向振动。假设力在空间中以分段恒定的方式变化。这种机械系统可以使用沿杆附接的压电致动器来实现。系统的动力学是通过积分微分关系方法解决变分问题来确定的。变分问题通过解析求解。为此，在时空网格上引入达朗贝尔类型的行波，以确定连续位移和动态势。后者与动量密度和应力有关。在终端时刻杆所存储的振动能量和控制动作产生的平均势能加权最小化的情况下提出控制问题。通过求解欧拉-拉格朗日方程可以明确地找到极值运动和相应的控制律。例如，研究了压电元件某些配置的控制能力。