An Integrated Lagrangian Modeling Method for Mechanical Systems with Memory Elements

Mechanical memory elements cannot be accurately modeled using the Lagrangian method in the classical sense, since these elements are nonconservative in the plane of their non-constitutive relationships, and the system differential equations are not self-adjoint and therefore do not allow a Lagrangian formulation. To overcome this problem, the integrated Lagrangian modeling method is introduced, in which the associated conventional energies in the system are replaced by the corresponding memory state functions of the memory elements. An example, a vehicle shimmy system equipped with fluid mem-inerters, is presented to verify the improvement of modeling accuracy of mechanical systems with memory elements via the integrated Lagrangian method. The simulation results show that under pulse and random excitation, using the Lagrangian method to model the system, the values of system response indicators exhibit significant errors ranging from $5.17\%$ to $24.54\%$ compared with the values obtained by the integrated Lagrangian method, namely, the accurate values. In addition, the influencing factors of the error and are discussed and the fractional-order memory elements and their modeling are also briefly generalized.