Abstract Background <p>Traditionally, mechanical memory elements are difficult to model accurately using the classical Lagrangian method due to their nonconservative behavior in the constitutive planes. To address this issue, Biolek et al. proposed a generalized Lagrangian modeling method in the field of electrical engineering. However, this approach has gained limited acceptance in mechanical engineering. </p> Methods <p>In contrast, this study demonstrates that the classical Lagrangian framework can be extended to model mechanical systems with memory elements through appropriate modifications. Instead of embedding displacement-dependent memory effects into energy state functions, the proposed method preserves the classical Lagrangian formulation for the linear conservative subsystem, while the contributions of memory elements are explicitly incorporated through their constitutive force relations within the same generalized-coordinate framework. Then, a vehicle shimmy system equipped with multiple memory elements (diamond-structured mem-springs and fluid-type mem-inerters) is used as a case study to evaluate response errors and influencing factors of the classical Lagrangian model under pulse and random excitations.</p> Conclusion <p> The results show that the classical Lagrangian model yields significant errors ranging from 1.39% to 20.25% in various response indicators compared to the improved Lagrangian method, further confirming the necessity of the proposed approach for modeling shimmy systems with memory elements.</p>