## 1 Introduction
From the perspective of concurrent design principles, the pre-search and development of dynamic systems for both motor and trailer vehicles must also consider the issue of strong/weak coupling interfaces. The structural design of the service car body system must ensure the 30-year service life of the aluminum alloy car body, and the safety of ESL (Equivalent Static Load) remains an open topic. It is assumed that the ESL sets are equivalent to the load cases specified by the industry codes or standards under some steady states or quasi-static states. The ESL sets are defined as the loads that produce the same displacements at each time step by static analysis as the ones provided by the transient analysis. The transient analysis, as shown in Equation (4), is based on the basic equations of the HSRS rigid-flex coupling system [16; 17]. \[\mathbf{M}\ddot{\xi}+\left(\dot{\mathbf{M}}-\frac{1}{2}\left[\frac{\partial\mathbf{M}}{ \partial\xi}\dot{\xi}\right]^{T}+\mathbf{D}\right)\dot{\xi}+\mathbf{K}\xi+f_{g}+\left[ \frac{\partial\mathbf{C}}{\partial\xi}\right]^{T}\mu=\mathbf{Q} \tag{4}\]  
where \(\xi=[q\quad\overline{q}]^{T}\), spanning across the generalized and modal spaces, i.e., MBS's generalized space variables \(q\) and flexible body's elastic modal space variables \(\overline{q}\). \(\mathbf{M}\) is the mass matrix of MBS including flexible bodies (e.g., the hypothesis of nine invariants in some analysis software such as ADAMS 2005), which is non-symmetric, non-positive definite, and non-invertible; \(\mathbf{K}\) and \(\mathbf{D}\) are the stiffness and damping matrices of the coupling system, respectively; \(f_{g}\) is the gravity obtained through the gravitational potential energies of (rigid- and flex-) entities by the partial derivation of \(\xi\); \(\mathbf{Q}\) is the external resultant force; \(\mu\) is an undetermined factor, which can form the constrained inner force term including Coriolis forces by multiplying the transposition of formula (5). \[C_{q}\equiv\frac{\partial\mathbf{C}}{\partial\xi} \tag{5}\]  
Applying the strong/weak duality problem of convex optimization, the literature [18; 19] believes that the structural dynamic response design based on ESL is not always convergent to obtain the optimum solution. However, the weak duality holds generally. If \(p^{*}\) is defined as the primal solution, and \(d^{*}\) as the dual solution, then \(d^{*}\leq p^{*}\) for the we ak duality. The quantity \(p^{*}-d^{*}\) is known as the duality gap, which can be a useful criteria for convergence. Unfortunately, for the strong dual solution, there is no similar criterion to ensure convergence. If we ignore the application premise of DVA techniques and even use the multiple DVA technical devices, a strong coupling interface will be formed between the equipment cabin and the aluminum alloy car body. Therefore, the aluminum alloy car body does not only need to be reinforced repeatedly, leading to exceed the empty axle load limit, but also causes the equipment cabin to have bolt loosen failure or skirt bracket cracking accidents. For this reason, the transaction strategy of flexible body to MBS interfaces is vital to the dynamic response design optimization of lightweight structures. If the couplings of flexible body to MBS interfaces are completely ignored, some analysis software [20] can only give the motion coupling relationship, completely losing the accurate analysis results of the complex constraint inner forces. For instances of two-axle freight wagons, the underframe is usually distorted and vibrated due to the roll and rock of the service car body [21]. If the rigidity of both end crossbeams is insufficient, vibration fatigue will occur between the traction beam and the end wall. Another example is that the long flat car used for the transportation of natural gas integrated tanks is reconstructed into a two-box container flat car, and the reinforcing structures are added to the longitudinal beams on both sides to reduce the deflection-span ratio.