Hinge clearance in the aircraft rudder system reduces the motion accuracy and stability of the transmission structure, thus affecting the maneuverability, stability and flight safety of the aircraft. Therefore, it is necessary to perform accurate dynamic modeling of rudder systems with hinge clearance and analyze their nonlinear characteristics to guide design and optimization.

The hinge-clearance contact force is modeled using the classical Hertz model and the modified Coulomb model implemented in Automatic Dynamic Analysis of Mechanical Systems (ADAMS). In this study, the rigid–flexible coupled dynamic model of a rudder system with hinge clearance considering a flexible rudder surface was developed using the virtual power principle and the mixed coordinate method. The fixed-step fourth-order Runge–Kutta integral solver was implemented in MATLAB for numerical simulations, and the correctness of the dynamic model was verified by comparing the obtained simulation results with ADAMS dynamic simulation results. Based on the dynamic model, numerical simulation experiments were performed to explore the influence of rigid–flexible coupling on the dynamic characteristics of the rudder system. Based on the theory of nonlinear dynamics, the bifurcation diagrams of a rigid body and a rigid–flexible coupled rudder system under different excitation frequencies were obtained. A specific frequency was selected to construct the phase trajectory diagram and the Poincaré map. The evolution of system dynamics caused by changes in the excitation frequency was analyzed, and the nonlinear coupling effect between the excitation frequency and structural flexibility was explored. The hinge clearance was increased to study its influence on the bifurcation diagrams of the rigid body system.

MATLAB numerical simulation results were overall consistent with the ADAMS dynamic simulation results. When only the first-order natural mode of the flexible rudder surface was considered, the response difference between the rigid–flexible coupled system and the rigid body system was very small. After the introduction of the flexible rudder, mainly the first two natural modes were excited. There was a sudden change in the bifurcation diagram of the rigid rudder system. The phase diagram and Poincaré map showed that the system was in a stable period-1 motion state in the low-frequency band, and that two attractors existed within the relative motion of the clearance. The system gradually transformed into a quasi-periodic motion state near the critical frequency of 21 Hz. Further, increased hinge clearance reduced the critical frequency of the rigid system. The rigid–flexible coupled rudder system resonated due to the excitation of the second-order mode, showing a resonance peak at 27.27 Hz, and the oscillation phenomenon was mitigated in the high-frequency band.

All rigid rudder systems exhibit a critical excitation frequency that is associated with the clearance size. In rudder systems, the presence of a flexible structure reduces stiffness, and a coupling nonlinear effect is observed between the excitation frequency and structural flexibility. These research results clarify the coupling law between the excitation frequency and structural flexibility, which provides a reference for optimizing the structural design of the steering gear and improving the dynamic stability of the rudder system. It is necessary to further explore the coupling nonlinear effects between excitation frequency, structural flexibility and clearance size.

Aerial refueling technology dramatically extends combat radius and flight endurance of fighter aircraft, serving as a crucial guarantee for success in modern warfare. The hose-drogue aerial refueling system has been widely adopted by most nations with aerial refueling capabilities. In aerial refueling operations using the hose-drogue system, the whipping phenomenon caused by excessive slack in the refueling hose is a critical factor that significantly affects both operational success and safety. Proper deployment and retraction of the hose can effectively prevent whipping. Accurate dynamic analysis of the hose during smooth towing and deployment/retraction is essential for ensuring safe and reliable aerial refueling.

The absolute nodal coordinate formulation (ANCF) is a flexible multibody dynamics modeling technique based on finite element theory and continuum mechanics principles. In ANCF, all nodal coordinates are defined in the global coordinate system, replacing the rotational coordinates typically used in conventional finite element methods with slope vectors. This approach not only yields higher accuracy in modeling flexible multibody systems but also performs well in scenarios with large deformations of flexible bodies. For modeling the fuel delivery hoses during steady towing and deployment/retraction, this study develops a dynamic model for variable-length three-dimensional beam elements using the principle of virtual work combined with ANCF. Equivalent stiffness terms and gyroscopic terms, both arising from variations in element length and potentially affecting system stability, are derived. When the element length is constant, the model can be simplified to the traditional ANCF with fixed-length elements. By applying a unified time-dependent function to set the lengths of all elements—where the undeformed lengths undergo simultaneous and identical changes—the method enables dynamic simulation of hoses at arbitrary non-zero lengths while reducing the overall degrees of freedom. Based on Green-Lagrangian strain theory, both axial and bending deformations of the hose are incorporated into the analysis. Additionally, internal damping forces are included via a damping coefficient.

Through a classical benchmark problem from ANCF studies, the validity of the developed dynamic model presented in this work has been rigorously verified. The simulation results demonstrate that under three distinct scenarios-fixed, extended, and shortened element lengths-the proposed dynamic model consistently achieves precise simulation of temporal morphological evolution in flexible hoses. The constant-length element cases are widely adopted as benchmark configurations in existing research, and the proposed model achieves accuracy comparable to previously reported results.

This study offers an effective methodology for creating a precise dynamic model of hose deployment and retrieval during aerial refueling. It provides a high-fidelity model for hose dynamics analysis and whiplash prevention. The model enables dynamic simulation of refueling hoses with arbitrary non-zero lengths. In practical aerial refueling operations, the refueling hose is subjected to various forces including aerodynamic forces, hydraulic forces from the fluid medium, and collision forces generated during docking. Building on the established dynamic model presented in this study, future research can extend the current work by incorporating these multi-physics interactions to conduct comprehensive dynamic simulations and analytical investigations of different phases within the aerial refueling process.

With advancements in modern flight vehicle design, an increased demand for maneuverability is observed, often requiring flight vehicles to operate under conditions involving a high angle of attack and large rudder angle. However, the prevalent linearized model of flight vehicle longitudinal dynamics in aerospace engineering is primarily tailored for launch vehicles, relying on assumptions such as small angle of attack and control surface deflection. Consequently, establishing a linearized model of flight dynamics applicable under conditions of the high angle of attack and large rudder angle is necessary.

This paper uses the small perturbation method to establish a linearized model of the rigid-flexible coupling flight dynamics for flight vehicles, which is then compared and analyzed with the commonly used model in aerospace engineering. Initially, the vector-form flight dynamic equations are projected onto the specified coordinate system, resulting in matrix-form scalar equations. Subsequently, the general linearized form of the dynamic equations in matrix form is derived, and the equation coefficients for the longitudinal dynamic model are obtained. High-precision linearization of the nonlinear expressions for inertia forces and moments owing to control surface deflection is performed, yielding linearized expressions that accurately describe the influence of the perturbation in the angle of attack and rudder angle on these forces and moments. A rigid-flexible coupling multibody system comprising a flexible body and rigid rudder is used as an example to compare simulation results of the linearized model derived in this paper, the traditional linearized model, and the original nonlinear model under the high angle of attack and large rudder angle, verifying the high simulation accuracy of the proposed model.

By comparing the coefficients of the dynamic equations derived in this paper with those in the commonly used model in aerospace engineering, we observe that when the angle of attack and the rudder angle are considered small, the coefficients of the dynamic equations obtained in this paper can be simplified to match those of the commonly used model in aerospace engineering. However, unlike the commonly used model in aerospace engineering, the linearized expressions for inertia forces and moments owing to control surface deflection in this paper include not only the increments of control surface deflection acceleration but also the increments of the rudder angle and angle of attack. Simulation results indicate that the linearized model derived in this paper produces results close to those of the nonlinear model than those of the traditional linearized model.

When the angle of attack and the rudder angle are small, the simulation results of the simplified linearized models are relatively close to those of the linearized model derived in this paper. However, under the high angle of attack and large rudder angle, a considerable discrepancy is observed between the simulation results of the linearized models that treat the angle of attack or rudder angle as small quantities and the linearized model derived in this paper. In such cases, treating the angle of attack or rudder angle as small quantities may lead to substantial errors. The research presented in this paper offers a high-precision linearized model that can be used for the guidance and control system design of a flight vehicle.