# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 1 Introduction  
Gearbox modeling is an important step in the design process of transmission systems. By solving mathematical models, the system's dynamic characteristics can be analyzed prior to prototyping, and the gearbox dynamic response characteristics can be effectively predicted for given input conditions. Then, effective suggestions regarding the design and control of transmission systems can be provided [1]. The dynamic response of a typical transmission system is affected by many uncertain factors. No mathematical model can account for all factors that affect transmission systems [2]; adding more components to the model increases the modeling cost and the model's complexity, resulting in time-consuming calculations or difficult solutions. Therefore, from the perspective of the calculation cost reduction and improvement of the system's design efficiency, efficient and accurate modeling methods are necessary for designing and evaluating gearboxes. Much research has been performed in the field of gearbox modeling and the dynamic response of gear trains. The Lagrange equation approach and the bond graph (BG) method [3] are common modeling techniques in the field of gear system modeling and simulation. When dealing with complex kinematic layouts, the BG model can intuitively represent the kinematic relationship between a system's parts, and an abstract dynamicequation is obtained using Lagrange's law [4]. Based on the causal relationship between the components of a BG model, the algebraic relationship between the flow variables and potential variables can be inferred for obtaining dynamic model equations [5]. So far, bond graph theory is widely used in the analysis of a variety of engineering systems. [PERSON] et al., used bond method to analyze the influence of different excitations on the vibration level of a helicopter's semi-active suspension [6]. [PERSON] [7] constructed the bond graph model of an exoskeleton actuator and extended it to the dynamic modeling of all types of wearable robot actuators. In the research of modeling and Simulation of gear trains, for example, [PERSON] [8; 9] modeled a four-speed automatic transmission using the BG theory methods, and used it for analyzing the system response dynamics in the case of wheels, for fully applied brakes and for the brake-off scenario [10]. The shift of a 10-gear gearbox was analyzed in [11], and an extra clutch was utilized for improving the shifting performance using the BG theory methods. [PERSON] [12] proposed a method for transforming device-mapping BG models into conceptual torque-path BGs. The BG method was used in [13] for optimizing the double-transition shift control, and different control strategies were proposed. A systematic design method was explored in [14] for synthesizing the configuration schemes of multi-row and multi-speed AT based on the lever analogy. [PERSON] et al. [15] considered three subsystems: a planetary gear train (PGT), clutch, and inertia rotor, proposed a standardized modeling process, and verified the effectiveness of this method on example gearboxes. A wind turbine gearbox was modeled in [16] using the BG method, to develop a framework for transmission chain modeling, and the parameters were synthesized in Monte Carlo simulations. The transmission path of vibrations in aero-engines was analyzed in [17] using the BG theory methods, to efficiently diagnose and minimize noise and vibration in the design stage. [PERSON] [18] discussed the effects of stiffness, damping, and inertia on the dynamic response of gear trains, based on the bond graph method. The existing research mainly focuses on linear models and on the dynamic response analysis of gear trains. The influencing factors that are considered in these models are relatively few, and the proposed modeling methods have strict requirements for configuring gear trains; in addition, the guidance modeling process is not universal.