Dynamic Analysis of Multibody Mechanical Systems

Dynamic analysis of multibody systems (MBSs) is critical in modern mechanical engineering. Multibody dynamics (MBD) enables the precise modeling and prediction of the behavior of dynamical systems composed of interconnected mechanical components. This analysis is essential for understanding the dynamic interactions within these components and predicting their behavior under various operating conditions. Engineers specialized in MBD can predict how such systems will perform in real-world scenarios since the modeling approach based on MBSs facilitates the accurate simulation of the dynamic behavior of mechanical systems. By mathematically modeling the interactions between mechanical components using proper equation sets, it is possible to reduce the reliance on extensive physical prototyping, saving time and resources. Through dynamic analysis carried out in a virtual environment, engineers with expertise in MBD can explore different configurations and materials to optimize the design of mechanical systems. This process enhances performance, efficiency, and safety, leading to superior product designs.

Multibody systems are formed of articulated systems composed of mechanical components. As a critical feature, kinematic joints constrain the motion of multibody mechanical systems. Nonlinear forces and nonlinear force fields, which generate complex dynamic behaviors, are also applied to the bodies that form a multibody system. Furthermore, multibody mechanical systems can be either rigid or deformable, leading to two entirely separate descriptions of motion, which are based on fundamental mathematical approaches that may eventually converge in a unified formulation. Multibody systems are generally mathematically modeled using nonlinear differential-algebraic equation (DAE) sets. Through the basic principles of analytical dynamics and continuum mechanics, as well as employing modern linear algebra techniques, the mathematical description of the motion of a multibody system can be systematically formulated into a nonlinear set of DAEs that must be solved numerically. Therefore, the multibody approach to analyzing constrained mechanical systems represents a fundamental tool for performing the optimal design and virtual prototyping of modern engineering systems.

The central goal of this Special Issue of Machines was to offer an overview of state-of-the-art research on the kinematic, kinetostatic, static, and dynamic analysis of multibody mechanical systems; on the development of effective and efficient algorithms to address these issues; and on the engineering applications that lie in this context. Consequently, the subjects covered in this Special Issue belong to a broad framework encompassing various fundamental aspects and engineering problems. The topics of interest for this Special Issue include modeling robotic systems within the framework of multibody dynamics, mathematical modeling of articulated mechanical systems, dynamic and vibration analysis approaches, experimental model validation techniques, finite-rotation and significant-deformation problems concerning rigid–flexible multibody systems, methodologies for formulating and solving the differential-algebraic equations of motion, nonlinear control techniques for controlling the dynamic behavior of multibody systems, developments in the finite element technologies, and advances in the kinematic and geometric description of the motion in general.

The current Special Issue of Machines encompasses 17 journal papers, all published in Machines between 2021 and 2024. Below is a summary of all the research papers contributing to the present Special Issue.

In [1], the authors discussed the importance of measuring cutting forces in machining processes and highlighted the limitations of traditional dynamometers at high frequencies. The paper introduces a newly designed Strain-Gauge-Based Dynamometer (SGBD) with a symmetrical structure capable of accurately measuring high-frequency dynamic forces in milling. The SGBD’s natural frequency is stabilized at 9.15 kHz, and it includes a data acquisition system for automatic force data collection. Tests showed that the SGBD is reliable, with minimal differences in cutting force measurements compared to a Kistler dynamometer, making it suitable for high-speed milling applications (Contribution 1).

In [2], the authors discussed the increasing use of multibody modeling and simulation in developing and testing mechanical systems, particularly Formula SAE cars, to avoid costly and time-consuming track testing. Advances in computer-aided engineering allow for evaluating a car’s dynamic behavior and optimizing its setup for peak performance. The paper focuses on a Formula SAE vehicle, detailing its multibody model and subassemblies like suspensions and powertrain. Simulated track testing is used to fine-tune the vehicle setup, achieving better performance in competitions, such as a 2 km/h speed increase in the skidpad event by adjusting tire camber angles (Contribution 2).

In [3], the authors presented a general closed-chain kinematic model for two-wheeled vehicles applicable to both bicycles and motorcycles. This model uses a multibody system approach to evaluate two methods for formulating wheel–road contact constraints: one based on vector cross-product geometry and another on surface parameterization of the front wheel. These methods are compared to study the contact geometry between the front wheel and the ground. The study also includes a detailed numerical analysis of the system kinematics and proposes an explicit formulation of the front assembly’s orientation using Euler angles. The results presented in the paper align well with the existing literature on vehicle kinematics and contact geometry (Contribution 3).

In [4], the authors developed a linear dynamic model for a front-loading washing machine, conceptualized with three moving rigid bodies, revolute joints, springs, dampers, and a prescribed rotational drum motion. In this work, Kane’s method was used to derive the equations of motion. The linear model helps to investigate modal and transient characteristics and the design of the washing machine. However, its reliability is limited to specific system parameters. The study identified these parameters and numerically investigated their ranges to ensure the model’s reliability (Contribution 4).

In [5], the authors investigated the detachment between a cam and a follower at various cam speeds and internal distances of the follower guide. Due to the follower’s nonlinear dynamics, the detachment was detected using the most prominent Lyapunov exponent, FFT power density function, and Poincare’s maps. Follower displacement and contact force were measured to identify detachment heights. In the paper, multi-degree-of-freedom spring–damper–mass systems enhanced dynamic performance and minimized detachment. The nonlinear response of follower displacement was analyzed under different conditions using SOLIDWORKS for numerical solutions and a high-speed camera for position tracking. Friction and impact were considered, and the peak nonlinear response was significantly reduced (15%, 32%, 45%, and 62%) with the multi-degree-of-freedom systems designed in the paper (Contribution 5).

In [6], the authors discussed the development of a new finite element called the ALE-RANCF, which combines features from ALE-ANCF and RANCF elements. Unlike ALE-ANCF, which cannot accurately describe rational cubic Bezier curves, the ALE-RANCF can. This new element maintains exact geometry and mechanics even when divided or merged, showing more minor deviations and oscillations than ALE-ANCF. Numerical examples demonstrate its feasibility and advantages over conventional finite elements (Contribution 6).

In [7], the authors discussed the emerging use of rovers with automatic and robotic systems in agriculture. This work highlights the importance of developing simulation models to design and test these rovers before creating prototypes. To this end, the authors proposed a simulation test rig using a multibody model to explore critical issues in rover design. Their simulations reveal significant differences between the two rover structures, particularly regarding energy savings, which is crucial for field operations. The modular simulation model proposed in the paper can also be adapted to other vehicle structures (Contribution 7).

In [8], the authors presented preliminary results on a new control architecture using Model Predictive Control (MPC) for Cable-Driven Parallel Robots (CDPRs). This work focuses on a three-degree-of-freedom robot in a suspended configuration, creating a cable-suspended parallel robot. The control scheme aims for accurate path tracking of the end-effector while maintaining positive cable tensions. To manage the nonlinear dynamics and reduce computational effort, a position-dependent MPC algorithm with an embedded integrator is used to compute optimal cable tensions. These tensions must stay within a feasible range to ensure that the cables pull the end-effector without breaking. The controller, though nonlinear, performs local linearization at each time step to simplify calculations. The optimal tensions are converted into motor torques via the inverse dynamic model of the servomotors. The control architecture is numerically validated on a spatial CDPR driven by three cables, with four different paths tested to showcase the controller’s features (Contribution 8).

In [9], the authors analyzed the core structure of a mine hoisting system known as a large headframe. They highlighted the limitations of traditional designs that only consider static analysis under load. This approach often leads to resonance issues in later applications. A new dynamic characteristic analysis and structural optimization method is proposed to address this. The technique involves modal analysis to determine the eigenfrequencies and vibration modes, revealing that the system’s multi-order eigenfrequencies are relatively close, making the headframe susceptible to resonance under alternating loads of similar frequencies. Harmonic response analysis shows that the vibration amplitude increases when the load frequency is near the first-order eigenfrequency, and the fourth- and fifth-order eigenfrequencies are very close, leading to severe resonance and potential structural damage. The paper proposes nine structural optimization schemes, with scheme nine identified as the optimal solution. This new method is significant for preventing resonance and ensuring the safety of mining operations (Contribution 9).

In [10], the authors proposed designing and controlling a footplate-based gait robot-assisted system for lower-limb actuators. Stroke often leads to disabilities in the lower-limb symmetrical gait pattern, making it difficult for patients to regain their usual walking ability without rehabilitation therapies. Footplate-based gait robot-assisted systems can facilitate practical training and track recovery progress, significantly reducing physiotherapy labor costs. This study simulates the dynamic and control aspects of a five-degree-of-freedom footplate-based gait robot-assisted system designed using the Stewart platform structure for stroke patient rehabilitation. The system’s effectiveness was evaluated using the clinical gait pattern of a healthy male, demonstrating its ability to simulate hip and knee flexion/extension and ankle dorsiflexion/plantar flexion to reproduce a typical symmetrical gait pattern. The results showed a mean percentage error of 6.70% when comparing the dynamic movement characteristics of the right and left thigh, leg, and foot to the clinical gait pattern, indicating the system’s accuracy and effectiveness in lower-limb actuation during simulation (Contribution 10).

In [11], the authors studied a Dual-Motor Precision Transmission Mechanism (DMPTM). The DMPTM aims to eliminate backlash while maintaining servo system stiffness. Existing models of DMPTM lack accuracy, hindering performance optimization and high-precision controller design. This paper uses the dead-zone model to describe backlash and friction using the Stribeck model, considering time-varying stiffness. A high-precision dynamic model of DMPTM is developed and validated through experiments in both time and frequency domains. The results show that the proposed model accurately represents the mechanism’s nonlinear characteristics, with Pearson correlation coefficients exceeding 99.41% for open-loop systems and 83.7% for closed-loop systems, outperforming existing models. The frequency response of the proposed model also closely matches the actual system, confirming its accuracy (Contribution 11).

In [12], the authors addressed the numerical difficulties in computing complex eigenvalues for damped multi-flexible-body problems. The proposed method handles arbitrary rigid body modes and algebraic constraints while leveraging Jacobians’ sparsity in large systems. A custom implementation of the Krylov–Schur method is introduced, demonstrating its robustness and accuracy across various complex test scenarios (Contribution 12).

In [13], the authors presented a closed-form dynamic model for a Two-Rotational and One-Translational (2R1T)-degree-of-freedom Parallel Kinematic Mechanism (PKM) with a hybrid rigid–flexible structure aimed at force-control applications. Using the Three-Prismatic–Prismatic–Spherical (3PPS) kinematic configuration and zero-torsion motion characteristics, the authors proposed a symbolic formulation approach to develop kinematic models for both forward and inverse kinematics. The model uses six joint variables—three active and three passive prismatic joints—as quasi-coordinates, incorporating the stiffness and deformation of the passive joints. A closed-form dynamic model is achieved by eliminating the passive joint variables through virtual work. This model was validated using commercial dynamic simulation software, and it was found that minimal flexure stiffness in driving directions is preferred for optimal performance (Contribution 13).

In [14], the authors carried out the modeling and control of an Unmanned Aerial Vehicle (UAV) for delivery purposes, integrating computer-aided design, multibody dynamic modeling, and motion control analysis. The UAV presented in this work, designed as a quadcopter with four arms connected to a central trunk, is systematically modeled using a multibody approach. For this purpose, SIMSCAPE MULTIBODY software is employed for dynamic analysis and control system design, starting from a CAD model created in SOLIDWORKS. The control system is developed through dynamic simulations in MATLAB/SIMULINK. The paper proposes propulsion systems and component design choices based on the UAV’s structural and functional needs. A vital contribution of the paper is the derivation of nonlinear algebraic constraint equations that describe the kinematics of the drone and impose nonholonomic constraints on relative angular velocity between rigid bodies. The control system uses cascaded Proportional–Integral–Derivative (PID) controllers to achieve various maneuvers that are essential for motion control. The paper details the tuning of PID controller parameters using MATLAB tools and dynamic simulations, demonstrating the high performance of the UAV system (Contribution 14).

In [15], the authors introduced a vibration model for a three-Prismatic–Revolute–Revolute (PRR) Planar Parallel Manipulator (PPM) with flexible links, using the linear transfer matrix method for multibody systems (MSTMM). It explores the dynamic characteristics of the PRR PPM, deriving the dynamic model and obtaining the transfer matrices and equations for each component and the entire system. The vibration characteristics are determined using MSTMM and verified through ANSYS simulation. The study also analyzes the relationship between natural frequencies and PPM configurations along a circular trajectory, showing that the system’s natural frequency varies with configurations, with similar trends in the first six orders. This modeling approach is efficient, accurate, and extendable to other parallel manipulators with flexible components (Contribution 15).

In [16], the authors presented a model for Underwater Vehicle Multi-Manipulator Systems (UVMMSs) by treating all components as part of a single system. It accounts for the forces exerted on the manipulators by the vehicle’s movement and the reaction forces on the vehicle from the manipulators. The model is developed using the Newton–Euler method through a mobile kinematic chain. Various numerical implementation approaches and simulation results are provided to show that the model accurately represents the interaction between the vehicle and the manipulators. This comprehensive model and its simulations are crucial for designing control strategies considering all system elements rather than ignoring interaction forces or treating the vehicle and manipulators as separate entities (Contribution 16).

In [17], the authors analyzed the dynamics and vibration of the deployment process for a replaceable interface mast. The rover’s mast, which carries precision instruments like cameras, is crucial for their performance. The replaceable interface mast addresses issues of a single structure and poor maintenance found in ordinary masts. The study focuses on the deployment process dynamics and vibration of this mast. Initially, the spatial motion of the reconfigurable rigid-body module was analyzed using the Absolute Nodal Coordinate Formulation (ANCF) and the Natural Coordinate Formulation (NCF), establishing a dynamic equation for the interface and reconfigurable module without external constraints. A dynamic model of the mast system deployment process was then created. The dynamic behavior of the replaceable interface mast was analyzed, and different system configurations, flexible interface geometric parameters, and driving rules were compared. A model of the mast system was built, and the deployment process was experimentally analyzed. Numerical solutions and experimental verification confirmed that the dynamic model accurately analyzes the deployment behavior dynamics of the replaceable interface mast, providing a reference for the design and behavior analysis of the mast system (Contribution 17).

As discussed in detail above by illustrating the content of each contribution to this Special Issue of Machines, MBD significantly reduces development costs and time by allowing virtual testing and iteration. The use of MBD techniques is instrumental in identifying potential failure points within a system and performing durability analyses of modern mechanical components. By analyzing the dynamic interactions between components, engineers can design more robust systems and implement preventive measures to mitigate the risk of failures. This proactive approach enhances the reliability and longevity of mechanical systems. The application of the modeling approach based on MBS spans multiple industries, including automotive, aerospace, robotics, and biomechanics. In the automotive sector, MBS designs suspension systems and analyzes vehicle dynamics. In aerospace, it aids in analyzing aircraft and spacecraft dynamics, ensuring optimal performance and safety. Therefore, the dynamic analysis of multibody systems is crucial for developing advanced control systems. By understanding the dynamic behavior of these systems, engineers can design control algorithms that improve stability and performance. This is particularly important in applications requiring precise control, such as robotics and automated machinery. Furthermore, utilizing MBD reduces the need for physical testing, which can be both costly and time-consuming. Virtual simulations enable rapid iteration and testing of different scenarios, leading to more efficient development processes. This approach saves resources and accelerates the time to market for new products.

In conclusion, dynamic analysis of multibody mechanical systems is an indispensable tool in modern engineering. Its ability to accurately predict system behavior, optimize designs, and prevent failures makes it essential for developing advanced engineering solutions. The widespread application of the modeling approach based on MBS across various industries underscores its importance in achieving high-performance, efficient, and reliable mechanical systems.

The guest editor of this Special Issue of Machines extends gratitude to the authors for their valuable and high-quality contributions; to the reviewers for their dedicated efforts and time in enhancing the submission and revision process; and to the publisher for their outstanding work and cooperation.