Artykuły w czasopismach: „Rigid flexible multibody- Dynamics”

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Chen, Gang, Weigong Zhang i Bing Yu. "Multibody dynamics modeling of electromagnetic direct-drive vehicle robot driver". International Journal of Advanced Robotic Systems 14, nr 5 (1.09.2017): 172988141773189. http://dx.doi.org/10.1177/1729881417731896.

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Collaborative dynamics modeling of flexible multibody and rigid multibody for an electromagnetic direct-drive vehicle robot driver is proposed in the article. First, spatial dynamic equations of the direct-drive vehicle robot driver are obtained based on multibody system dynamics. Then, the shift manipulator dynamics model and the mechanical leg dynamics model are established on the basis of the multibody dynamics equations. After establishing a rigid multibody dynamics model and conducting finite element mesh and finite element discrete processing, a flexible multibody dynamics modeling of the electromagnetic direct-drive vehicle robot driver is established. The comparison of the simulation results between rigid and flexible multibody is performed. Simulation and experimental results show the effectiveness of the presented model of the electromagnetic direct-drive vehicle robot driver.

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Duan, Yue Chen, Xia Li, Wei Wei Zhang, Guo Ning Liu i Ting Ting Wang. "Impact Dynamics of Flexible Multibody System Based on Continuous Contact Force Method". Applied Mechanics and Materials 744-746 (marzec 2015): 1628–34. http://dx.doi.org/10.4028/www.scientific.net/amm.744-746.1628.

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The impact dynamics of spatial multi-link flexible multibody system is studied based on the continuous contact force method (CCFM). According to the rigid-flexible coupling dynamic theory of flexible multibody system, the rigid-flexible coupling continuous dynamic equations of the system are established by using the recursive Lagrange method. The impact dynamic equations of the system are stylized derived on the use of CCFM basing on the nonlinear spring-damper model. The contact separation criterion is given to achieve the conversion and calculation of the dynamic model for the system at different stages. An impact dynamic simulation example for a two-link planar flexible multibody system is given, as well as the global dynamic response. The results show that the impact dynamic solving method based on CCFM can be used for the global impact dynamics of multi-link flexible multibody systems. The dynamic behavior of the system changes dramatically during the impact process. The large overall motion, the small deformation motion and the impact effect are coupled.

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Wang, Xiaoyu, Haofeng Wang, Jingchao Zhao, Chunyang Xu, Zhong Luo i Qingkai Han. "Rigid-Flexible Coupling Dynamics Modeling of Spatial Crank-Slider Mechanism Based on Absolute Node Coordinate Formulation". Mathematics 10, nr 6 (10.03.2022): 881. http://dx.doi.org/10.3390/math10060881.

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In order to study the influence of compliance parts on spatial multibody systems, a rigid-flexible coupling dynamic equation of a spatial crank-slider mechanism is established based on the finite element method. Specifically, absolute node coordinate formulation (ANCF) is used to formulate a three-dimensional, two-node flexible cable element. The rigid-flexible coupling dynamic equation of the mechanism is derived by the Lagrange multiplier method and solved by the generalized α method and Newton–Raphson iteration method combined. Comparison of the kinematics and dynamics response between rigid-flexible coupling system and pure rigid system implies that the flexible part causes a certain degree of nonlinearity and reduces the reaction forces of joints. The elastic modulus of the flexible part is also important to the dynamics of the rigid-flexible multibody system. With smaller elastic modulus, the motion accuracy and reaction forces become lower.

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Liu, Zhuyong, Jiazhen Hong, Jinyang Liu i Guanghao Xu. "58907 RIGID-FLEXIBLE COUPLING EFFECTS OF THE FLEXIBLE PLATE UNDERGOING LARGE OVERALL MOTION(Flexible Multibody Dynamics)". Proceedings of the Asian Conference on Multibody Dynamics 2010.5 (2010): _58907–1_—_58907–6_. http://dx.doi.org/10.1299/jsmeacmd.2010.5._58907-1_.

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Zhu, C. X., Yong Xian Liu, Guang Qi Cai i L. D. Zhu. "Dynamics Simulation Analysis of Flexible Multibody of Parallel Robot". Applied Mechanics and Materials 10-12 (grudzień 2007): 647–51. http://dx.doi.org/10.4028/www.scientific.net/amm.10-12.647.

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Take a kind of 3-TPT parallel robot as an example, the model of flexible multibody of parallel machine tool is built by using multibody dynamics simulation software ADAMS and finite element analysis software ANSYS. And dynamics equation of flexible body in spatial is also set up, after that the dynamics simulation is carried out. Then the simulation results of rigid bodies are compared with flexible ones, and the results show that the forces applied on flexible bodies appear high nonlinear, so the simulation results of flexible multibody system are more authentic, nicety and can reflect actual dynamics characteristic of parallel robot.

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Zakhariev, Evtim. "Nonlinear Dynamics of Rigid and Flexible Multibody Systems". Mechanics of Structures and Machines 28, nr 1 (2.08.2000): 105–36. http://dx.doi.org/10.1081/sme-100100614.

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Yu, Hua-Nan, Jing-Shan Zhao i Fu-Lei Chu. "Dynamic modeling of flexible multibody system using a meshing method". Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, nr 4 (10.05.2013): 611–31. http://dx.doi.org/10.1177/0954406213489444.

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Multi-rigid-body system dynamics can be used to investigate the dynamics of a mechanical system of rigid bodies while the finite element method is often utilized to model the quasi-static elastic deformations of an elastic structure. However, neither of these two methods can resolve the real dynamics of a mechanical system when both rigid displacements and elastic deformations coexist. Therefore, this article proposes a meshing method to simulate the mechanical system with uniform mass point movements. To split the specified solid structure into a set of regularly distributed dynamic units, one can assume that the mass density of the structure is evenly distributed within the whole concrete volume and the elasticity and damping of the material are isotropic. Then the whole solid structure of each component can be divided into a number of tetrahedrons the vertexes of which are the points with the mass parameters. The original distances between every pair of adjacent points are supposed to be identical, and the stiffness and the damping coefficients are introduced to formulate the internal and external dynamics of the adjacent mass points. To illustrate the correction and effectiveness of the method, the dynamics problems of a number of regular elastic bodies are investigated with large rigid displacements accompanying elastic deformations. Computer simulations demonstrate that this method is especially useful for real mechanical systems where the rigid displacements and elastic deformations coexist.

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Wang, Yi Ping, Wen Lei Sun i Qun Zhao. "Research on Dynamic Characteristics of 750KW Wind Turbine Flexible Blades". Applied Mechanics and Materials 34-35 (październik 2010): 1757–60. http://dx.doi.org/10.4028/www.scientific.net/amm.34-35.1757.

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It is necessary and fitting to use flexible multibody dynamics method to study the wind turbine blades. Because the characteristics of the blades will directly affect the whole wind turbine’s, and the results by using flexible multibody method are to agree with reality. Considering the anisotropic composite blades, it established the flexible blades model and rigid-flexible wind rotor model of the 750KW wind turbine by the software of ANSYS and ADAMS. Then fit the loads which are computed from BLADE to the wind rotor, analyze the dynamic characteristics of the rotor. It gets the dynamic features of the flexible blades and the rigid blades’ as a comparison, which will be useful to research on WTGS, and will supply reference data to blade trouble analysis and optimization design.

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Langlois, R. G., i R. J. Anderson. "A TUTORIAL PRESENTATION OF ALTERNATIVE SOLUTIONS TO THE FLEXIBLE BEAM ON RIGID CART PROBLEM". Transactions of the Canadian Society for Mechanical Engineering 29, nr 3 (wrzesień 2005): 357–73. http://dx.doi.org/10.1139/tcsme-2005-0022.

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A classical planar problem in forward flexible multibody dynamics is thoroughly investigated. The system consists of a damped flexible beam cantilevered to a rigid translating cart. The problem is solved using three distinctly different conventional approaches presented in roughly the chronological order in which they have been applied to flexible dynamic systems. First, a modal superposition formulation based on Bernoulli-Euler beam theory is developed. Second, an alternative solution is developed drawing exclusively on methods for rigid body dynamics combined with a knowledge of the theoretical modal behaviour of continuous beams. Third, a formulation based on the conventional finite element method using four-degree-of-freedom planar beam elements is adapted to include the rigid body motion of the cart. The relative merits of the three formulations are discussed and numerical simulation results generated using each of the three formulations are compared with each other and with a solution from a general-purpose flexible multibody dynamics formulation that is briefly outlined. The relative accuracy and efficiency of the methods and the challenges associated with generalizing each formulation are discussed.

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Qin, Wen Jie, D. W. Jia i Q. Y. Liu. "Multibody System Dynamics Simulation of Loads in Main Bearings of Crankshafts". Materials Science Forum 628-629 (sierpień 2009): 55–60. http://dx.doi.org/10.4028/www.scientific.net/msf.628-629.55.

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In this paper, as for the calculation of loads in main bearings in a crankshaft system, multibody system dynamics is used to simulate the dynamic characteristics of the system composed of flexible and rigid bodies, coupled with hydrodynamic lubrication analysis further. The multibody system model with flexible crankshaft of one V8 diesel engine is built in ADAMS software, in which the bearings are modeled as rigid constrained bearings and hydrodynamic bearings respectively. The resulted loads in main bearings using different models are compared. The results show that the deformation of crankshafts has great effect on the values of loads in main bearings, and the bearing loads in different directions tend to uniformity due to the hydrodynamic lubrication.

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Cyril, X., J. Angeles i A. Misra. "DYNAMICS OF FLEXIBLE MULTIBODY MECHANICAL SYSTEMS". Transactions of the Canadian Society for Mechanical Engineering 15, nr 3 (wrzesień 1991): 235–56. http://dx.doi.org/10.1139/tcsme-1991-0014.

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In this paper the formulation and simulation of the dynamical equations of multibody mechanical systems comprising of both rigid and flexible-links are accomplished in two steps: in the first step, each link is considered as an unconstrained body and hence, its Euler-Lagrange (EL) equations are derived disregarding the kinematic couplings; in the second step, the individual-link equations, along with the associated constraint forces, are assembled to obtain the constrained dynamical equations of the multibody system. These constraint forces are then efficiently eliminated by simple matrix multiplication of the said equations by the transpose of the natural orthogonal complement of kinematic velocity constraints to obtain the independent dynamical equations. The equations of motion are solved for the generalized accelerations using the Cholesky decomposition method and integrated using Gear’s method for stiff differential equations. Finally, the dynamical behaviour of the Shuttle Remote Manipulator when performing a typical manoeuvre is determined using the above approach.

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Yen, Chiaming, i Glenn Y. Masada. "Dynamic Analysis of Flexible Bodies Using Extended Bond Graphs". Journal of Dynamic Systems, Measurement, and Control 116, nr 1 (1.03.1994): 66–72. http://dx.doi.org/10.1115/1.2900682.

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An Extended Bond Graph (EBG) formulation is described for analyzing the dynamics of a flexible multibody system. This work extends the EBG method, which was originally developed for systems with small spatial motion, to rigid and flexible multibody systems exhibiting large overall motions. The development uses modular models for the elements so that complex system models can be derived by coupling these modules. The EBG formulation for moving reference frames is used to derive models of one-link and two-link flexible manipulator systems. This approach has several advantages over the Lagrangian and Newtonian methods, such as its ability to solve the forward and inverse dynamic problems using the same bond graph. Finally, the EBG formulations for cantilever beams and for multi-rigid body dynamic systems are shown to be special cases of the general EGBs for flexible bodies.

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Xue, Zhi-Peng, Ming Li, Yan-Hui Li i Hong-Guang Jia. "A Simplified Flexible Multibody Dynamics for a Main Landing Gear with Flexible Leaf Spring". Shock and Vibration 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/595964.

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The dynamics of multibody systems with deformable components has been a subject of interest in many different fields such as machine design and aerospace. Traditional rigid-flexible systems often take a lot of computer resources to get accurate results. Accuracy and efficiency of computation have been the focus of this research in satisfying the coupling of rigid body and flex body. The method is based on modal analysis and linear theory of elastodynamics: reduced modal datum was used to describe the elastic deformation which was a linear approximate of the flexible part. Then rigid-flexible multibody system was built and the highly nonlinearity of the mass matrix caused by the limited rotation of the deformation part was approximated using the linear theory of elastodynamics. The above methods were used to establish the drop system of the leaf spring type landing gear of a small UAV. Comparisons of the drop test and simulation were applied. Results show that the errors caused by the linear approximation are acceptable, and the simulation process is fast and stable.

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Tian, Qiang, Yanlei Sun, Cheng Liu, Haiyan Hu i Paulo Flores. "ElastoHydroDynamic lubricated cylindrical joints for rigid-flexible multibody dynamics". Computers & Structures 114-115 (styczeń 2013): 106–20. http://dx.doi.org/10.1016/j.compstruc.2012.10.019.

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Luo, Qing Guo, Xu Dong Wang, Zheng Bo Gong i Feng Wang. "Dynamics Simulation of the Crank and Connecting Rod Mechanism of Diesel Engine". Advanced Materials Research 354-355 (październik 2011): 438–41. http://dx.doi.org/10.4028/www.scientific.net/amr.354-355.438.

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Based on the virtual prototyping technology and flexible multibody system dynamic theory, the author founded the rigid-flexible coupling multibody dynamic analysis model of the crank and connecting rod mechanism of diesel engine, combining the means of 3D solid modeling, finite element analysis and multi-body dynamic simulation. Through the simulation, the dynamic loads in the working cycle of the mechanism are obtained, which provides reference for dynamic stress and fatigue analysis of the mechanism.

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Zhang, Lina, Xiaoting Rui, Jianshu Zhang, Guoping Wang, Junjie Gu i Xizhe Zhang. "A framework for establishing constraint Jacobian matrices of planar rigid-flexible-multibody systems". AIMS Mathematics 8, nr 9 (2023): 21501–30. http://dx.doi.org/10.3934/math.20231096.

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<abstract> <p>Constraint violation correction is an important research topic in solving multibody system dynamics. For a multibody system dynamics method which derives acceleration equations in a recursive manner and avoids overall dynamics equations, a fast and accurate solution to the violation problem is paramount. The direct correction method is favored due to its simplicity, high accuracy and low computational cost. This method directly supplements the constraint equations and performs corrections, making it an effective solution for addressing violation problems. However, calculating the significant Jacobian matrices for this method using dynamics equations can be challenging, especially for complex multibody systems. This paper presents a programmatic framework for deriving Jacobian matrices of planar rigid-flexible-multibody systems in a simple semi-analytic form along two paths separated by a secondary joint. The approach is verified by comparing constraint violation errors with and without the constraint violation correction in numerical examples. Moreover, the proposed method's computational speed is compared with that of the direct differential solution, verifying its efficiency. The straightforward, highly programmable and universal approach provides a new idea for programming large-scale dynamics software and extends the application of dynamics methods focused on deriving acceleration equations.</p> </abstract>

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Zahariev, E. V. "Earthquake dynamic response of large flexible multibody systems". Mechanical Sciences 4, nr 1 (20.02.2013): 131–37. http://dx.doi.org/10.5194/ms-4-131-2013.

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Abstract. In the paper dynamics of large flexible structures imposed on earthquakes and high amplitude vibrations is regarded. Precise dynamic equations of flexible systems are the basis for reliable motion simulation and analysis of loading of the design scheme elements. Generalized Newton–Euler dynamic equations for rigid and flexible bodies are applied. The basement compulsory motion realized because of earthquake or wave propagation is presented in the dynamic equations as reonomic constraints. The dynamic equations, algebraic equations and reonomic constraints compile a system of differential algebraic equations which are transformed to a system of ordinary differential equations with respect to the generalized coordinates and the reactions due to the reonomic constraints. Examples of large flexible structures and wind power generator dynamic analysis are presented.

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Zhao, Da Xing, Yong Yang, Wan Xu, Guo Long Ding i Ling Peng. "Rigid-Flexible Coupling-Based CNC Machine Tool Feed Drive System Co-Simulation". Advanced Materials Research 694-697 (maj 2013): 115–19. http://dx.doi.org/10.4028/www.scientific.net/amr.694-697.115.

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High-speed high-precision CNC processing technology plays a very important position in the CNC machining industry,machine disturbance,however,an important factor affecting the machining accuracy,the ball screw feed drive as an important part of the machine directly affects the operating characteristics of the machine.Ball screw drive system,Solidworks,Ansys,Adams the establishment of machine tool,rigid-flexible couping the multibody dynamics model,the dynamic characteristic curve of the machine.Compared with the rigid body model simulation results verify the feasibility of rigid-flexible coupling modeling method,optimized to provide the basis of the structural design of the machine components.

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Zhang, Xing, Qi Wei He i Xiang Yu. "Modeling and Dynamic Analysis for Rigid-Flexible Coupling Nonlinear Floating Raft Vibration Isolation System". Applied Mechanics and Materials 34-35 (październik 2010): 1994–98. http://dx.doi.org/10.4028/www.scientific.net/amm.34-35.1994.

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To analyze the influence of the nonlinearity of the floating raft vibration isolation system, the theory of flexible multibody dynamics was used here to model the rigid-flexible coupling nonlinear floating raft vibration isolation system. The dynamical characteristics of the model were studied with ADAMS software and some typical nonlinear phenomena, such as subharmonics, superharmonics and attractor coexistence, were observed through time domain analysis.

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Zhang, Xiao Ming, Yu Qing Wang i Jie Fang. "Dynamic Simulation of Crank-Connecting Rod-Piston Mechanism of Internal Combustion Engine Based on Virtual Prototype Technology". Applied Mechanics and Materials 143-144 (grudzień 2011): 433–36. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.433.

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A virtual prototype model of S195 diesel engine was established by using CAD, finite element analysis based on virtual prototype technology and dynamics theory of flexible multibody systems to study the kinetic discipline of the crank-rod mechanism,vibration of cylinder block caused by inertia force and the balance problem of inertia force.The rigid body and flexible body simulation were compared.The interaction force between components could be determined by flexible multibody dynamics analysis of the crankshaft system under the real working condition. The simulation results are accordant with those in true working state of S195 diesel engine. This research provides a feasible advanced method for design and development of diesel engine.

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Boutaghou, Z. E., Arthur G. Erdman i Henryk K. Stolarski. "Dynamics of Flexible Beams and Plates in Large Overall Motions". Journal of Applied Mechanics 59, nr 4 (1.12.1992): 991–99. http://dx.doi.org/10.1115/1.2894071.

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The dynamic response of flexible beams, plates, and solids undergoing arbitrary spatial motions are systematically derived via a proposed approach. This formulation is capable of incorporating arbitrary representation of the kinematics of deformation, phenomenon of dynamic stiffening, and complete nonlinear interaction between elastic and rigid-body dynamics encountered in constrained multibody systems. It is shown that the present theory captures the phenomenon of dynamic stiffening due to the transfer of the axial and membrane forces to the bending equations of beams and plates, respectively. Examples are presented to illustrate the proposed formulations.

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Skrinjar, Luka, Janko Slavič i Miha Boltežar. "Absolute Nodal Coordinate Formulation in a Pre-Stressed Large-Displacements Dynamical System". Strojniški vestnik - Journal of Mechanical Engineering 63, nr 7-8 (17.07.2017): 417. http://dx.doi.org/10.5545/sv-jme.2017.4561.

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The design process for dynamical models has to consider all the properties of a mechanical system that have an effect on its dynamical response. In multi-body dynamics, flexible bodies are frequently modeled as rigid, resulting in non-valid modeling of the pre-stress effect. In this research a focus on the pre-stress effect for a flexible body assembled in a rigid-flexible multibody system is presented. In a rigid-flexible assembly a flexible body is modeled with an absolute nodal coordinate formulation (ANCF) of finite elements. The geometrical properties of the flexible body are evaluated based on the frequency response and compared with the experimental values. An experiment including the pre-stress effect and large displacements is designed and the measured values of the displacement are compared to the numerical results in order to validate the dynamical model. The pre-stress was found to be significant for proper numerical modeling. The partially validated numerical model was used to research the effect of different parameters on the dynamical response of a pre-stressed, rigid-flexible assembly.

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Vukasovic, N., J. T. Celigu¨eta, J. Garci´a de Jalo´n i E. Bayo. "Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates". Journal of Mechanical Design 115, nr 2 (1.06.1993): 294–99. http://dx.doi.org/10.1115/1.2919191.

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In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

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Liu, Xiang, Jing-Shan Zhao i Zhi-Jing Feng. "Compliant dynamics of a rectilinear rear-independent system". Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, nr 5 (14.11.2016): 785–806. http://dx.doi.org/10.1177/0954406216631369.

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The rectilinear rear-independent suspension investigated in this paper could remain the wheel alignment parameters invariable in theory. However, its dynamics is much more complex than that of the existing suspensions because of its redundant constraints in structure. Considering the elasticity of the rectilinear rear-independent suspension, a rigid-flexible half-car dynamic model is established for the first time based on the discrete time transfer matrix method. At the same time, a rigid half-car dynamic model is established as a comparison. The natural frequency characteristics and dynamic response of the rectilinear rear-independent suspension under random road excitations are analyzed and compared with those of rigid half-car dynamic model. The results reveal that the suspension system has apparent influence to the dynamics of vehicle. The wheel alignment parameters will fluctuate within a narrow range which is mainly determined by the rolling vibration of vehicle. And the suspension system could reduce and filter the road excitations with high frequency and small amplitude. This provides a good effect on the ride comfort of vehicle. Dynamics analysis of the rectilinear rear independent suspension reveals that the proposed modeling approach could deal with the dynamics of rigid-flexible multibody systems with redundant constraints effectively.

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Lidström, P. "Instantaneous impulses in multibody dynamics". Mathematics and Mechanics of Solids 22, nr 4 (19.10.2015): 581–635. http://dx.doi.org/10.1177/1081286515598825.

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This paper considers instantaneous impulses in multibody dynamics. Instantaneous impulses may act on the multibody from its exterior or they may appear in its interior as a consequence of two of its parts interacting by an impact imposed by a unilateral constraint. The theory is based on the Euler laws of instantaneous impulses, which may be seen as a complement to the Euler laws for regular motions. Based on these laws, and specific continuum properties of the quantities involved, local balance laws for momentum and moment of momentum, involving instantaneous impulses and introducing the Cauchy impulse tensor, are derived. Thermodynamical restrictions on the impulse tensor are formulated based on the dissipation inequality. By stating a principle of virtual work for instantaneous impulses, and demonstrating its equivalence to Euler’s laws, Lagrange’s equations are derived. Lagrange’s equations are convenient to use in the case of multibody dynamics containing rigid as well as flexible parts. A central theme of this paper is the discussion of the interaction between parts of the multibody and their relation to geometrical and kinematical constraints. This interaction is severely affected by the presence of friction, which is notoriously difficult to handle. In a preparation for this discussion we first consider the one-point impact between two rigid bodies. The importance of the so-called impact tensor for this problem is demonstrated. In order to be able to handle the impact laws of Poisson and Stonge, an impact process, governed by a system of ordinary differential equations, is defined. Within this model phenomena, such as slip stop, slip start and slip direction reversal, may be handled. For a multibody with an arbitrary number of parts and multiple impacts, the situation is much more complicated and certain simplifications have to be introduced. Equations of motion for a multibody, consisting of rigid parts and in the presence of ideal bilateral constraints and unilateral constraints involving friction, are formulated. Unique solutions are obtained, granted that the mass matrix of the multibody system is non-singular, the constraint matrices satisfy specific full rank conditions and that the friction is not too high.

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Yoon, Ji Won, Kab Jin Jun i Tae Won Park. "Dynamic Analysis and Fatigue Life Prediction of a Very Flexible Component in Multibody System". Key Engineering Materials 321-323 (październik 2006): 1597–600. http://dx.doi.org/10.4028/www.scientific.net/kem.321-323.1597.

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Recently, the finite element absolute nodal coordinate formulation(ANCF) was developed for large deformation analysis of flexible bodies in multi-body dynamics. This formulation is based on finite element procedures and the general continuum mechanics theory to represent elastic forces. In this paper, a computational method, which predicts the dynamic and structural properties of a very flexible beam in a multibody system, is presented based on Euler-Bernoulli beam theory and ANCF. In order to consider the dynamic interaction between a continuous large deformable beam and a rigid multibody system, a combined system equations of motion was derived by adopting absolute nodal coordinates and rigid body coordinates. The efficiency and reliability of the computational results are verified by comparison with a commercial program. These methods can be applied for predicting the dynamic stress and fatigue life of the wire harness used in a robot system. The process of predicting the fatigue life using the proposed method in this paper may be applied to continuous mechanical parts of various dynamic systems.

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Shabana, A. A. "Dynamics of Flexible Bodies Using Generalized Newton-Euler Equations". Journal of Dynamic Systems, Measurement, and Control 112, nr 3 (1.09.1990): 496–503. http://dx.doi.org/10.1115/1.2896170.

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A force acting on a rigid body produces a linear acceleration for the whole body together with an angular acceleration about its center of mass. This result is in fact Newton-Euler equations which are used as basis for developing many recursive formulations for open loop multibody systems consisting of interconnected rigid bodies. In this paper, generalized Newton-Euler equations are developed for deformable bodies that undergo large translational and rotational displacements. The configuration of the deformable body is identified using coupled sets of reference and elastic variables. The nonlinear generalized Newton-Euler equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial coordinates as well as the assumed displacement field. A set of intermediate reference frames having no mass or inertia are introduced for the convenience of defining various joints between interconnected deformable bodies. The use of the obtained generalized Newton-Euler equations for developing recursive dynamic formulation for open loop deformable multibody systems containing revolute, prismatic and cylindrical joints is also discussed. The development presented in this paper demonstrates the complexities of the formulation and the difficulties encountered when the equations of motion are defined in the joint coordinate systems.

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Palomba, Ilaria, i Renato Vidoni. "Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion". Applied Sciences 9, nr 23 (28.11.2019): 5156. http://dx.doi.org/10.3390/app9235156.

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The dynamics of flexible multibody systems (FMBSs) is governed by ordinary differential equations or differential-algebraic equations, depending on the modeling approach chosen. In both the cases, the resulting models are highly nonlinear. Thus, they are not directly suitable for the application of the modal analysis and the development of modal models, which are very useful for several advanced engineering techniques (e.g., motion planning, control, and stability analysis of flexible multibody systems). To define and solve an eigenvalue problem for FMBSs, the system dynamics has to be linearized about a selected configuration. However, as modal parameters vary nonlinearly with the system configuration, they should be recomputed for each change of the operating point. This procedure is computationally demanding. Additionally, it does not provide any numerical or analytical correlation between the eigenpairs computed in the different operating points. This paper discusses a parametric modal analysis approach for FMBSs, which allows to derive an analytical polynomial expression for the eigenpairs as function of the system configuration, by solving a single eigenvalue problem and using only matrix operations. The availability of a similar modal model, which explicitly depends on the system configuration, can be very helpful for, e.g., model-based motion planning and control strategies towards to zero residual vibration employing the system modal characteristics. Moreover, it allows for an easy sensitivity analysis of modal characteristics to parameter uncertainties. After the theoretical development, the method is applied and validated on a flexible multibody system, specifically using the Equivalent Rigid Link System dynamic formulation. Finally, numerical results are presented and discussed.

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Rückwald, Tobias, Alexander Held i Robert Seifried. "Flexible multibody impact simulations based on the isogeometric analysis approach". Multibody System Dynamics 54, nr 1 (12.11.2021): 75–95. http://dx.doi.org/10.1007/s11044-021-09804-x.

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AbstractUsually detailed impact simulations are based on isoparametric finite element models. For the inclusion in multibody dynamics simulation, e.g., in the framework of the floating frame of reference, a previous model reduction is necessary. A precise representation of the geometry is essential for modeling the dynamics of the impact. However, isoparametric finite elements involve the discretization of the geometry. This work tests isogeometric analysis (IGA) models as an alternative approach in the context of impact simulations in flexible multibody systems. Therefore, the adaption of the flexible multibody system procedure to include IGA models is detailed. The use of nonuniform rational basis splines (NURBS) allows the exact representation of the geometry. The degrees of freedom of the flexible body are afterwards reduced to save computation time in the multibody simulation. To capture precise deformations and stresses in the area of contact as well as elastodynamic effects, a large number of global shape functions is required. As test examples, the impact of an elastic sphere on a rigid surface and the impact of a long elastic rod are simulated and compared to reference solutions.

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Zhang, Jian Hua, i Shou Shan Jiang. "Rigid-Flexible Coupling Model and Dynamic Analysis of Rocket Sled". Advanced Materials Research 346 (wrzesień 2011): 447–54. http://dx.doi.org/10.4028/www.scientific.net/amr.346.447.

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The Dynamics Analysis & Simulation of the Rocket Sled were done based on Multibody System Dynamics and Finite Element Analysis Theory. The most difficult work in the analysis was establishing the boundary conditions of the rocket sled. The paper made this kind of attempt. Then the relevant post processing figures and data were obtained, thereby providing the designer and manufacturer with detailed and reliable data. The conclusion is the simulation method is more effective than those before and the boundary conditions is correct and acceptable.

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31

Köthe, Alexander, i Robert Luckner. "Applying Eigenstructure Assignment to Inner-Loop Flight Control Laws for a Multibody Aircraft". CEAS Aeronautical Journal 13, nr 1 (21.12.2021): 33–43. http://dx.doi.org/10.1007/s13272-021-00549-z.

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AbstractUnmanned aircraft used as high-altitude platform system has been studied in research and industry as alternative technologies to satellites. Regarding actual operation and flight performance of such systems, multibody aircraft seems to be a promising aircraft configuration. In terms of flight dynamics, this aircraft strongly differs from classical rigid-body and flexible aircraft, because a strong interference between flight mechanic and formation modes occurs. For unmanned operation in the stratosphere, flight control laws are required. While control theory generally provides a number of approaches, the specific flight physics characteristics can be only partially considered. This paper addresses a flight control law approach based on a physically exact target model of the multibody aircraft dynamics rather than conventionally considering the system dynamics only. In the target model, hypothetical spring and damping elements at the joints are included into the equations of motion to transfer the configuration of a highly flexible multibody aircraft into one similar to a classical rigid-body aircraft. The differences between both types of aircraft are reflected in the eigenvalues and eigenvectors. Using the eigenstructure assignment, the desired damping and stiffness are established by the inner-loop flight control law. In contrast to other methods, this procedure allows a straightforward control law design for a multibody aircraft based on a physical reference model.

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32

Wang, Zhe, Qiang Tian i Haiyan Hu. "Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters". Nonlinear Dynamics 84, nr 2 (23.11.2015): 527–48. http://dx.doi.org/10.1007/s11071-015-2504-4.

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Cardona, A., M. Geradin i D. B. Doan. "Rigid and flexible joint modelling in multibody dynamics using finite elements". Computer Methods in Applied Mechanics and Engineering 89, nr 1-3 (sierpień 1991): 395–418. http://dx.doi.org/10.1016/0045-7825(91)90050-g.

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Zhang, Z. W., Y. X. Wu, J. G. Liu, W. Ren i M. H. Cao. "Research on the rigid-flexible multibody dynamics of concrete placing boom". Automation in Construction 67 (lipiec 2016): 22–30. http://dx.doi.org/10.1016/j.autcon.2016.03.009.

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Castellani, Michele, Jonathan E. Cooper i Yves Lemmens. "Flight Loads Prediction of High Aspect Ratio Wing Aircraft Using Multibody Dynamics". International Journal of Aerospace Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/4805817.

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A framework based on multibody dynamics has been developed for the static and dynamic aeroelastic analyses of flexible high aspect ratio wing aircraft subject to structural geometric nonlinearities. Multibody dynamics allows kinematic nonlinearities and nonlinear relationships in the forces definition and is an efficient and promising methodology to model high aspect ratio wings, which are known to be prone to structural nonlinear effects because of the high deflections in flight. The multibody dynamics framework developed employs quasi-steady aerodynamics strip theory and discretizes the wing as a series of rigid bodies interconnected by beam elements, representative of the stiffness distribution, which can undergo arbitrarily large displacements and rotations. The method is applied to a flexible high aspect ratio wing commercial aircraft and both trim and gust response analyses are performed in order to calculate flight loads. These results are then compared to those obtained with the standard linear aeroelastic approach provided by the Finite Element Solver Nastran. Nonlinear effects come into play mainly because of the need of taking into account the large deflections of the wing for flight loads computation and of considering the aerodynamic forces as follower forces.

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36

Jian, Shen, Han Feng, Chen Fang, Zhou Qiao i Pavel M. Trivailo. "Dynamics and modeling of rocket towed net system". Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 232, nr 1 (13.10.2016): 185–97. http://dx.doi.org/10.1177/0954410016673090.

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In order to study the complex dynamical behavior of the rocket towed net system, a three-dimensional model consisting of a rigid rocket model and a lumped mass net model is built based on the aerodynamics theory. The rocket towed net system model is solved by the fourth-order Runge–Kutta method in simulation. Simulation and experimental results show that the accuracies of rocket towed net system expanding distance were about 90% of the system length. With the comparison of simulation, a rigid multibody model and experimental results in rocket mass center trajectory, velocity, and pitch angle, the dynamical characteristics of rocket towed net system have been basically studied. It illustrates that the lumped mass model simulates the real rocket towed net system flying test better than the rigid multibody model. It also shows that the dynamical parameters of rocket towed net system flight have an impact on the system in the whole flying process. Constitutive model of flexible net mesh-belts can be considered in the future research studies.

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Choi, JaeWon, DuHyun Gong, Junhee Lee, Chongam Kim i SangJoon Shin. "Simulation of the flapping wing aerial vehicle using flexible multibody dynamics". International Journal of Micro Air Vehicles 13 (styczeń 2021): 175682932110433. http://dx.doi.org/10.1177/17568293211043305.

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An insect-type flapping wing micro aerial vehicle offers high aerodynamic efficiency and maneuverability in confined spaces. The complicated aerodynamic/structural behavior of flapping wing micro aerial vehicle, however, causes difficulties regarding the dynamic control and parametric design. This paper develops a moderately accurate numerical framework taking into account the passive motion of the main wings. Finite-element-based multibody dynamics and two-dimensional unsteady aerodynamics are combined to simulate the hover of a flapping wing micro aerial vehicle. In addition, flexible and rigid wings are compared through numerical simulation considering the flexibility. In terms of the average thrust, numerical simulation by fluid–structure interaction shows good agreements against the experimental results within 5% discrepancy.

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38

Samanta, B. "Dynamics of Flexible Multibody Systems Using Bond Graphs and Lagrange Multipliers". Journal of Mechanical Design 112, nr 1 (1.03.1990): 30–35. http://dx.doi.org/10.1115/1.2912575.

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A procedure is presented to study the dynamics of interconnected flexible systems using bond graphs. The concept of Lagrange multipliers, which are commonly used in analysis of constrained systems, is introduced in the procedure. The overall motions of each of the component bodies are described in terms of large rigid body motions and small elastic vibrations. Bond graphs are used to represent both rigid body and flexible dynamics of each body in a unified manner. Bond graphs of such sub-systems are coupled to one another satisfying the kinematic constraints at the interfaces to get the overall system model. Constraint reactions are introduced in the form of Lagrange multipliers at the interfaces without disturbing the integral causality in the subsystem models, which leads to easy derivation of system equations. The equations of motion and higher derivatives of the constraint relations are integrated to obtain the constraint reactions and the system response. The procedure is illustrated by an example system and results are in good agreement with those presented earlier.

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39

RASTI, A., i S. A. FAZELZADEH. "MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE". International Journal of Structural Stability and Dynamics 12, nr 06 (grudzień 2012): 1250049. http://dx.doi.org/10.1142/s0219455412500496.

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In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

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40

Wan, Lirong, Shuai Zhang, Zhaosheng Meng i Yunyue Xie. "Analysis of the Protection Performance of Face Guard for Large Mining Height Hydraulic Support". Shock and Vibration 2021 (12.03.2021): 1–16. http://dx.doi.org/10.1155/2021/6631017.

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With the increase of mining height, the problem of coal wall spalling in the working face gradually worsens. Hydraulic support and its face guard structure are the key pieces of equipment to restrain the coal wall spalling. However, at present, the hydraulic jack is mostly considered as rigid in the analysis of protection mechanism. This simplification cannot effectively reflect the true bearing state of the face guard. In order to improve the accuracy of analysis, this study considers the face guard jack as a flexible spring and establishes a rigid-flexible coupling analysis model of the face guard mechanism. First, based upon the multibody dynamics software ADAMS®, the multibody numerical model of the face guard of the hydraulic support was established. The influence of the two kinds of structures on the coal wall disturbance was analyzed and compared. Then, the rigid model was meshed. The hydraulic jacks were equivalent to the spring system, and the rigid-flexible coupling model was established. Based upon the application load on different positions of the rigid-flexible model, the load-bearing characteristics and hinge point force transfer characteristics of the two face guards were analyzed. The results show that the support efficiency of the integral type was higher than that of the split type. In the vertical support attitude, the dynamic disturbance of the coal wall, produced by the two kinds of face guards, was small. The four-bar linkage effectively improved the ultimate bearing capacity of the integral face guard. The results provide theoretical support for the design and optimization of the face guard.

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41

Lidström, Per. "Frames of reference in multibody dynamics". Mathematics and Mechanics of Solids 24, nr 1 (7.11.2017): 98–151. http://dx.doi.org/10.1177/1081286517731485.

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In this paper, a discussion is undertaken concerning the use of so-called floating frames of reference in the calculation of the kinetic and elastic energies of parts in a multibody system. The use of floating frames may simplify the calculation of the elastic energy, although sometimes at the expense of more elaborate expressions for the kinetic energy. These expressions may involve terms that couple the motion of the floating frame and the relative motion of the part. The choice of a floating frame may be arbitrary but in order to obtain as simple expressions as possible some care must be taken. When a (flexible) part is connected to a rigid part one may use a frame in which the rigid part is at rest. If so then one has, in general, to deal with coupling terms in the kinetic energy for the flexible part. There is one unique frame in which these coupling terms disappear. This frame is called the principal frame of reference. Relative to this frame the kinetic energy of the part is minimal compared to the kinetic energy relative to other frames. Two independent proofs of this property are presented. The principal frame is defined by the associated change of frame mapping. This mapping is given a full characterization. It may however be cumbersome to calculate the kinetic energy relative to the principal frame. A method for doing this is designated. A frame that has been given some attention in the literature is the principal axis frame of reference. In this paper, a full characterization of this frame and its relation to the principal frame is given. Two examples of an Euler–Bernoulli beam in rotational motion are presented and compared in the light of the theoretical findings of this paper. In conventional presentations of mechanics the Euclidean spaces associated with different frames of reference are taken to be identical. In this paper this assumption is abandoned and different frames of reference will correspond to different Euclidean spaces. From a conceptual point of view this is a natural step to take in order to increase clarity and generality. It automatically includes the dependence of the reference placement on the frame of reference. This approach has been analyzed in a previous paper by the present author. References to this paper will appear whenever needed for. Consequences of this approach are investigated in terms of transformation formulas for kinematical and dynamical quantities.

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42

Nakanishi, Toshikazu, Xuegang Yin i A. A. Shabana. "Dynamics of Multibody Tracked Vehicles Using Experimentally Identified Modal Parameters". Journal of Dynamic Systems, Measurement, and Control 118, nr 3 (1.09.1996): 499–507. http://dx.doi.org/10.1115/1.2801173.

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The mode shapes, frequencies, and modal mass and stiffness coefficients of multibody systems such as tracked vehicles can be determined using experimental identification techniques. In multibody simulations, however, knowledge of the modal parameters of the individual components is required, and consequently, a procedure for extracting the component modes from the mode shapes of the assembled system must be used if experimental modal analysis techniques are to be used with general purpose multibody computer codes. In this investigation, modal parameters (modal mass, modal stiffness, modal damping, and mode shapes), which are determined experimentally, are employed to simulate the nonlinear dynamic behavior of a multibody tracked vehicle which consists of interconnected rigid and flexible components. The equations of motion of the vehicle are formulated in terms of a set of modal and reference generalized coordinates, and the theoretical basis for extracting the component modal parameters of the chassis from the modal parameters of the assembled vehicle is described. In this investigation, the track of the vehicle is modeled as a closed kinematic chain that consists of rigid links connected by revolute joints, and the effect of the chassis flexibility on the motion singularities of the track is examined numerically. These singularities which are encountered as the result of the change in the track configuration are avoided by using a deformable secondary joint instead of using the loop-closure equations.

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43

Tian, Qiang, Jie Lou i Aki Mikkola. "A new elastohydrodynamic lubricated spherical joint model for rigid-flexible multibody dynamics". Mechanism and Machine Theory 107 (styczeń 2017): 210–28. http://dx.doi.org/10.1016/j.mechmachtheory.2016.09.006.

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Shao, Haoyuan, Zi Kan, Yifeng Wang, Daochun Li, Zhuoer Yao i Jinwu Xiang. "Dynamic Analysis and Numerical Simulation of Arresting Hook Engaging Cable in Carried-Based UAV Landing Process". Drones 7, nr 8 (13.08.2023): 530. http://dx.doi.org/10.3390/drones7080530.

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Carrier-based unmanned aerial vehicles (UAVs) require precise evaluation methods for their landing and arresting safety due to their high autonomy and demanding reliability requirements. In this paper, an efficient and accurate simulation method is presented for studying the arresting hook engaging arresting cable process. The finite element method and multibody dynamics (FEM-MBD) approach is employed. By establishing a rigid–flexible coupling model encompassing the UAV and arresting gear system, the simulation model for the engagement process is obtained. The model incorporates multiple coordinate systems to effectively capture the relative motion between the rigid and flexible components. The model considers the material properties, arresting gear system characteristics, and UAV state during engagement. Verification is conducted by comparing simulation results with experimental data from a referenced arresting hook rebound. Finally, simulations are performed under different touchdown points and roll angles of the UAV to analyze the stress distribution of the hook, center of gravity variations, and the tire touch and rollover cable response. The proposed rigid–flexible coupling arresting dynamics model in this paper enables the effective analysis of the dynamic behavior during the arresting hook engaging arresting cable process.

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45

Campanelli, Marcello, Marcello Berzeri i Ahmed A. Shabana. "Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems". Journal of Mechanical Design 122, nr 4 (1.09.1999): 498–507. http://dx.doi.org/10.1115/1.1289636.

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Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

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46

Ma, D., i H. M. Lankarani. "A Multibody/Finite Element Analysis Approach for Modeling of Crash Dynamic Responses". Journal of Mechanical Design 119, nr 3 (1.09.1997): 382–87. http://dx.doi.org/10.1115/1.2826359.

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Computer models of the human body are robust tools for gaining insight into the gross motion of ground vehicle or aircraft occupants and evaluating the loads and, deformations of their critical parts. The knowledge of occupant responses will help in the determination of the type and probable causes of injuries that may he sustained during a crash. An important aspect in crash analysis is how the large motion of the relatively rigid segments of an occupant, such as the limbs, and the small deformations of flexible segments, such as the spine column, are interrelated. To this end, a general methodology for kineto-static analysis of multibody systems with flexible structures undergoing large motion and structural deformations is developed. Rigid multibody dynamics is used to predict the gross motions and displacements at the boundaries between the relatively bulky (rigid) bodies and relatively flexible ones. A mixed boundary-condition finite-element analysis is formulated and solved at every numerical integration time to determine the corresponding reaction forces and moments at the boundaries and also the structural deformations. Based on this methodology, a multibody model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II anthropomorphic crash test dummy. The analytical results obtained are compared with the experimental results from the impact sled tests. Comparison of the results has shown better correlation between the analyses and the experiments compared with earlier studies.

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47

Zhao, Bo, Zhi-Nan Zhang i Xu-Dong Dai. "Modeling and prediction of wear at revolute clearance joints in flexible multibody systems". Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, nr 2 (18.04.2013): 317–29. http://dx.doi.org/10.1177/0954406213486384.

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This article proposes a numerical approach for the modeling and prediction of wear at revolute clearance joints in flexible multibody systems by integrating the procedures of wear prediction with multibody dynamics. In the approach, the flexible component is modeled based on the absolute nodal coordinate formulation. The contact force in the clearance joint is applied using the continuous contact force model proposed by Lankanrani and Nikravesh and the friction effect is considered using the LuGre friction model. The simulation of wear is performed by an iterative wear prediction procedure based on Archard’s wear model. The radial basis function neural network technique is employed to deal with the pin-on-disc experimental data for obtaining the wear coefficient used in the wear prediction procedure at different contact conditions. The comparison of the wear predicted at the clearance joint in the rigid and flexible planar slider-crank mechanisms demonstrates that the proposed approach can be used to model and predict wear at revolute clearance joints in flexible multibody systems, and the wear result predicted is slightly reduced after taking the flexibility of components into account.

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48

Juhasz, Ondrej, Roberto Celi i Mark B. Tischler. "Flight Dynamics Simulation Modeling of a Large Flexible Tiltrotor Aircraft". Journal of the American Helicopter Society 67, nr 2 (1.04.2022): 1–16. http://dx.doi.org/10.4050/jahs.67.022003.

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A high-order tiltrotor mathematical model is developed and validated against flight-test data for the XV-15 and simulations of a large civil tiltrotor (LCTR) concept. Rigid body and inflow states, as well as flexible wing and blade states are used in the analysis. Wing flexibility is important when modeling large aircraft where structural modes effect the frequency range of interest for flight control, generally 1–20 rad/s. Details of the formulation of the mathematical model are given, including derivation of structural, aerodynamic, and inertial loads. A novel "quasi-multibody" approach, based on numerical kinematics but without equations of constraints, allows the modeling of complex, flexible aircraft configurations in an easy to set up, and computationally very efficient manner. Assessments of the effects of wing flexibility are given. Flexibility effects are also evaluated by looking at the nature of the couplings between rigid body modes and wing structural modes.

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Guo, Yongbo, i Fansheng Wang. "Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode". Symmetry 12, nr 9 (10.09.2020): 1489. http://dx.doi.org/10.3390/sym12091489.

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Based on the rigid finite element method and multibody dynamics, a discrete model of a flexible conveyor belt considering the material viscoelasticity is established. RFE (rigid finite element) and SDE (spring damping element) are used to describe the rigidity and flexibility of a conveyor belt. The dynamic differential equations of the RFE are derived by using Lagrange’s equation of the second kind of the non-conservative system. The generalized elastic potential capacity and generalized dissipation force of the SDE are considered. The forward recursive formula is used to construct the conveyor belt model. The validity of dynamic equations of conveyor belt is verified by field test. The starting mode of the conveyor is simulated by the model.

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Yu, Zheng Ning, Wen Long Li i Yu Shan Zhao. "A New Dynamics Model of Structure-Changing Multibody Systems with Flexible Appendages". Advanced Materials Research 479-481 (luty 2012): 715–19. http://dx.doi.org/10.4028/www.scientific.net/amr.479-481.715.

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According to engineering requirements, structure of multibody systems would change and lead to obvious modifications in dynamics model obtained by traditional methods. In this paper, an existing method for rigid body systems with changing structures is extended to systems with flexible appendages, which is more common in engineering. Application of the new generalized approach to a typical 4-body system is actualized. Numerical simulation is carried out and results are the same as theoretical analysis, which indicate the availability and applicability of the new approach.

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