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Flexible multibody dynamics: Review of past and recent develop-ments
... The study of spacecraft dynamics with flexible accessories, such as spaceborne antennas and space manipulators, can also provide a certain reference for the study in this paper [11][12][13][14][15]. The multibody system is divided into multi-rigid body system and multi-flexible body system, the development of multi-rigid body dynamics theory has become mature today, and the multi-flexible body system is the majority in practical problems, and the theoretical method of multi-flexible body dynamics can be mainly divided into floating coordinate system method and absolute node coordinate method [16][17][18][19]. To establish the dynamic model of a rigid-flexible multi-body system, it is necessary to describe the rigid body motion and the deformation motion of the flexible body separately [20][21][22][23][24]. ...
... T 00 0 00 (16) Here, Tθ represents the transformation matrix from the Cartesian coordinate system to the curvilinear coordinate system, where θ denotes the angle between the coordinate axes of the curvilinear coordinate system. ...
... 16 quantity of flexible interfaces and reconfigurable rigid body modules in the mast system directly correlates with more drastic force variations at the deployment joint and nonlinear amplification of endpoint vibrations. Thinner interface shell thickness weakens the resistance to deformation, leading to increased endpoint vibrations and slower attenuation rates. ...
This paper studied the dynamic characteristics of the deployment process of the replaceable interface mast and the vibration generated at the end of the mast. The interface is discretized by the absolute nodal coordinate thin shell element with gradient reduction, and the three-dimensional motion of the reconfigurable rigid body module is described by the natural coordinate method. The dynamic equations of the interface and the reconfigurable module without external constraints are established by the principle of virtual work. Finally, the dynamic models of each part are assembled into the dynamic model of the mast system deployment process through the system constraint equation. According to the established dynamic model of the mast system deployment process, the dynamic behavior changes of the replaceable interface mast are analyzed. At the same time, the differences in the deployment behavior of the replaceable interface mast under different system configuration schemes, flexible interface geometric parameters, and different driving laws are studied and compared. It provides guidance for the scheme configuration of the mast system, the structural design of the interface, and the deployment motion planning of the mast. According to the physical prototype and the assembly of the mast system model, the deployment process of the replaceable interface mast is experimentally analyzed. It shows that the established dynamic model of the mast system can correctly analyze the dynamic characteristics of the deployment behavior of the replaceable interface mast, which provides a reference for the design and behavior analysis of the mast system.
... The most widely used approach is the floating frame of reference formulation (FFR). Many research computer codes and commercial simulation softwares are based on it [5]. ...
... This formulation uses two sets of coordinate to identify the configuration of any point on the deformable body. Reference coordinates to define the location and the orientation of the body reference in the inertial frame, and elastic coordinates to captures the body deformation with respect to the body local reference [3] [5]. However, the FFR formulation is restricted only for small deformation problems, because it uses infinitesimal rotations as nodal coordinates [6]. ...
... To solve large deformation problem in flexible multibody dynamics , a relatively recent approach was introduced. This approach is known in literature as the Absolute Nodal Coordinate Formulation (ANCF) [3] [5]. Unlike the FFR formulation, the location and the deformation of a material point on a flexible body when ANCF is used, are described directly in the global coordinate system, using a global element shape function, and nodal coordinates vector. ...
The finite element Floating Frame of Reference Formulation is the most widely used method in dynamic simulation of flexible multibody systems. However, this method is limited only for small deformation analysis. To solve large deformation problems a relatively recent approach called the Absolute Nodal Coordinate Formulation was introduced and have been a subject of many investigations. In this approach infinitesimal rotations are not used as nodal coordinates, only absolute displacements and global slopes of nodes are adopted as coordinates. In this work, the Absolute Nodal Coordinate Formulation is used to implement an efficient computational code for dynamic simulation and analysis of planar flexible multibody systems. For validation purpose, the implemented code is used to perform a dynamic simulation of the slider-crank mechanism with a flexible connecting rod.
... In view of the described complexities, employing a modeling approach that results in an optimized derivation algorithm for motion equations in terms of calculation complexity is important. Therefore, by comparing the computational complexity of dynamic formulations for deriving the motion equations [8], the extended recursive Gibbs-Appell (G-A) formulation is preferred over Euler-Lagrange (E-L) and Hamilton and Newton-Euler (N-E) equations as well as Kane's formulation. Book [9] derived the motion equations of a manipulator with N flexible links aided by the recursive E-L equations. ...
... In view of better operation of strain gauges, the strain gauge circuit is located on the wagon of prismatic joint in the nearest possible distance. To calculate the elastic deformation generated in the link during motion, the approach described in Ref. [8] is utilized: in view of boundary condition and placement of strain gauges on the link, a relation is fitted using its elastic deformation. Using the obtained data, the elastic deformation is calculated at each instant of time. ...
... Using the obtained data, the elastic deformation is calculated at each instant of time. Unlike Ref. [8], to obtain the link elastic deformation of this robot, the effective length of robot link varies with time. Consequently, the boundary conditions and obtained values at any moment should be recalculated. ...
In this paper, the dynamic model and vibration analysis of a flexible manipulator composed of N elastic links and robot actuators for which structural vibration is considered are investigated. In view of the concurrent linear and rotary motions of the link caused by revolute-prismatic joints, the interaction of joint’s structural vibration and link fluctuation is taken as an effective model parameter. Utilization of prismatic joint with hub in the manipulator’s structure in question, its significant length and low weight result in the operation of hub akin to a flexible link attached to revolute joint. To model the expressed hub oscillation with respect to the link, the assumed modes method and mode shapes of Euler-Bernoulli beam with independent generalized modal coordinate with respect to the link are employed. Noting the complexity of the present model relative to studied flexible manipulators, the recursive Gibbs-Appell formulation is used to derive the motion equations. Therefore, the dynamic equations of hub are of time-independent form, while these equations are obtained as time-variable for links. Although the obtained equations are simultaneously solved in the coupled form, the derived equations for a single-link flexible manipulator are simulated in three cases of (1) rigid hub, (2) elastic hub, and (3) elastic hub and flexible joints. The simulation results are compared with a similar experimental setup, indicating that the flexible manipulator model with elastic link, hub, and joints yields satisfactory results with an error of less than 1 mm and 1° in the longitudinal and rotational motions, respectively. Moreover, the results of lateral vibration show good accuracy, with the endpoint of the robot having perfect precision with just 1% tolerance.
... [8]). In order to also represent bending behaviour of cables their discretization using the finite segment method [12,15] or so called rigid finite elements [18] is possible. Other more complex approaches can utilize nonlinear 3D finite elements [4,20] or can employ elements based on the absolute nodal coordinate formulation (ANCF) [15]. ...
... In order to also represent bending behaviour of cables their discretization using the finite segment method [12,15] or so called rigid finite elements [18] is possible. Other more complex approaches can utilize nonlinear 3D finite elements [4,20] or can employ elements based on the absolute nodal coordinate formulation (ANCF) [15]. Many cable applications are influenced by contact forces including frictional behaviour. ...
... Generally, vector g(q,q, t) contains the sum of all generalized, potential and dissipative forces and appropriate other derivatives of a kinetic energy, while vector (q,q, t) follows from the differentiation of the constraint equations (see [15]). The approaches to the numerical solution of equation (22) used practically in this paper are summarized in [6]. ...
It is reasonable to develop models and to investigate the dynamic behaviour of systems composed of cables since cable vibration can have an important effect on the motion of these mechanical systems. This paper deals with the application of the nonlinear formulation for flexible body dynamics called the absolute nodal coordinate formulation (ANCF). It is used for modelling the systems composed of cables, pulleys, other rigid bodies and a motor with prescribed motion. The ANCF was chosen as a suitable approach, which that can allow to consider a detailed interaction of the cable and the pulley with its nonlinear dynamical behaviour. The ANCF uses absolute positions of nodes (reference vectors) and slopes (reference vector derivations) as a set of nodal coordinates. An in-house modelling tool in the MATLAB system was created based on the proposed modelling methodology and two case studies were performed. A simple system containing a pulley and a cable with two attached bodies was used in order to test the simulation tool based on the proposed modelling methodology with respect to different parameters. A more complex mechanical system composed of a driven weight joined with a motor by a cable led over a pulley was numerically and also experimentally investigated. The comparison of obtained numerical and experimental results shows sufficient agreement and proves that the proposed modelling approach can be used for dynamic analyses of such systems.
... Dynamics of flexible multibody systems has been vigorously studied over the last decades in connection with various application fields such as light weight robotic arm, rotating machinery, helicopter rotor with flexible blades, large flexible space structures, and so on. We can find hundreds of papers about the writing, solving, and simulation of the differential and algebraic equations governing the dynamics of flexible multibody systems [21,22]. We can distinguish methods used to build a validation model including nonlinear terms (centrifugal, Coriolis) [4,12] and methods used to derive a preliminary design model, which can be also used for control design [10,14]. ...
... conditions. Indeed, the Lagrange formulation of dynamics using the assumed mode method (AMM) or finite element method (FEM) leads to a generalized second-order differential equation [22,28]: ...
... Closed-loop multibody system Such mechanisms are commonly modeled through the Lagrange approach using Lagrange multipliers [22,24]. They are used to be denoted λ and correspond to the generalized constraint forces. ...
We present a new methodology to derive a linear model of flexible multibody system dynamics. This approach is based on the two-port model of each body allowing the model of the whole system to be built just connecting the inputs/outputs of each body model. Boundary conditions of each body can be taken into account through inversion of some input–output channels of its two-port model. This approach is extended here to treat the case of closed-loop kinematic mechanisms. Lagrange multipliers are commonly used in an augmented differential-algebraic equation to solve loop-closure constraints. Instead, they are considered here as a model output that is connected to the adjoining body model through a feedback. After a summary of main results in the general case, the case of planar mechanisms with multiple uniform beams is considered, and the two-port model of the Euler–Bernoulli beam is derived. The choice of the assumed modes is then discussed regarding the accuracy of the first natural frequencies for various boundary conditions. The overall modeling approach is then applied to the well-known four-bar mechanism.
... Flexible multibody dynamics refers to computational strategies employed to determine long time histories of motion, deformation, strain, and stress of a group of interconnected rigid and deformable components, which are subjected to large rigid body translations and rotations due to applied forces, constraints, contacts, and initial conditions [1,2]. ...
... Based on the degree of flexibility taken into account, one can distinguish between three classes of multibody systems 1 . The simplest are rigid multibody systems, which consist exclusively of rigid bodies. ...
Corotational formulations play an important role for flexible multibody dynamics systems, because they reflect the nature of many technical systems undergoing arbitrarily large rigid body motions but small deformations within each body. This paper defines flexible multibody dynamics and corotational formulations in this context. Furthermore, the «ingredients» and workflow of a flexible multibody dynamics simulation are briefly addressed for the reader less familiar with the topic. This part also points to major review papers and textbooks in the field, and embeds the unified formulations in the literature. The paper's main part presents state-of-the-art corotational flexible multibody dynamics formulations in a systematic and unified way. In this formulation part, the standard integral-based floating frame of reference formulation with modal reduction and with the conventionally employed lumped mass approximation is presented, and its drawbacks highlighted. Then, the so-called nodal-based, i.e., space-wise discretized, equations of motion are presented for several up-to-date nodal-displacement-based formulations within a unified framework. This approach clearly shows the equivalence of the presented formulations, and highlights the fact that the formulations differ only in the choice of degrees of freedom. Moreover, this contribution also intends to reduce the information and complexity within the scientific literature, since this unified framework allows the derivation of these formulations with significantly less effort.
... Moreover, it is implemented in some commercial software such as ADAMS and DADS. These methods are obtained with extending the methodology for rigid system by assuming that the global motion of structural members is composed of a rigid body motion and a small deformation [4,7,8]. The main limitation of these methods is that structural members should be linear elasticity in the rigid body frame, which leads to the nonlinear effects such as geometric stiffening cannot be considered. ...
... Φq (8) where F is the external load. The finite element coordinates used above are absolute coordinates, and the total motion (rigid-body motion and elastic deformation) is directly referred to an inertial frame. ...
The thermal stress due to temperature rise may increase the element strain and internal forces, which may lead to the failure of the deployment of deployable structure. However, few works are carried out to study the dynamic performance of the flexible multibody under different temperatures, especially for retractable roof structures. Based on this idea and formulation of finite element methods, the dynamic equations of the system considering the temperature effect are derived. Then two examples, a parallel four-bar linkage and a non-parallel four-bar linkage, are presented in this paper to study the influence of thermal loading on the structural behavior during the motion. The results show that the effect of temperature changes is slight for the parallel four-bar linkage, but the influence is significant for the non parallel four-bar linkage. It can be concluded that the former case may be more suitable for the deployable structures.
... The literature on multibody systems covers a high variety of topics. Among the most theoretically shifted works, we find the development of new formulations345, the study of how to introduce flexible bodies and joints [6,7], the analysis of contacts and impacts [8,9] or efficiency improvements101112. Other works have focused on practical applications such as vehicle dynamics131415, robotics [16,17] or biomechanics [18,19]. ...
... This allows the dynamics of complex systems to be addressed, with open or closed chains, in a more general way, ensuring that the conditions of the system are satisfied. The simultaneous resolution of the dynamics and constraint equations has its own techniques (see the references [3,5,6]). Among the papers found in the literature dealing with state observers under this framework (on which the present paper focuses), only those two studies, [14,27], include experiments based on real systems. ...
This article addresses the problems of online estimations of kinematic and dynamic states of a mechanism from a sequence of noisy measurements. In particular, we focus on a planar four-bar linkage equipped with inertial measurement units (IMUs). Firstly, we describe how the position, velocity, and acceleration of all parts of the mechanism can be derived from IMU signals by means of multibody kinematics. Next, we propose the novel idea of integrating the generic multibody dynamic equations into two variants of Kalman filtering, i.e., the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), in a way that enables us to handle closed-loop, constrained mechanisms, whose state space variables are not independent and would normally prevent the direct use of such estimators. The proposal in this work is to apply those estimators over the manifolds of allowed positions and velocities, by means of estimating a subset of independent coordinates only. The proposed techniques are experimentally validated on a testbed equipped with encoders as a means of establishing the ground-truth. Estimators are run online in real-time, a feature not matched by any previous procedure of those reported in the literature on multibody dynamics.
... As mechanical/aeronautical systems become increasingly complex, linear and standard multibody approaches (e.g., floating frames of reference or corotational formulations) are no longer suitable for treating highly flexible slender structures. Consequently, advanced aeroelastic environments depend on more sophisticated structural models such as ANCFs [25,26], geometrically exact beam formulations [27,28], and their variants [29,30]. This is when using the second group of integration schemes, the implicit methods, becomes mandatory. ...
This paper presents the analytical linearization of aerodynamic loads (computed with the unsteady vortex-lattice method), which is formulated as tangent matrices with respect to the kinematic states of the aerodynamic grid. The loads and their linearization are then mapped to a nonlinear structural model by means of radial-basis functions, allowing for a two-way strong interaction scheme. The structural model comprises geometrically exact beams formulated in a director-based total Lagrangian description, circumventing the need for rotational degrees of freedom. The structural model is spatially discretized into finite elements and temporally discretized with the help of an implicit scheme that identically preserves momenta and energy. The resulting nonlinear discrete equations are solved by applying Newton's method, requiring calculating the Jacobians of the whole aeroelastic system. The correctness of the linearized loads is then shown by direct comparison with their numerical counterparts. In addition, we employ our strongly coupled aeroelastic model to investigate the nonlinear static and dynamic behavior of a suspension bridge. With this approach, we successfully investigate the numerical features of the aeroelastic system under divergence and flutter conditions.
... There are various methods adopted to simulate the dynamics of these manipulators, which include the Finite Element Method (FEM), Assumed Mode Method (AMM), and Finite Segment Method (FSM) etc. An extensive literature review has been conducted, encompassing these techniques applied to flexible manipulators ranging from single-link to multi-link manipulator systems [1,2]. The most used techniques for the dynamics of flexible manipulators are FEM and AMM. ...
Flexible manipulators offer several advantages over rigid manipulators, including light-in-weight, high payload-to-weight ratio, lower power consumption, and the ability to operate at high speeds. However, these manipulators are susceptible to structural vibrations, which can be effectively suppressed by implementing an appropriate controller. The paper presents a performance evaluation of such a controller for tracking a complex trajectory that combines curved and linear paths, in contrast to existing literature-focused on pure linear or circular trajectories without considering any payload. Before controlling, the dynamics of a two-flexible manipulator system are modeled using a hybrid Euler-Lagrangian formulation and DeNOC matrices, with validation. After that, to achieve precise trajectory tracking while suppressing vibrations, a hybrid controller is designed, integrating command shaping techniques with a proportional-derivative (PD) feedback controller. The results demonstrate the effectiveness of command shaping and its comparison to unshaped input commands. The controller’s performance is evaluated using a semicircular trajectory within a 3-second timeframe, considering both payload and non-payload cases. The proposed control scheme effectively suppresses vibrations in both payload scenarios. Detailed analysis of tip deflections for both links with shaped and unshaped input commands is provided, along with the quantification of vibration suppression along the trajectory. The impact of different configurations and payloads on the natural frequency during trajectory tracking is presented. The study also shows the tracking of B-Splines trajectories, highlighting the requirement of higher gains for such trajectories and evaluating the effect of link flexibility by tracking error analysis. Vibration suppression is achieved by implementing the controller while tracking such trajectories.
... The traditional finite element model based on modal superposition cannot analyze the large deformation motion of shearer cables in the towing system [1]. Shabana [2] firstly proposed the Absolute Nodal Coordinates Formulation (ANCF). The ANCF method takes the position and slope of each node in the global coordinate system as generalized coordinates, the mass matrix is a constant array, and the generalized force expression is simple, which has a greater advantage in solving the nonlinear large deformation problem of shearer cables. ...
The development of intelligent and unmanned coal mining has put forward higher requirements on the service life and dynamic reliability of shearer cables. However, it is difficult to comprehensively consider the complexity of hosting conditions of coal mining working face and the dynamic characteristics of cables in different towing systems in the design and development of cables. The cables are periodized by pitch and have the same cross-sectional structure and properties. Based on the homogenization theory and volume average principle, the cable was assumed to be an orthotropic elastomer, and the tensile experimental method and finite element method were combined to calibrate the cable equivalent mechanical parameters. Based on the Absolute Node Coordinate Formulation (ANCF) method, the rigid-flexible coupled virtual prototype co-simulation model of shearer cable towing system was constructed to obtain the kinetic and kinematic parameters of each node of the cable and study the dynamic gradual change characteristics of the cable in different working areas. This research method has an important theoretical significance and engineering application value for the acquisition of dynamic characteristic parameters of shearer cables and the optimal design and dynamic reliability of cables.
... The use of multi-body codes for the analysis and optimization of mechanical system is a common practice both in industrial and academic sectors. Multi-body codes allows simulating whichever mechanical system being useful for example for the design, the health monitoring and the optimization process [1,2]. ...
Explicit simulation is an extensively tool used by automotive designer both for car-performance analysis in standard operating condition and both in drive-simulator, whom exploits the computational ability of this codes to operate in real-time. The computational capability of these simulation tools however implies a reduced set of information available in simulation results. Regards automotive sector, with a particular focus on suspension system, the confined number of information in structural terms (states of solicitation on components and constraint) can be a strong limitation in their massive use. In this context, the objective of this activity is to propose a calculation method, as simple as accurate, that foresees to characterize the suspension by an implicit multibody model, and then using the information already available in explicit multibody model (wheel motion and external forces at the contact patch) allows estimating whichever force in all points of the suspension. The results obtained with the proposed method were compared with those outcoming from ADAMS/Car supplying in all cases, good results.
... where J R is the inertia of the blades and the hub, τ represents the transmission ratio between the high-speed and the low-speed shaft, J G denotes the generator inertia, θ is the rotor azimuth angle, Q aero and Q G denotes the aerodynamic torque and the generator torque respectively. The floating platform motion in six Degrees of Freedom (DOFs) ξ = (x, y, z, ρ, β, σ) T , namely surge, sway, heave, roll, pitch, yaw, can be calculated according to the multi-body dynamics theory [16] M ...
Floating Offshore Wind Turbines (FOWTs) are more prone to suffer from faults and failures than bottom-fixed counterparts due to the severe wind and wave loads typical of deep water sites. In particular, mooring line faults may lead to unacceptably high operation and maintenance costs due to the limited accessibility of FOWTs. Detecting the mooring line faults is therefore critical, but the application of Fault Detection (FD) techniques has not been investigated yet. In this paper, an FD scheme based on a wave-excited linear model is developed to detect in a reliable way critical mooring line faults occurring at the fairlead and anchor ends. To reach the goal, a linear model of the FOWT is obtained by approximating the wave radiation and incident wave forces. Based on this model, an observer is built to predict the rigid rotor and platform dynamics. The FD scheme is thus implemented by comparing the Mahalanobis Distance of the observer prediction error against a probabilistic detection threshold. Numerical simulations in some selected fault scenarios show that the wave-excited linear model can predict the FOWT dynamics with good accuracy. Based on this, the FD scheme capabilities are demonstrated, showing that it is able to effectively detect two critical mooring line faults.
... The rotating hub-flexible rod presented herein is a typical example. Therefore, more reliable mechanical models and their solution strategies for these coupling mechanisms have been required for accurate speed control and accurate operation [1][2][3]. ...
Rotating hub-flexible rod system is a typical rigid-flexible coupling dynamic mechanism, which has a wide range of industrial applications. In this paper, a comprehensive nonlinear dynamical model of a rotating hub-flexible rod-concentrated mass system considering rigid-flexible coupling effect is established to study its dynamic properties. By employing the Hamilton principle and classical beam theory, a set of differential equations of motion are derived including the couplings of the elastic deformation of the rod and the rigid rotation of the hub. The additional centrifugal force, tangential force and Coriolis force due to the rigid-flexible coupling effect are elaborated. The derived governing partial differential equations are solved by the Galerkin method. The validity of the present model is verified by a comparative study. The tip motion trajectories of the rod for the prescribed rotation, the dynamic responses of the hub and the rod for an external torque acting on the hub and the dimensionless natural frequencies of the system for the steady-state rotation are graphically presented. The influences of parameters such as rotational speed ratio, concentrated mass ratio, concentrated mass location ratio and initial eccentricity ratio on the dynamics are discussed in detail.
... where J R is the inertia of the blades and the hub, τ represents the transmission ratio between the high-speed and the low-speed shaft, J G denotes the generator inertia, θ is the rotor azimuth angle, Q aero and Q G denotes the aerodynamic torque and the generator torque respectively. The floating platform motion in six Degrees of Freedom (DOFs) ξ = (x, y, z, ρ, β, σ) T , namely surge, sway, heave, roll, pitch, yaw, can be calculated according to the multi-body dynamics theory [16] M ...
Floating Offshore Wind Turbines (FOWTs) are more prone to suffer from faults and failures than bottom-fixed counterparts due to the severe wind and wave loads typical of deep water sites. In particular, mooring line faults may lead to unacceptably high operation and maintenance costs due to the limited accessibility of FOWTs. Detecting the mooring line faults is therefore critical, but the application of Fault Detection (FD) techniques has not been investigated yet. In this paper, an FD scheme based on a wave-excited linear model is developed to detect in a reliable way critical mooring line faults occurring at the fairlead and anchor ends. To reach the goal, a linear model of the FOWT is obtained by approximating the wave radiation and incident wave forces. Based on this model, an observer is built to predict the rigid rotor and platform dynamics. The FD scheme is thus implemented by comparing the Mahalanobis Distance of the observer prediction error against a probabilistic detection threshold. Numerical simulations in some selected fault scenarios show that the wave-excited linear model can predict the FOWT dynamics with good accuracy. Based on this, the FD scheme capabilities are demonstrated, showing that it is able to effectively detect two critical mooring line faults.
... The first point is a consideration of replacement of elastic links in a part. According to the theory of the flexible multibody dynamics [27], our analysis seamlessly can be extended to the flexible multibody dynamics, which may improve the efficacy to absorb the ground reaction force when grounding of the leg. The second point is the ground reaction analysis to require an introduction of the extra component as the ground surface in the formulation of the multibody dynamics with multiple parameters such as tenderness and friction. ...
... Such systems are robots or mechanisms that can deflect due to external loads or internal body forces, whose motion is described by means of kineto-elastodynamic models, hereafter referred to as dynamic models [17]. Several contributions can be found in the literature on the modeling of such systems as well as survey papers [18,19] and books [20,21]. The system elastic behavior is represented by continuous ordinary and partial differential equations. ...
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.
... Within the multibody approach, dynamics of a flexible structure is usually simulated applying floating frame of reference method for description of its motion as absolutely rigid body. This method is presented, for example, in (Shabana, 1997). ...
Two simulation techniques for analyzing flexible wheelset dynamics are presented. They are applied within multibody approach and implemented in "Universal mechanism" software. Equations of wheelset motion are derived using floating frame of reference and component mode synthesis methods. Modal analysis is carried out in external FEA software. Kinematics of a wheel profile is described taking into account flexible displacements of wheelset nodes. In the first techniques, Lagrangian approach is applied to obtain all terms of equation of motion including inertia forces. In the second one, Eulerian approach is simulated in the stage of integration of equation of motion. Non-rotating finite element mesh of the wheelset is considered using the interpolation of flexible displacements in the nodes. The first simulation results obtained using both approaches are presented. These results confirm correctness of the suggested techniques.
... Within the multibody approach, dynamics of a flexible structure is usually simulated applying floating frame of reference method for description of its motion as absolutely rigid body. This method is presented, for example, in (Shabana, 1997). ...
The computer simulation of railway wheelset (WS) dynamics taking into account elasticity allows analyzing WSs high frequency oscillations and vibroaccelerations of parts connected with them, modeling wheelsets strain, estimating the WS deformation impact upon force distributions in gears of locomotive drives, and also solving other problems. In the paper there are offered two approaches to the analysis of the dynamics of elastic WSs within the limits of which their finite element models with the rotating and non-rotating grid are under consideration. The WS kinematics is presented as a sum of its motion as an absolute solid together with the jointed coordinates and small elastic displacements related to modal coordinates. There are presented algorithms for the calculation of generalized forces of moving loads in the wheel/rail contact taking into account WS elasticity. The first results of modeling confirming the correctness of the methods offered are shown.
... [6][7] [8]. This away speed up manufacturing prototype and spread out component from industry component to the market which is a wrench point in the simulation of internal combustion engine [6][9][10] [11] [12]. ...
... An Absolute Nodal Coordinate Formulation (ANCF) cable/ beam element that considers the axial and bending strains is used to model the flexible cables. The ANCF formulation does not make small deformation assumptions (Shabana, 1997) by taking absolute displacements and gradients as generalized coordinates. The mass matrix is kept constant and reduces the nonlinearity of the kinetic equations (Shabana, 2015). ...
For simulating contact interactions and high displacement gradients between the cable and the saddle at the middle tower of tripletower suspension bridges, a cable element is developed by combining the absolute nodal coordinate formulation and the quasi-conforming technique. New curvature strains are developed and elastic forces are explicitly formulated for the cable elements. Thereafter, it is compared to the original one to verify its locking remedies. The numerical solutions using the element are compared to analytical results and solutions by the original element. Compared to the original, the proposed element suppresses the high-frequency disturbances in the velocity and acceleration curves. Using the element, the contact and sliding behavior between the cable and the saddle is analyzed by employing parameters obtained experimentally. The saddle’s mechanical and frictional performance subjected to different friction coefficients and unbalanced cable forces is investigated. The proposed model exhibits excellent accuracy in the prediction of the sliding force and the contact status between the cable and the saddle.
... Here, the intrinsic reference system effectively becomes the co-rotational system for each element and global geometric information is recovered via quaternion integration from root to tip. The review papers of Wasfy & Noor [15] and Shabana [16] further capture the large body of flexible multibody dynamic modelling literature. ...
This paper proposes a low-order geometrically exact flexible beam formulation based on the utilization of generic beam shape functions to approximate distributed kinematic properties of the deformed structure. The proposed nonlinear beam shapes approach is in contrast to the majority of geometrically nonlinear treatments in the literature in which element-based—and hence high-order—discretizations are adopted. The kinematic quantities approximated specifically pertain to shear and extensional gradients as well as local orientation parameters based on an arbitrary set of globally referenced attitude parameters. In developing the dynamic equations of motion, an Euler angle parametrization is selected as it is found to yield fast computational performance. The resulting dynamic formulation is closed using an example shape function set satisfying the single generic kinematic constraint. The formulation is demonstrated via its application to the modelling of a series of static and dynamic test cases of both simple and non-prismatic structures; the simulated results are verified using MSC Nastran and an element-based intrinsic beam formulation. Through these examples, it is shown that the nonlinear beam shapes approach is able to accurately capture the beam behaviour with a very minimal number of system states.
... In the past tentative models might deal with flexibility by lumped parameters, but, while certainly better than rigid body approaches, they were not really adherent with the behaviour of realistic components with distributed flexibility. The distributed flexible multibody dynamics is often approached in case of small flexible deformations by means of the finite-element modelling method in component mode synthesis techniques [1,2], which can deal with very detailed and computationally expensive models, though reduced of size by keeping the contributions of their lower frequency structural dynamics only. In these approaches the size of the motion equations can be high to include the distributed flexibility, otherwise the general description can be poor as the speeds are raised, when the structural dynamics of lightweight components is excited. ...
Flexible multibody dynamics is often approached in case of small flexible deformations
by means of the finite-element modelling method in component mode synthesis
techniques to reduce the size of the computations.
In this work, by means of proper shape functions bases, the elastic behaviour of slender
beams is approximated in its local axial and transversal displacement components. Attention is paid to the inertial contributions to enhance the driving function. An ideal quick return mechanism is here modelled with the meshless approach. The results of different modelling choices are collected, to show how this shape functions based formulation can handle the added flexibility by means of a small set of degrees of freedom and motion equations. Some aspects are detailed and discussed.
Proceedings DOI: 10.6092/unibo/amsacta/5963
... On the other hand, flexible manipulators offer several advantages such as higher speed, better energy efficiency, improved mobility and higher payload-to-arm weight ratio. At high operational speeds, inertial forces of moving components become quite large, leading to considerable deformation in the flexible links, generating unwanted vibrations [1][2][3]. It is therefore challenging task to achieve a high accuracy end-effector motion in flexible manipulators due to unwanted structural vibrations. ...
... where [M] and [K] are the global mass and stiffness matrices respectively, and Q is the global force vector. The mass matrix and stiffness matrices can be found in [37,38,33]. The kinetic energy is calculated by ...
We consider the problem of selecting among different computational models of various fidelity for evaluating the objective and constraint functions in numerical design optimization. Typically, higher-fidelity models are associated with higher computational cost. Therefore, it is desirable to employ them only when necessary. We introduce a reference error formulation that aims at determining whether lower-fidelity models (that are computationally cheaper) can be used in certain areas of the design space as the latter is being explored during the optimization process. The proposed approach is implemented using an existing trust region model management framework. We demonstrate the link between feasibility and fidelity and the key features of the proposed approach using the design example of a cantilever flexible beam subject to high accelerations.
... 4 Multi-body dynamics with flexible body 4.1 Analysis parameters Improvements in computing power and methods have given rise to techniques that now exist to couple MBD and structural analysis through flexible elements. Techniques for deformable MBD modeling and simulation outlined in publications by Shabana (1997), Bauchau (2011) and Cardona (2000) have been implemented in modern FE software packages with practical example results from Xingguo et al. (2007). Although Fig. 4 Joint modeling techniques and effects on analysis these concepts have existed for some time, no work has been performed for landing gear slave link systems. ...
Aircraft landing gear assemblies comprise of various subsystems working in unison to enable functionalities such as taxiing, take-off and landing. As development cycles and prototyping iterations begin to shorten, it is important to develop and improve practical methodologies to meet certain design metrics. This paper presents an efficient methodology that applies high-fidelity multi-disciplinary design optimization techniques to commercial landing gear assemblies, for weight, cost, and structural performance by considering both structural and dynamic behaviours. First, a simplified landing gear assembly model was created to complement with an accurate slave link subassembly, generated based of drawings supplied from the industrial partner, Safran Landing Systems. Second, a Multi-Body Dynamic (MBD) analysis was performed using realistic input motion signals to replicate the dynamic behaviour of the physical system. The third stage involved performing topology optimization with results from the MBD analysis; this can be achieved through the utilization of the Equivalent Static Load Method (ESLM). Lastly, topology results were generated and design interpretation was performed to generate two designs of different approaches. The first design involved trying to closely match the topology results and resulted in a design with an overall weight savings of 67%, peak stress increase of 74%, and no apparent cost savings due to complex features. The second design focused on manufacturability and achieved overall weight saving of 36%, peak stress increase of 6%, and an estimated 60% in cost savings.
... To obtain a reasonably accurate mathematical model of a flexible beam which undergoes large rotation, the number of degrees of freedom required in the finite element model can be very large [31]. This results in high computational effort for the solution of such systems. ...
In this paper a model order reduction technique for the dynamic simulation of beams undergoing large rotations is presented. The finite element model for the motion of such beams is based on the corotational formulation. The trajectory piecewise linear model order reduction (TPWLMOR) method with second order Krylov subspace is used to obtain the reduced order model from the finite element model. Improvements are suggested to improve the accuracy and the computational efficiency of the TPWLMOR model. Several numerical examples which include forced undamped and damped beams are presented to validate the proposed method.
... In the flexible multibody dynamic modeling approach, the equations of motion are developed using the relative nodal coordinate formulation (RNCF) with a floating frame of reference[4]. The component modes for the flexible body representation are obtained using the Craig-Bampton method[6]. ...
Axial piston type hydraulic pump is an important source of fluid power in hydraulic systems. In the present work, a methodology for modeling the dynamics of the pump in multibody framework considering component flexibility is studied. A multibody dynamic model of the pump is developed considering housing flexibility and pressure forces acting on the pistons. The flexibility of the housing is modeled using the relative nodal coordinate formulation (RNCF) with a floating frame of reference. The bearing forces, control piston force, swashplate motion and torque ripple are studied. The bearing and control piston force results are compared with the case of a rigid housing assembly. The experiments are conducted to measure the end cover vibration levels during operation of the pump. The vibration levels are also obtained from the simulation and compared with experiments. Numerical studies are conducted to better understand the influence of housing flexibility on the dynamics of the pump.
... In most differential equation, multi-orders are used to describe the characteristics of different objects in various scale. And in the history of maths, physics and engineering, there are many expressions composed with multiple components, such as plural and quaternion[3], which use linear superposition of the base band to represent, when vector[4] and tensor[5] are represented by a matrix form. The control equation[2] is also a common form of multi-leveled applications in particular, which is not composed with multiple dimension, but with different orders of a single dimension. ...
For a proper way of modelling and simulation, as well as the precise and completeness, and a new multi-level method named homogeneous tensor coordinate(HTC) was developed by improving the absolute nodal coordinate formulation(ANCF) and the homogeneous coordinates(HC). In HTC, a new form of 2×2 matrix was employed instead of angles, when the capacity of expression and calculation was increased at the same time. HTC has good performance in traditional rigid body mechanics, which especially suitable for multibody modelling. Therefore, the robotic engineering may benefit from this finding, and a new formulation for scientific research may also affect the physics although some mathematical problems are raised in matrix theory.
... Multibody dynamics was originally developed to deal with simple tree-like topologies composed of rigid bodies. As the need to model flexibility arose, the floating frame of reference formulation [1] was proposed. Unfortunately, as the magnitude of the elastic displacements increases, this formulation becomes increasingly inaccurate and the multibody dynamics community began to turn its attention to finite element based formulations. ...
As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism , focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity , resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available. * Multibody System Dynamics, 37(1): pp 29-48, 2016.
... Traditional finite element methods used to solve the flexible multibody system are based on the assumption that only small deformation and small rotation are happened in the components. Therefore, these methods will not provide proper dynamic results when the components of a system are subject to large rotation and large deforma- tion [2]. To solve the problem of large deformation and rotation , Shabana [3] proposed a new Absolute Nodal Coordinate Formulation (ANCF) to model the element of flexible components. ...
Based on the theory of Absolute Nodal Coordinate Formulation, this paper proposes a new dynamic computation method to solve the flexible multibody system with uncertain material properties (Young’s modulus and Poisson’s ratio) that may be induced by the material asymmetric distribution. Rather than traditionally considering an uncertain factor as one single variable in the whole system, the material properties vary continuously in the space domain so that they are described by the random field, which is then discretized to countable random variables using the expansion optimization linear estimation method. The uncertain response of the system is approximated by the Polynomial Chaos expansion, numerically implemented by a collocation method. We propose and prove an important theory that the collocation method provides the same results as the Gaussian quadrature formula if the roots of the corresponding orthogonal polynomials are used as the collocation points. As a result, the proposed method is a nonintrusive technique that does not modify the original solver but only adds a preprocess and a postprocess. The uncertain displacement of system is finally illustrated by ellipse and ellipsoid, which visually shows the uncertainty extent and correlation between different coordinates. The numerical examples show that the proposed method has almost an equivalent accuracy of Monte Carlo simulation but much higher efficiency.
... Significant efforts has been invested in the dynamic modelling of manipulators with flexible links since the 1970s [1]. Early investigations involved serial manipulators and kinematic decoupled parallel mechanisms with a detailed survey in [4] and [1]. Compared to serial manipulators and kinematic ki flexible on exist studies mechanisms, decoupled few parallel nematic coupled manipulators. ...
The modal analysis was carried out to investigate the proposed improved curvature based finite element method (ICFE) for parallel manipulator with flexible links. The flexible link was discretizatied with ICFE first, and with the proposed rigid-flexible coupling technique, the flexible displacement of the moving platform and passive links was obtained, then through Lagrange equation, the structural model was derived. At last, to investigate the accuracy and efficiency of the ICFE and rigid-flexible coupling technique, modal analysis of ICFE model with different nodes and comparison studies with CFE and the ABAQUS model was carried out.
... Other assumptions are: -small deformations: the relationship between strain and deformation can be considered linear; in addition axial and bending deformations can be considered as decoupled; -small link rotations: this assumption means the contribution of centrifugal and Coriolis terms in the motion equation can be neglected. Each link movement is described, using the so-called " floating frame of reference formulation" [7][8][9], by two sets of coordinates, describing respectively the large motion and the vibrations. The first set describes the rigid motion of the th j link, in terms of its absolute displacement and rotation, considering the motion of the reference system of this link ( ) ...
Concrete pump booms are subjected to vibrations that increase mechanical stress and shorten their lifespan. This paper aims to study the problem by considering the two subsystems, the boom and the concrete pump, that have the greatest effect on the phenomenon. The authors supply numerical and experimental tools that can analyze the problem in depth in all its complexity. First, the systems were investigated independently, to identify their individual aspects. Then a numerical model was created to reproduce the behavior of the whole system and the interaction between boom and pump. The result was a new powerful tool for investigating passive and active solutions for suppressing vibration.
... The flexible model is the basis for theoretical analysis. Various approaches included the flexible multi-body dynamics(FMD)12、the linear elasto-dynamics(KED)34、 substructure method[5] have been widely used. FMD can obtain accurate model, but the modeling and solving is very complex and mostly be used for geometric nonlinearity. ...
The analysis of dynamic characterization combined with dynamic modeling, simulation and experiment of the Delta robot is presented. The substructure displacement method, Ansys software and LMS dynamic testing equipment are respectively used for dynamic model, simulation and experiment. Meanwhile, the dynamic characterization data has been obtained in typical configurations. The results show that it is consistent in theoretical analysis and simulation and experiment test, so the validity of the analysis approach is verified.
... 等提出， 后经 Belytschko [5] 等的发展，主要应用于显式动力 分析中。对于梁、板及壳体结构的几何大变形问题， 由于其位移主要由整体刚体转动主导，CR 法在这 类单元上已有较好的应用。然而，由于固体的刚体 转动与纯变形难以直接分离，利用 CR 法求解固体 的大变形会有较大的困难。如果只是简单地假设一 个随体坐标来代替单元的刚体运动，则可能在变形 较大时产生附加的虚假应变，使单元产生不正常的 体积膨胀或收缩现象 [6] 。因此，采用 CR 法求解固 体的非线性问题时，通常需要引入特殊的处理技巧 或加入适当的人工参数才可以完成分析 [7][8] 。 面对上述方法在分析固体非线性问题时的困 难，美国普渡大学 Ting [9] 教授根据向量力学和数值 计算相结合的概念提出了一个新的构想。它以点值 作为结构行为的描述，同时引用广义向量力学作为 运动和变形的准则，从而发展了向量式结构与固体 力学，并已经成功地应用于钢桁架 [10] 、钢筋混凝土 框架 [11] 等结构的非线性问题的分析中。有限质点法 (finite particle method, 简称 FPM) [12] 也是以 Ting 的 概念为力学基础而发展起来的。与传统的以能量变 分原理为基础的有限单元法不同，有限质点法不再 以函数连续体的概念来描述分析对象，而是以真实 的物理模型为基础，将连续体在空间上离散为一系 列质点的集合，在时间上离散为一系列分段的途径 单元的集合，并在各时间段内直接用牛顿第二定律 取代连续体的偏微分方程来描述这些质点的运动。 目前该方法已应用于无穷小机构 [13] 和可展结构的 运动模拟 [14] 以及结构的倒塌破坏研究中 [ ...
The finite particle method (FPM) is a new developed method for numerical calculation, which is based on the vector mechanics and physical thoughts. It models the analyzed domain by a set of particles instead of mathematical function and continuous bodies adopted in traditional method, and thus the motion of each particle is directly formulated by Newton's second law. The formulations include a new description of kinematics called fictitious reverse motion to dissect rigid body and deformation displacement, and a set of deformation coordinates for each time increment to evaluate deformation and internal nodal forces. The explicit time integration is adopted to solve the equation of motion. Motions of all particles can describe the whole behavior. The fundamentals of PFM are presented first in this paper. Then, the formulations of the planar solid internal forces are derived. Finally, FPM is applied to the numerical calculations of geometric large deformations of planar solids. The results of numerical examples solved by self-designed program demonstrate that the presented method can achieve good accuracy and convergence. It also shows that FPM is quite effective and feasible for the analysis of planar solids undergoing large rotation and deflection.
... A numerical model, described in [12], was developed as a tool both for defining the control logic and for simulating boom behaviour. The boom kinematics were solved using the " floating frame of reference " formulation [13] to describe the large motion and the Finite Element Method (FEM) to model the segment flexibility. Using the Lagrange formulation, the boom's non-linear equation of motion was obtained ...
The independent modal control to suppress the vibration of nonlinear flexible structures is applied in this paper. Technological improvements in the mechanical field showed during the recent years have led to high-performance systems with low weight and, as a consequence, high flexibility and low damping. Here active control quickly bettered the traditional passive damping systems. The structure investigated in this paper is a multi-body flexible boom moved by hydraulic actuators. The nonlinear system dynamic was numerically modeled and a control strategy, based on the use of the same actuators, was developed. Finally a test rig was created to experimentally validate the proposed approach.
... The study on this issue has been pursued for quite a long time, and there are numerous experimental and theoretical results that have been published in hundreds of the research papers. See e.g., the review papers [1,2]. In the existing literature, researchers have proposed different approximations characterizing the vibration responses of the system [3,4]. ...
A rotating beam features a puzzling character in which its frequencies and modal shapes may vary with the hub's inertia and its rotating speed. To highlight the essential nature behind the vibration phenomena, we analyze the steady vibration of a rotating Euler-Bernoulli beam with a quasi-steady-state stretch. Newton's law is used to derive the equations governing the beam's elastic motion and the hub's rotation. A combination of these equations results in a nonlinear partial differential equation (PDE) that fully reflects the mutual interaction between the two kinds of motion. Via the Fourier series expansion within a finite interval of time, we reduce the PDE into an infinite system of a nonlinear ordinary differential equation (ODE) in spatial domain. We further nondimensionalize the ODE and discretize it via a difference method. The frequencies and modal shapes of a general rotating beam are then determined numerically. For a low-speed beam where the ignorance of geometric stiffening is feasible, the beam's vibration characteristics are solved analytically. We validate our numerical method and the analytical solutions by comparing with either the past experiments or the past numerical findings reported in existing literature. Finally, systematic simulations are performed to demonstrate how the beam's eigenfrequencies vary with the hub's inertia and rotating speed.
... There are different ways to model flexible multibody systems, see [32]. Two important approaches are nonlinear finite element methods and the floating frame of reference formulation. ...
Employing the floating frame of reference formulation in the topology optimization of dynamically loaded components of flexible multibody systems seems to be a natural choice. In this formulation the deformation of flexible bodies is approximated by global shape functions, which are commonly obtained from finite element models using model reduction techniques. For topology optimization these finite element models can be parameterized using the solid isotropic material with penalization (SIMP) approach. However, little is known about the interplay of model reduction and SIMP parameterization. Also securing the model reduction quality despite major changes of the design during the optimization has not been addressed yet. Thus, using the examples of a flexible frame and a slider-crank mechanism this work discusses the proper choice of the model reduction technique in the topology optimization of flexible multibody systems.
... In the last several decades, researchers have presented different formulations for flexible multibody dynamics. Among these formulations the floating frame of reference has been proven to be effective for small deformation and, therefore, it is widely used in computer simulation of flexible multibody systems [10]. In the floating frame of reference formulation, the frame is attached to the flexible body either by setting some of the deformation degrees-of-freedom to zero (three for 2D models and six for 3D models) or by employing the mean-axes conditions [5]. ...
One issue in the dynamic simulation of flexible multibody system is poor computation efficiency, which is due to high frequency components in the solution associated with a deformable body. Standard explicit numerical methods should take very small time steps in order to satisfy the absolute stability condition for the high frequency components and, in turn, the computational efficiency deteriorates. In this study, a hybrid integration scheme is applied to solve the equations of motion of a flexible multibody system for acheiving better computational efficiency. The computation times and simulation results are compared between the hybrid scheme and conventional methods. The results demonstrate that the efficiency of a flexible multibody simulation can be improved by using the hybrid scheme.
... Several studies have been done in order to define and present a precise mathematical model for flexible multi-body systems. These studies have started with investigation on a single flexible-link mechanisms, after with consideration flexible multi-body mechanisms in a planer and finally flexible mechanisms in the spatial environment [1][2][3][4][5]. The most used and adopted methods for modelling the flexible mechanism includes the usage of discretization methods such as finite element method (FEM) with the purpose of presenting dynamics models using a finite number of elastic degree-of-freedom. ...
In this paper, position control and reduction of vibration of a 3D flexible L-shape mechanism has been achieved through the synthesis of a constrained Model Predictive Control (MPC). A finite element model, based on the equivalent rigid link system (ERLS) theory, is used in order to describe accurately the dynamic behaviour of the system. The model has been validated through the experimental tests. In order to apply the constrained MPC control on the mechanism, a linearized model which takes gravity force into account is derived as well as a Kalman state estimator. The effectiveness and robustness of the control system has been evaluated and discussed through several tests. Furthermore, the performance of the MPC control has been compared with the performance of classical industrial control (PID).
... The method that defines position vector P inFig. 1 as Eq. 1 is called the " floating frame of reference . " (Shabana 1997 ...
This paper is concerned with the dynamic response analysis of an elastic boom when a floating crane lifts a heavy cargo. The objective is to evaluate the dynamic response of the floating crane and the heavy cargo, considering the effect of the boom's flexibility. Coupled surge, pitch, and heave motions are considered for the floating crane and the cargo, which are connected through an elastic boom and a wire rope. The floating frame of reference formulation and nodal coordinates is employed to model the boom as a flexible body. The equations of motion for the system are constructed based on the formulation of flexible multibody system dynamics. First, the dynamic response of the boom is investigated separately, and the response is compared with experimental data for verification. Second, a dynamic simulation of the boom with the floating crane and a heavy load is performed under regular wave conditions. Finally, the effects of the elastic boom on lifting cargo is discussed by comparing the simulation results between the elastic boom and a rigid boom.
The assessment of sliding and rocking responses in flexible structures subjected to ground vibrations presents complex challenges due to their high sensitivity to input motion, geometric variations, and dissipation issues. While numerous researchers have focused on studying flexible structures, the rocking response of these structures has often been overlooked, with examinations limited to models involving a single body rather than multiple bodies with intricate connection constraints. Consequently, the aim of this paper is to develop an analysis method capable of comprehensively describing the sliding and rocking motions of structures consisting of a combination of rigid blocks and flexible structures, as well as connection constraints, utilizing the Floating Frame of Reference Formulation (FFRF). The proposed analytical model was evaluated for its ability to predict the sliding and rocking responses of flexible structures laid on rigid blocks under external pulse excitation and initial conditions. By integrating the Finite Element Method into FFRF, the model enables the description of complex systems with diverse shapes and connections. Furthermore, we derive an impact equation that accounts for elastic deformation and develop an equation for calculating the coefficient of restitution, considering the effect of elastic displacement based on FFRF. This integration extends the simulation analysis to encompass flexible multi-body systems, accurately capturing energy dissipation during rocking impacts. To demonstrate and generalize the novel application of FFRF, time histories of sliding and rocking motion under pulse excitation and initial conditions were analyzed and presented. The results showcased the effectiveness and versatility of the proposed approach in capturing the intricate dynamics of sliding and rocking rigid blocks and flexible structures. Overall, this study contributes to the advancement of analytical methods for assessing the sliding and rocking behavior of multibody systems and the coefficient of restitution when elastic displacement was considered, providing valuable insights for design and analysis in various engineering applications.
The paper presents the derivation of a modal nonlinear structural model and its coupling with a nonlinear aerodynamic model for static aeroelastic analysis of highly deformable configurations. The structural analysis, intended for geometrically nonlinear structures, computes the deformations of a beamlike structure by dividing it into a few segments. Large deformations are treated as the sum of large, rigid-body displacements of the segment plus small, elastic deformation within the segment. The latter are computed using a modal approach, where the natural modes of each segment are calculated with augmented large fictitious masses to improve the coupling between segments. The use of a modal method in a large-deformation case introduces difficulties in the application of compatibility equations, which are explained and addressed in the paper. The nonlinear aerodynamic model is an equivalent strip theory, in which each section is assigned a database of aerodynamic coefficients. Both the aerodynamic and structural models are suitable for beam-like wing structures and are highly computationally efficient. The numerical examples include three loading cases that validate the implementation of the methodology and demonstrate its use for static aeroelastic applications.
3D printing makes the small and compact transmission system of flapping-wing micro aerial vehicle (FMAV) available for in-house designs and tests. However, the high flapping frequency requiring stringent strength integrity and fatigue longevity for the structure itself always serves as the major responsible reason for the structural failure of FMAV. In this paper, experiment as well as numerical computation efforts were acted in concert to gain an insightful understanding of the failure mechanism for the transmission structure of FMAV. Firstly, a FMAV structure was fabricated by 3D printing using Ultraviolet(UV) Cureable Resin. The isotropic constitutive behavior of the resin was evaluated and confirmed. Further, numerical computation model describing the mechanical behavior of FMAV structure was established and verified by experiments. Secondly, together with the computational structural model and constitutive model, the dynamic response of the transmission structure subject to high frequency flipping motion was described. Both simulation and experiments share the same failure mode and configuration and the dominate factor for the failure was found out to be the excessive bending deformation. Finally, design parametric study along with different boundary conditions aiming at the future ground testing were conducted to uncover the governing influences over the transmission structure integrity. Results may shed lights on the structural integrity design of FMAV and pave a new road for future advanced FMAV design, manufacturing and testing.
This paper reviews the state-of-the-art methods to perform structural optimization of flexible mechanisms. These methods are based on a system-based approach, i.e. the formulation of the design problem incorporates the time response of the mechanism that is obtained from a dynamic simulation of the flexible multibody system. The system-based approach aims at considering as precisely as possible the effects of nonlinear dynamic loading under various operating conditions. Also, the optimization process enhances most existing studies which are limited to (quasi-) static or frequency domain loading conditions. This paper briefly introduces flexible multibody system dynamics and structural optimization techniques. Afterwards, the two main methods, named the weakly and the fully coupled methods, that couple both disciplines are presented in details and the influence of the multibody system formalism is analyzed. The advantages and drawbacks of both methods are discussed and future possible research areas are mentioned.
The nonlinear dynamics of a flexible multibody system with interval clearance size in a revolute joint is investigated in this work. The system is modeled by using a unified mesh of absolute nodal coordinate formulation (ANCF), that is, the flexible parts are meshed via the finite elements of the absolute nodal coordinate formulation and the rigid parts are described via the ANCF Reference Nodes (ANCF-RNs). The kinetic models of all revolute joints are formulated by using ANCF Reference Node (ANCF-RN) coordinates. The influence of the Lund-Grenoble (LuGre) and the modified Coulomb's friction models on the system dynamics is comparatively studied. The Chebyshev tensor product (CTP) sampling method is used to generate the samples of the interval clearance size. With the purpose to maintain the continuous contact of the clearance joint, a modified extended delayed feedback control (EDFC) is used to stabilize the chaotic motion of the flexible multibody system. Finally, the dynamics of a planar slider-crank mechanism with interval clearance size in a revolute joint is studied, as a benchmark example, to check the effectiveness of the presented computation method and the modified extended delayed feedback control.
Position control and vibration damping of flexible link mechanisms are still challenging open issues in robotics. Finding solutions for these problems can lead to improvement in the operation and accuracy of the manipulators. In this paper the synthesis of robust controllers based on í µí°» ∞ loop shaping and í µí¼-Synthesis for both position control and vibration damping in a spatial flexible L-shape mechanism with gravity is presented. The design of the controllers is based on the evaluation of an uncertainty model which takes into account a ±20% uncertainty in the elasticity and mass density of the links. The response of each controller is tested also in the presence of external disturbances with the aid of highly accurate numerical simulations; furthermore, a comparison between the robust performances of synthesized controllers is presented in order to show the effectiveness of synthesized control systems.
This work is devoted to the validation of a computational dynamics approach previously developed by the authors for the simulation of moving loads interacting with flexible bodies through arbitrary contact modeling. The method has been applied to the modeling and simulation of the coupled dynamics of railroad vehicles moving on deformable tracks with arbitrary undeformed geometry. The procedure presented makes use of a fully arbitrary Lagrangian-Eulerian (ALE) description of the long flexible solid (track) whose mechanical properties may be captured using a dynamics-preserving selection of modes, e.g., via a Padé approximation of a transfer function. The modes accompany the contact interaction rather than being referred to a fixed frame, as it occurs in the finite-element floating frame of reference formulation. In the method discussed in this paper, the mesh, which moves through the long flexible solid, is defined in the trajectory coordinate system (TCS) used to describe the dynamics of the set of bodies (vehicle) that interact with the long flexible structure. For this reason, the selection of modes can be focused on the preservation of the dynamics of the structure instead of having to ensure the structure's static displacement convergence due to the motion of the load. In this paper, the validation of the so-called trajectory coordinate system/moving modes (TCS/MM) method is performed in four different aspects: (a) the analytical mechanics approach is used to obtain the equations of motion in a nonmaterial volume, (b) the resulting equations of motion are compared to the classical discretization procedures of partial differential equations (PDE), (c) the suitability of the moving modes (MM) to describe deformation due to variable-velocity moving loads, and (d) the capability of the finite nonmaterial volume to describe the dynamics of an infinitely long flexible body. Validation (a) is completely general. However, the particular example of a moving load applied to a straight beam resting on a Winkler foundation, with known semi-analytical solution, is used to perform validations (b)
This paper presents a new flexible multibody system (MBS) approach for modeling textile systems including roll-drafting sets used in chemical textile machinery. The proposed approach can be used in the analysis of textile materials such as lubricated polyester filament bundles (PFBs), which have uncommon material properties best described by specialized continuum mechanics constitutive models. In this investigation, the absolute nodal coordinate formulation (ANCF) is used to model PFB as a hyperelastic transversely isotropic material. The PFB strain energy density function is decomposed into a fully isotropic component and an orthotropic, transversely isotropic component expressed in terms of five invariants of the right Cauchy-Green deformation tensor. Using this energy decomposition, the second Piola-Kirchhoff stress and the elasticity tensors can also be split into isotropic and transversely isotropic parts. The constitutive equations are used to define the generalized material forces associated with the coordinates of three-dimensional fully parameterized ANCF finite elements (FEs). The proposed approach allows for modeling the dynamic interaction between the rollers and PFB and allows for using spline functions to describe the PFB forward velocity. The paper demonstrates that the textile material constitutive equations and the MBS algorithms can be used effectively to obtain numerical solutions that define the state of strain of the textile material and the relative slip between the rollers and PFB.
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