Flexible Mechanical Systems Research Articles

Dynamic Simulation and Collision Detection for Flexible Mechanical Systems with Contact Using the Floating Frame of Reference Formulation

Abstract Contact simulation is essential in modelling mechanical systems. The contact models require accurate geometric information, which is determined through collision detection methods. When the mechanical system includes flexible bodies such as structural components, the dynamic formulation and collision detection can be more challenging, as the geometric boundaries of such components keep changing during the simulation. The floating frame of reference (FFR) formulation is suitable for flexible systems with small deformation. In this work, a stable and efficient dynamic simulation method is introduced for flexible systems with contact based on the FFR formulation. In addition, a curve-based collision detection method is proposed, which is more consistent with the dynamic formulation and more efficient than common existing collision detection methods. Case studies of flexible beams and multibody systems are employed to demonstrate the performance of the proposed dynamic simulation and collision detection methods.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

Model-Based Co-Simulation of Flexible Mechanical Systems With Contacts Using Reduced Interface Models

Model-Based Co-Simulation of Flexible Mechanical Systems With Contacts Using Reduced Interface Models

Damping of Oscillations of a Rotary Pendulum System

This paper describes an innovative design based on the spectral approach of a novel shaper that eliminates frequency components that induce unwanted residual oscillations in various flexible mechanical systems, such as tower cranes or chain carousels, which are vital to many manufacturing and material-handling processes. However, their physical structure leads to flexible effects that limit their usefulness. Apart from the circular motion problem, control is provided by a single actuator, which makes it a so-called underactuated system. The input signal needs to be modified so that the spectral components from several interconnected degrees of freedom are considered together during shaper design, which increases the complexity of this task since one of its components induces nonlinear behavior. This means that traditional shaping techniques, based on linear theory, fail to provide good performance over the whole input range. The underdamped dynamics of the model and the effect of nonlinearities on the spectrum of the final signal are examined; the proposed method for application as a command shaping control technique is applied; and its effectiveness is analyzed by simulation and real-time implementation. The theoretical results verified on an experimental crane system confirm the expected oscillation phenomenon and show that the designed nonlinear shaper can reduce the payload swing significantly.

Adaptive control for a class of flexible mechanical systems with bending deformation and torsion deformation in the presence of nonlinear time-varying actuator faults and unknown control directions

Vibration control problem of a class of flexible mechanical systems with bending deformation and torsion deformation in the presence of nonlinear time-varying actuator faults and unknown control directions is considered in this paper. The dynamics model of this kind of flexible mechanical system, which includes two kinds of coupled vibration deformation and unknown nonlinear disturbances, is composed of some partial differential equations. In order to overcome the control difficulty brought by unknown control directions and time-varying nonlinear actuator faults, a Nussbaum function based adaptive boundary controller is designed to suppress the bending and twist vibration deformations of the mechanical system. The stability of the system is proved by Lyapunov direct method, and the effectiveness of the proposed control scheme is verified by numerical simulation at the end of this paper.

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

Case Study on the Impact of Flooding and Inundation on Pavement Performance

The experimental program was designed to discover the effectiveness of various treatments on mitigating pavement distress (mostly longitudinal cracking issues) caused by normal seasonal volumetric changes in expansive clays. Unfortunately, these objectives were compromised by subsequent flooding events. However, these flooding events offered rare opportunities to evaluate the impact of inundation on the performance of newly constructed pavement sections. Flood-induced pavement movement was monitored with cross-section surveys. Pavement condition data were obtained from regular pavement management activities conducted by the Louisiana Department of Transportation and Development’s Pavement Management System unit. The roadway pavements of test sections experienced significant movement because of the floods, but it is differential movement across the pavement that caused longitudinal cracks. The test section, with a geotextile-wrapped sand layer placed between the pavement structure and subgrade, had the best performance among all test sections with respect to longitudinal cracking; better performance was achieved when this sand layer was extended to the edge of the foreslope. This may be because the geotextile-wrapped sand layer created both a drainage system to minimize moisture fluctuations in the subgrade and a flexible mechanical system to absorb the swelling pressures from the underlying subgrade. Flood-induced structural damage was apparent by comparing falling weight deflectometer data of different test sections. Flood-induced heaves and depressions caused the ride quality of the roadway to deteriorate faster than expected. This may be because the prolonged inundation expedited the transition process of the pavement structure and materials from the newly constructed state to the natural equilibrium state.

Study on the symmetries and conserved quantities of flexible mechanical multibody dynamics

<abstract> <p>In this paper, in order to provides a powerful new tool for quantitative and qualitative analysis of dynamics properties in flexible mechanical multibody systems, the symmetry theory and numerical algorithms for preserving structure in modern analytical mechanics is introduced into flexible multibody dynamics. First, taking the hub-beam systems as an example, the original nonlinear partial differential-integral equations of the system dynamics model are discretized into the finite-dimensional Lagrange equations by using the assumed modal method. Second, the group analysis theory is introduced and the criterion equations and the corresponding conserved quantities of Noether symmetries are given according to the invariance principle, which provide an effective way for analytic integral theory of dynamic equations. Finally, a conserved quantity-preserving numerical algorithm is constructed by coordinates incremental discrete gradient, which makes full use of the invariance of conserved quantity to eliminate the error consumption for a long time. The simulation results show that the deeper mechanical laws and motion characteristics of flexible mechanical multibody systems dynamics can be obtained with the help of symmetries and conserved quantities, which can provide reference for more precise dynamic optimization design and advanced control of systems.</p> </abstract>

Stabilization of a nonlinear Euler–Bernoulli beam

In this work, we study the vibration control of a flexible mechanical system. The dynamic of the problem is modeled as a viscoelastic nonlinear Euler–Bernoulli beam. To suppress the undesirable transversal vibrations of the beam, we adopt a control at the right boundary of the beam. This control law is simple to implement. We prove uniform stability of the system using a viscoelastic material, the multiplier method and some ideas introduced in [20]. It is shown that a large range of rates of decay of the energy can be achieved through a determined class of kernels. Unlike most of the existing classes in the market, ours are not necessarily strictly decreasing.

Modeling, Design and Optimization of Flexible Mechanical Systems

Performance, efficiency and economy drive the design of mechanical systems and structures and has led lightweight engineering design to prominence [...]

Event-triggered vibration control for a class of flexible mechanical systems with bending deformation and torsion deformation based on PDE model

Most of the research on event-triggered control method is aimed at the traditional ordinary differential equation (ODE) system, which neglects the high-order mode of the system and lacks precision. There are also a few researches on event-triggered control for simple single-degree-of-freedom flexible mechanical systems. This paper focuses on the more complex multi-degree of freedom coupling flexible beams. There are two kinds of deformation of the flexible beam, namely bending deformation and torsion deformation. Based on the coupled partial differential equation (PDE) model of the flexible beam, an event-triggered control algorithm is designed in this paper to address the vibration control problem of the mechanical system. The stability and convergence of the closed-loop system are proved via Lyapunov’s direct method while the Zeno phenomenon is avoided. The simulation results clearly and directly show the effectiveness and superiority of this method, which are consistent with the theoretical analysis. Overall, the proposed event-triggered control method is feasible for PDE models and has good control effect while reduce signal transmission.

Augmented Perpetual Manifolds and Perpetual Mechanical Systems—Part I: Definitions, Theorem, and Corollary for Triggering Perpetual Manifolds, Application in Reduced-Order Modeling and Particle-Wave Motion of Flexible Mechanical Systems

Abstract Perpetual points in mechanical systems were defined recently. Herein, they are used to seek specific solutions of N-degrees-of-freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, the perpetual points proved that they form the perpetual manifolds, they are associated with rigid body motions, and herein these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem defining the conditions of the external forces applied in an N-degrees-of-freedom system led to a solution in the exact augmented perpetual manifold of rigid body motions is proven. In this case, the motion by only one differential equation is described; therefore, it forms reduced-order modeling (ROM) of the original equations of motion. Further on, a corollary is proven that for harmonic motion in the augmented perpetual manifolds, the system moves in dual mode as wave-particle. The developed theory is certified in three examples, and the analytical solutions are in excellent agreement with the numerical simulations. This research is significant in several sciences, mathematics, physics, and mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally, in mechanical engineering, the theory in all mechanical structures can be applied, e.g., cars, airplanes, spaceships, and boats, targeting only the rigid body motions.

Dynamics of soft mechanical systems actuated by dielectric elastomers

A computation methodology for studying the dynamics of flexible mechanical systems containing soft actuators of dielectric elastomers is originally proposed. The absolute nodal coordinate formulation (ANCF) is used to describe the rigid-body motions and large deformations of the flexible mechanical systems. A new viscoelastic solid element of ANCF is proposed for meshing the components of dielectric elastomers. The constitutive model of dielectric elastomers is deduced from the Helmholtz free energy according to thermodynamics, and embedded into the ANCF solid element. The proposed solid element is an 8-node hexahedra element with each node represented by 12 spatial nodal coordinates and 4 electrical nodal coordinates. Both the displacement and electric fields of the continuum system are discretized and described by the proposed element with high-order functions of interpolation. The element internal forces and their Jacobians are derived. Then dynamic equations of flexible multibody systems containing components of dielectric elastomers are established and solved numerically via the generalized-alpha algorithm of time integration. Finally, numerical examples are given to validate the proposed element and the computation methodology. The experimental test is also performed, and the test results are compared with the numerical results to further validate the proposed methodology.

Direct method for updating flexible multibody systems applied to a milling robot

Industrial robots are currently used in light milling operations for their low cost and large workspace compared with CNC machine tools. However, milling robots are prone to vibration instabilities (chatter) and process deviations since they are significantly less stiff than machine tools. As a result, robot dynamic response depends on its posture which represents a major challenge. This paper presents a direct method to update any multibody model, enclosing flexible rotational/translational or virtual joints with minimal tuning. The novel method allows determining the elastic parameters of the model based on a curve fitting of the frequency response functions measured at the tool tip. Fitting is fast and efficient as it occurs in the frequency domain without the need to transform the measured data into the model parameter space. It relies on a genetic algorithm followed by a deterministic procedure to ensure a refined solution of the identified global minimum. The method is firstly demonstrated and validated on a simulated flexible manipulator with three rotational joints. Its multibody model is built using minimal coordinates with known elastic parameters that the method recovers accurately. The new fitting algorithm is eventually applied to an actual industrial robot (KUKA KR90 R3100 robotic arm) resulting in the proper fit of its critical resonances. Posture dependency can also be tackled by considering multiple measurements in different poses within the same fitting procedure. Updating procedure was programmed in Matlab and made public so that it can be easily adapted to identify elastic parameters of other flexible mechanical systems.

Simultaneous path-following and vibration control for uncertain nonlinear flexible mechanical systems without dependency on oscillatory mathematical model

In this paper, a novel multiobjective algorithm for simultaneous path-following control (PFC) of multivariable flexible dynamical systems and maintaining boundedness of transverse vibrational effects is proposed. This method is specifically designed for control of complicated dynamical systems where obtaining precise mathematical models describing vibrational and translational dynamics is difficult or unattainable. To facilitate the control task, the proposed scheme is constructed such that it would be capable of stabilizing the closed-loop system solely using a rigid approximation of system dynamics while considering vibrational effects as unstructured uncertainties. This permits the designer to forgo the troublesome task of precisely modeling and numerically analyzing the flexible vibrational system, prioritizing important path-following and closed-loop stability characteristics. The proposed control algorithm incorporates a discrete sliding mode control (DSMC) scheme constructed on the basis of calculation of input bounds. This method does not introduce additional chattering effects as generation of control inputs is not directly dependent on switching functions. Unstructured uncertainties are estimated using one-step approximation technique which removes the need for employing difficult techniques used in the existing literature. The control algorithm is constructed in two distinct modes, the first of which considers appropriate path following as its objective. The second control mode halts the tracking task unless an assigned cost function resides in an acceptable interval, resulting in gradual reduction of intensity of vibrational effects. The control algorithm is numerically simulated in ANSYS® Mechanical APDL environment. The obtained results express the accuracy and efficiency of the control algorithm despite the use of considerably simpler procedure in comparison with existing works.

Fixed-order data-driven H∞ controller synthesis for flexible mechanical systems: Two-stage approach

In this paper, a two-stage method is introduced to design fixed-order data-driven [Formula: see text] controller for flexible mechanical systems. In the first stage of the proposed method, unknown parameters of anti-resonance filter that is added to the forward path of the control loop of the system to minimize resonant peaks, are calculated using frequency domain data obtained from open-loop system identification tests. In the second stage, a fixed-order data-driven [Formula: see text] controller is calculated by solving an optimization problem under convex [Formula: see text] constraints obtained based on the Nyquist diagram. With the proposed method, lower order controllers that meets the performance constraints of classical model-based [Formula: see text] problems can be synthesized without need of a parametric plant model. The method developed in this study is tested experimentally on a military stabilized platform and its performance is compared with a model-based [Formula: see text] controller design method.

Nonlinear model predictive acceleration synchronization technique for rigid-model-based tracking control of multi-DOF flexible systems

In this paper, the problem of stabilization and tracking control of uncertain flexible multivariable continuum mechanism systems with additional appendage considering boundedness of undesired vibrational effects are investigated. It is proposed that rather than controlling the entire dynamical system through installation of additional actuators on appendage, it is more desirable to stabilize the core system while considering undesired vibrational effect in the appendage as optimization cost. To this end, the nonlinear programming problem corresponding to a predictive control scheme is presented based on a cost defined according to magnitude of accelerations of undesired vibrations in the appendage. Stabilization of control algorithm is conducted based on demonstration of existence of feasible Lyapunov functions. Candidate Lyapunov functions are defined based on sliding functions analysis. The associated sliding functions are defined for core subsystems which correspond to degrees of freedom of dynamical system that are associated with reference signals. Undesired vibrations caused by other degrees of freedom corresponding to appendage are considered as segments of control cost in the optimization problem. To ensure the feasibility of control model in real-time applications, the control algorithm is obtained in discrete-time basis. Based on investigation of monotonically decreasing Lyapunov functions and expression of sliding functions as a combination of dynamical terms, reference inputs and control input terms, stabilizing constraints of closed-loop system are calculated. Similarly, based on combination of sliding surfaces with considered control and vibrational constraints, additional constraints for boundedness of vibrational terms in nonlinear programming problem are obtained. Numerical comparisons with a sliding control algorithm for core mechanical system and a reaching law method indicate improvements regarding reduction of undesired vibrational effects in appendage. Furthermore, regulation of appendage dynamics is conducted without installation of additional actuators, which reduces the implementation cost in comparison with a full-control scheme for core system and appendage.

Boundary stabilization of systems of high order PDEs arising from flexible robotics

Stabilization of a system of coupled PDEs of the fourth-order by means of boundary control is investigated. The considered setup arises from the classical Euler-Bernoulli beam model, and constitutes a generalization of flexible mechanical systems. A linear feedback controller is proposed, and using an abstract formulation based on operator semigroup theory, we are able to prove the well-posedness and the stability of the closed-loop system. The performances of the proposed controller are illustrated by means of numerical simulations.

Modeling and Analysis of Spatial Flexible Mechanical Systems With a Spherical Clearance Joint Based on the LuGre Friction Model

Abstract A computational methodology for modeling spatial flexible mechanical systems with stick-slip friction in a spherical clearance joint is presented. A modified three-dimensional (3D) absolute nodal coordinate formulation based shear deformable beam element with two nodes is proposed and employed to discretize the flexible components. To avoid locking problems, we employed an enhanced continuum mechanics approach to evaluate the beam element elastic forces. The strain components εyz, εyy, and εzz are approximated using linear interpolation to improve the computational efficiency while the loss of accuracy is acceptable. The contact and friction forces in a spherical clearance joint were evaluated by the hybrid contact and LuGre friction models, respectively. Three numerical examples are presented and discussed. A simple pendulum was utilized to prove the correctness of the modified beam element. A classical slider–crank mechanism was employed to validate the computational methodology. A spatial rigid–flexible slider–crank mechanism with a spherical clearance joint was used to investigate the effect of link flexibility and joint clearance on the dynamic behavior of mechanical systems. Using the LuGre friction model, we reproduced the Stribeck effect as it is expected in real world settings. The components with appropriate stiffness play the role of suspension for spatial mechanical systems with imperfect joints. The vibrations of the flexible components play an active role of intensifying the collision in kinematic joint with clearance.

A U-shaped microwave resonator for flexible mechanical sensors application.

A flexible mechanical deformation test system, which is a combination of the microwave and the flexible film, is proposed. The system consists of a U-shaped microwave resonator, coaxial transmission line, and PDMS flexible substrate. Microwave signals are generated from the vector network analyzer and loaded to the coaxial transmission line and then coupled to the U-shaped resonator through the coupling loop. The resonator works at the gigahertz band with the test range from 3 GHz to 40 GHz. The resonant frequency of the resonator can be adjusted by changing the shape of PDMS, and the related signals can be detected by using the S11 parameter of the microwave. The response time of the U-shaped resonator is less than 20 ms, and the sensitivity on the curved surface is 12.5 Hz/nm.