Vibro-impact Problems Research Articles

On Harmonic Balance Method-based Lagrangian contact formulations for vibro-impact problems

This article assesses two Harmonic Balance Method-based numerical Lagrangian strategies for the characterization of the periodic response of a mechanical system subject to unilateral contact constraints. The model used for evaluation is an academic rod model facing an obstacle that is firstly considered as equivalent to a stiffness (flexible) and then as a rigid obstacle. Nonlinear frequency response curves are obtained through an arc-length continuation procedure and confronted to the corresponding time integration simulations. An in-depth comparative analysis is conducted on the time signals obtained through both frequency domain strategies: (1) linear complementarity problem (LCP-HBM) and (2) dynamic Lagrangian frequency time (DLFT-HBM). The unilateral contact conditions are thoroughly investigated in the time domain and compared with the reference time integration simulations. Particular attention is paid to the relation between both methods and it is shown that the DLFT-HBM is asymptotically equivalent to the LCP-HBM. This study also shows that the oversampling of time signals inside alternating frequency–time procedures generates non-physical high harmonics. Finally, a convergence analysis is conducted in order to assess the influence of the different numerical parameters on the respect of the unilateral contact laws.

Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

Abstract In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

Experimental and numerical investigation of impacting cantilever beams: Second mode response

The dynamics of a cantilever beam undergoing vibro-impact, near its second linear natural frequency, is the focus of this paper. The flexible beam undergoes vibro-impact with two other flexible beams at its free end. Both experimental and computational methods are used for the study. From this it is shown that the computational model is able to capture the experimental vibro-impact behavior very well when the excitation amplitudes are reasonably high. Further, the presence of chaos through the period doubling route is demonstrated in the experimental studies. By using independent acceleration and velocity measurements, it is demonstrated that smoothing filters suggested in the literature for generating derivatives (acceleration from velocity) does not work very well for vibro-impact problems.

Existence and approximation for vibro-impact problems with a time-dependent set of constraints

We consider a discrete mechanical system subjected to perfect time-dependent unilateral constraints, which dynamics is described by a second order measure differential inclusion. The transmission of the velocity at impacts is given by a minimization property of the kinetic energy with respect to the set of kinematically admissible post-impact velocities. We construct a sequence of feasible approximate positions by using a time-stepping algorithm inspired by a kind of Euler discretization of the differential inclusion. We prove the convergence of the approximate trajectories to a solution of the Cauchy problem and we obtain as a by-product a global existence result.

A Solution Procedure for a Vibro-Impact Problem underFully Correlated Gaussian White Noises

A Solution Procedure for a Vibro-Impact Problem underFully Correlated Gaussian White Noises

Proper Orthogonal Decomposition of Flexible Clap and Fling Elastic Motions via High-Speed Deformation Measurements

Many complex unsteady mechanisms are thought to facilitate the high efficiency and agility commonly observed in small biological flyers. One of these, the flexible clap and fling maneuver, has not been extensively studied; an experimental characterization is the focus of this work. The clap–fling mechanism is approximated with a single flexible membrane flapping wing, replacing the symmetry plane between two wings with a splitter plate simulating the pair wing. This produces a complex vibro-impact aeroelastic problem, the deformation resulting from which is measured with a high-speed visual image correlation system. A low-dimensional representation of the ensuing large data set is obtained with proper orthogonal decomposition. The POD modes, and the relative importance of each, can help elucidate crucial mechanisms and relationships within the flapping system, and are computed for various membrane wing structures and flapping frequencies, with or without the presence of the splitter plate.

A POSITION-BASED TIME-STEPPING ALGORITHM FOR VIBRO-IMPACT PROBLEMS WITH A MOVING SET OF CONSTRAINTS

We consider a second-order measure differential inclusion describing the dynamics of a mechanical system subjected to time-dependent frictionless unilateral constraints and we assume inelastic collisions when the contraints are saturated. For this model of impact, we propose a time-stepping algorithm formulated at the position level and we establish its convergence to a solution of the Cauchy problem.

Limitations of Smoothening Functions for Automotive Vibro-Impact Problems

Nonlinear torsional models are used to analyze automotive transmission rattle problems and find solutions to reduce noise, vibration and dynamic loads. The torsional stiffness and inertial distribution of such systems show that the underlying mathematical problem is numerically stiff. In addition, the clearance nonlinearities in the gear meshes introduce discontinuous functions. Both factors affect the efficacy of time domain integration and smoothening functions are widely used to overcome computational difficulties and improve the simulation. In this paper, alternate smoothening functions are studied for their influence on the numerical solutions and their impact on global convergence and computation times. In particular, four smoothening functions (arctan, hyperbolic-cosine, hyperbolic-tan and quintic-spline) are applied to a five-degree-of-freedom generic torsional system with two backlash (clearance) elements. Each function is assessed via a global convergence metric across an excitation map (a design of experiment). Regions of the excitation map, along with multiple solutions, are studied and the implications to assessing convergence are critically examined. It is observed that smoothening functions do not lead to better convergence in many cases. The smoothening parameter needs to be carefully selected, or over-smoothened solutions may be found. The system studied is representative of a typical automotive rattle problem and it was found that benefits were limited from applying such smoothening functions.

Limitations of Smoothening Functions for Automotive Vibro-Impact Problems

Nonlinear torsional models are used to analyze automotive transmission rattle problems and find solutions to reduce noise, vibration and dynamic loads. The torsional stiffnes s and inertial distribution of such systems show that the underlying mathematical problem is numerically stiff. In addition, the clearance nonlinearities in the gear meshes introduce discontinuous functions. Both factors affect the efficacy of time domain in tegration and smoothening functions are widely used to overcome computational difficulties and improve the simulation. In t his paper, alternate smoothening functions are studied for their influence on the numerical solutions and their impact on global convergence and computation times. In particular, four smoothening functions (arctan, hyperbolic-cosine, hyperbolic-tan an d quintic-spline) are applied to a five-degree-of-freedom g eneric torsional system with two backlash (clearance) elements. Each function is assessed via a global convergence metric across an excitation map (a design of experiment). Regions of the excitation map, along with multiple solutions, are studied and the implicat ions to assessing convergence are critically examined. It is obser ved that smoothening functions do not lead to better convergence in many cases. The smoothening parameter needs to be carefully selected, or over-smoothened solutions may be found. The system studied is representative of a typical automotive rattle pr oblem and it was found that benefits were limited from applyin g such smoothening functions.

A new measure of efficiency for model reduction: Application to a vibroimpact system

The aim of this paper is to propose a new way to measure the efficiency of the proper orthogonal decomposition (POD) to construct a reduced-order model in Structural Dynamics. It investigates the efficiency of three reduced-order models for a vibroimpact problem: (1) PODdir-basis, which is the basis constructed using the direct method of the proper orthogonal decomposition; (2) PODsnap-basis, which is the basis constructed using the snapshot method of the POD; and (3) LIN-basis, which is the basis composed by the normal modes of the associated linear (LIN) conservative system. The efficiency is measured in terms of (1) number of elements to represent the dynamics with a given precision and (2) computational cost to simulate the time response within a given precision.

A proximal-like method for a class of second order measure-differential inclusions describing vibro-impact problems

We are interested in the study of discrete mechanical systems subjected to frictionless unilateral constraints. The dynamics is described by a second order measure-differential inclusion for the unknown positions, completed by a Newton's impact law describing the transmission of the velocities when the constraints are saturated. By using another formulation of the problem at the velocity level, we introduce a time-stepping algorithm, inspired by the proximal methods for differential inclusions, and we prove the convergence of the approximate solutions to a solution of the Cauchy problem.

Random vibrations with strongly inelastic impacts: Response PDF by the path integration method

This paper presents a study of SDOF stochastic vibroimpact problems using numerical path integration (PI). This is a challenging problem due to discontinuities in the state space paths of displacement and velocity response. It is shown that by introducing a suitable transformation of the state space variables, PI can be much simplified, and very accurate numerical results can be obtained. This is verified by comparison with extensive Monte Carlo simulation results.

Near-elastic vibro-impact analysis by discontinuous transformations and averaging

We show how near-elastic vibro-impact problems, linear or nonlinear in-between impacts, can be conveniently analyzed by a discontinuity-reducing transformation of variables combined with an extended averaging procedure. A general technique for this is presented, and illustrated by calculating transient or stationary motions for different harmonic oscillators with stops or clearances, and self-excited friction oscillators with stops or clearances First- and second-order analytical predictions are derived, and shown to agree to estimated accuracy with results of numerical simulation.

Asymptotics for some vibro-impact problems with a linear dissipation term

Given γ ⩾ 0 , let us consider the following differential inclusion, (S) x ¨ ( t ) + γ x ˙ ( t ) + ∂ Φ ( x ( t ) ) ∋ 0 , t ∈ R + , where Φ : R d → R ∪ { + ∞ } is a lower semicontinuous convex function such that int ( dom Φ ) ≠ ∅ . The operator ∂ Φ denotes the subdifferential of Φ. When Φ = f + δ K with f : R d → R a smooth convex function and K ⊂ R d a closed convex set, inclusion (S) describes the motion of a discrete mechanical system subjected to the perfect unilateral constraint x ( t ) ∈ K and submitted to the conservative force − ∇ f ( x ) and the viscous friction force − γ x ˙ . We define the notion of dissipative solution to (S) and we prove the existence of such solutions with conservation (resp. loss) of energy at impacts. If γ > 0 and Φ | dom Φ is locally Lipschitz continuous, any dissipative solution to (S) converges, as t → + ∞ , to a minimum point of Φ. When Φ is strongly convex, the speed of convergence is exponential. Assuming as above that Φ = f + δ K , suppose that the boundary of K is smooth enough and that the normal component of the velocity is reversed and multiplied by a restitution coefficient r ∈ [ 0 , 1 ] while the tangential component is conserved whenever x ( t ) ∈ bd ( K ) . We prove that any dissipative solution to (S) satisfying the previous impact law with r < 1 is contained in the boundary of K after a finite time. The case r = 1 is also addressed and leads to a qualitatively different behavior.

Algorithme de type ‘sweeping process’ pour un problème de vibro-impact avec un opérateur d'inertie non trivial

We consider a mechanical system with a finite number of degrees of freedom submitted to a perfect unilateral constraint. We assume that the inertia operator is nontrivial and the impact law consists in the transmission of the tangential component of the velocity and the reflexion of its normal component which is multiplied by the restitution coefficient e ∈ [ 0 , 1 ] . By adopting the measure-differential formulation of J.J. Moreau, a velocity-based time-stepping method is developed, reminiscent of the catching-up algorithm for sweeping processes. We prove that the numerical solutions converge to a solution of the vibro-impact problem. To cite this article: R. Dzonou et al., C. R. Mecanique 335 (2007).

Non-standard finite-difference methods for vibro-impact problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

CONTINUOUS DEPENDENCE ON DATA FOR VIBRO-IMPACT PROBLEMS

We consider vibro-impact problems, i.e. mechanical systems with a finite number of degrees of freedom subject to frictionless unilateral constraints. The dynamics is described by a second-order measure differential inclusion completed by an impact law of Newton's type. Motivated by the computation of approximate solutions, we study in this paper the continuous dependence of solutions on data. When several constraints can be active at the same time, continuity on data does not hold in general and an example of such a behavior is presented. We then propose a criterion involving the geometry of the active constraints along the limit trajectory which ensures continuity on data.

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh–Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.

Problèmes de vibro-impact : étude de la dépendance par rapport aux données

We consider a vibro-impact problem i.e. a mechanical system with a finite number of degrees of freedom, subject to perfect unilateral constraints. We assume that the transmission of velocities at impacts is described by a Newton's law with a restitution coefficient e∈[0,1]. Motivated by the computation of approximate solutions, we study the dependence of the solutions on data. For a large class of problems (single or multi-constrained systems, weak regularity of data), we obtain a criterion, involving the geometry of the constraints at impacts, which ensures continuity with respect to the data. To cite this article: L. Paoli, C. R. Acad. Sci. Paris, Ser. I 339 (2004).