Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints

We consider the dynamics of impact oscillators with multiple degrees of freedom subject to more than one motion limiting constraint or stop. A mathematical formulation for modeling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom system are considered, along with definitions of the impact map for multiply constrained systems. We consider sticking motions that occur when a single mass in the system becomes stuck to an impact stop, and discuss the computational issues related to computing such solutions. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. Numerical examples of sticking orbits for this system are shown and we discuss identifying the region, S in phase space where these orbits exist. We use bifurcation diagrams to indicate differing regimes of vibro-impacting motion for two different cases; firstly when the stops are both equal and on the same side (i.e. the same sign) and secondly when the stops are unequal and of opposing sign. For these two different constraint configurations we observe qualitatively different dynamical behavior, which is interpreted using impact mappings and two-dimensional parameter space

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Periodic sticking motion in a two-degree of freedom impact oscillator

Periodic sticking motions can occur in vibro-impact systems for certain parameter ranges. When the coefficient of restitution is low (or zero), the range of periodic sticking motions can become large. In this work the dynamics of periodic sticking orbits with both zero and non-zero coefficient of restitution are considered. The dynamics of the periodic orbit is simulated as the forcing frequency of the system is varied. In particular, the loci of Poincaré fixed points in the sticking plane are computed as the forcing frequency of the system is varied. For zero coefficient of restitution, the size of the sticking region for a particular choice of parameters appears to be maximized. We consider this idea by computing the sticking region for zero and non-zero coefficient of restitution values. It has been shown that periodic sticking orbits can bifurcate via the rising/multi-sliding bifurcation. In the final part of this paper, we describe three types of post-bifurcation behavior which occur for the zero coefficient of restitution case. This includes two types of rising bifurcation and a border orbit crossing event

Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator

We consider the rising phenomena which occur in sticking solutions of a two-degree of freedom impact oscillator. We describe a mathematical formulation for modelling such a systems during both free flight and during sticking solutions for each of the masses in the system. Simulations of the sticking solutions are carried out, and rising events are observed when the forcing frequency parameter is varied. We show how the time of sticking reduces significantly as a rising event occurs. Then within the sticking region we show how rising is qualitatively similar to the multi-sliding bifurcation for sliding orbits

Galerkin-Ivanov transformation for nonsmooth modeling of vibro-impacts in continuous structures

This work deals with the modeling of nonsmooth vibro-impact motion of a continuous structure against a rigid distributed obstacle. Galerkin's approach is used to approximate the solutions of the governing partial differential equations of the structure, which results in a system of ordinary differential equations (ODEs). When these ODEs are subjected to unilateral constraints and velocity jump conditions, one must use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a computationally demanding task. Ivanov proposed a nonsmooth transformation for a vibro-impacting multi-degree-of-freedom system subjected to a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions; however, he did not explore its application for simulating distributed collisions in spatially continuous structures. We adopt Ivanov's transformation to deal with multiple unilateral constraints in spatially continuous structures. Also, imposing the velocity jump conditions exactly in the modal coordinates is nontrivial and challenging. Therefore, in this work we use a modal-physical transformation to convert the system from modal to physical coordinates on a spatially discretized grid. We then apply Ivanov's transformation on the physical system to simulate the vibro-impact motion of the structure. The developed method is demonstrated by modeling the distributed collision of a nonlinear string against a rigid distributed surface. For validation, we compare our results with the well-known penalty approach

Scenarios in the experimental response of a vibro-impact single-degree-of-freedom system and numerical simulations

In this paper, possible scenarios within the experimental dynamic response of a vibro-impact single-degree-of-freedom system, symmetrically constrained by deformable and dissipative bumpers, were identified and described. The different scenarios were obtained varying selected parameters, namely peak table acceleration A , amplitude of the total gap between mass and bumpers G and bumper’s stiffness B. Subsequently, using a Simplified Nonlinear Model results in good agreement with the experimental outcomes were obtained, although the model includes only the nonlinearities due to clearance existence and impact occurrence. Further numerical analysis highlighted other scenarios that can be obtained for values of the parameters not considered in the experimental laboratory campaign. Finally, to attempt a generalization of the results, suitable dimensionless parameters were introduced

Direct and inverse problems encountered in vibro-impact oscillations of a discrete system

International audienceWe study direct and inverse problems that arise in the vibro-impact oscillations of a discrete system. Specifically, we examine a class of systems with two coordinates undergoing single-or double-sided impacts; however, the presented techniques are sufficiently general to apply to systems with multiple impacts. The analytical methods employed are a nonlinear normal mode (NNM)-type analysis and a boundary value problem (BVP) formulation, and enable the computation of various branches of bifurcating periodic solutions with different impacting characteristics. Additional insight on the dynamics of these systems is obtained by direct integrations of the equations of motion and by numerical Poincare´aps. It is found that the vibro-impact systems considered possess rich nonlinear dynamics, including vibro-impact localized and nonlocalized time-periodic motions, complicated bifurcation structures giving rise to new types of single-and double-sided impacting motions, mode instabilities, and chaotic responses. We also formulate inverse vibro-impact problems, whereby, we seek the class of dynamical systems that produce specified orbits in the configuration plane. The solutions of the inverse problems are generally non-unique, since they can be reduced to underdetermined sets of algebraic equations with multiple infinities of unknowns. Numerical applications are provided to demonstrate the techniques and validate the analytical results

Bi-stability induced by motion limiting constraints on boring bar tuned mass dampers

This paper investigates the effect of displacement constraints on the attenuation performance of tuned mass dampers (TMDs) used in boring and turning applications. A simplified piecewise- smooth mechanical model is investigated through time domain simulations and hybrid periodic orbit continuation, first under harmonic excitation, then under regenerative cutting load. A quasi-frequency response function is derived for impacting TMDs through composition of different families of period-1 orbits, then an acceptability map for turning is formulated based on the appearance of cutting-edge contact-loss and fly-over events. The bi-stable domain boundaries are determined through two parameter continuation of contact-loss grazing events. It is shown that in both cases arising rigid body collisions can significantly hinder TMD damping performance and lead to resonance problems or machine tool chatter

Application of nonsmooth modelling techniques to the dynamics of a flexible impacting beam

Non-smooth modelling techniques have been successfully applied to lumped mass-type structures for modelling phenomena such as vibro-impact and friction oscillators. In this paper, the application of these techniques to continuous elements using the example of a cantilever beam is considered. Employing a Galerkin reduction to form an N -degree-of-freedom modal model, a technique for modelling impact phenomena using a non-smooth dynamics approach is demonstrated. Numerical simulations computed using the non-smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side. A method for dealing with sticking motions when numerically simulating the beam motion is presented. In addition, choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed