Nonlinear Contact Force Research Articles

Transient Chaos in Platform-vibrator with Shock

Platform-vibrator with shock is widely used in the construction industry for compacting and molding large concrete products. Its mathematical model, created in our previous work, meets all the basic requirements of shock-vibration technology for the precast concrete production on low-frequency resonant platform-vibrators. This model corresponds to the two-body 2-DOF vibro-impact system with a soft impact. It is strongly nonlinear non-smooth discontinuous system. This is unusual vibro-impact system due to its specific properties. The upper body, with a very large mass, breaks away from the lower body a very short distance, and then falls down onto the soft constraint that causes a soft impact. Then it bounces and falls again, and so on. A soft impact is simulated with nonlinear Hertzian contact force. This model exhibited many unique phenomena inherent in nonlinear non-smooth dynamical systems with varying control parameters. In this paper, we demonstrate the transient chaos in a vibro-impact system. Our finding of transient chaos in platform-vibrator with shock, besides being a remarkable phenomenon by itself, provides an understanding of the dynamical processes that occur in the platform-vibrator when varying the technological mass of the mold with concrete. Phase trajectories, Poincaré maps, graphs of time series and contact forces, Fourier spectra, the largest Lyapunov exponent, and wavelet characteristics are used in numerical investigations to determine the chaotic and periodic phases of the realization. We show both the dependence of the transient chaos on the control parameter value and the sensitive dependence on the initial conditions. We hope that this analysis can help avoid undesirable platform-vibrator behaviour during design and operation due to inappropriate system parameters, since transient chaos may be a dangerous and unwanted state of a vibro-impact system.

Force Reconstruction at Mechanical Interfaces Using a Modal Filtering Decomposition Approach

Traditional approaches to analyzing nonlinear systems often involve assuming a specific model form and model order for the nonlinearity. These nonlinearities in a system could instead be modeled as a nonlinear force applied to an underlying linear system. Force reconstruction methods can be used to recreate this nonlinear force from measured data, allowing the system change to be quantified. A modal based force reconstruction technique has proven capable of recreating force inputs at unmeasured locations and has been successfully applied to nonlinear intermittent external contact forces in previous studies. To extend these findings, a two-beam system was designed such that intermittent contact occurs near the joint locations, resulting in correlated forces. Previous work with this methodology has not addressed correlated loading conditions, so this work extends these techniques to the localization of correlated forces beyond measured points. The methodology is first demonstrated using analytical case studies, then performed experimentally. Acceleration responses are obtained from a sparse measurement grid which does not include the contact points, so the forces are reconstructed at unmeasured locations. Estimates of the nonlinear contact forces are calculated using the force reconstruction process, which are then compared to the measured force values. Cases are presented which address intermittent contact, loosening of preloaded connections, and the reconstruction of multiple connections simultaneously. Reconstructing correlated forces beyond measured points allows for the ability to locate and estimate forces without needing to assume the locations of these forces prior to acquiring data, extending the range of applications for these techniques.

A Study on Clearance Effects on Dynamic Responses of Robot Manipulator

Clearances caused by assemblage, manufacturing errors and wear, affect inevitably the dynamic responses of mechanisms such as robot manipulator. In this study, the effects of clearance on a robot manipulator system are investigated numerically. The contact behavior along normal and tangential direction of clearance joint is described by a nonlinear contact force model and a modified Coulomb friction model respectively. Then, the dynamics equations of the robot manipulator system are established considering joint clearance. In order to investigate the effects of clearance on dynamic performances of practical mechanism, a planar robot manipulator system on a spacecraft system with a revolute clearance joint is used as the apply example. Four case studies for various clearance sizes are implemented to investigate and discuss the effects of joint clearance. The simulation results indicate that clearance joints have severe effects on the dynamic performances of mechanism system and the impact in clearance joints represented by contact force models must be considered in dynamics analysis and design of mechanism system. The simulation results in this work can predict the effects of clearance on robot manipulator system preferably and it is the basis of precision analysis, robust control system design of robot manipulator system.

Parameter optimization for torsion spring of deployable solar array system with multiple clearance joints considering rigid–flexible coupling dynamics

In this paper, four novel evaluation indices and corresponding hierarchical optimization strategies are proposed for a deployable solar array system considering panel flexibility and joint clearance. The deployable solar array model consists of a rigid main-body, two panels and four key mechanisms, containing torsion spring mechanism, closed cable loop mechanism, latch mechanism and attitude adjustment mechanism. Rigid and flexible components are established by Nodal Coordinate Formulation and Absolute Nodal Coordinate Formulation, respectively. The clearance joint model is described by nonlinear contact force model and amendatory Coulomb friction model. The latch time, stabilization time, maximum contact force and impulse sum of the contact force of the solar array system are selected as the four novel evaluation indices to represent the complex dynamic responses of a deployable solar array with clearance joints instead of the lock torque widely used in conventional works. To eliminate the gross errors caused by the nonlinear and nonsmooth mechanical properties, a hierarchical optimization strategy based on an adaptive simulated annealing algorithm and a nondominated sorting genetic algorithm is adopted for the solar array system with clearance joints. Results indicate that the effects of panel flexibility on the evaluation index responses and design optimization of the solar array system cannot be neglected. Besides, increasing the weight factor of the stabilization time index of the rigid system may compensate for the differences in optimal results of the rigid–flexible coupling system. That may provide some references for optimization design of deployable space mechanisms considering clearance joints.

Kinematic accuracy and nonlinear dynamics of a flexible slider-crank mechanism with multiple clearance joints

Clearance joint appears in the multibody system and it could cause the uncertainty of motion. A general method for dynamic modelling and analysis of slider-crank mechanism with multiple clearance joints is presented and discussed in this work. The contact-impact characteristics of clearance joint are described by the nonlinear continuous contact force model. With flexible link taken into account, the dynamic characteristics of slider-crank mechanism with multiple clearance joints are analyzed based on the presented model. Then, the dynamic response of a flexible slider-crank mechanism is measured to clarify how to clearance joints can cause the vibration phenomenon of multibody system. Moreover, the stability and chaos analysis are conducted with the purpose of performing a parametric study, which quantifies the effects of clearance size and rotational speed on dynamic behavior of multibody system with multiple clearance joints. The numerical results indicate that the dynamic behavior of a flexible slider-crank mechanism depends on the characteristics of clearance joints.

Modelling and simulation of active and passive seat suspensions for vibration attenuation of vehicle occupants

Literature on biodynamic modelling of the occupant in a vehicle associates riders’ comfort with human-seat interaction forces and vibration attenuation capabilities of the suspension system. Though the superiority of active suspension systems over passive ones is well established in literature, quantification of this superiority by using the best possible passive suspension system has not been reported. This work attempts to do this. It integrates a nonlinear cushion seat contact force model and a 12 degrees of freedom two-dimensional seated human body model supported by an inclined backrest with a full vehicle model through the seat suspension system. The passive and active proportional integral derivative suspension system parameters are obtained by simultaneously minimizing the seat effective amplitude transmissibility, cushion contact force and head motion using a multi-objective genetic algorithm in MATLAB-SIMULINK co-simulation. The time responses of cushion contact force, head vertical and fore-aft motion are studied for two road profiles—random and bump. Comparative analyses were also done with regard to internal forces, absorbed power and the effect on vehicle chassis. Human body response to different grades of road roughness and vehicle speeds were investigated. The results establish the clear superiority of the active system in all aspects with rise in suspension travel and acceleration of vehicle chassis in the vertical direction for a random road profile. A parameter sensitivity analysis allowed us to identify spring stiffness as the component which needs greatest care during fabrication.

A New Estimation of Nonlinear Contact Forces of Railway Vehicle

The core part of any study of rolling stock behavior is the wheel-track interaction patch because the forces produced at the wheel-track interface govern the dynamic behavior of the whole railway vehicle. It is significant to know the nature of the contact force to design more effective vehicle dynamics control systems and condition monitoring systems. However, it is hard to find the status of this adhesion force due to its complexity, highly non-linear nature, and also affected with an unpredictable operation environment. The purpose of this paper is to develop a model-based estimation technique using the Extended Kalman Filter (EKF) with inertial sensors to estimate non-linear wheelset dynamics in variable adhesion conditions. The proposed model results show the robust performance of the EKF algorithm in dry, wet/rain, greasy, and fully contaminated track conditions in traction and braking modes of a railway vehicle. The proposed model is related to the other works in the area of wheel-rail systems and a tradeoff exists in all conditions. This model is very useful in condition monitoring systems for railway asset management to avoid accidents and derailment of a train

A Coupled Discrete‐Finite Element Method for Shear Strength Analysis of Geogrid‐Reinforced Railway Ballast

This paper presents a coupled discrete‐finite element method for the investigation of shear strength of geogrid‐reinforced ballast by direct shear tests and pull‐out tests. The discrete element method (DEM) and finite element method (FEM) are employed to simulate ballast and geogrid, respectively. Irregularly shaped ballast particles are modeled with clumps, and the nonlinear contact force model is used to calculate contact force between particles. Continuum geogrid is modeled by a two‐node beam element with six degrees of freedom. A contact algorithm based on the static equilibrium is proposed at the geogrid‐ballast contact surface. The simulation results indicate that shear strengths increase with the installation of geogrid. Moreover, ballast particle displacements and nominal volumetric strains are analyzed to provide a microscopic view on the mechanism of the reinforcement effect of geogrid.

Stochastic dynamic analysis of vehicle-bridge coupled systems with nonlinear Hertz contacts by explicit time-domain method

ABSTRACT Vibration analysis of vehicle-bridge coupled systems has received considerable attentions over the past two decades. This work is dedicated to the stochastic dynamic analysis of vehicle-bridge coupled systems with nonlinear Hertz wheel-rail contacts subjected to random track irregularities. An explicit time-domain method (ETDM) is proposed for efficient nonlinear dynamic analysis of vehicle-bridge coupled systems with a dimension-reduced iteration scheme focusing on the nonlinear wheel-rail contact forces. The high-efficient ETDM is further utilised to implement the numerous nonlinear dynamic analyses required in the Monte-Carlo simulation (MCS) for obtaining the response statistics of the vehicle-bridge coupled systems. A numerical example involving a single-wheel vehicle traversing a simply-supported beam with nonlinear Hertz wheel-rail contacts considering random track irregularities is presented to demonstrate the computational accuracy and efficiency of the proposed method, and an engineering application involving a four-wheel vehicle moving across the Egongyan suspension bridge is further analysed to indicate the effectiveness of the proposed method for large-scale engineering problems.

Nonlinear dynamics analysis of shift manipulator for robot driver considering multiple revolute clearance joints and variable load

Clearance joints and variable load have a great influence on the dynamic performance of shift manipulator for robot driver. In this article, a dynamics model of clearance joint and a simulation method of variable load are proposed to analyze the influence of clearance joint and variable load on the dynamic performance of shift manipulator. Firstly, a hybrid nonlinear contact force model is established to describe the normal contact force of revolute clearance joint, and a modified Coulomb friction model is used to describe the tangential contact force of revolute clearance joint. On this basis, a virtual prototype dynamics model of shift manipulator with multiple revolute clearance joints is established. Then a power transmission model of shift manipulator including the variable load model at the end of shift manipulator is established. The time-varying and nonlinear characteristics of the load in gear engaging direction are simulated by neural network. Finally, the influences of different clearance dimension and variable load on the dynamic performance of shift manipulator are analyzed. Experiment and simulation results show the validity of the presented analysis, which provides a basis for further structure optimization of shift manipulator.

Impact Behavior of a Rotating Rigid Body with Impact and Viscous Friction

The impact between a rotating link and a solid flat surface is considered. For the impact, we consider three distinct periods: elastic period, elastoplastic period, and restitution period. A Hertzian contact force is considered for the elastic period. Nonlinear contact forces developed from finite element analysis are used for the remaining two phases. The tangential effect is taken into account considering a friction force that combines the Coulomb dry friction model and a viscous friction function of velocity. Simulations results are obtained for different friction parameters. An experimental setup was designed to measure the contact time during impact. The experimental and simulation results are compared for different lengths of the link.

Upgraded Reduction Technique for Dynamic Analysis of Structures with Friction Contact

A recent reduction technique for the nonlinear forced response analysis of structures with contact interfaces is upgraded. The evaluations of the existing method (Dual formulation), which is based on dual Craig–Bampton method, show its considerable off-line computational time saving because of no matrix manipulation, whereas the resulting reduced model is less accurate than Rubin’s in approximating the full finite element model. The accuracy of this formulation is significantly improved with a low additional computational cost in the proposed upgraded formulation. That is obtained by consistent projection of all structural matrices on a reduction basis made up of two sets of vectors: free-interface normal modes and residual flexibility attachment modes. Assembly approach can be considered as dual because the interface forces are kept as generalized coordinates in opposition to primal methods, where interface displacements are retained. To reduce the computational cost, the nonlinear governing equations are also solved in frequency domain using the multiharmonic balance method and alternating frequency time method. Contact elements are introduced between adjacent interface nodes to find the nonlinear contact forces. The performance of the new formulation is demonstrated through two numerical models with friction contact.

Nonlinear response analysis for a dual-rotor system supported by ball bearing

Rolling element bearings are used as main supports and are key sources of vibrations for aero-engine rotor systems. Mainly including the inner and outer rings, cages, and balls or rollers, the complicated mechanical structures of rolling element bearings exhibit nonlinear behaviors due to the bearing clearance, nonlinear Hertzian contact forces, and defects. Due to imbalanced rotations, which are unavoidable, a wide variety of nonlinear behaviors can be expected in dual-rotor systems. Thus, in this study, we focus on the nonlinear behaviors of a dual-rotor system supported by a rolling element bearing. The nonlinear characteristics, such as the time-varying stiffness and primary resonance behaviors affected by the ball bearing parameters, are discussed. A dual-rotor system supported by a ball bearing was built while considering the nonlinear factors from the 4# ball bearing (at left end of the high-pressure rotor system). The nonlinear governing equations were established using the finite element method. Using the numerical methods, the effects of the bearing clearance on the time-varying stiffness and primary resonance behaviors are analyzed in detail. The results indicate that the mean values of the equivalent stiffness of the system and the rotation speed for forced resonance decrease as the clearance of the ball bearing increases. Periods of “zero-stiffness” occur when Ω1 (the rotation speed of the high-pressure rotor system) is less than 800 rad/s, and the periods of “zero-stiffness” over one period grow as the bearing clearance is increased. Meanwhile, in the case of a small or even zero clearance of a ball bearing, the amplitude–frequency curves still exhibit “hard characteristics” in the forced resonance domain, and hysteresis and jump resonance occur. All of the results can be used to establish base-line behaviors for system status determination, prediction, and control and to eliminate vibrations.

Modelling and simulation of an integrated human-vehicle system with non-linear cushion contact force

Driver comfort is related to human-seat interaction forces and the seat suspension system. For a human body subjected to vertical seat vibration, the head exhibits vertical as well as fore-aft motion, which leads to high forces in the neck. In order to capture these effects, a detailed 12 degrees of freedom two-dimensional seated human body model with inclined backrest support is developed to study direct and cross-axis seat to head transmissibility. The human model is then integrated with a non-linear cushion-human interaction model. The cushion parameters are obtained using a genetic algorithm and a global criterion-based scheme to minimize the least square difference between experimental contact force-time signal and analytical results. Subsequently, the human body model, along with a non-linear seat cushion model, are incorporated into a full vehicle model. Seat suspension parameters are obtained by minimizing the seat effective amplitude transmissibility and human body comfort factors. Both random road excitation and sudden bumps on the road have been studied. The influence of the human body's position in the vehicle and parameter sensitivity with respect to cushion-human interaction force, vertical and fore-aft head acceleration are investigated for multi-compression damper with an inclined damper-seat suspension system. The angle of inclination of the damper has been found to be the parameter with the highest sensitivity. These results and the modelling methodology are expected to lead to a better way of analyzing biodynamic response of the driver.

Skidding dynamic performance of rolling bearing with cage flexibility under accelerating conditions

Rolling bearings are widely used in rotating machinery and its dynamic performance plays a significant effect on the stability, reliability and even safety of the machine. Presence of the skidding is likely to cause damages to the surfaces of the inner race, the outer race, the rolling elements, and/or the cage. In this paper, a skidding dynamics model of a rolling bearing is established to reveal the skidding characteristics. In the dynamics model, the cage is discretized into several segments having the same number of the rolling elements so as to introduce the flexibility of the cage represented by springs connecting the adjacent segments. And the nonlinear contact forces between the rolling elements and inner\\outer races and the cage, the corresponding friction forces, and the gravity and centrifugal forces of the rolling elements are considered comprehensively. The simulation results are compared with the traditional dynamics model with rigid cage. By using the dynamics model, effects of the radial load, the inner ring acceleration, and the connect stiffness between the mass blocks of the cage on the bearing skidding are investigated. The results indicate that the rolling element skidding generally happens at the time instant that the rolling element is entering or leaving the loaded region. The smaller radial load or the higher angular acceleration of inner ring are the primary causes of the bearing skidding. In addition, increasing the connect stiffness could effectively alleviate the skidding phenomenon. Therefore, the proposed bearing dynamics model is proven to be capable of making a reasonable prediction on the bearing skidding with a more realistic flexible cage.

Fundamental Contact Dynamics Model of a Crawler-Type Vehicle Simulation

In this paper, I established a single crawler model with a dynamic contact force considered. I modeled each crawler track using dynamic contact model. By modeling each crawler track, I hope to be able to analyze trafficability of a crawler type vehicle in various road surfaces. First, I think that I established a single crawler model. I modeled a single crawler using Hunt-Crossley model and Coulomb model that handle a nonlinear contact force model. And also, I built a prototype single crawler for comparing the experimental examination with the simulation examination. In advance, the running over limit of the prototype single crawler was analyzed geometrically, and I bore out that it could run over to 163 mm. Therefore, I conducted the experiment examination with step height of 160 mm and 165 mm and obtained trajectory data that were allowed to run over and were not allowed to run over. From the result of both the experimental examination and the simulation examination, I bore out the validity of our single crawler model. I described about our single crawler model, the simulation examination and the experimental examination in this paper.

An improved nonlinear dynamic reduction method for complex jointed structures with local hysteresis model

An improved nonlinear dynamic reduction with the harmonic balance method is developed to approximate the steady-state vibration responses of the complex jointed structures having local hysteresis nonlinearity. The modal superposition method combined with the static stiffness compensation is used to degrade the mode truncation effects on the transfer functions by using only one concerned dominant mode. The local nonlinearity transformation reduction is used to decrease the dimensions of nonlinear algebraic equations and iteration matrix by defining the local nonlinear contact forces as the iteration vector. The transfer functions only related to the nonlinear joints are extracted to connect the local nonlinear dynamic responses with the local nonlinear contact forces and external excitations. The local nonlinear contact forces are consequently updated. The discrepancy between the given local nonlinear contact forces and updated ones is used to construct the residual error functions to update the iteration vector to obtain the nonlinear solutions. Comparison with the generalized modal superposition and full order methods in the literature is performed to validate the proposed method by using a lap-type bolted joint beam system and a complex jointed structure. The good agreement of comparison results of two examples validates the proposed method, and indicates a higher computational efficiency. The effective construction of the transfer functions plays a critical role in predicting the steady-state nonlinear dynamic responses of the complex jointed structures, and consumes a lot of computational costs.

A New Theoretical Model to Study the Closing Bounce Characteristics of the Electromagnetic Relay Under Capacitive Loads

Contact bounce is a common phenomenon in the closing process of the electromagnetic relay under capacitive loads. The investigations on the contact characteristics are mainly conducted by experimental methods. However, the experimental method is extremely complicated and time-consuming in structural design, and it cannot clearly reveal the mechanism of contact bounce. Therefore, a new theoretical model is proposed to investigate the dynamic characteristics of the electromagnetic relay under capacitive loads. The model allows for simulating dynamic process and collecting bounce parameters in a simple and effective way compared with conventional simulation and experimental methods. The physical mechanism of contact bounce for the electromagnetic relay is presented. The nonlinear contact force between the fixed and moving contacts is evaluated according to the Kelvin–Voigt viscoelastic contact model. The equations of motion considering the coupling of electromagnetic and mechanical fields are established. The theoretical results obtained in this study are validated by comparison with experimental data. The influences of the structural parameters, coil current, surge current, and contact resistance on the contact bounce are analyzed. The numerical results well demonstrate that the contact bounce can be effectively weakened by adjusting the parameters of contact spring and the mass of the moving contact.

Impact of a Multiple Pendulum with a Non-Linear Contact Force

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing.

Dynamics Analysis of Spatial Landing-Gear Mechanism with Hinge Clearance and Axis Deviation

The hinge clearance and rotation axis deviation of aircraft landing-gear retraction mechanism are unavoidable due to factors, such as manufacture, assembly, wear, and wing deformation. Such negative factors affect the dynamic response of the mechanism itself, which is more serious for spatial mechanisms. To study this effect, both numerical and experimental investigations on a spatial landing-gear mechanism are carried out in this study. To build the numerical model, the nonlinear contact force model and modified Coulomb friction model are adopted in the hinge clearance of the landing-gear mechanism. Considering the flexibility effect, some key moving parts, such as sidestay links, are flexibly processed to deal with the axis deviation problems. Based on the proposed model, the influences of clearance size and axis deviation value on the dynamic behaviors of spatial landing-gear mechanism are studied. Both the test and simulation results indicate that the clearance and axis deviation have little effect on the actuation response of the retraction mechanism if they are within reasonable ranges. Nevertheless, the axis deviation is greatly influential to the load on structure and the friction in hinge, whereas the clearance mitigates the adverse effects on the motion of mechanism caused by the axis deviation to some extent.