Hamid M. Lankarani

A contact force model with hysteresis damping for impact analysis of multibody systems

A continuous contact force model for the impact analysis of a two-particle collision is presented. The model uses the general trend of the Hertz contact law. A hysteresis damping function is incorporated in the model which represents the dissipated energy in impact. The parameters in the model are determined, and the validity of the model is established. The model is then generalized to the impact analysis between two bodies of a multibody system. A continuous analysis is performed using the equations of motion of either the multibody system or an equivalent two-particle model of the colliding bodies. For the latter, the concept of effective mass is presented in order to compensate for the effects of joint forces in the system. For illustration, the impact situation between a slider-crank mechanism and another sliding block is considered.

Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.

Influence of the contact–impact force model on the dynamic response of multi-body systems

Abstract This work deals with contact—impact force models for both spherical and cylindrical contact surfaces. The incorporation of the friction phenomenon, based on the Coulomb friction law, is also discussed together with an effective computational strategy, which includes the automatic step size selection procedure. Impacts within a revolute clearance joint in a basic slider—crank mechanism are used as an example to compare the different contact force models. The collision is a prominent phenomenon in many multi-body systems such as mechanisms with intermittent motion, kinematic discontinuities, and clearance joints. As a result of an impact, the values of the system state variables change very fast, eventually looking like discontinuities in the system velocities and accelerations. The impact is characterized by large forces that are applied and removed in a short time period. The knowledge of the peak forces developed in the impact process is very important for the dynamic analysis of multi-body systems and it has consequences in the design process. The model for the contact—impact force must consider the material and geometric properties of the colliding surfaces, consider information on the impact velocity, contribute to an efficient integration, and account for some level of energy dissipation. These characteristics are ensured with a continuous contact force model, in which the deformation and contact forces are considered as continuous functions.

Dynamic Response of Multibody Systems with Multiple Clearance Joints

A general methodology for the dynamic modeling and analysis of planar multibody systems with multiple clearance joints is presented. The inter-connecting body components that constitute a real joint are modeled as colliding bodies, which dynamic behaviors are influenced by geometric, physical and mechanical properties of the contacting surfaces. A continuous contact force model, based on the elastic Hertz theory, together with a dissipative term, is used to evaluate the intra-joint contact forces. The incorporation of the friction, based on the classical Coulomb’s friction law, is also included. The suitable contact force models are embedded into the dynamic equations of motion for the multibody system. A simple mechanical system with multiple clearance joints is used to demonstrate the accuracy and efficiency of the presented approach and to discuss the main assumptions and procedures adopted. The effects of single versus multiple clearance joints are discussed.Copyright

Dynamic behaviour of planar rigid multi-body systems including revolute joints with clearance

Abstract This paper deals with a general methodology to assess the influence of the clearance size and the friction coefficient on the dynamic response of planar rigid multi-body systems including revolute joints with clearance. When there is a clearance in a revolute joint, impacts between the journal and the bearing can occur, and consequently, local deformations take place. The impact is internal and the response of the system is performed using a continuous contact force model. The friction effect because of the contact between joint elements is also included. The dynamic response of the systems is obtained numerically by solving the constraint equations and the contact-impact forces produced in the clearance joint, simultaneously with the differential equations of motion and a set of initial conditions. Numerical results for two simple mechanisms with revolute clearance joints are presented and discussed. In the present work, the clearance size and friction effects are analysed separately. Through the use of Poincaré maps, both periodic and chaotic responses of the systems are observed. The results predict the existence of the periodic or regular motion at certain clearance sizes and friction coefficients and chaotic or non-linear in other cases.

Dynamics of Multibody Systems With Spherical Clearance Joints

This work deals with a methodology to assess the influence of the spherical clearance joints in spatial multibody systems. The methodology is based on the Cartesian coordinates, being the dynamics of the joint elements modeled as impacting bodies and controlled by contact forces. The impacts and contacts are described by a continuous contact force model that accounts for geometric and mechanical characteristics of the contacting surfaces. The contact force is evaluated as function of the elastic pseudo-penetration between the impacting bodies, coupled with a nonlinear viscous-elastic factor representing the energy dissipation during the impact process. A spatial four bar mechanism is used as an illustrative example and some numerical results are presented, being the efficiency of the developed methodology discussed in the process of their presentation. The results obtained show that the inclusion of clearance joints in the modelization of spatial multibody systems significantly influences the prediction of components’ position and drastically increases the peaks in acceleration and reaction moments at the joints. Moreover, the system’s response clearly tends to be nonperiodic when a clearance joint is included in the simulation.Copyright

The Head Injury Criterion (HIC) functional

The mathematical structure of the Head Injury Criterion (HIC) functional is described. Necessary conditions for the extremal solution of the HIC functional are presented, along with results characterizing the critical points of the functional. A program written in Borland C is developed to construct the HIC functional, and various characteristics of the HIC are determined from the constructed functional, including the average HIC value, HIC variance, and higher-order HIC moments. By investigating the form of the HIC functional, it is possible to extract structural information by the computation of the moments of the functional. Sample functional plots for experimental acceleration pulses are also provided.

Translational Joints With Clearance in Rigid Multibody Systems

A computational methodology for dynamic description of rigid multibody systems with translational clearance joints is presented and discussed in this work. Over the past years, extensive work has been done to study the dynamic effect of the revolute joints with clearance in multibody systems, in contrast with the little work devoted to model translational joints with clearance. In a joint with translation clearance, there are many possible ways to set the physical configuration between the slider and guide, namely: (i) no contact between the two elements, (ii) one corner of the slider in contact with the guide surface, (iii) two adjacent slider corners in contact with the guide surface, and (iv) two opposite slider corners in contact with the guide surfaces. The proposed methodology takes into account these four different situations. The conditions for switching from one case to another depend on the system dynamics configuration. The existence of a clearance in a translational joint removes two kinematic constraints from a planar system and introduces two extra degrees of freedom in the system. Thus, a translational clearance joint does not constrain any degree of freedom of the mechanical system but it imposes some restrictions on the slider motion inside the guide limits. When the slider reaches the guide surfaces, an impact occurs and the dynamic response of the joint is modeled by contact-impact forces. These forces are evaluated here with continuous contact force law together with a dissipative friction force model. The contact-impact forces are introduced into the systems equations of motion as external generalized forces. The proposed methodology is applied to a planar multibody mechanical system with a translational clearance joint in order to demonstrate its features.

Joint Clearances With Lubricated Long Bearings in Multibody Mechanical Systems

Proper modeling of joint clearance is of great importance in the analysis and design of multibody mechanical systems. The clearance may be due to wear or imperfection in manufacturing. When there is no lubricant in the clearance solid-to-solid contact occurs. The impulse due to contact between the links is transmitted throughout the system. The presence of a lubricant avoids such contact, as the hydrodynamic forces developed by the lubricant film support the loads acting on the bodies and prevent the bodies from coming into contact. In this paper, an analysis of revolute joint clearances in multibody mechanical systems with and without lubricant is presented. Squeeze as well as viscous effects are considered utilizing the hydrodynamic theory of lubrication in long bearings. Unlike the traditional machine design approach, the instantaneous lubricant forces are the unknown and evaluated in terms of the known geometrical position and velocity of the journal and bearing. In the case of analysis of a joint clearance with no lubricant, a modified Hertzian relation is used to model the impact or contact between the journal and bearing, which includes a hysteresis damping term to account for the energy dissipation during impact. The methodology is applied for the analysis of a slider-crank mechanism having a clearance in the piston pin. The simulations are carried out with and without lubricant and the results are compared, It is shown that the lubricant results in a steady motion with fewer peaks in the required cranking moment for the system.

Frictional impact analysis in open-loop multibody mechanical systems

Analysis of impact problems in the presence of any tangential component of impact velocity requires a friction model capable of correct detection of the impact modes. This paper presents a formulation for the analysis of impact problems with friction in open-loop multibody mechanical systems. The formulation recognizes the correct mode of impact; i.e., sliding, sticking, and reverse sliding. Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of the Newton’s hypothesis are avoided. The formulation is developed by using a canonical form of the system equations of motion using joint coordinates and joint momenta. The canonical momentum-balance equations are solved for the change in joint momenta using Routh’s graphical method. The velocity jumps are calculated balancing the accumulated momenta of the system during the impact process. The impact cases are classified based on the pre-impact positions and velocities, and inertia properties of the impacting systems, and expressions for the normal and tangential impulse are derived for each impact case. The classical problem of impact of a falling rod with the ground (a single object impact) is solved with the developed formulation and verified. Another classical problem of a double pendulum striking the ground (a multibody system impact) is also presented. The results obtained for the double pendulum problem confirms that the energy gain in impact analysis can be avoided by considering the correct mode of impact and using the Poisson’s instead of the Newton’s hypothesis.