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Mechanism and Machine Theory

Research paper

A continuous contact force model for impact analysis in multibody dynamics

Abstract

A new continuous contact force model for contact problems with complex geometries and energy dissipations in multibody systems is presented and discussed in this work. The model is developed according to the Hertz law, and a hysteresis damping force is introduced for modeling the energy dissipation during the contact process. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, an approximate dynamic equation for the collision system is proposed, achieving a good approximation of the system dynamic equation. An approximate function relating the deformation velocity and indentation depth is determined based on the approximate dynamic equation and is utilized to calculate the energy loss due to the damping force. Then, a primary formula of the hysteresis damping parameter is derived by combining energy balance and the law of conservation of linear momentum. The new model is developed by modifying the primary formula through nondimensional analysis. The comparison with published experimental data and the analysis of the simulation data of eight different continuous contact models reveal the capability and high precision of the new model as well as the effect of the geometry of the contacting surfaces on the dynamic system response.

Introduction

Contact-impact phenomena frequently occur in multibody systems, mainly due to the clearances between bodies and joints [1], [2], [3], [4], [5], [6], [7]. Proper modeling of a contact-impact phenomenon is very important for an accurate description of the dynamic behavior of multibody systems [8,9]. In the past few years, research on impact analysis in multibody systems has increased significantly [1,9,10]. The contact-impact process is characterized by an extremely short duration, a large contact force, fast energy dissipation, and great changes in the body velocities [11]. The modeling of contact-impact problems relies heavily on several factors, such as the topological properties of contacting surfaces, material characteristics, initial velocities and friction [12]. To describe the dynamic behavior of a multibody system, several crucial aspects of the impact modeling need to be taken into account, including the change in velocity before and after the impact, the duration time, the peak contact force and the indentation depth [9].

The earliest model of impact is the coefficient of restitution, which can describe the changes in the velocity and energy before and after the impact. There are different definitions of the coefficient of restitution [13,14]. Based on the speed before and after the impact, Newton's definition is the most popular and commonly used among them [10]. The restitution coefficient, which is relatively easy to measure experimentally [15], provides a concise description of the impact phenomenon. However, the details of the contact force and deformation in the process of collision cannot be described by the coefficient of restitution, except for the velocity change and the energy loss before and after the impact.

The second approach is the nonsmooth method, in which the duration of the collision is ignored, and the impact is assumed to occur instantaneously [16]. There are two ways to treat contact-impact problems in a multibody system, namely, the linear complementarity problem (LCP) [17,18] and differential variational inequality (DVI) [19,20]. Compared with the coefficient of restitution, the nonsmooth approach can calculate the contact force with a relatively efficient calculation; however, this approach is not valid for the modeling of impact duration due to the instantaneous assumption [9,[21], [22], [23]].

The third approach is named the compliant continuous contact force model, because the contact force is described as a continuous function of the indentation depth (relative deformation). Time histories of velocities contact forces and deformations, and duration time can be described by this method. The nonlinear Hertz contact model proposed in 1880 has provided an important basis for fundamental research on contact mechanics [24]. In recent years, several continuous contact models considering the energy dissipation and impact duration have been proposed on the basis of the Hertz model [25,26], such as the influential models proposed by Hunt and Crossley [27], Lankarani and Nikravesh [21], and Flores et al. [11]. Other continuous contact models, such as the Herbert and McWhannell model [28], the Lee and Wang model [29], the Gonthier et al. model [30], the Zhiying and Qishao model [31], the Kuwabara and Kono model [32], the Gharib and Hurmuzlu model [33], the Hu and Guo model [12], the Shen et al. model [34], the Safaeifar and Farshidianfar model [35], the Yu et al. model [36], the Poursina and Nikravesh model [37], the Carvalho and Martins model [38] and the Jian et al. model [39], have been developed. In these models, the hysteresis damping factor is derived as a function of the restitution coefficient, which is relatively easy to measure [15]. The strengths and weaknesses of these models have been discussed in the literature [1,9], especially the application limitations [40].

A new continuous contact force model, which is inspired by the work of Flores et al. [10] and Hu and Guo [12], is proposed in this paper. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, an approximate dynamic equation for the collision system is proposed for the first time, achieving a good approximation of the system dynamic equation. The new model is established via nondimensional analysis and is likely to solve the contact problems concerning complex geometries. The comparison with experimental data and the analysis of simulation data show a very high accuracy of the new model, which will be useful for multibody system dynamics. The remainder of the paper is organized as follows. Section 2 covers the fundamentals of the continuous contact force models. Then, the energy loss associated with the restitution coefficient is described in Section 3. Section 4 demonstrates the construction of the new contact model, and the model is verified in Section 5 through the comparison with experimental data and the analysis of simulation data of eight different contact models. Numerical examples and conclusions are presented in Sections 6 and 7, respectively.

Section snippets

General issues regarding the continuous contact force models

As shown in Fig. 1a, two solid objects (with masses mi and mj ) with a direct-central impact, the deformation taking place in the local contact zone, is divided into two phases: the compression phase and the restitution phase. At the initial time of impact t ( − ) , the objects have velocities v i ( − ) and v j ( − ) ; then, the deformation increases until the relative normal deformation between the contacting bodies reaches the maximum δm at time tm, and the objects reach the same velocity vij . After that,

The energy loss associated with the restitution coefficient

For a more accurate model, the energy change during the collision process needs to be considered [48]. The description of the energy change utilizes the coefficient of restitution, which is relatively easy to measure [15] but cannot describe the contact force, deformation, etc. of the collision process.

The coefficient of restitution, denoted cr , is defined [10,12] as c r = − δ ˙ ( + ) δ ˙ ( − ) where δ ˙ ( − ) = v i ( − ) − v j ( − ) and δ ˙ ( + ) = v i ( + ) − v j ( + ) are the initial relative velocity and the relative separating velocity

General issues regarding the construction of the new model

Although the coefficient of restitution can describe the changes in energy and velocity before and after the collision concisely, some important details of the contact-impact process cannot be obtained, as mentioned above. Previous studies have shown that continuous contact models can depict these details if the hysteresis damping factor λ is fixed by the measured value of the restitution coefficient. As one continuous contact model was adopted to simulate the contact-impact process, the

Validation and comparison of the new model

In this section, the new contact-impact model described in Eqs. (57) and (58) is verified through the comparison with published experimental data and the analysis of simulation data. The validation and comparison are carried out from the following aspects: consistency of post- and pre-restitution coefficients, contact duration time T and peak contact force Fmax , which are crucial for the description of the contact-impact behavior of a multibody system [9]. Seven different compliant continuous

Numerical examples

To conduct a more detailed analysis of the contact model proposed in this study and evaluate the influences of the contacting surface geometries on the system dynamic responses, the contact-impact of a free-falling object is simulated. As shown in Fig. 10, an elastic body made of PTFE is released from the initial position under the influence of gravity only and collides with the ground, which is assumed to be rigid and stationary. The basic parameters of the simulation are given in Table 4 [10,

Conclusions

A new continuous contact force model is proposed for impact analysis in multibody dynamics with complex geometries and with energy dissipation. The main difficulty of this research is that it is almost impossible to obtain an analytical solution from the system dynamic equation. An approximate dynamic equation is developed by introducing equivalent velocity. The analysis and simulations illustrate that the approximate dynamic equation with properly chosen equivalent velocities can accurately

Declaration of Competing Interest

This is a pure academic research paper. There is not any conflict of interests to any person/ organizations/groups.

Acknowledgements

This research was supported by the National Key Research and Development Plan of China [grant number 2019YFB1309600]; the National Natural Science Foundation of China [grant numbers 11702294 and 51775002]; the Joint Program of Beijing Municipal Foundation and Education Commission [grant number KZ202010009015]; and the Beijing Natural Science Foundation [grant number 3194047]. The author would like to thank the anonymous reviewers for their insightful comments and suggestions on an earlier draft

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