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Bi-potential method applied for dynamics problems of rigid bodies involving friction and multiple impacts

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Abstract

This study aims to extend the application of the bi-potential method to solve the dynamics problems of rigid bodies involving friction and multiple impacts. The key issue is the determination of the impact impulsion. The impact bi-potential function and the formula for calculating the bi-potential coefficient of rigid bodies are presented. The effects of the friction and normal restitutive coefficients on the results are analyzed. Extension from discrete granular systems to multibody systems is realized. Further, the internal software FER/Mech is used for realizing dynamic simulation of the entity object, and the comparison with the experimental video verifies the correctness of the proposed method. Results show that the achieved improvements of this study are more accurate. In the discrete granular systems, the numerical solutions of the proposed method are in good agreement with those of another numerical method. But in the multibody systems, the numerical solutions obtained by the proposed method are more stable. It is noteworthy that the bi-potential method not only does not increase the degree of freedom of the system, but also makes the programming simple and numerically robust, thus reducing the computing cost.

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All data included in this study are available from the corresponding author on reasonable request.

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Acknowledgements

We gratefully acknowledge the financial support of the National Youth Science Foundation of China (Grant No. 1200229 0) and the National Natural Science Foundation of China (Grant No. 11770 20241).

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Authors and Affiliations

  1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China

    Ling Tao, Yan Li, Zhi-Qiang Feng, Yi-Jin Cheng & Hui-Jian Chen

  2. LMEE Univ-Evry, Université Paris-Saclay, Evry, France

    Zhi-Qiang Feng

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  1. Ling Tao
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  2. Yan Li
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Correspondence to Zhi-Qiang Feng.

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Appendix

Appendix

Fig. 20
figure 20

Energy consistency of the simulation results: a Example 1; b Example 2; and c Example 3

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Fig. 21
figure 21

Example 4: a model of planar ball from Ref. [66]; b comparison of energy consistency between the proposed method and Ref. [66]

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The Newton’s restitutive hypothesis may increase the energy in certain cases; thus, this paper adopts the model proposed by Moreau [32], and the energy diagrams of the three examples are shown in Fig. 20. In Examples 1 and 2, when the ball is in impact, the kinetic energy decreases sharply, the potential energy reaches the minimum, and the total energy is lost due to impacts and friction forces. In Example 3, there are two cases when the kinetic energy reaches its minimum. Case 1 is that Points 1 and 3 both hit the cylinder. Case 2 is that Point 2 is in impact and the spring is compressed until it does not change. The kinetic energy increases at Stages 2 and 4, the potential energy increases at Stages 1 and 3. During Stage 1, when Point 3 is in impact, \({\nu _y}\) and \({\omega _M}\) are zero, but \({\omega _S}\) is not zero until Point 1 is in impact. Thus, the kinetic energy is decreasing to zero, while the potential energy increases at this stage. During Stage 2, y increases in the negative direction, and \({\omega _S}\) continues to evolve in a clockwise direction. Here, the kinetic energy increases, but the potential energy decreases. When Point 2 is in impact, y and \({\varphi _M}\) remain constant, \({\omega _S}\) decreases to zero and \({\varphi _S}\) will reach its maximum values in the clockwise direction at Stage 3. Obviously, the kinetic energy decreases, while the potential energy increases until the spring is totally compressed. During Stage 4, the woodpecker rotates anticlockwise from \({\omega _S} = 0\). When it rotates to a certain position, the sleeve begins to fall. Therefore, the kinetic energy increases and the potential energy decreases. It is clear that there is a loss of energy, not an increase in energy, in the events of impact for all examples. In order to more accurately verify the correctness of the proposed method, a fourth example of multipoint impacts is added here. Chatterjee et al. [66] propose the rigid bodies constraints and Coulomb’s friction law to solve impact problems, and they use the global Stronge’s energetic coefficient of restitution to deal with the energy loss during the impact process. Thus, the energy consistency of the fourth example is shown in Fig. 21. When the restitutive coefficient is 0.715 in the proposed method, the energy diagrams of the two methods are in good agreement. In this example, there are two impact points at the same time. The simulation results show the dissipation of energy by impact and friction forces.

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Tao, L., Li, Y., Feng, ZQ. et al. Bi-potential method applied for dynamics problems of rigid bodies involving friction and multiple impacts. Nonlinear Dyn 106, 1823–1842 (2021). https://doi.org/10.1007/s11071-021-06916-z

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