# Impact Dynamics Analysis of Mobile Mechanical Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Impact analysis for two bodies on a free-fall drop, which come in contact together, and
- (2)
- Impact study for a multi-body system from an internal combustion engine mechanism.

## 2. Impact Analysis of a Link with Planar Surface

#### 2.1. Analytical Method

_{1}; J

_{0}is the mass inertia momentum depending on bar rotation center, O and $\theta $ is the bar rotation angle:

#### Velocities

_{A}= y

_{A}(t); $\stackrel{\xb7}{y}=\frac{dy}{dt}$—the point speed at which it will consider that it will actuate the impact force, i.e., speed $\stackrel{\xb7}{y}={v}_{A}$ (point A speed); y

_{1}—a positive real variable, which can be considered to be the free length of y component displacement; k—stiffness which corresponds to the interaction between the bar and the contact surface; e—is a positive real variable which specifies the deformation characteristics exponent of the applied force; c

_{max}—is a non-negative, double-precision variable that specifies the maximum damping coefficient; d—is a positive double-precision variable that specifies the secondary penetration at which ADAMS/Solver applies full damping.

_{i}, R

_{j}—represent the radii of each body in contact; ν

_{i}, ν

_{j}—represent the transversal contraction coefficients for bodies i and j brought in contact together; E

_{i}, E

_{j}—represent the longitudinal elasticity modulus for body i and body j, respectively.

- if y ≥ y
_{1}, there will be no penetration at the level of the contact surface, and in this case the impact force will be equal to zero (penetration p = 0); - if y < y
_{1}, penetration occurs at the end closer to the i marker, and the impact force is >0 (penetration p = y_{1}− y); - when p < d, the damping instantaneous coefficient is a cubic STEP function of penetration, labeled p;
- when p > d, the instantaneous damping coefficient will be c
_{max}.

_{1}. Practically, this function enters when the considered bodies collide, otherwise, it will be equal to zero. Also, it will consider that the force in the mathematical expression of (14) has two components:

- a spring or stiffness component and a damping or viscous component:

- damping (viscous component), which is a function of the speed of penetration.

_{0}is equal to zero for this aplication. By comparing Figure 5 and Figure 6, the mathematical expression (21) becomes:

Algorithm 1 Programming Sequence for Time Intervals |

for t from 0 by 0.0001 to 0.6 doif $\left(y\right)\le \left({y}_{1}-d\right)then$$F:=k\xb7{\left({y}_{1}-y\right)}^{e}-vy\xb7{c}_{\mathrm{max}};$ print $\left(t,y,F\right)$; $\mathrm{elif}\left({y}_{1}-d\right)y$$\mathrm{and}y{y}_{1}$ then $F:=k\xb7{\left({y}_{1}-y\right)}^{e}-vy\xb7\left({c}_{\mathrm{max}}+a\xb7{P}^{2}\xb7(3-2\xb7P)\right);$ print $\left(t,y,F\right)$ else$F:=0;$ print $\left(t,y,F\right)$ end ifend do; |

_{1}= 0.8; damping coefficient: c

_{max}= 1; penetration depth: d = 0.01; stiffness: k = 48,049; force exponent: e = 2.2. By considering the diagram in Figure 7, it can be noted that the impact force will have a maximum value of 250 N, and in Figure 8 the viscous component has a maximum value around 3000 N, and these values correspond to c = 1. If we modify the damping coefficient at a value of c = 6, the force stiffness component will reach a value of 450 N, according to Figure 9, and the viscous component reported in Figure 10 will reach a value of 5 × 10

^{4}N.

_{1}= 0.897; damping coefficient: c

_{max}= 6; penetration depth: d = 0.01; stiffness: k = 48,049; force exponent: e = 2.2.

#### 2.2. Impact Modeling with MSC Adams

^{−3}. In addition, the recorded harmonics as amplitude and number obtained in the case of impact forces variation are generated mainly by the damping phenomenon.

^{−4}.

#### 2.3. Impact Analysis with Finite Element Method

^{−4}which corresponds to the hypothesis of large deflections. Time variation diagrams were identified for the following parameters: prismatic body stress, computed by considering a local coordinate system solitary with this component and dynamic parameters, which define the impact phenomenon between the bar and the prismatic body, namely, penetration depth, friction stress, contact pressure and contact force. For this model, the contact force is represented in Figure 21 and Figure 22.

^{−4}. With reference to the previous case, one it can note a decrease of amplitude for the resultant elastic displacement and an increased number of harmonics for all the monitored kinematic and dynamic parameters after the first impact. Thus, the contact force records a maximum value during the impact of 433. N and afterwards, the values will be small due to the damping phenomenon (Figure 23). Considering the diagrams reported in the first case (case a), it can be observed that after the impact, all the monitored parameters reach absolute values at the same time instants, whereas different variation forms are reported. By having in sight the fact that in all the modeling cases the bar motion in a free fall drop records high amplitudes, the finite element analysis was setup under the large deflections hypothesis. Thus the used finite elements support the contact modeling under this hypothesis. This setup was done in order to evaluate in a correct manner the bar component motion before and during the impact phenomenon.

#### 2.4. Impact Modeling with Abaqus

## 3. Dynamic Analysis of the Impact in the Case of a Crank-Connecting Rod Mechanism

- time variation functions are defined for the interest marker displacement and velocity (the marker that materializes the force point application, i.e., marker 13, placed in the joint rotation center);
- the impact function will be created based on the following kinematic and dynamic parameters: displacement, dy_ MARKER_i, velocity, vy_ MARKER_i, for marker i, stiffness k, exponential coefficient e, damping c, penetration d, as follows:

#### 3.1. Impact Dynamic Analysis with Rigid Elements

_{m}represents the engine torque (T

_{m}—reported in diagrams); M

_{0}is the initial torque; ω

_{0}is an initial angular velocity; WZ represents the resulting angular velocity during the engine torque.

^{−3})

^{−3})

^{−3})

^{−3})

^{5}Nmm and the minimum value around −8.25 × 10

^{5}Nmm.

_{0}is required. In this case, the maximum and minimum values of the engine torque and the impact forces components change, resulting in 8.63 × 10

^{6}Nmm for the engine torque, the value of the component along the y-axis being around 2.7 × 10

^{5}N and the value of the component along the z axis being of 64,925 N.

#### 3.2. Impact Dynamic Analysis with Deformable Elements

#### 3.2.1. Mathematical Considerations

- Evaluating the natural frequencies and vibration modes;
- Elastic dynamic analysis.

_{p}is the part from the modal matrix corresponding to the node P translational degrees of freedom.

_{p}matrix dimension is 3 × M, where M represents the number of vibration modes. The modal coordinates q

_{i}, (i = 1, …, M) are the flexible body generalized coordinates. The angular deformations vector was obtained in a similar mode as the translation deformation vector, using a modal superposition, similar to the following expression:

_{p}is the modal matrix part corresponding to the rotational degrees of freedom of node P. The Φ* matrix dimension is 3XM, where M is the number of modes of vibration [18].

#### 3.2.2. Numerical Simulation

_{0}= 10 rad/s, M

_{0}= −100,000 Nmm.

^{−3}), with the following values of the dynamic parameters: k = 20,000, e = 0.1, c = 10, d = 10

^{−3}.

^{4}Hz and for vibration mode no.10 with a natural frequency of 1.061 × 10

^{4}Hz.

- modal-dynamic analysis with calculating the engine torque for the mechanism actuation;
- kinematic and dynamic parameters analysis of the generated contact during the impact between the bolt and the connecting rod head.

## 4. Results and Conclusions

- The mathematical model corresponding to Equation (16) is compatible with the Adams software theory, in relation to the impact phenomenon analysis. A Maple program sequence was derived, which can be numerically processed based on the created mathematical models by considering two major impact force components, namely, the stiffness component and the viscous component. These component variations with time were presented in Figure 7 and Figure 8.
- The dynamic parameters were processed via the Maple program by considering Equation (16) with the numerical values indicated in Figure 7 and these corresponds to the virtual models obtained with the Adams software (Figure 16 and Figure 17), with Ansys software (Figure 24) and Abaqus software (Figure 28), respectively.
- For the analytical models and numerical processing of the imported models with the Adams software, the bar component motion was monitored in a time interval equal to 0.6 s.
- By modifying the damping coefficient value to c = 6, it can be remarked that the impact force maximum value will be modified, as well, along the number of contacts between the bar component and the prismatic body. At the same damping coefficient, by keeping the contact conditions and the integration step of the equations system, which dictate the bar component motion, one can see that the impact force values are not compatible with these cases, namely, analytical case (Figure 9), numerical simulation with the Adams software (Figure 16) and the finite element analysis (Figure 24 and Figure 28).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ambrósio, J.A.C. Impact of rigid and flexible multibody systems deformation description and contact models. In Virtual Nonlinear Multibody Systems; Springer: Dordrecht, The Netherlands, 2003; pp. 57–81. [Google Scholar]
- Khulief, Y.A.; Haug, E.J.; Shabana, A.A. Dynamic Analysis of Large Scale Mechanical Systems with Intermittent Motion; Technnical Report No. 83-10; University of Iowa, College of Engineering: Iowa City, IA, USA, 1983. [Google Scholar]
- Hunt, K.H.; Grossley, F.R.E. Coefficient of Restitution Interpreted as Damping in Vibroimpact. ASME J. Appl. Mech.
**1975**, 42, 440–445. [Google Scholar] [CrossRef] - Ahmed, S.; Lankarani, H.M.; Pereira, M.F.O.S. Frictional impact analysis in open-loop multibody mechanical systems. J. Mech. Des.
**1999**, 121, 119–127. [Google Scholar] [CrossRef] - McCoy, M.l.; Lankarani, H.; Moradi, R. Use of simple finite elements for mechanical systems impact analysis based on stereomechanics, stress wave propagation, and energy method approaches. J. Mech. Sci. Technol.
**2011**, 25, 783–795. [Google Scholar] [CrossRef] - Tempelman, E.; Dwaikat, M.M.S.; Spitás, C. Experimental and Analytical Study of Free-Fall Drop Impact. Testing of Portable Products. Exp. Mech.
**2012**, 52, 1385–1395. [Google Scholar] [CrossRef] [Green Version] - Kantar, E.; Erdem, R.; Ozgur, A. Nonlinear Finite Element Analysis of Impact Behavior of Concrete Beam. Math. Comput. Appl.
**2011**, 16, 183–193. [Google Scholar] [CrossRef] [Green Version] - Chen, C.-R.; Wu, C.-H.; Lee, H.-T. Determination of Optimal Drop Height in Free-Fall Shock Test Using Regression Analysis and Back-Propagation Neural Network. Shock Vib. J.
**2014**. [Google Scholar] [CrossRef] [Green Version] - Corbin, N.A.; Hanna, J.A.; Royston, W.R.; Warner, R.B. Impact-induced acceleration by obstacles. New J. Phys.
**2018**, 20, 1–8. [Google Scholar] [CrossRef] - Brun, P.-T.; Audoly, B.; Goriely, A.; Vella, D. The surprising dynamics of a chain on a pulley: Lift off and snapping. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2016**, 472, 20160187. [Google Scholar] [CrossRef] [PubMed] - Dankowicz, H.; Fotsch, E. On the analysis of chatter in mechanical systems with impacts. Procedia IUTAM
**2017**, 20, 18–25. [Google Scholar] [CrossRef] - King, H.; White, R.; Maxwell, I.; Menon, N. Inelastic impact of a sphere on a massive plane: Nonmonotonic velocity dependence of the restitution coefficient. Europhys. Lett.
**2011**, 93, 14002. [Google Scholar] [CrossRef] - Muller, P.; Heckel, M.; Sack, A.; Poschel, T. Complex velocity dependence of the coefficient of restitution of a bouncing ball. Phys. Rev. Lett.
**2013**, 110, 254301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yan, M.; Wang, T. Finite Element Analysis of Cylinder Piston Impact Based on ANSYS/LS-DYNA. In Proceedings of the 2012 International Conference on Mechanical Engineering and Material Science (MEMS 2012), Hong Kong, China, 27–28 July 2012. [Google Scholar]
- Žmindák, M.; Pelagić, Z.; Pastorek, P.; Močilan, M.; Vybošťok, M. Finite Element Modelling of High Velocity Impact on Plate Structures. Procedia Eng.
**2016**, 136, 162–168. [Google Scholar] [CrossRef] [Green Version] - Wittenburg, J. Dynamics of Systems of Rigid Bodies. Leitfäden der Angewandten Mathematik und Mechanik; Springer: Berlin/Heidelberg, Germany, 1977. [Google Scholar]
- Wehage, R.A. Generalized Coordinate Partitioning in Dynamic Analysis of Mechanical Systems. Ph.D. Thesis, University of Iowa, Iowa, IA, USA, 1980. [Google Scholar]
- Haug, E.J.; Wu, S.C.; Yang, S.M. Dynamics of Mechanical Systems with Coulomb Friction, Stiction, Impact and Constraint Addition and Deletion, I. Mech. Mach. Theory
**1986**, 21, 401–406. [Google Scholar] [CrossRef] - Lankaram, M.H.N.; Kravesh, P.E. Application of the Canonical Equations of Motion in Problems of Constrained Multibody Systems with Intermittent Motion. In Proceedings of the 14-th ASME Design Automation Conference, Advances in Design Automation, Kissimmee, FL, USA, 25–28 September 1988; Volume 14, pp. 417–423. [Google Scholar]
- Flores, P.; Ambrósio, J.; Claro, J.P.; Lankarani, H.M. Contact-impact force models for mechanical systems. In Kinematics and Dynamics of Multibody Systems with Imperfect Joints; Springer: Berlin/Heidelberg, Germany, 2008; pp. 47–66. [Google Scholar]
- Flores, P.; Ambrosio, J.; Claro, J.C.P.; Lankarani, H.M. Influence of the contact—impact force model on the dynamic response of multi-body systems. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn.
**2006**, 220, 21–34. [Google Scholar] [CrossRef] [Green Version] - Khemili, I.; Romdhane, L. Dynamic analysis of a flexible slider–crank mechanism with clearance. Eur. J. Mech. A Solids
**2008**, 27, 882–898. [Google Scholar] [CrossRef] - Craig, R.R.; Bampton, M.C.C. Coupling of substructures for dynamics analyses. AIAA J.
**1968**, 7, 1313–1319. [Google Scholar] [CrossRef] [Green Version]

**Figure 14.**Variation diagrams for impact force and elastic displacement (c = 1) vs. time (detailed view).

**Figure 17.**Force, displacement and speed variation diagrams of marker no: 25 vs. time (detailed view at first contact—t = 0.1824 s).

**Figure 19.**Detailed view of the impact force and elastic displacement c = 6 during first impact vs. time.

**Figure 34.**The position vector to a deformed point P’ on a flexible body relative to a local body reference system and to the fixed (ground) reference system.

**Figure 35.**Snapshots during virtual simulations of the considered mechanism with a deformable connecting rod (MARKER_13 Node 610 location).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dumitru, S.; Constantin, A.; Copilusi, C.; Dumitru, N.
Impact Dynamics Analysis of Mobile Mechanical Systems. *Mathematics* **2021**, *9*, 1776.
https://doi.org/10.3390/math9151776

**AMA Style**

Dumitru S, Constantin A, Copilusi C, Dumitru N.
Impact Dynamics Analysis of Mobile Mechanical Systems. *Mathematics*. 2021; 9(15):1776.
https://doi.org/10.3390/math9151776

**Chicago/Turabian Style**

Dumitru, Sorin, Andra Constantin, Cristian Copilusi, and Nicolae Dumitru.
2021. "Impact Dynamics Analysis of Mobile Mechanical Systems" *Mathematics* 9, no. 15: 1776.
https://doi.org/10.3390/math9151776