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There are a few numerical simulation methods available for impact problems. However, most numerical results are not validated experimentally. The goal of this paper is to examine how well the simulation results correspond to the physical reality. In this work, normal and oblique impacts of a hemispherical-tip rod on a square plate are investigated both numerically and experimentally. In the numerical approach, finite element method is used to discretize the contact bodies to describe the deformation precisely combined with the floating reference frame method to describe the rigid motion. In the experimental study, strain gauges and Laser Doppler Vibrometers are employed to measure the high-frequency impact responses. Detailed comparative studies between numerical and experimental results are performed. In the case of normal impact, great attention is given to investigate the influence of finite element mesh size on the simulation accuracy and a “Prediction-Refinement” discretization strategy is proposed for obtaining a mesh which is optimal for impact dynamics. In the case of oblique impact, the influence of Coulomb’s friction coefficient is investigated additionally. It shows that the numerical results are in good agreement with the experimental results for both normal and oblique impacts.

For accurate modeling of multibody systems with impacts, there are two main ingredients that should be basically chosen while developing a contact formulation: one is the scheme to describe the kinematics and deformation of contact bodies and the other is the technique to enforce the nonpenetration constraint during contact.

The numerical approaches for kinematics and deformation description in contact/impact analysis can be divided into two categories: computational contact mechanics based on finite element method (FEM), as shown in, for example, [

The other influential factor on the numerical result of dynamic simulation is the appropriate approach to describe impact. For contact/impact dynamics of discretized elastic body, two typical methods are presented to model contact: penalty method and Lagrangian method. A brief overview of these two formulations is presented as follows. In penalty method, the contact force is defined by a force function of local penetration at each contact point, as a spring-damper model. Various forms of force function are presented, developed, and used by a number of researchers [

For the experimental investigations of impact problems, the main difficulty is that the impact duration is very short and the impact responses have extremely high frequencies, so they are very difficult to measure. Early experimental investigation mainly focused on transient strain response and the measurement instruments are strain gauges [

In this work, a three-dimensional impact of a hemispherical-tip rod on a square plate, including normal impact and oblique impact, is investigated both numerically and experimentally. In the numerical approach, finite element method combined with the floating reference frame method is employed to describe the rigid-flexible coupling motion. To exclude the influence of several manually defined parameters and enforce the impenetrability constraint strictly, the Lagrangian method is applied to model the contacts. In the experimental study, strain gauges and LDVs are employed to measure the high-frequency impact responses. Great attention is given to investigate the influence of mesh size on the simulation accuracy, aiming to get a finite element mesh technique which ensures a convincing numerical solution. In our work, a “Prediction-Refinement” discretization strategy is proposed for obtaining a mesh which is optimal for impact dynamics, in the sense that the computational efforts involved are minimal under the accuracy requirement. In the case of oblique impact, the influence of Coulomb’s friction coefficient is investigated. The numerical results show good agreement with the experimental results for both normal and oblique impacts if the spatial discretization is conducted appropriately.

In multibody dynamics with contacts/impacts, the rigid motion of flexible body is described by floating reference frame, and the deformation is modeled by finite element nodal coordinates or modal coordinates. This section provides the derivation of the kinematics and equations of motion of contact/impact problem in multibody system.

As shown in Figure _{i} is discretized by lumped mass FEM; therefore, the mass of body is distributed to each finite element node. The inertial reference frame is represented by

Kinematics of a single flexible body.

Deriving (

Applying the principle of virtual power for the whole body, the dynamic equations of motion can be written as follows:

Substituting (

There are several approaches to model the contact in the flexible multibody dynamics, including analytical solution, penalty method, and Lagrange multiplier method. Analytical solution such as St. Venant’s contact theory is limited to objects with simple geometry. In the penalty method, the contact force is defined by a force function of local penetration at the contact point, because the nonpenetration constraint is not precisely satisfied during the contact process; the accuracy of numerical simulation depends on the choice of the coefficient of contact stiffness. In contrast, the Lagrange multiplier method for contact modeling, where constraint equations are appended to dynamic equations to be solved together, reflects the nonpenetration condition without manually defined parameters. In this section, using Lagrange multiplier method, the contact constraint equations and equations of motion of system are derived.

In FEM, mostly the node-to-segment approach is used to discretize the contact surfaces, as shown in Figure

Node-to-segment contact pair.

The position of point

When contact occurs, according to the nonpenetration condition in the contact point, the locking of the free motion in a normal direction is described by the following constraint equation on position level:

According to (

In the case of stick, the motion in the tangent direction is also locked. This results in the following constraints on velocity and acceleration level:

In (

If the contact pair is sliding, the friction force on body

To describe the stick-slip phenomenon, a related transition law can be written as follows:

Assume, at a certain moment, there are

A schematic diagram of the normal impact experiment is shown in Figure

Schematic diagram of the normal experiment.

Experimental setup.

The strains are measured with strain gauges. As shown in Figure

For the measurement of displacements and velocities, two Laser Doppler Vibrometers (LDVs) of type PSV-300F, made by Polytec GmbH, are used. The vibrometer utilizes an interferometric technique to measure vibrational signals. The measurement range of velocity is ±10 m/s, and the resolution reaches 10^{−6} m/s. The back point of the rod and the center point of the plate are measured in the normal impact experiment. The material and geometrical parameters of the two colliding bodies are tabulated in Table

Geometrical and material parameters.

Steel rod | Aluminum plate | |
---|---|---|

Radius (mm) | 10 | — |

Length (mm) | 800 | 250 |

Width (mm) | — | 250 |

Thickness (mm) | — | 5 |

Young’s modulus (Gpa) | 205.8 | 59.1 |

Poisson ratio | 0.28 | 0.32 |

Density (kg/m^{3}) |
7727 | 2627 |

For simulating contacts using FEM, the modeling of contact itself as well as the consideration of the resulting wave propagation is closely related to the spatial discretization. Therefore, in order to obtain the convergence solution in FEM, the rod/plate case is simulated with different meshes and the simulation results are compared with the measurements.

For the main part of the two bodies, to get an accurate representation of wave propagation using the FEM, the smallest interesting wave length

In the aluminum plate, the longitudinal and the lower transversal wave speeds can be written as follows:

To determine the value of the highest frequency

Amplitude-frequency curve of velocity of point P1.

For the discretization of the contact region, additional considerations are required. To sufficiently represent the deformation and stress distribution in the contact region, the contact region must be discretized in a much smaller size than the element length required for the wave propagation, as shown in Figure

Spatial discretization of the contact bodies.

As the contact radius is about 0.49 mm, to ensure that a certain amount of nodes are in contact, the element length in the contact region should be less than 0.49 mm. To investigate the influence of element size in the contact region, three meshes are used to discretize the bodies. In mesh 1, the contact element size is set to 0.4 mm amd mesh refinements are performed for meshes 2 and 3 in the contact region, while the elements in the main part keep unchanged. The parameters of the three meshes are listed in Table

Element length of meshes 1–6.

Mesh | Element length of main part (mm) | Contact element length (mm) | |
---|---|---|---|

Rod | Plate | ||

Number 1 | 25 | 15 | 0.4 |

Number 2 | 25 | 15 | 0.2 |

Number 3 | 25 | 15 | 0.1 |

Number 4 | 12 | 8 | 0.2 |

Number 5 | 8 | 5 | 0.2 |

Number 6 | 4 | 2 | 0.1 |

Figure _{1} of the plate. Small deviation is visible for the simulated response of mesh 1 compared with that of mesh 2 and the results of mesh 2 and mesh 3 are nearly the same. However, Figure

Velocity of point P1: (a) simulation results with mesh refinements in contact region; (b) comparison between simulation using mesh 3 and experiment.

As shown in Table

Velocity of point P1: (a) simulation results of further mesh refinements; (b) comparison between simulation using mesh 5 and experiment.

The other comparisons between simulation using mesh 5 and experiment, including velocity of point P2 and strain of point S2, are shown in Figure

Comparisons between convergent simulation and experiment: (a) velocity of point P2; (b) strain of point S2.

In this section, the rod-plate impact problem is simulated using 6 meshes and finally the numerical result agrees well with the experimental result. It means that the finite element mesh is influential to the simulation result. Here, a “Prediction-Refinement” mesh strategy is shown in Figure

The “Prediction-Refinement” discretization strategy for impact dynamics.

To investigate the tangential motion during the contact, an oblique impact experiment is performed. The experimental equipment is the same as in the normal case. The vertical view of the experimental setup is shown in Figure

Vertical view of the oblique impact experiment.

In the simulation, the mesh size used here is the same as the mesh 5 in the normal case. The friction model is implemented to represent the tangential force, and the friction coefficient is set as 0.35 according to measurement. In order to investigate the influence of the friction coefficient on the tangential motion, the friction coefficients of 0.15 and 0.55 are also simulated.

Figure

Comparisons between simulations using different friction coefficients and experiment: (a) velocity of point Q1; (b) velocity of point Q2.

In Figure

In this paper, the impacts between a steel rod and an aluminum plate are studied numerically and experimentally. In the numerical approach, FEM is used to discretize the contact bodies to describe the deformation precisely combined with the floating reference frame method to describe the rigid motion. The Lagrangian method is applied to model the normal contacts to enforce the impenetrability constraints strictly and exclude the influence of some manually defined parameters such as contact stiffness in penalty method. To model the tangential friction, the tangential contact is also modeled using Lagrangian method and the sliding friction is described by Coulomb’s friction law. In the experimental study, strain gauges and LDVs are employed to measure the high-frequency impact responses.

Firstly, the normal impact case is investigated, and detailed comparative studies between the numerical and experimental results show that the simulation accuracy is greatly influenced by the spatial discretization. This work presents a “Prediction-Refinement” discretization procedure to obtain a mesh which is optimal for impact dynamics using FEM, in the sense that the computational efforts involved are minimal under the accuracy requirement. For the discretization of the contact region, the contact radius can be predicted by an analytical method such as Hertz model in advance. The expected contact region has to be discretized by sufficient elements to ensure that a number of nodes are in contact in order to describe the contact accurately. The discretization of the main part is also essential to the numerical results. Because the impact leads to very high-frequency vibration in the whole body, the required mesh size for correct computation of the wave propagation can be calculated from the lowest wave speed and the highest considered frequency. Then the mesh should be refined both in the contact region and in the noncontact region until the numerical result reaches convergence. The simulation coincides with the measurement only when the contact bodies are discretized appropriately.

In the case of oblique impact, the experiment shows that friction has a measurable influence on the tangential motion. The value of friction coefficient greatly affects the resultant responses. Good agreements between the simulations and measurements can be achieved if the friction coefficient is chosen reasonably.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research work is supported by the National Science Foundation of China (no. 11132007 and no. 11202126), for which authors are grateful.