## Introduction

In multibody systems, contact between bodies is a very important thing, in fact many system include contact between bodies and could not be analyzed without using contact modelling.

In Adams contacts can be modelled using two different approaches:

- The continuous method (or impact method): the impact is finite, but has a short duration, the forces are generated by spring-damper elements and depend on the penetration between the geometries of the two bodies.
- The instantaneous method (or restitution method): the impact has an infinitesimal duration, the positions and orientation of the bodies don’t change and the impulsive forces and torques have constant directions.

When the contact between bodies is considered in the instantaneous case, it is necessary to introduce a restitution coefficient Ԑ, which defines the type of contact. If Ԑ=1 there is an ideal elastic collision in which there is a conservation of the mechanical energy. If Ԑ=0 there is an ideal plastic collision in which there is a complete dissipation of the mechanical energy and the bodies after the impact remain exactly attached at the contact point. In reality there is always 0< Ԑ <1 because there are no perfectly elastic impact since part of the energy possessed by the bodies is dissipated.

## Objectives

The purpose of the project is to analyze the contact between two bodies with the multibody software MSC Adams and to analyze the differences between the contact made with the two different methods: continuous and instantaneous, in the case of Ԑ=1, Ԑ=0.5, Ԑ=0.

The analyzed bodies have known characteristics and also the analytical results are known for all the types of contact.

Dynamic simulation were made to determine the variations of velocity and the impulses on the contact point (f) on the constraint (r) and on the center of mass (g)

The analyzes made are dynamic analyzes when the solver error and the step size varies.

The solver used is SI2.

## The modelling problem

The characteristics of the two bodies are in the following table:

corpo 1 | Corpo 2 | |

Mass | 0.5 kg | 1 kg |

Inertia | 1/12 kg*m^2 | 2/5 kg*m^2 |

It was decided to use a 1 m side cube and a 1 m diameter sphere.

The first body (cube) is constrained to the ground through a cylindrical constrain (along z-axe) in the point ar1={0.5,0,0}, while the second body (sphere) it is not constrained and hits the body 1 along y-axe which is also the normal direction at the contact point. Contact takes place at the following coordinates with respect to the center of mass of body 1 and body 2 respectively: a_{f1}={ -0.5,-0.5,-0.5} e a_{f2}={0,1,0}.

The first body has an initial zero velocity , while the second body has an initial velocity of 1 m/s along y-axe.

To model the contact, the contact between sphere and plane was used, attaching a plane on the lower surface of the cube as seen in the following figure.

The type of contact considered are three:

- Elastic impact (Ԑ=1);
- Intermediate impact (Ԑ=0.5);
- Plastic impact (Ԑ=0).

The impulsive forces have been calculated as follows:

Adams’ post processor was used to perform the integral of the force. An example is given, all the other impulses have been calculated in the same way:

The expected analytical results are shown below.

The meaning of the symbols used is illustrated:

1: cube;

2: sphere;

f: contact point;

g: center of mass;

R': impulsive constraint reaction in the cylindrical constraint;

T': impulsive constraint torque in the cylindrical constraint;

F': impulsive force in the contact point.

The units of measurement are:

Velocity: [m/s];

Angular velocity: [rad/s];

Impulsive force: [N*s];

Impulsive torque: [N*m*s];

## Simulations and analysis of results

As for the simulation setting, the GSTIFF solver was used with SI2 formulation.

The simulations were performed by varying the step size and varying the solver error. The following cases were considered:

Step size: 0.0022; 0.0026; 0.0028;

Solver error: 10^-3; 10^-4; 10^-6; 10^-8.

As regards the contact, the following settings have been used:

**Elastic impact**

In the case of elastic impact, the following settings were used respectively for continuous and instantaneous contact:

The results obtained for the continuous method are summarized in the following table:

Analyzing the results, one immediately notes that in the continuous case the variations of velocity are very close to the analytical values, for all the combinations of step size and errors used and in many cases they coincide, while the impulses are similar to those expected except in the case of the larger step size 0.0028.

The results obtained for the instantaneous method are summarized in the following table:

Analyzing the results, one immediately notes that the results are less accurate than the continuous case.

Velocity are not determined accurately in all cases, and still have a maximum error of 15% expect one case where the error is bigger.

As for the impulses, they are obviously wrong except that in a case where they are about twice the expected value, this is obtained for a step size of 0.0026 and a solver error of 10^-3.

**Intermediate impact**

In the case of intermediate impact, the following settings have been used for continuous and instantaneous contact respectively:

The results obtained for the continuous method are summarized in the following table:

Analyzing the table, one immediately notice that the velocity variation are very close to the analytical values for all the combination of step size and solver error used. As regards the impulses they are determined more accurately in the case of the error 10^-3, while as the error increases the impulses tend to decrease their value.

The results obtained for the instantaneous method are summarized in the following table:

Analyzing the results, one immediately notice that the results are less accurate than the continuous case. The variations of velocity are not determined accurately in all cases, but still have a maximum error of 34%, decreasing the solver error there are less accurate results.

As for the impulses, they are obviously wrong, expect in cases of error equal to 10^-3 and 10^-4 and for smaller step size where they are more accurate.

**Plastic impact**

In the case of plastic impact, the instantaneous model does not respect the physic of the problem because the model doesn’t see the variation of the angular velocities of the sphere and also going to monitor the relative displacement between the two bodies at the contact point, ideally after the impact the two bodies should remain united, while it is seen that after contact the bodies separate.

To keep the bodies attached after the impact, a spherical constraint was used, in this way the bodies after the impact remain connected in the contact point.

The results obtained by activating the spherical constraint at the moment of impact are as follows:

The speed variations are different from those expected with an error of 50% and the impulse of the contact force is 3 orders of magnitude greater than the expected value.

3 gcon constraints were used to satisfy the speed constraints.

Activating them at the moment of impact the results are as follows.

The results are the similar to the previous case.

An attempt was also made to activate the constraint after contact, therefore at a time equal to 0.1012s.

In this case, the variation of velocities are close to the true value and the impulse are similar to the true value except the impulse F’_{y} that is very high.

**Continuous method**

As far as continuous contact is concerned, the following settings have been used:

Initially I tried to use high damping without activating any constraints.

In this way the physics of the problem is not respected because there is no variation in the angular velocity of the sphere and the bodies do not remain in contact after the impact.

To keep the bodies attached after the impact, a spherical constraint was used, in this way the bodies after the impact remain connected in the contact point, however the variations in angular velocity have a strange trend in fact instead of remaining constant after the impact, they vary their value until they stabilize at another value. They have the same trend as the instantaneous case.

So it was decided to try introducing 3 gcon custom constraints to satisfy the speed constraint.

By introducing the 3 gcon constraints that go to enforce the speed constraints linked to the plastic impact, it is seen that the two bodies remain united after the impact, but also in this case the values do not return.

An attempt was also made to activate the constraint after contact, therefore at a time equal to 0.1012s.

The results are similar to the instantaneous case.

In the plastic case, by varying the step size and the solver error, the results are very similar, therefore only one case has been reported.

## Conclusion

Analyzing the results you immediately notice a greater accuracy of the continuous model compared to the instantaneous model, as regards the velocity increases they are determined by both methods quite accurately both in the case of elastic impact and in the intermediate one while as regards the impulses you can immediately see a better accuracy of the continuous method compared to the instantaneous one and there are worse results when the solver error decreases and the step size increases.

While as regards the plastic case only in one case the results obtained are similar to the analytical values, this was achieved by activating the spherical constraint or the gcon constraint at the end of the impact before the bodies separate.

**Script in Adams**

The following scripts were used to make the simulations:

- Elastico_cont: elastic contact simulation with continuous method;
- Elastico_rest: elastic contact simulation with instantaneous method;
- Intermedio_cont: intermediate contact simulation with continuous method;
- Intermedio_res: intermediate contact simulation with instantaneous method;
- Plastico_cont: simulation of the plastic contact with continuous method without activation of constraints;
- Plastico_cont_gcon: simulation of the plastic contact with continuous method with activation of the gcon constraint;
- Plastico_rest: plastic contact simulation with instantaneous method without constraint activation;
- Plastico_rest_sferico: simulation of the plastic contact with instantaneous method with activation of the spherical constraint;
- Plastico_rest_gcon: simulation of the plastic contact with instantaneous method with activation of the gcon constraint;