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# Coefficient of Restitution

## Lumped Mass Model

A material coefficient of restitution can be separated into two components:

• a tangential coefficient of restitution (Rt)
• a normal coefficient of restitution (Rn)

The normal coefficient of restitution for a material is defined as the negative ratio of the outgoing velocity to incoming velocity.

R = - V outgoing / V incoming

The outgoing normal velocity is in the opposite direction as the incoming velocity.

The tangential coefficient of restitution is defined as the ratio of the outgoing velocity to incoming velocity if “Consider rotational velocity” is not selected under Project Settings.

R = V outgoing / V incoming

The outgoing tangential velocity is in the same direction as the incoming velocity.

A perfectly elastic material will have a normal coefficient of restitution of 1. An object striking this material will rebound back with the same speed. A perfectly inelastic material will have a coefficient of restitution of 0. This means that an object striking this material will rebound with zero velocity, i.e. it will come to a halt. All real materials have coefficients of restitution between 0 and 1.

The tangential coefficient of restitution is generally equal to or larger than the normal coefficient of restitution.

In RocFall only materials have coefficients of restitution (rocks do not).

## Rigid Body Model

Only the normal coefficient of restitution is considered in the Rigid Body engine unless the Use tangential CRSP damping check box is selected on the General page of Project Settings. The normal coefficient of restitution is defined as the square root of the negative ratio of the outbound energy to the incoming energy.

Rn2 = - Energy out / Energy in

Impact geometry (angle and velocities) and the dynamic coefficient of friction determine the modes of the impact. Outgoing velocities are calculated using impulses. Normal impulses are determined using Rn as defined above. Tangential impulses (Pt) are proportional to the normal impulses (Pn).

Pt ~ m * Pn

Friction is the only source of energy loss in the tangential direction unless Consider tangential CRSP damping is selected under Project Settings.

## Rocscience Coefficients of Restitution Table

Recent research indicates that appropriate values of coefficient of restitution depend on the analysis method - Lumped Mass or Rigid Body. Rigid Body coefficients of normal restitution are significantly lower than the corresponding Lumped Mass values (Bar et.al., 2016).

Coefficients of Restitution (Lumped Mass)

Coefficients of Restitution (Lumped Mass and Rigid Body)

These tables are the extent of the data that is available from Rocscience. Unfortunately, we do not have a more comprehensive list than this one. For further information, you can review the References listed at the end of this help system or search the internet for more values, but values are generally difficult to find. Unfortunately, coefficients of restitution are not always easy to determine, and the analysis can be sensitive to the values you choose.

## Note on Lumped Mass vs. Rigid Body

In general, the coefficients for Lumped Mass models will be different from the coefficient used in a Rigid Body model. Although related to some degree, the definitions of the normal coefficients are different for the two methods. This has been affirmed by recent research. Therefore, we are separating the recommended values into two separate tables. In view of the lack of data for the Rigid Body method, we recommend that you run Rigid Body method to study the shape effects before finalizing the analysis with the Lumped Mass method.