# Damping¶

In a dynamic mechanical simulation damping, that means energy dissipation out of the system, usually has two main reasons.

## Numerical Damping¶

Numerical damping is connected to the numerical time integration scheme that is used to predict the temporal evolution of the system. In SimScale there are non-dissipative (no damping) as well as dissipative time schemes available. The main disadvantage of dissipative time schemes is that they reduce the accuracy of the solution, as the energy conservation can not be ensured. On the other hand energy dissipation is often needed to reduce un-physical oscillations of the system and even enable the convergence of the solution. Often the user has to make a compromise between accuracy and robustness.

## Material Damping¶

The material damping has its origin in the physical behavior of the material. Damping (and thus energy dissipation) is observed due to internal friction of the material. Several models are available on SimScale to mimic this behavior.

### Rayleigh Damping¶

Rayleigh damping, which is also known as proportional viscous damping, assumes that the damping is proportional to the vibrating velocity. With the damping matrix $$\mathbf{C}$$ the system equation reads as:

$\mathbf{M} \ddot{\vec{u}}+\mathbf{C}\dot{\vec{u}}+\mathbf{K}\vec{u} = \vec{f}.$

The damping effect of the Rayleigh damping is controlled by two parameters: $$\alpha_{K}$$ and $$\beta_{M}$$. The viscous damping matrix $$\mathbf{C}$$ has the form:

$\mathbf{C} = \alpha_{K}\mathbf{K} + \beta_{M}\mathbf{M}$

### Hysteretic Damping¶

Hysteretic Damping, also known as structural damping. Here the damping is assumed to be proportional to the displacement. The damping effect is controlled by the hysteretic damping coefficient $$\kappa$$ of the material. The system equation is then:

$\mathbf{M} \ddot{\vec{u}}+\mathbf{K}(1+i\kappa)\vec{u} = \vec{f}.$