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# Damping

In a dynamic simulation, damping means energy dissipated out of the system. Damping in FEA is used for two main reasons: numerical damping and material damping. We will go through them in this document.

## Defining Damping in the Workbench

Damping is only available for two analysis types: Dynamic and Harmonic. The configuration is made in the Materials section:

## Numerical Damping

Numerical damping in FEA is connected to the numerical time integration scheme (implicit and explicit) 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 energy conservation can not be ensured.

On the other hand, energy dissipation is often needed to reduce unphysical 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 in SimScale to mimic this behavior.

### Rayleigh Damping Model

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:

$$\mathbf{M} \ddot{\vec{u}}+\mathbf{C}\dot{\vec{u}}+\mathbf{K}\vec{u} = \vec{f} \tag{1}$$

Where,

• $$\mathbf{M}$$ is the mass matrix
• $$\mathbf{C}$$ is the damping matrix
• $$\mathbf{K}$$ is the stiffness matrix
• $$\vec{f}$$ is the force vector
• $$\ddot{\vec{u}}, \dot{\vec{u}}, \vec{u}$$ are the resulting acceleration, velocity and displacement vectors

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

$$\mathbf{C} = \alpha\mathbf{K} + \beta\mathbf{M} \tag{2}$$

where $$\alpha$$ is stiffness-proportional damping coefficient $$[seconds]$$ and $$\beta$$ is mass-proportional damping $$[1/seconds]$$.

### Hysteretic Damping Model

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:

$$\mathbf{M} \ddot{\vec{u}}+\mathbf{K}(1+i\kappa)\vec{u} = \vec{f} \tag{3}$$

Important

The hysteretic damping model is only available for harmonic analysis.

Last updated: May 25th, 2021