What is damping?
Damping is an influence within or upon a dynamic system that has the effect of reducing, restricting or preventing its movements. In physical systems, damping is produced by processes that dissipate the energy. Damping forces come from several sources simultaneously, such as energy loss during hysteretic loading, viscoelastic material properties, contact friction and so on.
You can find a great blog about the concept and basic understanding here:
Sources of Damping
Generally, we can have damping resulted from:
1. Material nonlinearity ⇒ inelastic dissipation
2. Internal friction ⇒ material behavior
3. External friction ⇒ joint behavior
Why we use damping in Abaqus?
We take profit of damping in Abaqus to accurately models the energy loss in a dynamic system. Damping is also beneficial when modeling very fast phenomenon and noisy dynamics. It will play a critical role to have a meaningful solution.
So, there are two main reasons for adding damping to a model: to limit numerical oscillations or to add physical damping to the system.
How we can model damping in Abaqus?
The most widely used damping form called viscous damping, in which the damping force is velocity proportional. In fact, the complex damping may be proportional to the square of velocity, such as the damping force of solid motion in liquid, and sometimes it even has nothing to do with speed, such as the friction at the supports.
Two primary aspects for defining damping are available in Abaqus:
Structural damping (imaginary stiffness), used in frequency domain dynamics and in mode-based transient dynamics.
How we can enter data for damping in Abaqus?
In general, Abaqus has five categories of damping definition sources:
1. Material damping
You specify damping when defining material.
2. Element damping
Includes contributions from complex spring elements, dashpot elements and connector elements (using connector damping)
3. Global damping
Assumes the damping coefficient is constant in all materials. It is essentially a crude approximation to improve performance (preliminary design).
4. Modal damping
Applies only to mode-based linear dynamic analyses, so it can only be used in Abaqus/Standard.
5. Artificial damping
Damping associated with the time integration method (numerical damping).
Here, we only focus on material damping in Abaqus. Other types will discuss later in future articles.
Material Damping in Abaqus
In most cases, the material model itself may provide damping in the form of plastic dissipation or viscoelasticity. For many applications such damping may be adequate. Another option is to use Rayleigh damping or structural damping.
In Abaqus, material damping can be defined for both direct-integration (nonlinear, implicit or explicit) and mode-based (linear) dynamic analyses.
Forms of Material Damping
♦ Rayleigh damping (Viscous damping)
♦ Structural damping
♦ Viscoelasticity (for viscoelastic materials)
Some models do not have any dissipation sources (an example is a linear system with chattering contact, such as a pipeline in a seismic event). In such cases it is often desirable to introduce some general damping. Abaqus provides Rayleigh damping for this purpose. It provides a convenient abstraction to damp lower (mass-dependent) and higher (stiffness-dependent) frequency range behavior.
In Rayleigh model, damping is linearly proportional to mass M and stiffness K matrices. A damping matrix C is added to the system. We define C as:
C matrices are generated for each material in the model. We define the Rayleigh damping factors (alpha, beta) as material properties in Abaqus:
Supposing that only Rayleigh damping is present in the model and all the materials use identical Rayleigh damping factors, we can find a relation between the effective system damping ratio and the Rayleigh damping parameters. According to the orthogonality between mass matrix and stiffness matrix, for each natural frequency of the system ωi, the effective damping ratio is equal to:
Reminder. Damping Ratio………………………………………………………………………………………………..
The fraction of critical damping or damping ratio is defined as:
And natural frequency is:
Where c, k and m are damping, stiffness and mass of each mode, respectively.
Thus, mass proportional damping dominates when the frequency is low, and stiffness proportional damping dominates when the frequency is high:
Structural Damping (Abaqus/Standard)
Assumes that the damping forces are proportional to the forces caused by stressing of the structure and are opposed to the velocity:
Where F and I are the damping force and the forces caused by stressing of the structure for every DOF, respectively. User provide the s value when defining material in Abaqus/CAE:
Damping in real structures is frequency dependent. Therfore, this is best suited for frequency domain dynamic procedures.
The damping forces due to structural damping are intended to represent frictional effects (as distinct from viscous effects). Thus, structural damping is suggested for models involving materials that exhibit frictional behavior (hysteretic damping) or where local frictional effects are present throughout the model, such as dry rubbing of joints in a multi-link structure.
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. The term viscous implies that they deform slowly when exposed to an external force. They cannot return to their first configuration. It is due to dissipation of energy. Elastic materials strain when stretched and immediately return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties:
Viscoelasticity model in Abaqus is useful for materials in which dissipative losses primarily caused by viscous (internal damping) effects. The model must be combined with an elastic material model. Viscoelasticity cannot be combined with any of the plasticity models.
Defining viscoelasticity in Abaqus
In Abaqus, the viscoelasticity can be defined as:
* Time domain viscoelasticity
Viscoelasticity is a function of time for transient analysis. It describes isotropic rate-dependent material behavior. We can use a Prony series formulation to enter data:
** Frequency domain viscoelasticity
Viscoelasticity is a function of frequency for steady-state small-vibration analyses. This analysis is limited to Abaqus/Standard.
For example, many applications of elastomers involve dynamic loading in the form of steady-state vibration, and often in such cases, the dissipative losses in the material must be modeled to obtain useful results.