Alpha damping and Beta damping are used to define Rayleigh damping constants α and β. The damping matrix [C] is calculated by using these constants to multiply the mass matrix [M] and stiffness matrix [K]:

[C] = α[M] + β[K]

The ALPHAD and BETAD commands are used to specify α and β, respectively, as decimal numbers. The values of α and β are not generally known directly, but are calculated from modal damping ratios, ξ_i. ξ_i is the ratio of actual damping to critical damping for a particular mode of vibration, i. If ω_i is the natural circular frequency of mode i, α and β satisfy the relation

ξ_i = α/2ω_i + βω_i/2

In many practical structural problems, alpha damping (or mass damping) may be ignored (α = 0). In such cases, you can evaluate β from known values of  ξ_i and ω_i, as

β = 2 ξ_i/ω_i

Only one value of β can be input in a load step, so choose the most dominant frequency active in that load step to calculate β.

To specify both α and β for a given damping ratio ξ, it is commonly assumed that the sum of the α and β terms is nearly constant over a range of frequencies (see Figure 1.1: Rayleigh Damping). Therefore, given ξ and a frequency range ω₁ to ω₂, two simultaneous equations can be solved for α and β:

Figure 1.1: Rayleigh Damping

Alpha damping can lead to undesirable results if an artificially large mass has been introduced into the model. One common example is when an artificially large mass is added to the base of a structure to facilitate acceleration spectrum input. (You can use the large mass to convert an acceleration spectrum to a force spectrum.) The alpha damping coefficient, which is multiplied by the mass matrix, will produce artificially large damping forces in such a system, leading to inaccuracies in the spectrum input, as well as in the system response.

Beta damping and material damping can lead to undesirable results in a nonlinear analysis. These damping coefficients are multiplied by the stiffness matrix, which is constantly changing in a nonlinear analysis. Beta damping is not applied to the stiffness matrices generated by contact elements. The resulting change in damping can sometimes be opposite to the actual change in damping that can occur in physical structures. For example, whereas physical systems that experience softening due to plastic response will usually experience a corresponding increase in damping, a Mechanical APDL model that has beta damping experiences a decrease in damping as plastic softening response develops.

Damping for contact elements and MPC184 constraint and joint elements is restricted. See element descriptions for details.

Material-dependent damping enables you to specify alpha damping (α) or beta damping (β) as a material property (MP,ALPD, MP,BETD, TB,SDAMP,,,,ALPD, and TB,SDAMP,,,,BETD).

Both linear material property commands (MP,ALPD and MP,BETD) and both data table commands (TB,SDAMP,,,,ALPD and TB,SDAMP,,,,BETD) can be used to define damping for elements as a whole. To define unique Rayleigh damping coefficients for section materials, use the data table commands.

The data table commands can be frequency-, temperature-, or time-dependent (TBFIELD). Frequency-dependent properties are supported for full harmonic analyses only.

Damping for contact elements and MPC184 constraint and joint elements is restricted. See element descriptions for details.

For more information about material-dependent damping, see Material Damping in the Material Reference.

The damping ratio is the simplest way of specifying damping in the structure. It represents the ratio of actual damping to critical damping, and is specified as a decimal number with the DMPRAT command. Use MP,DMPR to define a material dependent damping ratio.

Structural damping enables you to incorporate hysteretic behavior due to internal material friction by specifying a coefficient on the stiffness matrix. In this type of damping, the damping force is proportional to the displacement rather than the velocity as in the other damping options. Both constant structural damping (DMPSTR) as well as material-dependent structural damping (MP,DMPS and TB,SDAMP with TBOPT = STRU) are supported.

A constant structural damping coefficient does not produce damping with a constant ratio to critical in a full harmonic analysis.

Damping for contact elements and MPC184 constraint and joint elements is restricted. See element descriptions for details.

Mode Dependent Damping Ratio gives you the ability to specify different damping ratios for different modes of vibration. It is specified with the MDAMP command and is available only for the spectrum and mode-superposition method of solution (transient dynamic and harmonic analyses).

Viscoelastic materials have a frequency-dependent complex modulus in the harmonic domain. The imaginary component of the complex modulus, also called the loss modulus, results in a material damping matrix that is added to any other forms of damping defined in the analysis.

Element damping involves using element types having viscous damping characteristics, including but not limited to COMBIN14, COMBIN37, and COMBIN40.