Rayleigh Damping

The formulation presented in the previous sections does not take into account the damping mechanisms that arise in time-dependent systems. Damping is the dissipation of energy in the solid structure due to a combination of different phenomena, including molecular interaction within the material.

In dynamic problems, the contribution of damping forces can be taken into account by including a velocity-dependent damping term, $c \dot{u}$ , in the equation of motion (Eqn. (4460)):

Figure 1. EQUATION_DISPLAY

ρ \ddot{u} + c \dot{u} - \nabla\cdot σ - b = 0

(4574)

which assumes a linear relationship between the damping force and velocity. The weak form of Eqn. (4574) can be constructed and discretized as described in the previous section, leading to the general discretized equation for a linear elastic damped system:

Figure 2. EQUATION_DISPLAY

M \ddot{u} (t) + C \dot{u} (t) + K u (t) = f^{e x t}

(4575)

The damping matrix in each element is:

Figure 3. EQUATION_DISPLAY

\begin{matrix} C_{M N} = \int_{V_{0}} N_{M} c {I N}_{N} d V \end{matrix}

(4576)

As damping is a complex combination of different phenomena, the damping matrix is often approximated using Rayleigh damping, which models the damping matrix as a linear combination of the stiffness and mass matrices:

Figure 4. EQUATION_DISPLAY

C = τ_{K} K + f_{M} M

(4577)

You can determine the scalar coefficients $τ_{K}$ and $f_{M}$ from a desired modal damping factor and the knowledge of the first two eigenvalues of the undamped system.

The eigenvalue problem of the undamped system is:

Figure 5. EQUATION_DISPLAY

(K - ω_{i}^{2} M) x_{i} = 0

(4578)

where $ω_{i}$ and $x_{i}$ are the eigenvalues and eigenvectors, respectively:

Figure 6. EQUATION_DISPLAY

X = \begin{matrix} [x_{1} & \dots & x_{i} & \dots & x_{n}] \end{matrix}, Ω = [\begin{matrix} ω_{1}^{2} \\ ⋱ \\ ω_{i}^{2} \\ ⋱ \\ ω_{n}^{2} \end{matrix}]

(4579)

Since $K$ and $M$ are symmetric matrices, the eigenvectors are orthogonal. The eigenvectors are normalized with respect to the mass matrix $M$ . After solving the undamped eigenvalue problem Eqn. (4578), the equations for the damped system can be decoupled by multiplying Eqn. (4575) by $X^{T}$ :

Figure 7. EQUATION_DISPLAY

\begin{matrix} X^{T} M X = I \\ X^{T} K X = Ω \\ X^{T} C X = Γ \end{matrix}

(4580)

It is possible to write the solution of the damped system, $u$ , as a linear combination of the (undamped) eigenvectors $x_{i}$ :

Figure 8. EQUATION_DISPLAY

u = X g (t)

(4581)

where $g (t)$ are the generalized, or modal, degrees of freedom. Eqn. (4575) then reduces to the diagonal form:

Figure 9. EQUATION_DISPLAY

I \ddot{g} (t) + Γ \dot{g} (t) + Ω g (t) = q (t)

(4582)

with the generalized loads:

Figure 10. EQUATION_DISPLAY

q (t) = X^{T} f^{e x t} (t)

(4583)

Assuming frequency-proportional damping, with modal damping factors $ζ_{i}$ , the diagonal matrix $Γ$ becomes:

Figure 11. EQUATION_DISPLAY

Γ = [\begin{matrix} 2 ζ_{1} ω_{1} \\ ⋱ \\ 2 ζ_{i} ω_{i} \\ ⋱ \\ 2 ζ_{n} ω_{n} \end{matrix}]

(4584)

Each row of Eqn. (4582) is then:

Figure 12. EQUATION_DISPLAY

{\ddot{g}}_{i} (t) + 2 ζ_{i} ω_{i} {\dot{g}}_{i} (t) + ω_{i}^{2} g_{i} (t) = q_{i} (t)

(4585)

The Rayleigh damping coefficients $τ_{K}$ and $f_{M}$ can now be determined by comparing the damping term in Eqn. (4585) with Eqn. (4577):

Figure 13. EQUATION_DISPLAY

2 ζ_{i} ω_{i} = f_{M} + τ_{K} ω_{i}^{2}

(4586)

For example, you can write Eqn. (4586) for the first two fundamental eigenfrequencies $ω_{1}$ and $ω_{2}$ :

Figure 14. EQUATION_DISPLAY

\begin{matrix} τ_{K} = \frac{2 ζ_{1} ω_{1} - 2 ζ_{2} ω_{2}}{ω_{1}^{2} - ω_{2}^{2}} \\ f_{M} = 2 ζ_{1} ω_{1} - \frac{2 ζ_{1} ω_{1} - 2 ζ_{2} ω_{2}}{ω_{1}^{2} - ω_{2}^{2}} ω_{1}^{2} \end{matrix}

(4587)

with $τ_{K} > 0$ and $f_{M} > 0$ .

If you assume a uniform modal damping factor $ζ$ for both frequencies, $τ_{K}$ and $f_{M}$ become:

Figure 15. EQUATION_DISPLAY

\begin{matrix} τ_{K} = \frac{2 ζ}{ω_{1} + ω_{2}} \\ f_{M} = 2 ζ ω_{1} (1 - \frac{ω_{1}}{ω_{1} + ω_{2}}) \end{matrix}

(4588)

A typical modal damping factor is $ζ = 0.02$ .

An even simpler choice is to restrict the Rayleigh damping to stiffness proportional damping, by assuming $f_{M} = 0$ and $τ_{K} > 0$ , and tune the parameter with the fundamental eigenfrequency:

Figure 16. EQUATION_DISPLAY

2 ζ ω = τ_{K} ω^{2}

(4589)

If you express the eigenfrequency $ω$ as a function of the period $T$ :

Figure 17. EQUATION_DISPLAY

ω = \frac{2 π}{T}

(4590)

$τ_{K}$ reduces to:

Figure 18. EQUATION_DISPLAY

\frac{ζ T}{π} = τ_{K}

(4591)

If you then choose the time-step for the stiffness proportional damping $τ_{K} = Δ t$ , the modal damping factor becomes:

Figure 19. EQUATION_DISPLAY

ζ = \frac{Δ t}{T} π

(4592)

A time-step of $Δ t = T / 100$ , with the choice of $τ_{K} = Δ t$ , gives a modal damping factor of:

Figure 20. EQUATION_DISPLAY

ζ = \frac{π}{100} = 0.0314

(4593)