Normal Modes

A normal mode is an undamped, sinusoidal motion pattern over time at a fixed frequency. The structure remains in dynamic equilibrium and undergoes a free vibration. The normal modes and natural frequencies of a structure govern its dynamic response.

In Solid Mechanics, normal modes and natural frequencies are often called eigenmodes and eigenfrequencies, as they are calculated from an eigenvalue problem. To derive the eigenvalue problem, consider a converged solution of the general nonlinear equations of motion:

Figure 1. EQUATION_DISPLAY

r (u) = M \ddot{u} (t) + f (u (t), \dot{u} (t)) - p (u (t)) = 0

(4643)

where $M$ is the mass matrix, $u (t)$ , $\dot{u} (t)$ , and $\ddot{u} (t)$ are the displacement, velocity, and acceleration vectors respectively, $f (u (t), \dot{u} (t))$ is the internal force vector and $p (u (t))$ is the external load vector.

For a perturbation $Δ u$ of the displacement $u$ , Eqn. (4643) becomes:

Figure 2. EQUATION_DISPLAY

r (u + Δ u) = M [\ddot{u} + Δ \ddot{u}] + f (u + Δ u) - p (u + Δ u)

(4644)

Linearizing Eqn. (4644) in

u

gives:

Figure 3. EQUATION_DISPLAY

r (u + Δ u) = M \ddot{u} + M Δ \ddot{u} + f (u, \dot{u}) + \frac{\partial f}{\partial u} Δ u - p (u) - \frac{\partial p}{\partial u} Δ u

(4645)

where the velocity-dependent internal force

f (\dot{u})

is neglected. The resulting tangent stiffness matrix is then:

Figure 4. EQUATION_DISPLAY

K = \frac{\partial f}{\partial u} - \frac{\partial p}{\partial u}

(4646)

where $\frac{\partial f}{\partial u}$ is the stiffness from the internal forces including geometric and material nonlinearities and $\frac{\partial p}{\partial u}$ is the stiffness from the follower forces.

With the assumption of a converged solution (that is,

r (u) = 0

u

), the equilibrium equation at

u + Δ u

becomes:

Figure 5. EQUATION_DISPLAY

M Δ \ddot{u} + K Δ u = 0

(4647)

Assuming that the perturbation is a free harmonic oscillation:

Figure 6. EQUATION_DISPLAY

Δ u = x_{k} \sin (ω_{k} t)

(4648)

Eqn. (4647) is then:

Figure 7. EQUATION_DISPLAY

(K - ω_{k}^{2} M) x_{k} \sin ω_{k} t = 0

(4649)

As the term

\sin ω_{k} t

is nonzero for an arbitrary t, Eqn. (4649) reduces to:

Figure 8. EQUATION_DISPLAY

(K - ω_{k}^{2} M) x_{k} = 0

(4650)

This is the eigenvalue problem for normal modes. For positive definite and symmetric matrices

K

and

M

, the eigenvalues

λ_{k} = ω_{k}^{2}

and the eigenvectors

x_{k}

are real. Furthermore the eigenvectors of distinct eigenvalues

i \neq j

are orthogonal with respect to the mass and stiffness matrices:

Figure 9. EQUATION_DISPLAY

x_{i}^{T} M x_{j} = 0

(4651)

Figure 10. EQUATION_DISPLAY

x_{i}^{T} K x_{j} = 0

(4652)

The eigenvalue problem has

k = 1, ..., n

distinct eigenvalues and eigenvectors if the matrices

K [n, n]

and

M [n, n]

have rank

n

. The magnitude of the eigenvectors are arbitrary and can be normalized with respect to the mass matrix:

Figure 11. EQUATION_DISPLAY

x_{k}^{T} M x_{k} = 1

(4653)

Figure 12. EQUATION_DISPLAY

x_{k}^{T} K x_{k} = ω_{k}^{2}

(4654)

The eigenvalues and eigenfrequencies are defined as:

Figure 13. EQUATION_DISPLAY

λ_{k} = ω_{k}^{2}

(4655)

Figure 14. EQUATION_DISPLAY

ω_{k} = \sqrt{λ_{k}}

(4656)

Figure 15. EQUATION_DISPLAY

f_{k} = \frac{ω_{k}}{2 π}

(4657)

where Eqn. (4655) is the $k^{t h}$ eigenvalue ( $\frac{{r a d}^{2}}{\sec^{2}}$ ), Eqn. (4656) is the $k^{t h}$ circular eigenfrequency ( $\frac{r a d}{\sec}$ ), and Eqn. (4657) is the $k^{t h}$ eigenfrequency ( $\frac{1}{\sec} = H z$ ).

The eigenvalue problem is solved using the Krylov-Schur method in the SLEPc open source library and is used to compute the smallest and largest eigenvalues in magnitude. In most transient problems, the lowest eigenfrequencies are dominant in the structural response. The higher eigenfrequencies and eigenvectors become dependent on the mesh size and are therefore inaccurate. Generally, at least five elements are necessary to represent a half sine wave of an eigenmode with engineering accuracy. In impact problems the higher eigenfrequencies are of interest to check the validity of the solution.