modal analysis

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Modal analysis is a set of mathematical techniques for determining the modal characteristics of a dynamical system. Modal analysis can be used to determine the resonant frequencies and vibration responses of a structure and is closely related to failure analysis. The technique often assumes a system to behave linearly as an approximation so that matrix algebra can be used, for which computers are especially well suited. This assumption turns out to be pretty good, in fact, since stable equilibrium configurations correspond to local minima in potential energy, which can be approximated locally by parabolas, the hallmarks of the simple harmonic oscillator.

Example

Consider three masses m_1 = m\,, m_2 = M\,, and m_3 = m\, constrained to move along a single direction, say, the x\, axis, with m_1\, and m_3\, on either side of m_2\,. Now join m_1\, and m_3\, separately to m_2\, with two springs having spring constants equal to k\,. Let the three generalized coordinates of the masses be x_1\,, x_2\, and x_3\,, each of which are zero at the equilibrium positions of the masses.

The total kinetic energy of the system is

T = \frac{1}{2} m \dot{x}_1^2 + \frac{1}{2} M \dot{x}_2^2 + \frac{1}{2} m \dot{x}_1^3\,.

From Hooke's law, the total potential energy of the system is

V\, \frac{1}{2}k(x_2-x_1)^2 \frac{1}{2}k(x_3-x_2)^2\,
\frac{1}{2}k x_1^2 + \frac{1}{2}k x_2^2 + \frac{1}{2}k x_3^2 - k x_1 x_2 - k x_2 x_3\,

So the Lagrangian for the system is

L = \frac{1}{2} m \dot{x}_1^2 + \frac{1}{2} M \dot{x}_2^2 + \frac{1}{2} m \dot{x}_1^3 - \frac{1}{2}k x_1^2 - \frac{1}{2}k x_2^2 -\frac{1}{2}k x_3^2 + k x_1 x_2 + k x_2 x_3\,.

Now, assuming there aren't any damping forces present, we can obtain the equations of motion from

\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}_i}\right) - \frac{\partial L}{\partial x_i} = 0\,.

This results in a set of three equations:

m \ddot{x}_1 - kx_1 + kx_2 = 0\,,
M \ddot{x}_2 - 2 kx_2 + k x_1 + kx_3 = 0\,,
m \ddot{x}_3 - kx_3 + kx_2 = 0\,.

Now, we let x = [x_1, x_2, x_3]^T\, and \ddot{x} = [\ddot{x}_1, \ddot{x}_2, \ddot{x}_3]^T\,, and define the following matrices:

\mathbf{M} = \left( \begin{matrix} m & 0 & 0 \\ 0 & M & 0 \\ 0 & 0 & m \end{matrix} \right)\,,
\mathbf{K} = \left( \begin{matrix} k & -k & 0 \\ -k & 2k & -k \\ 0 & -k & k \end{matrix} \right)\,.

Here \mathbf{M}\, is the mass matrix and \mathbf{K}\, is the stiffness matrix, although in general these matrices may be in units other than that of mass or stiffness. With these definitions, we may write our equations of motion as:

\mathbf{M} \ddot{x} - \mathbf{K}x = 0\,.

Now we may make a series of substitutions in order to simplify calculations. First, we define the matrix square root. The square root of a matrix \mathbf{A}\,, if defined, is \mathbf{A}^{\frac{1}{2}}\, such that \mathbf{A}^{\frac{1}{2}} \mathbf{A}^{\frac{1}{2}} = \mathbf{A}\,.

Let x = \mathbf{M}^{-\frac{1}{2}} q\,, or equivalently let q = \mathbf{M}^{\frac{1}{2}}x\,. In our example we do not need to use diagonalization to calculate \mathbf{M}^{-\frac{1}{2}}\, because \mathbf{M}\, is already a diagonal matrix. Therefore substitute into the equation of motion to obtain

\mathbf{M} \mathbf{M}^{-\frac{1}{2}} \ddot{q} - \mathbf{K} \mathbf{M} \mathbf{M}^{-\frac{1}{2}} q = 0\,.

Left multiply by \mathbf{M}^{-\frac{1}{2}}\, to obtain

\mathbf{M}^{-\frac{1}{2}} \mathbf{M} \mathbf{M}^{-\frac{1}{2}} \ddot{q} - \mathbf{M}^{-\frac{1}{2}} \mathbf{K} \mathbf{M}^{-\frac{1}{2}} q = 0\,,

or

\ddot{q} - \mathbf{M}^{-\frac{1}{2}} \mathbf{K} \mathbf{M}^{-\frac{1}{2}} q = 0\,.

Now, let \tilde{\mathbf{K}} = \mathbf{M}^{-\frac{1}{2}} \mathbf{K} \mathbf{M}^{-\frac{1}{2}}\,, which is the modified stiffness matrix. At this point we obtain

\ddot{q} - \tilde{\mathbf{K}}q = 0\,.

This is an eigenvalue problem and can be solved directly, but to further simplify the problem, we can use some more substitution. First, we must diagonalize \mathbf{K}\,, i.e. find a matrix \mathbf{P}\,, such that \mathbf{P}^T\mathbf{K} \mathbf{P} = \mathbf{D}\, where \mathbf{D}\, is a diagonal matrix. Then let q = \mathbf{P}r\, so that

P \ddot{r} - \mathbf{KP} r = 0\,,

or,

\ddot{r} - \mathbf{D} r = 0\,.

With the problem simplified thusly, we can easily express the solutions to the equations of motion, since by changing coordinates from x_i\, to q_i\, and then to r_i\,, we've managed to decouple the systems completely (\mathbf{D}\, is diagonal), so that we may write:

\ddot{r}_i - \lambda_i r_i = 0\,.

Writing \lambda_i = \omega_i^2\,, where \lambda_i\, is the i^{th}\, eigenvalue of \mathbf{K}\, and diagonal element of \mathbf{D}\,, and where \left\{\omega_i\right\}\, is the set of natural or resonant frequency of the system. We can solve each of these equations independently, so that r_i = A_i \cos(\omega_i t + \phi_i)\,. Any general set of coordinates in which the system oscillates at a single frequency in each coordinate is called a set of normal coordinates. Thus \left\{r_i\right\}\, represent the normal coordinates for the system. We can get back our original coordinates by applying the transformations in reverse:

x = \mathbf{M}^{-\frac{1}{2}} \mathbf{P} r\,.

The matrix \mathbf{S} = \mathbf{M}^{-\frac{1}{2}} \mathbf{P}\, is called the matrix of modal shapes, since it encodes characteristic shape of each mode. Writing x = \mathbf{S}r\,, and denoting the column vectors of \mathbf{S}\, as S_i\, we get that

x = A_1 S_1 cos(\omega_1 t + \phi_1) + A_2 S_2 cos(\omega_2 t + \phi_2) + ...\,.

Thus, the net displacement of each mass, the components of x\,, are linear superpositions of the displacements brought about by each mode of oscillation. For instance, in our example of three masses connected to each other with springs, we would find that one mode for which S_1 = [ -1,0,1 ]^T\,. This mode is antisymmetric with the outer masses oscillating 90^\circ\, out of phase with one another, and the inner mass remaining stationary. This mode has its own frequency \omega_1\, and is independent of and uninfluenced by the other modes.

In order to solve the problem in its entirety, then, we need determine the values of A_i\, and \phi_i\,. This can be done as long as the initial conditions for the system are given, or any other independent conditions that fix these constants. Normally the initial conditions are given in terms of the system's initial displacements and velocities.

In the above development we used an approach that allowed us to find both the modal shapes and natural frequencies. If we were only after the frequencies \omega_i\, we would have used the equation \mathbf{M}\ddot{x} - \mathbf{K}x = 0\, directly, by assuming that x_i\, was of the form A_i \cos(\omega_i t + \phi_i)\,. In this method we consider each mode separately, by assuming that the amplitudes of all the other modes are zero (by some appropriate initial conditions). We can then substitute into the above equation to obtain

(\omega_i^2 \mathbf{M} + \mathbf{K})x = 0\,.

Since non-trivial solutions require that \det(\omega_i^2\mathbf{M} + \mathbf{K}) = 0\,, we can find \left\{\omega_i\right\}\, as the roots of the characteristic equation.

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