Modal Frequencies theoretical background

The mathematical process behind determining natural frequencies and the corresponding mode shapes through finite element analysis is complex. To make the theory easier to understand, we will consider a very simple linear spring-mass system with one degree of freedom (DOF). That is, the mass is only free to move in a single direction.

In the following image, m is the mass, and k is the stiffness of the spring:

dof spring mass system diagram

Hooke's law defines the force (F) required to displace a spring of stiffness (k) a distance (X):

(1) F = k·X

Newton's second law of motion, specifies the relationship between force (F), mass (m), and acceleration (a) for our single DOF system:

(2) F = m·a

Combining these two equations, we have a relationship between mass (m), acceleration (a), spring stiffness (k), and displacement (X):

(3) m·a = k·X

The following equation gives us the angular frequency of oscillation (ω), in radians/second, for this simple linear spring mass system:

(4) ω = (k/m)1/2

There are 2π radians per vibration cycle. Therefore, the next equation gives us the natural frequency of oscillation (fn), in Hertz:

(5) fn = (k/m)1/2 / (2·π)

In an actual finite element model, we have 3D motion and many DOF due to all of the nodes and elements in the mesh.

Note: We do not consider damping in the natural frequency calculation. Damping is defined as a decrease in the oscillation amplitude due to energy being drained from the system. Energy is lost in overcoming frictional, viscous, or other resistive forces. The amount of damping that occurs in typical manufacturing materials (such as steel, aluminum, and concrete) is small. So, the natural frequency without damping is typically not far from the actual natural frequency. Damping decreases the vibration frequency relative to an undamped system.

In a continuous system, there is an infinite number of vibrating modes. However, in a finite element model, there is a finite number of DOF, and therefore a finite number of vibration modes. Determining the vibration frequencies and mode shapes for these complex 3D systems involves matrix operations, eigenvalues, and eigenvectors.

Equation (4) is an eigenvalue and eigenvector problem with a finite number of solutions. The angular frequency (ω) is a scalar quantity. For any particular solution, all DOFs oscillate at the same angular frequency. However, the amplitude of the vibration (δ) is a vector quantity. In our complex system, δ is analogous to the displacement (X) in our single DOF example. The different DOFs oscillate with different amplitudes, given by the components of δ. This amplitude variation gives us the mode shape.

The proper definition of eigenvalues and eigenvectors, and the mathematical operations used to solve a modal analysis, are beyond the scope of this topic.