Vibration and DOFs, modes, and mode shapes

A structure has the same number of modes of vibration as the number of degrees-of-freedom (DOFs). Each mode of vibration is characterized by one resonance (or natural) frequency and one mode shape.

A mode shape generally represents the displacement of a point on the structure relative to all other points.

A free vibration of a structure is a linear combination of all modes of vibration of the structure.

A structure with a single DOF

A single DOF structure has one mode of vibration and is characterized by:

• A lumped mass constituting one point on the structure.

• A structural deflection along one axis.

• One resonance frequency.

The mode shape of a single DOF structure is difficult to visualize since the structure has only one point. It is sufficient to think of such a mode shape as simply the oscillation of a lumped mass along one axis. Since there are no other points on the structure, there are no other possible relationships of motion.

A structure with two DOFs

A structure with two points that can only move along one axis has two DOFs, thus two modes of vibration.

To illustrate such a structure, the structure in the following graphics is composed of two lumped masses, two springs and two dampers.

In the first mode of vibration, it is possible for both points (masses) on the structure to oscillate in the same direction at the same time. The two masses are moving in-phase one with respect to each other as seen in the following graphics. This first mode is characterized by a relatively low frequency and a relatively high amplitude.

Mode 1	Response of either mass for mode 1

In the second mode of vibration, the points oscillate in opposite directions. They are out-of-phase with one another. Close inspection of the relative positions of the dampers in the following graphics reveals that the two masses are moving in opposite directions at the same time. Mode 2 has a higher frequency and lower amplitude than mode 1.

Mode 2	Response of either mass for mode 2

The response function for mode 1 approximates the behavior of either point in the system at the low resonance frequency whereas the response function for mode 2 approximates the behavior at the high resonance frequency.

It is easier to visualize the mode shapes of such a structure because the points are either oscillating in-phase or out-of-phase relative to each other. At the low resonance frequency, the in-phase relationship is most apparent, and at the high resonance frequency, the out-of-phase relationship is most apparent.

The real pattern of displacement of either of the two points at virtually any frequency results from these two different displacements acting on the structure simultaneously, where the actual response of the structure is composed of modes 1 and 2.

Looking at the frequency response function for each separate mode of vibration, it is easy to see how a third frequency response function, the one you would expect to obtain from the actual response of the structure, is likewise the sum of the frequency responses for each separate mode.

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A structure with 3 DOFs

A structure with 3 DOFs has three modes of vibration. Suppose you have a diving board divided into three segments as shown, and suppose each segment is represented by one node that has one DOF.

In mode 1, all three nodes oscillate around their points of equilibrium in phase with each other. This is the mode shape that is most apparent when a diver jumps off the end of the diving board.

Mode 1

The diving board, however, is also vibrating in modes 2 and 3. These modes are marked by out-of-phase relationships between the various nodes.

Mode 2
Mode 3

Again, as the resonance frequency increases, the response magnitude decreases. Compare the frequencies and magnitudes of all three modes for the responses at node 1.

	Mode shape	Response at node 1
Mode 1
Mode 2
Mode 3

When you add the responses for all three modes, a more accurate picture of the true behavior of node 1 during excitation emerges. The corresponding frequency response plot shows that all three modes are present at increasingly higher frequencies.

Mode 1 + Mode 2 + Mode 3 =		►
	Response at node 1	FRF at node 1

The response of the diving board at nodes 2 and 3 is similar to node 1. The only thing changing is the amplitude of displacement of nodes 2 and 3.

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