Mode shape

Vibration of a circular disc in two dimensions with a pinned boundary condition along the entire outer edge. The disc is vibrating at a single mode. See other modes.

In the study of vibration in Engineering, a Mode Shape describes the expected curvature (or displacement) of a surface vibrating at a particular Mode. To determine the vibration of a system, the mode shape is multiplied by a function that varies with time, thus the mode shape always describes the curvature of vibration at all points in time, but the magnitude of the curvature will change. The Mode Shape is dependent on the shape of the surface as well as the boundary conditions of that surface.

Modes

A Mode of vibration is characterized by a modal frequency and a mode shape, and are numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one valley) it would be vibrating in mode 2.

In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have a radial coordinate and an angular coordinate. If you measured from the center outward along the radial coordinate you would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the antisymmetric (also called Skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2-1 or 1-2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).

Each mode is entirely independent of all other modes. Thus all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes (with lower modes having greater amplitude).

Since the lower modes vibrate with greater amplitude, they cause the most displacement and stress in a structure. Thus they are called Fundamental Modes.

Nodes

A static representation of the above mode shape, with nodal lines shown in pale green.

In a one dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times.

When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about 1/3 of the way from the center to the edge, and the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In a real system these lines would equal zero exactly, as shown to the right.

Vibration

Typically, an object will vibrate at several modes at once, thus the total displacement will be a superposition of the mode shapes of the individual modes. Each mode is multiplied by a different time function, such that all modes vibrate at a different frequency.

For example, a beam might have a mode shape of:

$sin({\frac {\pi nx}{L}})$

Where n is the mode number, x is the distance from a given end of the beam, and L is the overall length.

The time function may look like:

$sin({\frac {\pi nt}{T}})$

Where t is time and T is the period of vibration.

Thus the vibration for a given mode is given by:

$sin({\frac {\pi nx}{L}})sin({\frac {\pi nt}{T}})$

Since the total vibration of the beam is given by the superposition of all modes, the total vibration for our example system is given by:

$\sum _{n=1}^{\infty }sin({\frac {\pi nx}{L}})sin({\frac {\pi nt}{T}})$

See also

• Wave equation
• Eigenfunction
• Vibrations of a circular drum
• Cushioning
• Critical Speed
• Damping
• Mechanical resonance
• Modal Analysis
• Noise, Vibration, and Harshness
• Quantum vibration
• Random vibration
• Shock
• Simple Harmonic Oscillator
• Structural Dynamics
• Torsional vibration
• Vibration isolation

References

• Blevins, Robert D. Formulas for natural frequency and mode shape
• Tzou, H. S. & Bergman, L. A. Dynamics and Control of Distributed Systems