Experimental Modal Analysis of a Tennis Racket

The content of this page was originally posted on November 25, 2014.

The animated vibrational mode shapes on this page were collected by one of my students, Kritika Vayur in November 2014 as part of the work toward her M.Eng. in Acoustics degree.

Experimental Modal Analysis (EMA) is an experimental method for extracting vibrational mode shapes, resonance frequencies, and damping rates for a structure. The experiment involves tapping the structure (in this case, a tennis racket) with a special hammer that has a force transducer in the tip to measure the impact force as a function of time. The resulting vibration response of the structure to this impact is measured with a small accelerometer attached to the structure. A two-channel frequency analyzer is used to capture the force and acceleration time signals and transform them to frequency spectra. The ratio of acceleration output to force input is called a Frequency Response Function, and is a complex quantity (in the frequency domain) whose real and imaginary parts provide the frequencies and the damping rates, and the amplitude and phase information necessary to determine the mode shapes. Using a roving impact technique, the accelerometer location is fixed, and the hammer impacts are applied at various locations on the structure - for the tennis racket data below, we used approximately a 1-inch grid spacing. The resulting collection of Frequency Response Function data is post-processed using STAR Modal software to curve fit the FRF data, resulting in the animated shapes below.

The racket in this experiment is a Donnay Forumla hollow graphite racket. The racket was freely supported by rubber bands, as this provides a boundary condition closest to the hand-held condition.^[1,2]

Structural Vibration Modes of the Frame

A tennis racket exhibits two distinct types of vibrational behavior:

• flexural bending and torsional vibrations involving the frame and handle
• membrane modes of the string bed

The first set of modes (flexural bending and torsional) primarily involve the motion of the frame, with the string-bed acting as an extension of the frame - just as if the strings were a solid flat surface. The flexural bending and torsional modes have considerable influence on the perception of feel in the hands of the player.

Membrane Modes of the String Bed

The second type of vibrational modes in a tennis racket involve only the string-bed; the frame is essentially fixed in place while the strings move together in a manner very similar to that of a circular membrane. In fact, the usual manner of identifying the modes of the string-bed is to classify them according to the number of circular and diameter nodes. The modal identification (m,n) indicates the number of diameters (m) and the number of circles (n). String-bed membrane modes contribute to the sound that the racket makes when it strikes a ball. Several of the string-bed modes - especially the (0,1) and (0,2) modes - are important to determining the ball-racket dwell time and racket power. However, since the string-bed membrane modes do not involve vibration of the handle, they have very little influence on the perception of feel in the hands of the player. These string-bed modes will be strongly damped by the popular string dampers (which - by the way - have absolutely no effect on reducing vibration in the handle).