Acoustics and Vibration Animations

Daniel A. Russell, Graduate Program in Acoustics, The Pennsylvania State University

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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:

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.

1st Bending - 146 Hz

The lowest frequency vibrational mode of a tennis racket is a flexural bending mode in which the entire racket (frame + strings) vibrates much like a free-free bar. There is a node (a location that does not vibrate) in the handle, approximately 5 inches from the butt end, and another nodal line across the width of the head, close to the middle of the head. The location of this nodal line in the head region is important for defining the location of the sweet spot. This vibrational mode is often also associated with the sensation of sting in the hands of the player.

1st Torsional - 382 Hz

The torsional mode is a twisting of the racket head and handle, with a nodal line down the center of the long axis of the racket. The nodal line across the width of the racket head is in nearly the same location as the nodal line for the first bending mode, so that the positions of the nodes for the 1st bending and torsional modes are key in defining the region of the sweet spot on the racket. Torsional modes are excited strongly by impacts away from the central axis of the racket, and may contribute to the impact felt at the wrist for bat hits.

2nd Bending - 411 Hz

The second bending mode has a node about 2-3 inches from the butt end of the handle. There are two nodal lines across the width of the head, but the center of the string-bed on the head exhibits motion. Because this vibrational mode has a frequency well within the range where the human hand is most sensitive (100-600 Hz), this mode also contributes to the sensation of feel.

3rd Bending - 862 Hz

The third bending mode has two nodal lines across the racket head in the string region, and two nodes in the handle.

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).

Membrane (0,1) - 582 Hz

The (0,1) membrane mode has one circular node - the outer rim of the string-bed where it attaches to the frame. The entire string-bed moves as a single system. The frequency of this mode shape is highly dependent on the tension in the strings, and this shape (and its frequency) are directly related to the power of the racket, since the period of vibration (inverse of the frequency) strongly affects the dwell-time of the ball as it impacts the racket.

Membrane (0,2) - 1397 Hz

The (0,2) membrane mode has two circles, the outer rim at the frame, and a second circular node so that the inner and outer portions of the string-bed oscillate with opposite phase. This mode has a high enough frequency (short enough period) that it probably does not contribute much to racket-ball interaction.

Membrane (1,1)a - 936 Hz

The (1,1) membrane has one diameter and one circle (the outer rim). In a uniform circular membrane, the (1,1) mode is actually a pair of degenerate modes, with the orientation of the diagonal rotated by 90 degrees. In an oval-shaped membrane, as is the case for the tennis racket head, the two orientations are not degenerate but have different frequencies. The lower of the pair, denoted (1,1)a has - in this case - the diameter as a diagonal node line across the string bed.

Membrane (1,1)b - 996 Hz

The higher frequency of the (1,1) split pair for the oval-shaped string bed has the nodal diameter lying along the long axis of the racket head. Because both of the (1,1) orientations have a nodal line passing through the center of the head, they do not play a very important role for impacts near the sweet spot. These modes will contribute to the "ping" sound that you hear when you tap the racket on the strings.

Membrane (2,1)a - 1300 Hz

The (2,1) mode has two diameters and one circle. This mode shape would also be degenerate for a circular membrane, but is split for a tennis racket because of the oval shape of the head. The lower frequency of this pair, designated (2,1)a, has one nodal diameter along the long axis of the racket, and the other at right angles across the width.

Membrane (2,1)b - 1301 Hz

The higher frequency (2,1)b mode has two diameters, making an "X" on the string-bed. Because the (2,1) modes have nodal lines crossing the center of the head, they will not be excited by impacts at the sweet spot in the center of the head.

Membrane (3,1)a - 1546 Hz

The final pair of membrane modes are the (3,1) pair with three nodal diameters and six alternating regions where the string bed oscillates with opposite phase. The mode with the lower frequency has a nodal diameter aligned with the long axis of the racket.

Membrane (3,1)b - 1578 Hz

The higher frequency mode of the (3,1) pair has a nodal diameter across the width of the racket head.


  1. Brody, American Journal of Physics
  2. G.H. Banwell, J.R. Roberts, B.J. halkon, S.J. Rothberg, and S. Mohr, "Understanding the Dynamical Behavior of a Tennis Racket under Play Conditions," Experimental Mechanics,, 54: 517-537 (2014).