Dynamics: Free Vibration

A musical box

a) A musical box                                                          b) the core of the musical box
Fig. 15-14: A musical box and its core (The models are provided by Prof. B Zhuang, Zhejiang University, China)
Fig. 15-15: Cantilever beams with different lengths and sections in the music box (mm) [15.6]

A musical box is a device that produces music using mechanical vibration. Fig. 15-14a shows one of many decorative music boxes which are readily available. The core of the music box is the unit show in Fig. 15-14b.

A spring is used to rotate a music tube, converting the potential energy stored in the spring into the kinetic energy which drives the rotation of the tube. Raised points on the tube displace cantilever metal bars causing them to vibrate and generate sound. Different geometries (lengths and cross-sections) of the bars (see Fig. 15-15) provide different natural frequencies of the bars, generating different music notes. The distribution of the raised points on the tube is designed to create a particular music tune when the tube rotates. The music unit shown in Fig. 15-14b has eighteen metal bars generating eighteen different musical notes.

A given musical note relates to a particular natural frequency of the vibrating body. Table 15-4 gives the relationships between some natural frequencies and music notes.

Table 15-4: Relations between natural frequency and music note [15.6, 15.7]

Natural frequency (Hz)








Music note

(Middle) C







Fig. 15-16 shows part of the keyboard of a piano. The main reference point on a piano is known as Middle C. This is the white note located approximately in the centre of the keyboard and immediately to the left of a pair of black keys. Striking the Middle C key, the sound generated corresponds to a frequency of 262 Hz. The next white key, D, to the right of the Middle C key, produces a sound corresponding to a frequency of 294 Hz.

Fig. 15-16: The keyboard of a piano