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where fn = natural frequency of the system, (Hz)

8 = deflection of the spring under the weight of the mass — 8. in inches and 8cm in centimeters

This provides a convenient way of remembering the natural frequency of a spring mass system. A one-inch deflection spring is a 3-Hz oscillator, and a one-centimeter deflection spring is a 5-Hz oscillator. These simple oscillators appear over and over in various forms throughout architectural acoustics.

Air Spring Oscillators

When air is contained in an enclosed space, it can act as the spring in a spring mass system. In the example shown in Fig. 6.2, a frictionless mass is backed by a volume of air. When the mass moves into the volume, the pressure increases, creating a force that opposes the motion. The spring constant of the enclosed air is derived from the equation of state

PVy = constant

differentiating

Figure 6.2 A Frictionless Mass and an Air Spring

The rate of change of pressure with volume is v P

from which we can obtain the spring constant for a trapped air volume of depth h

Y PSdV y PS2 Y PS

and the natural frequency is

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