Structures which are subjected to dynamic loading, particularly aircraft, vibrate or oscillate in a frequently complex manner. An aircraft, for example, possesses an infinite number of natural or normal modes of vibration. Simplifying assumptions, such as breaking down the structure into a number of concentrated masses connected by weightless beams ( lumped mass concept), are made but whatever method is employed the natural modes and frequencies of vibration of a structure must be known before flutter speeds and frequencies can be found. We shall discuss flutter and other dynamic aeroelastic phenomena in Chapter 28 but for the moment we shall concentrate on the calculation of the normal modes and frequencies of vibration of a variety of beam and mass systems.

10.1 Oscillation of Mass/Spring Systems

Let us suppose that the simple mass/spring system shown in Fig. 10.1 is displaced by a small amount x ₀ and suddenly released. The equation of the resulting motion in the absence of damping forces is

(10.1)

Figure 10.1: Oscillation of a mass/spring system.

where k is the spring stiffness. We see from Eq. (10.1) that the mass, m, oscillates with simple harmonic motion given by

(10.2)

in which ? ² = k/m and ? is a phase angle. The frequency of the oscillation is ?/2 ? cycles per second and its amplitude x ₀. Further, the periodic time of the motion, that is the time...