## M

FIGURE 2.134 (a) Beam in flexural vibration and (b) equilibrium of beam segment in vibration.

2.13.7 Distributed Mass Systems

Although many structures may be approximated by lumped mass systems, in practice, all structures are distributed mass systems consisting of an infinite number of particles. Consequently, if the motion is repetitive, the structure has an infinite number of natural frequency and mode shapes. The analysis of a distributed-parameter system is entirely equivalent to that of a discrete system once the mode shapes and frequencies have been determined, because in both cases the amplitudes of the modal response components are used as generalized coordinates in defining the response of the structure.

In principle, an infinite number of these coordinates are available for a distributed-parameter system, but in practice, only a few modes, usually those of lower frequencies, will make a significant contribution to the overall response. Thus, the problem ofa distributed-parameter system can be converted to a discrete system form in which only a limited number of modal coordinates are used to describe the response.

### 2.13.7.1 Flexural Vibration of Beams

The motion of the distributed mass system is best illustrated by a classical example of a uniform beam with of span length L and flexural rigidity EI and a self-weight of m per unit length, as shown in Figure 2.134a. The beam is free to vibrate under its self-weight. From Figure 2.134b, dynamic equilibrium of a small beam segment of length dx requires that

0X 0t2 V 7

where and

Substituting these equations into Equation 2.391 gives the equation of motion of the flexural beam:

04y m 02y

Equation 2.394 can be solved for beams with given sets of boundary conditions. The solution consists of a family of vibration mode with corresponding natural frequencies. Standard results are available in

TABLE 2.6 Frequencies and Mode Shapes of Beams in Flexural Vibration kn EI in = - A -T Hz

EI = flexural rigidity (N m2)

Boundary condition

Pinned-pinned

Fixed-fixed

0 0