Free vibration 197 b1 π2 for a simply supported beam. From Chapter 7 it is known that the maximum displacement of the uniform beam, listed in Table 15.2, subjected to self-weight can be expressed in a unified form as follows: 4 m gL ∆ d1 EI
(15.33)
where g is gravity, m is the distributed mass of the beam along its length L and d1 is a constant dependent on the boundary conditions. For example, d1 5/384 for a simply supported beam. The constant d1 for the beams with four common boundary conditions are given in Table 15.2. Table 15.2 Coefficients for single-span beams
b1 d1 A1(m1/2/s) B1(m/s2)
Simply supported beam
Fixed end beam
Propped cantilever beam
Cantilever beam
9.87 0.0130 0.561 0.315
22.4 0.00260 0.570 0.325
15.4 0.00542 0.565 0.320
3.52 0.125 0.621 0.385
It can be noted that equations 15.32 and 15.33 contain the same term EI, through which a relationship between the fundamental natural frequency f and the maximum displacement ∆ due to the self-weight of a beam can be established. Equations 15.32 and 15.33 can be expressed as: 4 4π2 mL4 2 m gL EI f EI d 1 b21 ∆
leading to: b1 f d 1g 2π
∆1 A ∆1
b12d1g 1 1 2 B1 2 ∆ 2 4π f f
1
(15.34)
(15.35)
As the constants b1 and d1 are given, the coefficients A1 and B1 can be calculated for the beams listed in Table 15.2 and have units of m1/2/s and m/s2 respectively. The corresponding values of A1 and B1 are given in Table 15.2 using g 9.81m/s2. Thus the displacement ∆ is measured by m in the above two equations. The fundamental natural frequency of a beam can be calculated using Table 15.2 and equation 15.34 if the maximum displacement of the beam due to its self-weight is known. Alternatively, the maximum displacement of a beam can be estimated using Table 15.2 and equation 15.35 if the fundamental natural frequency is available either from calculation or vibration measurement. Examples 15.3 and 15.4 illustrate the applications.