## L

3.4.6.2 Coupling Vibration

When a plane structure is subjected to dynamic excitations, its constituent members may have longitudinal and flexural vibrations. The vibration modes and frequencies may be coupled. Coupling vibration means that when a structural system is in vibration, all the constituent members vibrate in the same frequency for both the longitudinal and flexural motions. On the other hand, uncoupling vibration implies that longitudinal and flexural motions are independent of each other so the vibration mode of a system depends on whether the mode is associated with longitudinal or flexural frequency but not affected by both. Whether a structure is in coupling or uncoupling motion depends on the structural configuration, distribution of the structure's mass and stiffness, and modeling.

The structure is analyzed as rigid frame with one d.o.f. in rotation shown in Figure 3.24b and then analyzed as an elastic frame with three d.o.f. shown in Figure 3.24c. The two-bar frame shown in Figure 3.24a is used for illustration. Member properties are A = 232.26 cm2, E = 206.84 GN/m2, g = 76.973 kN/m3,1 = 10114.423 cm4, and h = 8.6196 m.

(A) Find the natural frequencies of flexure only, p, and of coupling effect, p', for the first five modes by considering a wide range of slenderness ratios for the two identical members: L/R = 20, 40, 60, and 80; where the radius of gyration, R, is constant based on the given cross-section and L is changed. (B) Study the influence of longitudinal frequency parameter on the coupling frequencies by letting the longitudinal dynamic stiffness be replaced by static stiffness as AE/L; find the pseudo-coupling frequencies p'', and compare them with p' obtained in A. p'/p, p''/p', and p''/p versus L/R are plotted in Figure 3.25 to 3.27, respectively. These three figures reveal that (1) p and p' are, respectively, upper and lower bounds of the frequencies; (2) the coupling effect on frequencies becomes more significant for higher modes and smaller slenderness ratios; and (3) the pseudo-coupling approach may be used for lower modes.

3.4.7 Effects of Elastic Media, Torsion, and Axial Force on Vibration

The dynamic-stiffness formulation is also developed for investigating the effect of elastic media on longitudinal and flexural vibrations, axial force on flexural vibration for both Bernoulli-Euler and

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