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3.4.6 Coupling of Longitudinal and Flexural Vibration 3.4.6.1 Longitudinal Vibration and Stiffness Coefficients

Consider element dx of a longitudinal bar shown in Figure 3.23a; the equilibrium equation of the element is

where u is the longitudinal displacement and N is the axial force in tension. Since the axial force can be expressed in terms of area, as N — AEe — AE0u/ 0x, Equation 3.80 becomes

02u_ 2 02u dt2 0x2

where a — AE/m. Using the separation of variables and substituting u — X(x)g(t) into Equation 3.81

d2X p dx2

of which the solutions are

X = CM sin kx + C2 cos kx; g = dM sin pt + d2 cos pt

a where k

X and g are the shape function and time function, respectively. Based on the following boundary conditions shown in Figure 3.23b, the dynamic-stiffness coefficient can be derived as [1, pp. 213-214]

Note that when k approaches zero (i.e., p = 0) as the static case, the stiffness coefficients become

sym.

0 0

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