Recall that the bending moment is proportional to the local curvature and so d2v

where EI may be regarded as the effective bending stiffness of the beam at a particular cross section. For homogeneous and isotropic beams, E is Young's modulus and I is the cross-sectional area moment of inertia about the z axis for a particular cross section. Substitution of Eq. (2.207) into Eq. (2.206) and of the resulting equation into Eq. (2.204) yields the partial differential equation of motion for a nonuniform beam as

In the following sections, we will be treating the special case of free vibration for which q(x, t) — 0. Also, for simplicity we will specialize the equations for the case of spanwise uniformity of all properties, for which the parameter a can be defined as a constant such that

0 0

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