## Jo i even but

The zero initial velocity requires that

Multiplication by sin(jjrx/£) and integration will result in determining that A,- — 0 for all /. These results can be summarized by noting that = 0 for all even values of / except

The Cj s can be determined by substitution of the odd generalized coordinates back into the equations of motion,

so that the complete string displacement becomes oc

The first term is thus the response due to the initial displacement, and the sum over the odd-indexed modes is the response due to the forcing function.

### 2.2 Uniform Beam Torsional Dynamics

Now that the fundamental aspects of structural dynamics analysis have been considered for the uniform string problem, these concepts will be applied to the dynamics of beam torsional deformation. The beam has many more of the characteristics of typical aeronautical structures. Indeed, high-aspect-ratio wings and helicopter rotor blades are frequently idealized as beams, especially in preliminary design. Even for low-aspect-ratio wings, although a plate model is more realistic, the bending and torsional deformation can be approximated by use of beam theory with adjusted stiffness coefficients.

In an effort to retain a level of simplicity that promotes tractability, the torsional rigidity of St. Venant theory, denoted GJ, will be taken as given. For homogeneous and isotropic beams, G denotes the shear modulus and / is a constant that depends only on the geometry of the cross section. For such beams / can be determined by solving a boundary-value problem over the cross-sectional area, which requires finding the cross-sectional warping caused by torsion. Although analytical solutions for this problem are available for some simple cross-sectional geometries, solving for the cross-sectional warping and torsional stiffness is not a trivial exercise in general. For nonhomogeneous, anisotropic beams one may also use the symbol GJ to denote an effective torsional rigidity, which can be determined by solving a far more involved boundary-value problem over the cross-sectional area.

### 2.2.1 Equation of Motion

The beam will be considered initially to have nonuniform properties along the x -axis, which is chosen to coincide with the elastic axis for the beam. In our idealized model for beams, this axis is assumed to be straight and, by definition, corresponds to the locus of the cross-sectional shear centers. This choice of x -axis structurally uncouples torsion and transverse displacements caused by bending for isotropic beams. For composite beams, this choice of x-axis uncouples torsion and transverse shear deformation, but torsion and transverse displacements may remain coupled depending on whether the beam has bending-torsion elastic coupling. For a further simplification, we assume that cross-sectional mass centroids lie along the elastic axis, in which case transverse motions due to bending are inertially uncoupled from torsional motion. The elastic twisting deflection, 6, will be positive in a right-handed sense about this axis as illustrated in Fig. 2.11. In contrast, the twisting moment, denoted by T, is the structural torque, that is, the resultant moment of the tractions on a cross-sectional face about the elastic axis. Recall that an outward-directed normal on the positive x face is directed to the right, whereas an outward-directed normal on the negative x face is directed to the left. Thus, the direction of a positive torque tends to rotate the positive x face in a direction that is positive in the right-hand sense and the negative x face in a direction that is negative in the right-hand sense, as depicted in Fig. 2.11. This will affect the boundary conditions, as noted below.

Letting plpdx be the polar mass moment of inertia about the x-axis of the differential beam segment in Fig. 2.12, one can obtain the equation of motion by equating the resultant twisting moment on both faces of the segment to the rate of change of the segment's angular momentum about the elastic axis. This yields

9 T 926

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