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Since the dependencies on x and t have been separated across the equality, each side must equal a constant, say a4. The resulting ordinary differential equations then become

For a ^ 0, the general solution to the second (the time-dependent) equation can be written as in the cases for the string and beam torsion, namely

where it is clear from the second ofEqs. (2.213) that

For a/0, the general solution to the spatially dependent equation can be obtained by presuming a solution of the form

Substitution of this assumed form into the fourth-order differential equation for X(x) yields

which can be factored to

which indicates a general solution of the form

X(x) — C i exp(iax) + Ci exp(—i ax) + C 3 exp(ax) + C4 exp(—ax). (2.219)

Rewriting the exponential functions as trigonometric and hyperbolic sine and cosine functions yields an alternative form of the general solution as

X(x) — D\ sin(co) + /J2 cos(«-V) + />, sinh(ax) + /J4 cosh(ax). (2.220)

Eventual determination of the constants D¡(i =1,2,3, and 4) and a will require specification of appropriate boundary conditions. To facilitate this procedure this last solution form can be rearranged to provide in some cases a slight advantage in the algebra, so that

X(x) — E\ [sin(«x) + sinh(«_v)] + /:2[sin(«_v) — sinh(«_v)]

+£,3[cos(a.v) + cosh(ax)] + £4[cos(ax) — cosh(ax)].

To complete the solution the constants A and B can be determined from the initial deflection and rate of deflection of the beam. The remaining constants, C, , D¡, or E¡ (i — 1, 2, 3, and 4), can be evaluated from the boundary conditions, which must be imposed at each end of the beam. As was true for torsion, the very important special case of a — 0 is connected with rigid-body modes for beam bending and is addressed in more detail in Section 2.3.4.

2.3.3 Boundary Conditions

For the beam bending problem it is necessary to impose two boundary conditions at each end of the beam. A boundary condition can be any linear, homogeneous relation involving the beam deflection and one or more of its partial derivatives. Although it is not a mathematical requirement, the particular combination of conditions to be specified at a beam end should represent a physically realizable constraint. The various derivatives of the beam deflection can be associated with particular beam states at any arbitrary point along

0 0