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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

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