7.B8 Boundary conditions: 1. 2. a2w (0) = 0 ax2 3. 4. Applying the first two boundarv conditions to equation for W results in the following equation: W C[cos k x + cosh k x] 7 E[sin k x + sinh k x] Applying boundary conditions 3. and 4. to this equation results in the following two simultaneous equations: C[-cos k + cosh k] + E[-sin k + sinh kl = 0 C[sin k + sinh kl + E[-cos k + cosh k] = 0 In order to obtain a nontrivial solution the determinent of these equations must be equal to zero, therefore: [-co's k + cosh kl 2 - [sinh 2 k - sin 2 kl This equation reduces to the following equation for frequency: cos k cosh k = 1 The solution to this equation possesses a double root at k = 0, which corresponds to the two rigid body modes--vertical translation and rotation about the space colony's center of gravity. The lowest natural frequency for bending is given by k = 4.73 which is obtained from the approximate solution: w then becomes: k. " (i+l/2)n l kl = ~ = 4. 73 w =22.37a
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