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Duke Vuong MEAE 7010 Homework # 4 5. Suppose that the uniform string considered in Section 5.2 is unrestrained from transverse motion at the end x = 0, and is attached at the end x = l to a yielding support of modulus k, the ends o the string being constrained against appreciable movement parallel to the axis of rotation. Show that the nth critical speed is given by n n T l where cot n n and T . kl Solution: From the differential equation (9) of the form, d2y y 0 dx 2 with the general solution that satisfies the end condition y (0) 0 (at x = 0) y C sin x . The second end (x = l) is attached to a yielding support of modulus k y (l ) ly (l ) with y ( x) C cos x . We have sin l l cos l . With the dimensionless parameter , of the form l The characteristic values of are given by n n2 l2 Per equation (8), the constant parameter is 2 T . Thus, the corresponding critical speeds are given by n n T l . 22. Reduce each of the following differential equations to the standard form d dy p q r y 0 dx dx (a) x d2y dy 2 x y 0 2 dx dx Solution: Setting a pe a10 dx q a2 p a0 r a3 p. a0 With a0 x a1 2 a2 x a3 1 . We have p x2 r x. q x2 Therefore, the reduced equation has the form d 2 dy 2 x x x y 0 dx dx