Oo

2n it

and where the Kronecker symbol Sjj — 1 for i — j and Sjj — 0 for i ^ j.

1. Consider a uniform circular rod of length I, torsional rigidity GJ, and mass moment of inertia per unit length pj. The beam is clamped at the end x — 0 and it has a concentrated inertia Ic at its other end where x — I.

(a) Determine the characteristic equation that can be solved for the torsional natural frequencies for the case in which Ic — pjIt,, where C is a dimensionless parameter.

(b) Verify that the characteristic equation obtained in part (a) approaches that obtained in the text for the clamped-free uniform rod in torsion as t, approaches zero.

(c) Solve the characteristic equation obtained in part (a) for numerical values of the first four eigenvalues, a,I, i — 1.2, 3, and 4, when t; — 1.

(d) Solve the characteristic equation obtained in part (a) for the numerical value ofthe first eigenvalue, a\l, when t; — 1, 2, 4, and 8. Make a plot of the behavior of the lowest natural frequency versus the value of the concentrated inertia. Note that (a \ i )2 versus ( is the same thing in terms of dimensionless quantities.

8. Consider a clamped-free beam undergoing torsion.

(a) Prove that the free-vibration mode shapes are orthogonal, regardless of whether the beam is uniform.

(b) Given that the kinetic energy is

show that K can be written as

where M, is the generalized mass of the z'th mode and is the generalized coordinate for the z'th mode.

(c) Given that the potential energy is the internal (i.e., strain) energy, that is,

2 Jo \dxj show that P can be written as j oo

P= -^M,^,2, z /=i where &>,- is the natural frequency.

(d) Show that, for a uniform beam and for <pt(() = 1 as given in the text, Mi — plplfor all i.

9. Consider a free-free beam undergoing torsion.

(a) Given the mode shapes found in the text, find an expression for P in terms of GJ, I, and the generalized coordinates.

(b) Given the mode shapes found in the text, find an expression for K in terms of plp,l, and the time derivatives of the generalized coordinates.

(c) Substitute your results from parts (a) and (b) into Lagrange's equations and identify the resulting generalized masses.

Answers:

10. Consider a clamped-free beam undergoing bending.

(a) Prove that the free-vibration mode shapes are orthogonal, regardless of whether the beam is uniform.

(b) Given the kinetic energy as

show that K can be written as

where M, is the generalized mass of the / th mode, and is the generalized coordinate for the / th mode.

(c) Given that the potential energy is the internal (i.e., strain) energy, that is,

show that P can be written as

^ 00 z ;=i where &>, is the natural frequency.

(d) Show that, for a uniform beam and for <p(x ) as given in the text, M, = ml for all i.

11. Consider a uniform beam with the boundary conditions shown in Fig. 2.35 undergoing bending vibration.

(a) Using the relations derived in the text, plot the square of characteristic value, (ail)2, which is proportional to the fundamental frequency, versus k from 0 to 100. Check your results versus those given in Fig. 2.37.

(b) Plot the fundamental mode shape for values of k of 0.01, 0.1, 1, 10, and 100. Suggestion: Use Eq. (2.263). You can check your results for k — 1 with those given in Fig. 2.36, and your results for k = 100 will not differ very much from those of Fig. 2.38 in which k — 50.

12. Find the free-vibration frequencies and plot the mode shapes for the first five modes of a beam of length I, having bending stiffness EI and mass per unit length m, that is free at its right end and that has the sliding condition (see Fig. 2.27) at its left end. Normalize the mode shapes to have unit deflection at the free end, and determine the generalized mass for the first five modes.

Answers: a>o — 0, u>\ — 5.59332^EI/(ml4), a>2 = 30.2258^EI/(ml4), = 74.6389^EI/(mi4), co4 = 138.791/E7J(mFj; M0 = ml and Af,- = ml/4 for i — 1,2 oo. As a sample of the mode shapes, the first elastic mode is plotted in Fig. 2.41.

Figure 2.41 First elastic mode shape for sliding-free beam. (Note: the "zeroth" mode is a rigid-body translation mode.)

13. Consider the beam in Problem 12. Add to it a translational spring restraint at the left end, having spring constant k — kEI/I3. Find the first three free-vibration frequencies and mode shapes for the cases in which k takes on values of 0.01, 1, and 100. Plot the mode shapes, normalizing them to have unit deflection at the free end.

Answers'. Sample results: A plot versus k of («,/: )2 for i — 1, 2, and 3 is shown in Fig. 2.42, and the first mode shape for k — 1 is shown in Fig. 2.43.

14. Consider a beam that at its left end is clamped and at its right end is pinned with a rigid body attached to it. Let the mass moment of inertia of the rigid body be given by Ic — /unl\

(a) Find the first two free-vibration frequencies for values of // equal to 0.01, 0.1, 1, 10, and 100. Comment on the variation of the natural frequencies versus [i.

(b) Choose any normalization that is convenient, and plot the first mode shape for these same values of [i. Comment on the variation of the mode shapes versus [i.

Figure 2.42 Variation versus k of (a; t )2 for i =1,2, and 3, for a beam that is free on its right end and has a sliding boundary condition spring-restrained in translation on its left end.

Figure 2.42 Variation versus k of (a; t )2 for i =1,2, and 3, for a beam that is free on its right end and has a sliding boundary condition spring-restrained in translation on its left end.

Answers:

(a) Sample result: o>, = 1.99048^EI/(mlA) for // = 1.

(b) Sample result: The first mode shape for // = 1 is shown in Fig. 2.44.

15. Consider a uniform cantilever beam of length t, bending rigidity EI, and mass per unit length m. Until time t — 0 the beam is undeflected and at rest. At time t — 0 a transverse concentrated load of magnitude F cos(Qr) is applied at x = I.

(a) Write the generalized equations of motion.

(b) Determine the total beam displacement v(x. t ) for time t > 0.

(c) For the case when Q = 0, determine the tip displacement of the beam. Ignoring those terms that are time dependent (these would die out in a real beam because of dissipation), plot the tip displacement versus the number of mode shapes retained in the solution up to five modes. Show the static tip deflection from elementary beam theory on the plot. (This part of the problem illustrates how the modal representation can be applied to static response problems. )

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