Page 351 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

338 Introduction to the Finite Element Method Chap. 10

10-2 A tapered rod is modeled as two uniform sections, as shown in Fig. PlO-2, where
EA^ = 2£>42 and = 2 m 2 - Determine the two natural frequencies in longitudinal
vibration.

i/z i / Z ----^ Figure PlO-2.
10-3 Set up the equation for the free-free vibration of a uniform rod of length /, using
three axial elements of length //3 each.
10-4 Assuming linear variation for the twist of a uniform shaft, determine the finite
element stiffness and mass matrices for the torsional problem. The problem is
identical to that of the axial vibration.
10-5 Using two equal elements, determine the first two natural frequencies of a fixed-free
shaft in torsional oscillation.
10-6 Using two uniform sections in torsional vibration, describe the finite element relation­
ship to the 2-DOF lumped-mass torsional system.
10-7 Figure PlO-7 shows a conical tube of constant thickness fixed at the large end and
free at the other end. Using one element, determine the equation for its longitudinal
vibration.

Figure PlO-7.

10-8 Treat the tube of Fig. PlO-7 as a two-element problem of equal length in longitudinal
vibration.
10-9 Determine the equation for the tube of Fig. PlO-7 in torsional vibration using
(a) two elements and (b) A^-stepped uniform elements.
10-10 The simple frame of Fig. PlO-10 has pinned joints. Determine its stiffness matrix.

Figure PlO-10.
10-11 In the pinned truss shown in Fig. PlO-11, pin 3 is fixed. The pin at 1 is free to move in
a vertical guide, and the pin at 2 can only move along the horizontal guide. If a force

P is applied at pin 2 as shown, determine U2 and ¿j in terms of P. Calculate all