∗18.3 Determine the natural vibration frequencies and modes of a simply supported uniform beam, idealized as an assemblage of two finite elements (Fig. P18.3), using (a) the consistent-mass matrix, and (b) the lumped-mass matrix. Compare these results with the exact solutions obtained in Example 17.1.*Denotes that a computer is necessary to solve this

problem.

18.4 Determine the natural vibration frequencies and modes of a uniform beam clamped at both ends, idealized as an assemblage of two finite elements (Fig. P18.4), using (a) the consistent mass matrix, and (b) the lumped-mass matrix. Compare these results with the exact solutions obtained by solving Problem 17.1.]

18.5 Determine the natural vibration frequencies and modes of a uniform beam clamped at one end and simply supported at the other, idealized as an assemblage of two finite elements (Fig. P18.5), using (a) the consistent-mass matrix, and (b) the lumped-mass matrix. Compare these results with the exact solutions obtained by solving Problem 17.2.

∗18.6 Figure P18.6 shows a one-story, one-bay frame with mass per unit length and second moment of cross-sectional area given for each member. The frame, idealized as an assemblage of three finite elements, has the three DOFs shown if axial deformations are neglected in all elements. (a) Using influence coefficients, formulate the stiffness matrix and the consistent-mass matrix. Express these matrices in terms of m, E I, and h. (b) Determine the natural vibration frequencies and modes of the frame; express rotations in terms of h. Sketch the modes showing translations and rotations of the nodes.

18.8 Repeat part (a) of Problem 18.6 starting with the stiffness and mass matrices for each element and using the direct assembly procedure of Section 18.5.

4.7 Using the classical method for solving differential equations, derive Eq. (4.4.2), which describes the response of an

undamped SDF system to a linearly increasing force; the initial conditions are u(0) = ˙u(0) = 0.

4.9 (a) Determine the maximum response of a damped SDF system to a step force.

b) Plot the maximum response as a function of the damping ratio.

4.10 The deformation response of an undamped SDF system to a step force having finite rise time is given by Eqs. (4.5.2) and (4.5.4). Derive these results using Duhamel’s integral.

4.14 Determine the response of an undamped system to a rectangular pulse force of amplitude po

and duration td by considering the pulse as the superposition of two step excitations (Fig. 4.6.2).

4.15 Using Duhamel’s integral, determine the response of an undamped system to a rectangular pulse force of amplitude po and duration td .