Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 647

10.26 Use the central difference method to solve Problem 10.25. Find the critical time step.
10.27 Use the Houbolt method to solve Problem 10.25.
10.28 Use the Newmark method to solve Problem 10.25.
10.29 Use the Wilson- θ method to solve Problem 10.25.

Ritz Vectors
10.30 Use the Ritz vector method to solve Problem 10.25. Assume the damping matrix is
proportional to the mass matrix, and is equal to 0.000625 M . Condense the system
equations from 6 DOF to 1, 2, 3, 4, 5, and 6 DOF, respectively. Compare the displace-
ment at c in the X direction with the results of Problems 10.27, 10.28, or 10.29.
10.31 The amplitude of the time-varying load acting on the fixed-pinned beam of Chapter
7, Fig. 7.1 varies linearly over the span. Let the time variation be sinusoidal so that
the external load may be written as F represents the spatial

distribution. The beam is 1600 mm long and has a square 60 mm

( s ) sin t , where F ( s )

× 60 mm cross-

section. The density of the beam is 7860 kg/m 3

and its modulus of elasticity is 200

GPa. Also, p 0 = 2 kN/m.

(a) Assemble the global structural matrices and apply the boundary conditions to
obtain 7

7 global stiffness and consistent mass matrices for the beam with 4
elements of equal length.
(b) Find the natural frequencies and the mode shapes of the beam using the matrices
of part (a).
(c) Let the loading frequency

×

be equal to one-third the lowest natural frequency
of the beam. Calculate the displacement of the midpoint of the beam using the
modal properties found in part (b).
(d) Determine 3 3 stiffness and mass matrices for the beam using 3 Ritz vectors.

(e) Calculate the 3 natural frequencies corresponding to the matrices found in part
(d) and compare with the results of part (b).
(f ) Calculate the displacement of the midpoint of the beam using the modal prop-
erties of the matrices of part (d) and compare with the results of part (c).

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