Chapter 13

(a) Can a diagonal coefficient in a consistent mass matrix ever be negative? Explain.

Buckling of a tapered column is to be studied. Each element of the column is tapered. In which element matrices \(\left([\mathbf{k}]\right.\) or \(\left[\mathbf{k}_{g}\right]\) ) does the effect of taper appear, and how is it to be included?

Construct a 4 by 4 matrix \(\left[k_{\sigma}\right]\) for a uniform beam element, analogous to Eq. 14.2-9, by using the quadratic displacement field \(w=(1-\xi) w_{1}+\) \(\xi w_{2}+(1-\xi) \xi L\left(\theta_{1}-\theta_{2}\right) / 2\), where \(\xi=x / L\)

Consider a particle that is allowed to free-fall from at-rest initial conditions. under its own weight due to gravity. If the particle has mass- proportional damping, then the equation governing its velocity \(v\) is \(\dot{v}+\beta v=g\), where \(g\) is the acceleration due to gravity. (Remark: The same equation governs the velocity of a particle allowed to sink in a viscous fiuid where \(\beta\) is related to a fluid viscosity.) (a) Determine the analytic solution for \(v\). (b) Consider the ratio of the damped velocity to the undamped velocity (i.e., \(\left.v_{\text {undamped }}=g t\right)\) for values of \(t\) of about 1 second. Does this ratio offer guidelines on what values of \(\beta\) are permissible without excessively damping rigid-body modes?

Consider the following stiffness and mass matrices: $$ [\mathrm{K}]=\left[\begin{array}{rr} 2 & -2 \\ -2 & 5 \end{array}\right] \quad[\mathrm{M}]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$Exact eigenvalues and eigenvectors for axial vibration are \(\lambda_{1}=1, \lambda_{2}=6\), \(\\{\overline{\mathbf{D}}\\}_{1}=[2,1]^{T}\), and \(\\{\overline{\mathbf{D}}\\}_{2}=[1 \quad-2\rfloor^{T} .\) Consider approximate eigenvectors \(\left.\begin{array}{|ll}\text { [1.7 } 1.0\end{array}\right]^{T}\) and \([1.2-2.0]^{T}\) and show that the Rayleigh quotient provides an accurate estimate of \(\lambda_{1}\) and \(\lambda_{2}\)

culation \(M=E I[\mathbf{B}\\}\\{\mathrm{d}\\}\), where \([\mathbf{B}]\) is based on a cubic field and \(\\{\mathrm{d}\\}=\) \(\left[\begin{array}{llll}0 & 0 & w_{2} & \theta_{2}\end{array}\right]^{T}\). Compare this \(M\) with that obtained by statics; that is, \(M=0.1 L-P w_{2}\), where \(P=-30.0 \mathrm{~N}\) in this case

Consider axial vibrations of a uniform bar of length \(L\) and mass \(m=\rho A L\), free at one end and fixed at the other. Using two-node bar elements, model the bar first by one element, then by two elements of equal length \(L / 2\). In each case, compute the lowest natural frequency using (a) the consistent mass matrix [m]. (b) the lumped mass matrix \([\mathrm{m}]\). (c) the average mass matrix \(([\mathrm{m}]+[\mathrm{m}]) / 2\). The exact lowest natural frequency is \(\omega_{1}=(\pi / 2 L) \sqrt{E / \rho}\).

The stiffness, consistent mass, and optimally lumped mass matrices for the unsupported, uniform, three-node quadratic bar element shown are $$ \frac{A E}{3 L}\left[\begin{array}{rrr} 7 & -8 & 1 \\ -8 & 16 & -8 \\ 1 & -8 & 7 \end{array}\right], \frac{\rho A L}{30}\left[\begin{array}{rrr} 4 & 2 & -1 \\ 2 & 16 & 2 \\ -1 & 2 & 4 \end{array}\right], \frac{\rho A L}{6}\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ (a) For axial vibration, determine the three natural frequencies and mode shapes using the consistent mass matrix. (b) Repeat part (a) using the optimally lumped mass matrix. (c) What is the physical significance of the lowest frequency and mode for parts (a) and (b)? (d) The exact frequencies of an unsupported continuous bar of length \(L\) are \(\omega_{n}=(n \pi / L) \sqrt{E / \rho} ; n=0,1,2, \ldots .\) What are the percentage errors of the frequencies computed in parts (a) and (b)? Problem 13.19

Show that the mode acceleration method reduces to the mode displacement method if the structure moves freely-that is, with \(\left\\{\mathrm{R}^{e \times t}\right\\}=\\{0\\}\).

Many methods of solving large eigenproblems require factoring either the stiffness matrix or a combination of the stiffness and mass matrices (e.g.) the determinant search and subspace iteration methods). Factoring requires approximately \(n_{\mathrm{eq}} b^{2} / 2\) operations (i.e., multiplications) where \(n_{\mathrm{eq}}\) is the number of equations and \(b\) is the semibandwidth. For full matrices, the number of operations is about \(n_{e q}^{3} / 6 .\) Consider a system of 5000 equations with \(b=500\). If this system of equations is partitioned into \(m\) master and \(s\) slave d.o.f., what must \(m\) be so that factoring the condensed (full) system is no more expensive than factoring the original (banded) system? What if \(b=100\) instead?

A massless and flexible string of length \(2 L\) hangs from the ceiling. It carries two particles, each of mass \(m\), one at the middle and the other at the lower end. (a) What is the horizontal deflection of a small horizontal force \(Q\) applied to the lower end? (b) What are the natural frequencies of vibration and the mode shapes?

The string shown is under tension \(T\) and has mass \(\rho\) per unit length. Use. [M] and \(\left[\mathbf{K}_{\sigma}\right]\) matrices associated with a cubic lateral-displacement field. Omit the conventional stiffness matrix \([\mathbf{K}]\). Solve for the natural frequencies and mode shapes of small- displacement lateral vibrations. (The exact fundamental frequency is \(\omega_{1}^{2}=\pi^{2} T / 4 \rho a^{2}\).) (a) Use one element. Nonzero d.o.f. are then \(\theta_{1}\) and \(\theta_{2}\). (b) Use two elements and impose symmetry about the center. Nonzero d.o.f. to be used are then \(\theta_{\text {end }}\) and \(w_{\text {center }}\).

The forward and backward Euler direct integration methods are defined by \(\begin{array}{ll}\\{\mathbf{D}\\}_{n+1}=\\{\mathbf{D}\\}_{n}+\Delta t\\{\dot{\mathbf{D}}\\}_{n} & \text { forward Euler } \\\ \left.\\{\mathbf{D}\\}_{n+1}=\\{\mathbf{D}\\}_{n}+\Delta t \dot{\mathbf{D}}\right\\}_{n+1} & & \text { backward Euler }\end{array}\) Are-these methods explicit or implicit?

Model a simply supported beam by a single element. Let \(L=1.0 \mathrm{~m}, A=\) \(0.0002 \mathrm{~m}^{2}, E I=300.0 \mathrm{~N} \cdot \mathrm{m}^{2}\), and \(\rho=2100.0 \mathrm{~kg} / \mathrm{m}^{3}\). Impose symmetry (and reduce the problem to a single d.o.f.) by setting \(\theta_{2}=-\theta_{1}\). (a) Determine the fundamental frequency \(\omega_{1}\) if there is no axial force. (b) Determine the axial force that makes the frequency \(347 \mathrm{rad} / \mathrm{sec}\). (c) Determine the frequency if the axial force is \(1200 \mathrm{~N}\) in compression.

A particle of unit mass is supported by a spring of unit stiffness. There is no damping and no external load. Thus \(k=m=\omega=1\). At time \(t=0\), the particle has zero displacement, zero acceleration, but unit velocity. Use. the central-difference method, Eq. 13, \(10-5\), to compute displacement versus time over five time steps. Use a \(\Delta t\) of (a) \(1.0\), (b) \(\sqrt{2.0}\), (c) \(2.0\), and (d)

A particle of unit mass is supported by a spring of unit stiffness. There is no damping and no external load. Thus \(k=m=\omega=1\). At time \(t=0\), the particle has zero displacement, zero acceleration, but unit velocity. Use. the central-difference method, Eq. 13, \(10-5\), to compute displacement versus time over five time steps. Use a \(\Delta t\) of (a) \(1.0\), (b) \(\sqrt{2.0}\), (c) \(2.0\), and (d)

Consider Problem \(13.52\) again, in which the central-difference method (Eq. 13.10-5) is applied to a spring-mass system for which \(k=m=\omega=1\). Now use \(\Delta t=\sqrt{3.96}\) and start the algorithm using \(u_{0}=0\) and \(u_{-1}=-1\). Follow the motion for at least ten cycles, and observe that the computed amplitude displays "beating" but no net growth.

The uncoupled equations produced by a modal analysis (Section 13.6) have a lower \(\omega_{\max }\) than \(\omega_{\max }\) of the full system. Hence, in integrating the uncoupled equations, what are the relative merits of explicit and implicit methods? How does the specific choice of modal method affect your answer?