Chapter 10: Problem 31
Solve problem \(10.30\) if the string is released from the given configuration with the velocity \(v_{0}(x)=c_{0} w_{0}(x)\), where \(c_{0}\) is a constant.
Chapter 10: Problem 31
Solve problem \(10.30\) if the string is released from the given configuration with the velocity \(v_{0}(x)=c_{0} w_{0}(x)\), where \(c_{0}\) is a constant.
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Get started for freeConsider the free flexural vibrations of a simply supported uniform elastic beam of length \(L\), bending stiffness \(E I\), radius of gyration \(r_{G}\) and mass per unit length \(m\), and let it be represented mathematically using Rayleigh beam theory. (a) Establish the modal boundary conditions for the structure. \((b)\) Derive the frequency equation for the beam. (c) Determine the first three natural frequencies and modal functions for a beam with \(r_{G} / L=0.1\). Plot the modal functions.
Consider the free flexural vibrations of a simply supported uniform elastic beamcolumn of length \(L\), bending stiffness \(E I\), and mass per unit length \(m\) that is subjected to a constant compressive end load \(P_{0}\). (See Example 9.12.) Determine the first three natural frequencies and modal functions for the case where the applied load is half the static buckling load. Plot the modal functions.
Consider the free flexural vibrations of a uniform elastic beam of length \(L\), bending stiffness \(E I\) and mass per unit length \(m\) that is clamped at both edges, as shown, and is represented mathematically using Euler-Bernoulli theory. (a) Establish the modal boundary conditions for the structure. \((b)\) Derive the frequency equation for the beam. \((c)\) Determine the first three natural frequencies and modal functions. Plot the modal functions. Fig. \(\mathbf{P} 10.13\) Fig. P10.12 \(E I, m\)
A simply supported Euler-Bernoulli beam of length \(L\), bending stiffness \(E I\) and mass per unit length \(m\) is deflected by a static load and held in the configuration \(w_{0}(x)=a x\left(L^{3}-2 L x^{2}+x^{3}\right)\), where \(a=q_{0} / 24 E I\) is a constant. Determine the amplitudes and phase angles for the free vibration response of the structure if it is released from rest when in this configuration.
Consider free torsional vibration of a uniform circular elastic rod of length \(L\), torsional stiffness \(G J\) and mass per unit length \(m\), that is free at its left end and fixed at its right end. (a) Establish the modal boundary conditions for the structure. (b) Derive the frequency equation for the rod. (c) Determine the first three natural frequencies and modal functions. Plot the first three modes.
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