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Chapter 10: Problem 19

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.

Short Answer

Expert verified
Question: Determine the first three natural frequencies and modal functions for a simply supported uniform elastic beam-column subjected to a compressive end load, \(P_0\), where the applied load is half the static buckling load.

Step by step solution

01

Calculate the static buckling load

The static buckling load, \(P_{cr}\), can be calculated using the Euler formula for a simply supported beam: \(P_{cr} = \dfrac{\pi^2 EI}{(L)^2}\)
02

Determine the compressive end load, \(P_0\)

Since the applied load is half the static buckling load, we can calculate the compressive end load as: \(P_0 = \dfrac{1}{2} P_{cr}\)
03

Write down the governing equation

The governing equation for free vibrations of a beam-column can be written as: \(E I \dfrac{\text{d} ^4 w}{\text{d} x^4} - P_0 \dfrac{\text{d} ^2 w}{\text{d} x^2} + m \dfrac{\text{d} ^2 w}{\text{d} t^2} = 0\)
04

Apply boundary conditions and derive the equation for natural frequencies

For a simply supported beam, the boundary conditions are: 1. \(w(0) = 0\) 2. \(\dfrac{\text{d}^2 w}{\text{d}x^2}(0) = 0\) 3. \(w(L) = 0\) 4. \(\dfrac{\text{d}^2 w}{\text{d}x^2}(L) = 0\) By applying these boundary conditions, we can transform the governing equation into an eigenvalue problem and derive the equation for natural frequencies: \(\tan(\dfrac{\alpha L}{2}) = \dfrac{\alpha^2 L^2}{4P} - \dfrac{P}{\alpha^2 L^2}\) where \(\alpha=\dfrac{\omega L}{\sqrt{EI/m}}\), \(P = \dfrac{2P_0}{\pi^2} \ge 1\), and \(\omega=\dfrac{2 \pi}{T}\) is the circular natural frequency with period \(T\).
05

Find the first three natural frequencies

By solving the equation for natural frequencies \(\tan(\dfrac{\alpha L}{2}) = \dfrac{\alpha^2 L^2}{4P} - \dfrac{P}{\alpha^2 L^2}\), numerically or graphically, for the first three positive roots \(\alpha_s\), \(s=1,2,3\), we can obtain the corresponding natural frequencies as follows: \(\omega_s = \dfrac{\alpha_s \sqrt{EI/m}}{L}\), \(s=1,2,3\)
06

Calculate the modal functions

Using the values of \(\alpha_s\) obtained in the previous step, the modal functions \(w_s(x)\) can be calculated as follows: \(w_s(x) = C_1 \cosh(\alpha_s x) + C_2 \sinh(\alpha_s x) + C_3 \cos(\alpha_s x) + C_4 \sin(\alpha_s x)\), where \(C_1\), \(C_2\), \(C_3\), and \(C_4\) are constants that can be found using the boundary conditions.
07

Plot the modal functions

Finally, plot the modal functions \(w_1(x)\), \(w_2(x)\), and \(w_3(x)\). This can be done by using any mathematical software or by drawing them manually, keeping in mind the boundary conditions and the shape of the trigonometric functions involved.

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Most popular questions from this chapter

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.

Consider 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.

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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