Find the first three natural frequencies (you may use software to solve equation) of vibration of a beam, simply supported at one end and free at x=L, with EI = const and \(\rho\) = const
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The equation of motion for a beam can be derived using the Euler-Bernoulli beam theory. Show moreā¦
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Deal with transverse vibrations of the uniform beam of this section, but with various end conditions. In each case show that the natural frequencies are given by the formula in Eq. (19), with $\left\{\beta_{n}\right\}_{1}^{\infty}$ being the positive roots of the given frequency equation. Recall that $y=y^{\prime}=0$ at a fixed end, $y=y^{\prime \prime}=0$ at a hinged end, and $y^{\prime \prime}=y^{(3)}=0$ at a free end (primes denote derivatives with respect to $x$ ). The ends at $x=0$ and $x=L$ are both hinged; the frequency equation is $\sin x=0$, so that $\beta_{n}=n \pi$.
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Find the characteristic frequencies for sound vibration in a rectangular box (say a room) of sides $a, b, c$. Hint : Separate the wave equation in three dimensions in rectangular coordinates. This problem is like Problem 3 but for three dimensions instead of two. Discuss degeneracy (see Problem 3).
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Derive the frequency equation for the transverse vibration of a uniform beam resting on the springs at both ends, as shown in the figure below.The springs can deflect vertically only and the beam is in the horizontal position p.E,A,1 For this problem, you do not have to find the determinant that would yield the frequency equation. You only need to apply the four boundary equations and get the four C,C2,C3,and c4. It is the determinant of this 4 x 4 matrix that yields the frequency equation.
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