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Chapter 4: Problem 4

A long slender rod is vibrating in its fundamental longitudinal mode. It is desired to raise its frequency 1 percent by drilling holes of equal depth in each end of the rod. The holes are to have an area equal to one-fourth the cross-sectional area of the rod. How deep should the holes be drilled? How will their presence affect the first few overtones?

Short Answer

Expert verified
Answer: The holes should be drilled to a depth of 0.005 times the rod's length to raise the fundamental frequency by 1 percent.

Step by step solution

01

Define the problem variables

Let's define the given parameters: - \(f_0\): original fundamental frequency of the rod - \(A\): cross-sectional area of the rod - \(A_h\): area of the holes, which is one-fourth of the rod's area (\(A_h = \frac{A}{4}\)) - \(h\): depth of the holes to be determined
02

Calculate the new frequency

To raise the fundamental frequency by 1 percent, we need to find the new frequency \(f_1\). Since we want a 1% increase, we have: $$f_1 = f_0 (1 + 0.01) = 1.01 f_0$$
03

Relate the new frequency to the rod's geometry and properties

With the holes drilled, the effective length of the rod decreases. Let \(L\) be the original length of the rod and \(L'\) be the effective length after drilling the holes. The relation between the original frequency \(f_0\), new frequency \(f_1\) and effective length \(L'\) can be expressed as: $$\frac{f_1}{f_0} = \frac{L}{L'}$$ Then, we can substitute the value of \(f_1\) from Step 2: $$\frac{1.01 f_0}{f_0} = \frac{L}{L'}$$
04

Find the effective length \(L'\)

Solving the equation above for \(L'\): $$L' = \frac{L}{1.01}$$
05

Relate the hole's depth to the effective length

The total length of the rod is now reduced by the sum of the depths of both holes. The holes have the same depth, so we can represent this situation as: $$L' = L - 2h$$
06

Solve for the depth of the holes \(h\)

Now, we'll substitute the expression for \(L'\) from Step 4 into the equation from Step 5: $$\frac{L}{1.01} = L - 2h$$ Rearranging the equation to solve for the hole's depth \(h\): $$ h = \frac{L - \frac{L}{1.01}}{2} = \frac{0.01L}{2} = 0.005L$$ So, the holes should be drilled to a depth of 0.005 times the rod's length to raise the fundamental frequency by 1 percent.
07

Discuss the effect on the first few overtones

The presence of the holes affects the fundamental frequency and the overtone frequencies similarly because they contribute to the overall stiffness of the rod. The first few overtones will also experience an increase in their frequencies, as the effective length of the rod is reduced. The exact increase in the frequency for each overtone depends on the mode shape, boundary conditions, and material properties of the rod. However, we can expect a similar 1 percent increase in their frequencies.

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Most popular questions from this chapter

Show that the frequencies for the transverse vibrations of a bar of length \(l\) clamped at one end and free at the other may be found from the equation \(\cosh m \cos m=-1\), in which \(m=n l .\) Verify that the smallest root is \(m_{1}=1.875\).

Consider a medium in which \(c_{\text {phase }}=A \omega^{n}\), where \(A\) and \(n\) are constants. Show that \(c_{\text {eroup }}=c_{\text {pasae }} /(1-n)\). For what values of \(n\) is the dispersion normal? Anomalous?

Two long waveguides having different characteristic impedances \(Z_{1}\) and \(Z_{2}\) are joined by a quarter-wave section having the characteristic impedance \(\left(Z_{1} Z_{2}\right)^{1 / 2}\). A wave \(\xi_{1}=\) \(A_{1} e^{i\left(\omega t-x_{1} x\right)}\) is traveling toward the coupling section. Assume that there is a reflected wave \(\xi_{1}=\check{B}_{1} e^{i\left(\omega t+x_{1} x\right)}\), that there are waves going in both directions in the coupling section, and that there is a wave \(\xi_{2}=X_{2} e^{i\left(\omega t-x_{1} x\right)}\) transmitted into the other waveguide. By satisfying boundary conditions at the two junctions, show that \(\breve{B}_{1}=0\) and that \(\breve{A}_{2}=A_{1}\left(Z_{1} / Z_{2}\right)^{1 / 2}\), which is the value one may get very easily from (4.2.10) by power considerations alone.

Show how to define a characteristic impedance for torsional waves on a round rod or tube and express the power \((4.5 .13)\) in terms of this impedance.

A positive-going sinusoidal wave travels along an exponential horn. Show that the power transmitted is the same at all points.
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