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Chapter 4: Problem 4

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.

Short Answer

Expert verified
Question: Define the characteristic impedance for torsional waves on a round rod or tube and express the power (4.5 * 13) in terms of this impedance. Answer: The characteristic impedance (Z) for torsional waves in a round rod or tube is defined as \(Z = \frac{T}{\omega} = J\frac{\partial^2\theta}{\partial t^2}\frac{1}{\frac{\partial\theta}{\partial t}}\). The power (4.5 * 13) can be expressed in terms of this impedance as \(P = \frac{Z(\frac{\partial\theta}{\partial t})^2}{J}\).

Step by step solution

01

Understanding Torsional Waves

Torsional waves are mechanical waves that travel through a material by twisting it around its axis. In the case of a round rod or tube, the axis is the line passing through the center of the rod, perpendicular to its circular cross-section. The particles in the material oscillate in a circle around the axis, resulting in a twisting motion.
02

Derive the Characteristic Impedance for Torsional Waves

The characteristic impedance (Z) of torsional waves in a round rod or tube can be defined as the ratio of the angular velocity (ω) of the torsional wave to the torsional displacement (θ). To derive the characteristic impedance, we need to consider the torsional wave equation: \(T = J\frac{\partial^2\theta}{\partial t^2}\) Where T is the torque applied on the round rod or tube, J is the polar moment of inertia of the rod's cross-section, and θ is the torsional displacement. Now, we can express the angular velocity (ω) of the torsional wave as: \(\omega = \frac{\partial\theta}{\partial t}\) By dividing both sides of the torsional wave equation by ω, we get: \(\frac{T}{\omega} = J\frac{\partial^2\theta}{\partial t^2}\frac{1}{\frac{\partial\theta}{\partial t}}\) Thus, the characteristic impedance (Z) can be defined as: \(Z = \frac{T}{\omega} = J\frac{\partial^2\theta}{\partial t^2}\frac{1}{\frac{\partial\theta}{\partial t}}\)
03

Expressing Power in Terms of the Characteristic Impedance

The power transmitted by a torsional wave can be calculated using the formula: \(P = T\omega\) Since we want to express the power in terms of the characteristic impedance (Z), we can substitute the value of T from our previous step: \(P = \frac{Z\omega^2}{J} = \frac{Z(\frac{\partial\theta}{\partial t})^2}{J}\) Now, we have a given power of 4.5 * 13, so let's plug this value into the equation: \(P = 4.5*13 = \frac{Z(\frac{\partial\theta}{\partial t})^2}{J}\) We don't have the specific values for Z, J, and ω in this problem. However, the important part of this exercise was to express the power (4.5 * 13) in terms of the characteristic impedance for torsional waves on a round rod or tube, which we have achieved with the above equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Torsional Wave Equation
The study of waves in various forms is an essential part of physics and engineering. Among them, torsional waves represent a type of mechanical wave that propagates through twisting motion around an axis. This type of motion is particularly relevant when analyzing the behavior of solid structures like rods or tubes.

When delving into torsional waves, the foundational piece is the torsional wave equation, which mathematically describes how the wave propagates through a medium. This equation is a crucial tool for understanding how energy is transferred in the form of torque, \(T\), within the structure. It is expressed as:
\[T = J\frac{\partial^2\theta}{\partial t^2}\]
Here, \(J\) represents the polar moment of inertia of the cross-section, and \(\theta\) is the torsional displacement, or the angle through which a point in the medium has rotated from its original position. By solving this equation, one can predict the behavior of torsional waves under various conditions and design structures to withstand them.
Torsional Wave Power Transmission
The concept of power transmission in the context of torsional waves encompasses how energy is transferred by these waves from one point to another within a material. The ability to calculate and understand this power is indispensable in areas like mechanical engineering, where it helps in designing and evaluating the performance of drive shafts and other rotating machinery.

The power transmitted by a torsional wave can be calculated using the formula:
\[P = T\omega\]
where \(P\) is the power, \(T\) is the torque, and \(\omega\) is the angular velocity, the rate at which the angle \(\theta\) changes with time. This relationship is central to understanding how efficiently a system can transmit rotary power and is a primary concern when ensuring that structures can handle the operating stresses they are exposed to.
Angular Velocity in Torsional Waves
Angular velocity plays a pivotal role in the analysis of torsional waves. It describes the speed at which a point within the medium rotates about the axis of the wave spread. In mathematical terms, angular velocity, denoted as \(\omega\), is defined as the first derivative of the torsional displacement \(\theta\) with respect to time:
\[\omega = \frac{\partial\theta}{\partial t}\]
This concept is intimately connected to the energy carried by the torsional wave; the faster the angular velocity, the higher the energy involved. Understanding angular velocity is key to grasping how torsional waves affect materials and helps in predicting the stresses that parts under torsion will experience. This knowledge is particularly important in the design of mechanical systems where rotational movements occur, ensuring that each component can withstand the expected operational demands.

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Most popular questions from this chapter

Show that in the first-order approximation, the relative shift in period \(T=2 \pi / \omega\) due to the mass of the spring is $$ \frac{T_{1}-T_{0}}{T_{0}}=\frac{1}{6} \frac{m}{M}=\frac{\pi^{2}}{6}\left(\frac{t_{1}}{T_{0}}\right)^{2} $$ where \(t_{l}=2 l / c_{s}\) is the time required for a longitudinal wave in the spring to travel from \(M\) to the point of support and back again. Thus, show that the familiar limit of negligible spring mass is equivalent to the limit \(t_{0}^{2} \ll T_{0}{ }^{2}\).

Show that a longitudinal wave on a slender rod exerts an average force $$ \bar{F}_{x}=\bar{E}_{1}=\frac{1}{c} \bar{P}=c g_{x} $$ when it is absorbed. Hint: See Sec. 1.11.

A slender rod of area \(S_{1}\), density \(\rho_{1}\), and Young's modulus \(Y_{1}\) is attached to a second rod of area \(S_{2}\), density \(\rho_{2}\), and Young's modulus \(J_{2}\). Longitudinal waves of frequency \(\omega\) travel along the first rod and are partly reflected and partly transmitted at the discontinuity. Using the procedure folloned in Sec. 1.9, find expressions for the complex amplitude reflection and transmission coefficients and the related power reflection and transmission coefficients.

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?

An exponential horn whose area is given by (4.3.3) makes an excellent impedance transformer. Let us investigate the properties of such a horn whose length is an integral number of half-wavelengths, so that \(k_{1} l=n \pi\). In the notation of Sec. \(4.2\), the wave on the horn is $$ \xi=e^{-\alpha x}\left(\breve{A}_{4} e^{-j \boldsymbol{\kappa}_{1} \boldsymbol{z}}+\breve{A}_{-} e^{j \boldsymbol{\mu}_{1} \boldsymbol{x}}\right) e^{j \omega t} . $$ Compute the characteristic impedance \(\breve{Z}(x)=F / \xi\) at any position \(x\) on the horn. Set it equal to the load impedance \(\bar{Z}_{t}\) at \(x=l=\pi / n k_{1}\) and show that the impedance at \(x=0\) is then \(\breve{Z}(0)=\left(S_{0} / S_{l}\right) \breve{Z}_{l}\), where \(S_{0}\) is the area of the horn at \(x=0\) and \(S_{l}\) is the area at \(x=l\).
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