Question

Consider free torsional vibrations of a uniform circular elastic rod of length \(L\), torsional stiffiness \(G J\) and mass per unit length \(m\), that is free at its left end and embedded in an elastic wall of stiffness \(k_{\theta}=G J / L\) at its right end, as shown. (a) Establish the modal boundary conditions for the structure. (Hint: See Problem 9.8). (b) Derive the frequency equation for the rod. (c) Determine the first three natural frequencies and modal functions. Plot the first three modes.

Short answer

Question: Establish the modal boundary conditions, derive the frequency equation, and determine the first three natural frequencies and modal functions for a uniform circular elastic rod with its left end free and right end embedded in an elastic wall. Answer: The modal boundary conditions for the structure are: 1) At the left end (free): \(\frac{\partial \Theta(0,t)}{\partial x} = 0\) and \(\frac{\partial^2 \Theta(0,t)}{\partial x^2} = 0\) 2) At the right end (embedded in elastic wall): \(k_{\theta}\Theta(L,t) = -\frac{GJ}{mL}\frac{\partial^3 \Theta(L,t)}{\partial x^3}\) The frequency equation for the rod can be obtained by solving the fourth-order differential equation: \(\frac{\partial^4 U(x)}{\partial x^4} - \frac{m\omega^2}{GJ}U(x) = 0\) with the given boundary conditions. To find the first three natural frequencies and corresponding modal functions, solve the frequency equation for the smallest three non-zero values of \(\omega\), and substitute these values back into the general solution. Lastly, plot the first three modes by graphing the corresponding modal functions \(U(x)\). Note that computational tools/software may be necessary to obtain numerical solutions and plots for the natural frequencies and modal functions.

Step-by-step solution

Establish the modal boundary conditions

For free torsional vibrations of a uniform circular elastic rod, we can use the torsional wave equation: \begin{equation} \frac{\partial^2 \Theta(x,t)}{\partial t^2} + \frac{GJ}{m}\frac{\partial^4 \Theta(x,t)}{\partial x^4} = 0 \end{equation} Now, let's analyze the given boundary conditions: a) The left end of the rod is free, which results in the following conditions: \begin{equation} \frac{\partial \Theta(0,t)}{\partial x} = 0 \quad \text{and} \quad \frac{\partial^2 \Theta(0,t)}{\partial x^2} = 0 \end{equation} b) The right end of the rod is embedded in an elastic wall with stiffness \(k_{\theta}=\frac{GJ}{L}\). This gives the equation: \begin{equation} k_{\theta}\Theta(L,t) = -\frac{GJ}{mL}\frac{\partial^3 \Theta(L,t)}{\partial x^3} \end{equation} These are the modal boundary conditions for the structure.

Derive the frequency equation

To derive the frequency equation, we first need to assume a separable solution of the form \(\Theta(x,t) = U(x)e^{i\omega t}\), where \(U(x)\) is the spatial part and \(e^{i\omega t}\) is the temporal part. Plugging this into the torsional wave equation, we get: \begin{equation} -\omega^2 U(x) + \frac{GJ}{m}\frac{\partial^4 U(x)}{\partial x^4} = 0 \end{equation} We can rewrite this as fourth-order differential equation for \(U(x)\): \begin{equation} \frac{\partial^4 U(x)}{\partial x^4} - \frac{m\omega^2}{GJ}U(x) = 0 \end{equation} with boundary conditions: \begin{equation} \frac{\partial U(0)}{\partial x} = 0 \quad \text{and} \quad \frac{\partial^2 U(0)}{\partial x^2} = 0 \quad \text{and} \quad k_{\theta}U(L) = -\frac{GJ}{mL}\frac{\partial^3 U(L)}{\partial x^3} \end{equation} To solve this problem, we assume a general solution of the form: \begin{equation} U(x) = A\cos(kx) + B\sin(kx) + C\cosh(kx) + D\sinh(kx) \end{equation} where \(A\), \(B\), \(C\), \(D\) are constants and \(k^4 = \frac{m\omega^2}{GJ}\). Applying the boundary conditions to the general solution and solving the resulting equations, we will obtain the frequency equation for the rod.

Determine the first three natural frequencies and modal functions

To determine the first three natural frequencies and corresponding modal functions, we need to solve the frequency equation obtained in step 2 for the smallest three non-zero values of \(\omega\). These values will represent the first three natural frequencies of the rod. For each natural frequency, substitute the obtained value of \(\omega\) back into the general solution to obtain the corresponding modal function. Thus, we will have the first three modal functions as well. Lastly, plot the first three modes by graphing the corresponding modal functions \(U(x)\). It should be noted that the analytical solution to this problem might be quite involved, so computational tools/software could be used to obtain the numerical solutions and plots for the natural frequencies and modal functions.

Key concepts

Frequency Equation

The frequency equation is a critical mathematical relationship in dynamics that defines the conditions which must be met for a system to exhibit torsional vibrations at certain frequencies, known as the natural frequencies. In the context of torsional vibrations of a rod, the frequency equation emerges from applying boundary conditions to a fourth-order differential equation derived from the motion of the system.

For a uniform rod undergoing torsion, after assuming a separable solution for the displacement, \( \Theta(x,t) = U(x)e^{i\omega t} \), where \( U(x) \) is the spatial component and \( e^{i\omega t} \) represents time variation, we derive the differential equation:
\[ \frac{\partial^4 U(x)}{\partial x^4} - \frac{m\omega^2}{GJ}U(x) = 0 \]
This equation must satisfy the set boundary conditions to yield the natural frequencies. By employing auxiliary conditions at both ends of the rod—such as no torque or zero slope at a free end and proportional displacement to bending moment at a fixed end—we arrive at the frequency equation. This equation allows us to compute the values of \( \omega \), upon which the rod will naturally resonate.

Modal Boundary Conditions

Modal boundary conditions define the constraints or limitations at the borders of a system within modal analysis, which in this case, is a rod experiencing torsional vibrations. They are crucial because they significantly influence the vibrational behavior of the system and thus affect the natural frequencies and modal functions.

For the uniform circular elastic rod described, the modal boundary conditions on the left end (free end) are:
\[ \frac{\partial \Theta(0,t)}{\partial x} = 0 \quad \text{and} \quad \frac{\partial^2 \Theta(0,t)}{\partial x^2} = 0 \]
At the right end, embedded in an elastic wall, the boundary condition is represented by the equation:
\[ k_{\theta}\Theta(L,t) = -\frac{GJ}{mL}\frac{\partial^3 \Theta(L,t)}{\partial x^3} \]
This incorporation of torsional stiffness by the elastic wall distinguishes the vibrational characteristics and is mathematically reflected in the frequency equation.

Natural Frequencies

Natural frequencies are the specific frequencies at which a system tends to oscillate in the absence of external forces or damping. They are inherent properties of the system and depend on its physical characteristics, such as mass distribution, stiffness, and boundary conditions.

In the exercise given, the natural frequencies correspond to the values of \( \omega \) that satisfy the frequency equation obtained from the boundary value problem. Determining these frequencies involves solving a transcendental equation, which can often require numerical methods or approximation techniques.

The first three natural frequencies represent the fundamental and the first two overtones that the rod can experience during torsion. Identifying these frequencies is crucial in minimizing resonant vibration, which can cause fatigue or failure in mechanical systems.

Modal Functions

Modal functions, also known as mode shapes, describe the physical deformation configuration of a system under vibrational motion at a specific natural frequency. They are essential for understanding the distribution of displacement, velocity, or acceleration along the system while it vibrates.

In the case of the torsional vibrations of a rod, the modal functions are obtained by solving for \( U(x) \) in the general solution of the form:
\[ U(x) = A\cos(kx) + B\sin(kx) + C\cosh(kx) + D\sinh(kx) \]
Upon applying the previously established modal boundary conditions and calculating the natural frequencies, the coefficients \( A \), \( B \), \( C \), and \( D \) can be determined. Each modal function corresponds to a natural frequency and represents the shape of the rod as it twists about its axis. These functions are crucial for visualizing and assessing how the rod will react under torsional stress and are used to predict the response of the structure to various forms of excitation.