Question

A uniform elastic rod of length \(L\), membrane stiffness \(E A\) and mass per unit length \(m\) is attached to a rigid base at its left end, as shown. Determine the steady state motion of the rod if a motor causes the base to undergo the prescribed motion \(\chi_{x}(t)=h_{0} \sin \Omega t\).

Short answer

Based on the given information about the steady state motion of a uniform elastic rod, provide a short answer that explains the solution procedure and the resulting motion equation. The steady state motion of a uniform elastic rod subjected to a prescribed motion at its base can be found by setting up the Euler-Bernoulli beam equation, applying given boundary conditions, assuming a solution, and using the prescribed motion to find the steady state motion. The resulting motion equation is: \(\chi_{(x,t)}=h_{0}\sin\left(\frac{\pi x}{L}\right)e^{-i\Omega t}+\sum_{n=2}^{\infty}C_{n}\sin\left(\frac{n\pi x}{L}\right)e^{-i\Omega t}\). However, the coefficients \(C_{n}\) for \(n\geq2\) are not easy to determine, and advanced techniques such as the method of separation of variables or the method of matched asymptotic expansions are required for a complete solution.

Step-by-step solution

Set up the Euler-Bernoulli beam equation

The Euler-Bernoulli beam equation, which describes the bending of the elastic rod under the influence of an external force, can be written as: \(\frac{\partial^2}{\partial x^2}\left(E A \frac{\partial^2 \chi_{(x,t)}}{\partial x^2}\right) = m \frac{\partial^2 \chi_{(x,t)}}{\partial t^2}\), where \(\chi_{(x,t)}\) is the displacement of the rod at position x and time t.

Apply boundary conditions

Since the rod is attached to a rigid base at its left end (x = 0), the following boundary conditions apply: 1. Displacement at the left end: \(\chi_{(0,t)}=h_{0} \sin \Omega t\) 2. Slope at the left end: \(\frac{\partial \chi_{(x,t)}}{\partial x}|_{x=0}=0\) 3. Moment at the right end (x = L): \(\frac{\partial^2 \chi_{(x,t)}}{\partial x^2}|_{x=L}=0\) 4. Shear force at the right end (x = L): \(\frac{\partial^3 \chi_{(x,t)}}{\partial x^3}|_{x=L}=0\)

Assume a solution and apply the prescribed motion

Assuming a solution of the form: \(\chi_{(x,t)}= e^{i(K x - \Omega t)}\) where \(K\) is the wavenumber and \(i\) is the imaginary unit, we plugging this assumed solution into the Euler-Bernoulli beam equation and solve for \(K\): \(-K^4+ \frac{m\Omega^2}{E A}K^2 = 0\) \(K^2=0 \Rightarrow \text{or} \Rightarrow K^2 = \frac{m\Omega^2}{E A}\). Since we are interested in the steady state motion due to the prescribed motion at the base, we ignore the trivial solution \(K^2=0\) and use \(K^2=\frac{m\Omega^2}{E A}\) for further calculations. Now, we can rewrite the assumed solution as: \(\chi_{(x,t)}= e^{i\left(\pm \sqrt{\frac{m\Omega^2}{E A}}x - \Omega t\right)}\)

Apply boundary conditions and obtain the steady state motion

By applying the boundary conditions given in Step 2, we will find the coefficients of the assumed solution: 1. Displacement at the left end: \(\chi_{(0,t)}=h_{0} \sin \Omega t \Rightarrow e^{0}=1\) 2. Slope at the left end: \(\frac{\partial \chi_{(x,t)}}{\partial x}|_{x=0}=0 \Rightarrow \pm i\sqrt{\frac{m\Omega^2}{E A}}=0\) Since the second boundary condition cannot be satisfied by the assumed solution, we need to rewrite the solution in a more general form by considering an infinite sum of harmonic terms: \(\chi_{(x,t)}=\sum_{n=1}^{\infty}C_{n}\sin\left(\frac{n\pi x}{L}\right)e^{-i\Omega t}\) By substituting the first boundary condition (displacement at left end) into the modified solution, we get: \(h_{0} \sin \Omega t =\sum_{n=1}^{\infty}C_{n}\sin\left(\frac{n\pi (0)}{L}\right)e^{-i\Omega t} \Rightarrow C_{1}=h_{0}\) The steady state motion of the rod can then be given as: \(\chi_{(x,t)}=h_{0}\sin\left(\frac{\pi x}{L}\right)e^{-i\Omega t}+\sum_{n=2}^{\infty}C_{n}\sin\left(\frac{n\pi x}{L}\right)e^{-i\Omega t}\) However, the coefficients \(C_{n}\) for \(n\geq2\) are not easy to determine due to the complex nature of the boundary condition. Therefore, this problem requires more advanced techniques such as the method of separation of variables or the method of matched asymptotic expansions to solve completely.

Key concepts

Elastic Rod Vibration

Understanding how an elastic rod vibrates is essential in the field of structural engineering and material science. The vibrations of such rods are characterized by their response to external stimuli, such as the motion of a motor at one end.

In the given exercise, we examine the steady state motion of a uniform elastic rod fixed at one end and subjected to sinusoidal motion at the base. The Euler-Bernoulli beam equation plays a crucial role in describing the vibrational behavior of the rod. This equation models the bending that occurs due to an external force and is given by \[\frac{\partial^2}{\partial x^2}\left(E A \frac{\partial^2 \chi_{(x,t)}}{\partial x^2}\right) = m \frac{\partial^2 \chi_{(x,t)}}{\partial t^2}\], where through various boundary conditions, the resulting vibrations can be analyzed.

To ensure an easily understandable content, it is vital to remember that physical interpretation of the vibrational modes and natural frequencies is the motion and speeds at which the rod will 'naturally' oscillate when disturbed.

Boundary Conditions

Boundary conditions are crucial for solving differential equations, as they define the constraints that the solution must satisfy at the boundaries of the domain. For the elastic rod attached to a rigid base, several specific conditions apply, all of which must be taken into account to determine the rod's response to vibration.

The imposed boundary conditions for this exercise include:

• Fixed displacement at the left end due to the motor's movement.
• No slope at the left end implies that there is no initial angular deflection.
• No bending moment and shear force at the free end of the rod, which is consistent with it being unsupported.

Applying these conditions allows students to focus on suitable solution forms that satisfy physical constraints. While solving these problems, visualizing the boundary conditions as physical constraints - like imagining the rod's fixed and free ends - can aid in grasping the concept.

Steady State Motion

Steady state motion refers to the consistent, ongoing behavior of a system after any initial transients have settled. It is the 'end behavior' or the regular pattern of vibration that establishes over time due to sustained external forcing - in this case, the motion caused by the motor at the base.

In the provided solution process, finding the steady state motion involves assuming a solution form that reflects the continuous sinusoidal input from the motor and applying boundary conditions to refine this assumed solution. This method provides an equation characterizing the motion of every point on the rod over time. It is important to recognize that the steady state solution ignores transient behaviors that may occur shortly after the motor starts but provides a simplified view of the system's long-term response. Visualizing the rod's motion as a continuous wave, influenced by the base's prescribed movement, can help students conceptualize what steady state motion entails.