Problem 8 - Advanced Engineering Mathematics

Question

A uniform semi-infinite elastic beam moving along the \(x\) -axis with a constant velocity \(-v_{0}\) is brought to a stop by hitting a wall at time \(t=0\). See FitURE 1523. The longitudinal displacement \(u(x, t)\) is determined from $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0 \\ &u(0, t)=0, \quad \lim _{x \rightarrow \infty} \frac{\partial u}{\partial x}=0, t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{i=0}=-v_{0}, x>0 \end{aligned} $$ Solve for \(u(x, t)\).

Step-by-Step Solution

Answer

The displacement function is \(u(x, t) = v_0t\) for \(at < x\), and 0 otherwise.

Step 1: Identify the wave equation

Recognize that the given equation \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \) is a wave equation, and it represents the motion of the semi-infinite beam.

Step 2: Apply Boundary Conditions

Use the boundary conditions provided: \(u(0, t) = 0\) and \(\lim_{x \rightarrow \infty} \frac{\partial u}{\partial x} = 0\) to define behavior at specific points. The first condition means the displacement at the wall (\(x = 0\)) is zero, and the second implies no strain at the far end.

Step 3: Apply Initial Conditions

Utilize the initial conditions: \(u(x, 0) = 0\) and \(\left. \frac{\partial u}{\partial t}\right|_{t=0} = -v_0\). These specify that initially (at \(t=0\)), the beam hasn't displaced, but is moving with a velocity \(-v_0\).

Step 4: Solve the equation using Method of Characteristics

Recognize the form of a standard wave equation solution: \( u(x, t) = f(x - at) + g(x + at) \), where \(f\) and \(g\) are determined from boundary and initial conditions. The displacement can be expressed in terms of waveforms moving in opposite directions.

Step 5: Determine the functions f and g

Substitute the initial conditions into the general solution form to find \(f\) and \(g\). The condition \(u(x, 0) = 0\) gives \(f(x) + g(x) = 0\), implying \(g(x) = -f(x)\). The velocity condition \(\left. \frac{\partial u}{\partial t}\right|_{t=0} = -v_0\) implies the derivative \(-af'(x) + ag'(x) = -v_0\). Substituting \(g(x) = -f(x)\) gives \(-2af'(x) = -v_0\), leading to \(f'(x) = \frac{v_0}{2a}\).

Step 6: Integrate to find f(x)

Integrate \(f'(x) = \frac{v_0}{2a}\) to find \(f(x) = \frac{v_0 x}{2a} + C\). Apply the boundary condition \(u(0, t) = 0\) which implies \(f(-at) + g(at) = 0\), resulting in \(C=0\). Thus, \(f(x) = \frac{v_0 x}{2a}\).

Step 7: Express solution using f and g

Using \(g(x) = -f(x)\), the displacement becomes \( u(x, t) = \frac{v_0}{2a} ((x - at) - (x + at)) = \frac{-v_0}{2a}(-2at) = v_0t \) for \(at < x\) and 0 otherwise due to conditions.

Key Concepts

Boundary Conditions

Boundary conditions are critical in understanding wave equations, especially in problems involving physical boundaries like walls or ends of a beam. In this problem:

• The condition \( u(0, t) = 0 \) indicates that at the point where the beam hits the wall, the displacement \( u \) is zero. This is a typical scenario in wave problems where an obstacle or fixed point is encountered. As the beam hits the wall, the point itself does not move; hence, the displacement there is zero.
• The second boundary condition, \( \lim_{x \rightarrow \infty} \frac{\partial u}{\partial x} = 0 \), means that as we move farther away from the wall, the strain (or the change in displacement per unit length) tends to zero. This implies that there are no waves affecting the beam far away from the wall, recognizing it as a semi-infinite beam.

These boundary conditions help describe how the wave behaves both at the fixed point and at infinity. They are essential in solving the wave equation since they provide the constraints needed to find a unique solution.

Initial Conditions

Initial conditions describe the state of the system at the starting time (usually \( t = 0 \)). In this problem, they play a significant role in determining how the elastic beam begins moving and deforming.

• The condition \( u(x, 0) = 0 \) tells us that at the initial time \( t = 0 \), the displacement everywhere along the beam is zero. This reflects that, initially, the structure of the beam has not yet deformed under the impact.
• Furthermore, the condition \( \left. \frac{\partial u}{\partial t} \right|_{t=0} = -v_0 \) implies the initial velocity of the beam. It was moving with a constant velocity \( -v_0 \) before hitting the wall. This indicates that right before the wall impact, the beam was in motion but immediately stops at \( t = 0 \).

These initial conditions are used to initialize the wave equation solution, allowing us to compute the subsequent motion or deformation of the beam over time.

Method of Characteristics

The method of characteristics is a powerful tool for solving partial differential equations like the wave equation given in this exercise. It involves transforming the wave equation into a simpler form that can be handled more easily.
The wave equation solution has a standard form \( u(x, t) = f(x - at) + g(x + at) \). This denotes two waveforms:

• One moving forward in space with speed \( a \), denoted by \( f(x - at) \).
• Another moving backward with speed \( a \), denoted by \( g(x + at) \).

By using the initial and boundary conditions, we can determine the explicit forms of \( f \) and \( g \) that satisfy these constraints. In this problem, we found that \( g(x) = -f(x) \) based on boundary considerations and solved for \( f(x) \) using derivatives and integration.
This method simplifies the process of finding a solution that fits both the physical and mathematical aspects of the problem.

Elastic Beam Motion

In this exercise, the wave equation represents the motion of an elastic beam moving along an axis, needing consideration of how elastic materials behave when subjected to external forces, like hitting a wall.

• An elastic beam, under the influence of forces, follows the wave equation, which is a mathematical representation of the propagation of waves through a medium. The equation given \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \) describes how the wave, or the motion, disperses along the length of the beam as time progresses.
• The motion at different points of the beam is represented by the displacement \( u(x, t) \), showing how each part of the beam is displaced over time.
• Since the beam impacts a wall, the boundary conditions (powerfully illustrated in the problem) stipulate physical constraints, such as no movement at the wall (\( x = 0 \)) and no strain at infinity.

Understanding elastic beam motion with wave equations is important in fields like mechanical and civil engineering, where materials' response to external forces determines structural integrity.