Question

The vibration of a beam with clamped ends is governed by $$ \begin{gathered} \epsilon^{2} \frac{d^{4} u}{d x^{4}}-\frac{d^{2} u}{d x^{2}}=\lambda^{2} u \\ u(0)=u(1)=u^{\prime}(0)=u^{\prime}(1)=0 \end{gathered} $$ Determine a first-order expansion for small \(\epsilon\) for \(u\) and \(\lambda\).

Short answer

For small \( \epsilon \), the first-order expansion gives \( u = A\sin(n\pi x) \) and \( \lambda = n\pi \).

Step-by-step solution

Assume Expansions for Small ε

Let's assume that both \( u \) and \( \lambda \) can be expanded in powers of \( \epsilon \). Therefore, assume:- \( u = u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots \)- \( \lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + \cdots \)These expansions allow us to solve the problem iteratively by plugging these expressions into the given differential equation using small terms of \( \epsilon \).

Substitute Expansions into Differential Equation

Substituting the assumed expansions into the differential equation:\[\epsilon^{2} \frac{d^{4}}{d x^{4}}(u_0 + \epsilon u_1) - \frac{d^{2}}{d x^{2}}(u_0 + \epsilon u_1) = (\lambda_0 + \epsilon \lambda_1)^2 (u_0 + \epsilon u_1)\]Next, expand the right side and equate terms of similar orders in \( \epsilon \).

Solve the Leading Order Equation

For leading order terms (i.e., \( \epsilon^0 \)), consider:\[ -\frac{d^2 u_0}{d x^2} = \lambda_0^2 u_0 \]This is a classical Sturm-Liouville problem subjected to boundary conditions:\[ u_0(0) = u_0(1) = u'_0(0) = u'_0(1) = 0 \]

Solution to the Leading Order Equation

For a non-trivial solution, \( \lambda_0 \) can be determined as:\[ \lambda_0 = n\pi \]where \( n = 1, 2, 3, \ldots \) as it satisfies the boundary conditions of a standard vibrating problem. The corresponding solution \( u_0 \) becomes:\[ u_0 = A\sin(n\pi x) \]where \( A \) is a constant to be defined by the problem specifics but may remain arbitrary at this stage.

Solve the First Order Corrected Equation

Now consider the \( \epsilon^1 \) terms by substituting into the differential equation:\[\frac{d^{4}u_0}{d x^{4}} - \frac{d^{2} u_1}{d x^{2}} = 2 \lambda_0 \lambda_1 u_0 + \lambda_0^2 u_1\]Using the solution \( u_0 = A\sin(n\pi x) \), solve for \( u_1 \) using the homogeneous solutions available within the defined boundary conditions.

Determine First Order Expansion for λ and u

In many problems of this type, symmetry and orthogonality conditions imply that no inhomogeneous solution exists that does not trivialize part of \( u_1 \). Determine consistent corrections such as \( \lambda_1 = 0 \) and confirm linearity of edge constraints, leading to potential constant or null contribution of \( u_1 \).

Conclusion

To first order, corrections suggest:- \( u = u_0 + \epsilon u_1 = A\sin(n\pi x) + O(\epsilon) \)- \( \lambda = n\pi + O(\epsilon^2)\) for leading small terms.The result is consistent with keeping the major terms and underlines symmetry of errors in \( u_1 \) for small \( \epsilon \).

Key concepts

Boundary Value Problems

When dealing with boundary value problems, the goal is to find a solution to a differential equation that satisfies certain conditions, or boundaries, at multiple points. For instance, if a beam is clamped at both ends, the displacement at these ends should be zero. In mathematical terms, these conditions influence how the solutions behave over a specified interval, usually from point 0 to 1, as seen in our exercise.
Boundary value problems are quite common in physics and engineering as they model real-world situations where initial and end conditions are known, such as vibrations of beams or heat distribution.

• Homogeneous boundary conditions: These occur when the specified function values at the boundaries are zero, as in our example where both the function value and its derivative are zero at the endpoints.
• Non-homogeneous boundary conditions: Here, the function values at the boundaries are not zero, requiring different methods to find solutions.

Understanding boundary value problems is crucial for solving and analyzing systems that involve constraints, such as clamped beams or wires in vibrational analysis.

Sturm-Liouville Theory

Sturm-Liouville theory provides a framework for solving certain types of boundary value problems, particularly those that can be expressed in a standard form. This theory is valuable because it transforms differential equations into a systematic process of finding eigenvalues and eigenfunctions, which are essential for many physical systems.
In our exercise, the leading order equation, a second-order differential equation, was solved using the principles of Sturm-Liouville theory. The solution involved finding eigenvalues, denoted as \( \lambda_0 = n\pi \), which correspond to the natural frequencies of vibrations.

• Eigenvalues and eigenfunctions: These are solutions that help describe the vibrational modes of a system. The eigenvalues correspond to the frequencies, while the eigenfunctions describe the mode shapes.
• Orthogonality: Sturm-Liouville theory also provides conditions where the eigenfunctions are orthogonal, aiding in expanding functions using series of these functions, simplifying complex boundary value problems.

This theory allows complex boundary value problems to be broken down into solvable parts, which helps in further mathematical analysis, making it indispensable for engineers and physicists.

Vibrational Analysis

Vibrational analysis examines how objects vibrate, especially those with mechanical constraints like beams or membranes. It's key in designing structures, machinery, or any system subjected to dynamic forces. In this context, vibrational analysis involves determining how different modes or frequencies of vibration occur.
In our example, the beam's vibration was characterized by solutions to a differential equation with particular boundary conditions. The expansion for small \( \epsilon \) revealed insights into the system's vibrational properties. This process starts with simpler models and refines them to match more complex, real systems.

• Mode shapes: These are the patterns in which systems like beams vibrate. In our problem, these modes were described by sine functions, \( A\sin(n\pi x) \).
• Naturally occurring frequencies: Each mode shape corresponds to specific frequencies \( n\pi \), telling how often parts of the system oscillate.

Understanding vibrational analysis is crucial for the safe, efficient design of structures, as it helps predict and control the dynamic response of systems.

Differential Equations

Differential equations involve mathematical equations that relate a function with its derivatives. They are indispensable in modeling various physical phenomena, from motion to electrical circuits and heat flow. In the realm of boundary value problems and vibrational analysis, differential equations describe how systems transition or react to changes over time and space.
In our example, the main differential equation described how the beam's displacement and its derivatives are related, incorporating a small parameter \( \epsilon \) that captures more subtle dynamics of the vibration.

• Perturbation methods: In problems like the given exercise, perturbation methods are employed to find approximate solutions by expanding the functions and analyzing terms up to desired accuracy, often using powers of a small parameter.
• Iteration: By substituting assumed solutions into the equation, unwanted terms are balanced, allowing the solution of even complex, nonlinear differential equations iteratively.

Differential equations remain a fundamental tool in mathematics for predicting and understanding behaviors of continuous systems under various conditions.