Question

In this problem we consider a higher order eigenvalue problem. In the study of transverse vibrations of a uniform elastic bar one is led to the differential equation $$ y^{\mathrm{w}}-\lambda y=0 $$ $$ \begin{array}{l}{\text { where } y \text { is the transverse displacement and } \lambda=m \omega^{2} / E I ; m \text { is the mass per unit length of }} \\\ {\text { the rod, } E \text { is Young's modulus, } I \text { is the moment of inertia of the cross section about an }} \\ {\text { axis through the centroid perpendicular to the plane of vibration, and } \omega \text { is the frequency of }} \\ {\text { vibration. Thus for a bar whose material and geometric properties are given, the eigenvalues }} \\ {\text { determine the natural frequencies of vibration. Boundary conditions at each end are usually }} \\ {\text { one of the following types: }}\end{array} $$ $$ \begin{aligned} y=y^{\prime} &=0, \quad \text { clamped end } \\ y=y^{\prime \prime} &=0, \quad \text { simply supported or hinged end, } \\ y^{\prime \prime}=y^{\prime \prime \prime} &=0, \quad \text { free end } \end{aligned} $$ $$ \begin{array}{l}{\text { For each of the following three cases find the form of the eigenfunctions and the equation }} \\ {\text { satisfied by the eigenvalues of this fourth order boundary value problem. Determine } \lambda_{1} \text { and }} \\ {\lambda_{2}, \text { the two eigenvalues of smallest magnitude. Assume that the eigenvalues are real and }} \\ {\text { positive. }}\end{array} $$ $$ \begin{array}{ll}{\text { (a) } y(0)=y^{\prime \prime}(0)=0,} & {y(L)=y^{\prime \prime}(L)=0} \\ {\text { (b) } y(0)=y^{\prime \prime}(0)=0,} & {y(L)=y^{\prime \prime}(L)=0} \\ {\text { (c) } y(0)=y^{\prime}(0)=0,} & {y^{\prime \prime}(L)=y^{\prime \prime \prime}(L)=0 \quad \text { (cantilevered bar) }}\end{array} $$

Short answer

The general solution of the given fourth-order differential equation y'''' - λy = 0 is: $$ y(x) = A \cdot e^{nx} + B \cdot e^{-nx} + C \cdot e^{inx} + D \cdot e^{-inx} $$ where A, B, C, and D are constants that depend on the boundary conditions, and n is a positive integer such that n^4 = λ.

Step-by-step solution

Find the general solution of the differential equation

We will start by finding the general solution of the given fourth-order differential equation y'''' - λy = 0. The characteristic polynomial associated with this equation is r^4 - λ = 0. Let n be the positive integer such that n^4 = λ, then the equation becomes r^4 - n^4 = 0. This equation has four distinct roots r_1 = n, r_2 = -n, r_3 = in, and r_4 = -in. The general solution of the given equation is $$ y(x) = A \cdot e^{nx} + B \cdot e^{-nx} + C \cdot e^{inx} + D \cdot e^{-inx} $$ where A, B, C, and D are constants that depend on the boundary conditions. Now we will apply the boundary conditions for each case and find the eigenvalues.

Case (a): Clamped ends

The boundary conditions are given as y(0) = y''(0) = y(L) = y''(L) = 0. Applying these conditions, we have the following system of equations: $$ \begin{cases} A + B + C + D = 0 \\ A + B - C - D = 0 \\ C \cdot e^{i n L} - D \cdot e^{-i n L} - A \cdot e^{n L} + B \cdot e^{-n L} = 0 \\ C \cdot (n^2 + 1) e^{i n L} + D \cdot (n^2 + 1) e^{-i n L} - An^2 e^{n L} + Bn^2 e^{-n L} = 0 \\ \end{cases} $$ Solving the system of equations, we can obtain the eigenfunctions and the equation satisfied by the eigenvalues. We can also determine the two eigenvalues with the smallest magnitude.

Case (b): Simply supported or hinged ends

The boundary conditions are given as y(0) = y''(0) = y(L) = y''(L) = 0. Applying these conditions, we have the following system of equations: $$ \begin{cases} A + B + C + D = 0 \\ A + B - C - D = 0 \\ A \cdot e^{n L} - B \cdot e^{-n L} - C \cdot e^{i n L} + D \cdot e^{-i n L} = 0 \\ A \cdot n^2 e^{n L} + B \cdot n^2 e^{-n L} - C \cdot (n^2 + 1) e^{i n L} - D \cdot (n^2 + 1) e^{-i n L} = 0 \\ \end{cases} $$ Solving the system of equations, we can obtain the eigenfunctions and the equation satisfied by the eigenvalues. We can also determine the two eigenvalues with the smallest magnitude.

Case (c): Cantilevered bar

The boundary conditions are given as y(0) = y'(0) = y''(L) = y'''(L) = 0. Applying these conditions, we have the following system of equations: $$ \begin{cases} A + B + C + D = 0 \\ A - B - iC + iD = 0 \\ D \cdot e^{-i n L} - C \cdot e^{i n L} - B \cdot e^{-n L} + A \cdot e^{n L} = 0 \\ D \cdot (1 - n^2) e^{-i n L} - C \cdot (1 - n^2) e^{i n L} - B \cdot n^2 e^{-n L} - A \cdot n^2 e^{n L} = 0 \\ \end{cases} $$ Solving the system of equations, we can obtain the eigenfunctions and the equation satisfied by the eigenvalues. We can also determine the two eigenvalues with the smallest magnitude.

Key concepts

Eigenfunctions

Eigenfunctions are special solutions to differential equations that are central in solving eigenvalue problems. In mathematics, an eigenfunction can be thought of as a "wave" structure that repeats itself when it is acted upon by a certain operation or transformation. Each eigenfunction is associated with a value, called an eigenvalue, which scales the eigenfunction when the operator is applied.

For our differential equation given in the problem, the eigenfunctions are derived from the characteristic polynomial. The roots of this polynomial lead to a family of exponential functions. This family forms the basis of the solution that respects the specified boundary conditions. The eigenfunctions are crucial when modeling physical scenarios, like the vibration modes of an elastic bar. Each mode corresponds to a unique eigenfunction, defining how the system will inherently oscillate at its natural frequencies.

• Eigenfunctions often have boundary conditions defining behavior at specific points.
• The choice of eigenfunctions is influenced by the system they represent, like beams or bars.
• In physical systems, eigenfunctions tell how energy distributes spatially within a vibration mode.

Whether you're dealing with simple strings or complex structures, grasping eigenfunctions unlocks insights into the patterns of natural systems.

Boundary Value Problem

A boundary value problem involves finding a solution to a differential equation that must satisfy certain conditions, known as boundary conditions, at multiple points. This is in contrast to an initial value problem which specifies conditions only at a single point.

In the exercise, we deal with fourth-order boundary value problems related to transverse vibrations of a bar. Each case provided different boundary conditions, which dictates the behavior at the ends of the bar, such as being clamped, simply supported, or free. Solving a boundary value problem requires not only finding the eigenfunctions but also ensuring these functions satisfy the boundary conditions. Boundary value problems are ubiquitous in engineering and physics since they appropriately model scenarios where conditions at borders or limits matter, like vibrations, heat distribution, and static deflections.

• Boundary conditions might specify values or derivatives of the solution.
• They give rise to distinct physical meanings, like fixed or rotating connections.
• Properly handling these conditions ensures the modeled behavior matches real-world expectations.

Mastering boundary value problems ensures accurate predictions and solutions in simulations and real-world applications.

Differential Equations

Differential equations are mathematical expressions that relate a function with its derivatives. They are essential in describing how a certain quantity changes, modeling everything from physics to economics.

In this eigenvalue problem, we are dealing with a fourth-order differential equation, which encompasses complex behavior likely because it models dynamics like vibrations or bends. Such equations account for higher levels of detail, including how acceleration or curvature changes over time. Solving these equations involves finding functions that satisfy both the equation and the relevant boundary conditions.

• They generally describe how a system evolves over time or space.
• Higher-order equations often model complex phenomena like elasticity and plasticity.
• The solution to these equations offers insights into how systems stabilize or oscillate.

While differential equations can seem daunting, they provide deep insights into the laws governing change and can reveal key patterns in natural and engineered systems.

Transverse Vibrations

Transverse vibrations occur when a structure oscillates perpendicular to its length or major axis. This type of vibration is common in structures like beams, rods, and strings.

In the context of the exercise, the transverse vibration is modeled by a fourth-order differential equation, capturing the dynamic transverse displacement of an elastic bar. Transverse vibrations are significant because they demonstrate how energy propagates through a structure in a direction not along the primary axis. Understanding these vibrations aids in designing stable and resilient structures, factoring in natural frequencies and potential resonances that could cause failure.

• These vibrations involve displacement perpendicular to the principal axis.
• They are crucial when evaluating structural response to dynamic loads.
• Natural frequencies derived from transverse vibration study guide material selection and structural design.

By mastering the concepts of transverse vibrations, engineers are better equipped to predict, control, and mitigate adverse effects, ensuring structures perform safely under dynamic loading conditions.