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3.1 Introduction The second civil engineering problem we will study is to estimate the deflection y(x) of a beam under uniform loading, where the ends of the beam held in place (see Figure 2). With some simplifying assumptions, the deflection of a beam supported at x = 0 and z = L can be modeled by the boundary value problem (BVP) I y" S :y+ EI 2EI (x-L), 0≤x≤L, y(0) = 0, y(L) = 0, (1) where y(x) is the deflection of the beam, L is the length of the beam, Q is the intensity of the uniform load, E is the modulus of elasticity, S is the stress at the endpoints, and I is the central moment of inertia. Our goal is to find an approximate solution for (1) using Matlab's ode 45 but first we need to introduce second order initial value problems (IVPS) and the method of linear shooting.
S y(x) Figure 2: Deflection of a beam. x
3.2 Second Order Initial Value Problems We have seen plenty of first order initial value problems (IVPs), but you may not have seen many second order ones yet. In general, a second order IVP looks like y" =f(t,y,y), y(to) = y₁, y' (to) = y2- Notice that a second order IVP has two initial conditions; one for y and one for y', both given at the same initial value of the independent variable (usually time), to. Written in this form, the solution to (2) cannot be directly approximated using one of Matlab's standard built-in ODE solvers, such as ode 45. However, since Matlab's solvers are designed for systems of first order IVPs, we would like to write the IVP in (2) without using any second derivatives. To do this, define the new function z(t) = y' (t), so that the original second order IVP can be written as a system of two first order IVPS: y (to) [~ ] - [ f(t.,3,2) ]; y, z) In this form, the solution can be approximated using one of Matlab's built-in ODE solvers. = (2) Y1 Y2
3.3 Linear Shooting The linear shooting method is used to find an approximate solution to a linear second order boundary value problem (BVP), which has the form y" = p(x)y' +q(x)y+r(x), a ≤x≤ b, y(a) = a, y(b) = 3. (3) Here, p(x), g(x), and r(r) are continuous on [a, b], and g(x) > 0 on [a, b]. The BVP is guaranteed to have a unique solution, but in its current form we don't have the tools to solve it, even approximately. We learned in Section 3.2 how to transform a second order IVP into a system of first order IVPs, but (3) is not an IVP at all. Notice that instead of having two conditions (y and y') on the solution at the same the point (an IVP), the BVP has two conditions on the solution, y, but at different points. To apply the linear shooting method, we temporarily set aside the BVP in (3), and instead consider two closely related and cleverly chosen IVPS: y" = p(x)y' +q(x)y+r(x), y" = p(x)y' +q(x)y, a ≤x≤ b, a≤x≤ b, y(a)= a, y(a) = 0, y'(a) = 0, y'(a) = 1. (5) Approximate solutions to (4) and (5) can be found using ode 45 because they are both second order IVPs (see Section 3.2). Now, let y₁ (2) denote the solution to (4) and let y2 (2) denote the solution to (5). It turns out that y₁ and 32 can be combined in a certain way to recover the solution to the original BVP given in (3). Define the new function y(x) by y(x) = y₁ (x) + Cy₂(x). (6) It can be shown that, for the right choice of the constant C (see problem 2b), y(x) satisfies the ODE and boundary conditions given in (3). Once C' is known, we can use ode 45 to find approximate solutions to the IVPs in (4) and (5), combine them according to (6), and the result is an approximate solution to the original second order BVP given in (3).
A Matlab function called linearShooting.m has been provided, and is currently configured to solve a specific BVP of the form given in (3) (you can run this code to see what it does). By modifying the "INPUTS" section of the code, the approximate solution for any BVP of the form given in (3) can be found and plotted. 3.4 Task Set B 1. Classify the ODE for the deflection of a uniform beam, given in (1). What is the order of the ODE? Is it linear? If so, is it homogeneous or nonhomogeneous? Are the coefficients constant or variable? 2. (a) Explain how Eq. (6) is a manifestation of the nonhomogeneous principle. (b) Given that y₁ (x) and y2 (2) are solutions to (4) and (5), respectively, find a value for the constant C so that (6) is the solution to the BVP given in (3). 3. (a) Set z(x) = y'(r), and rewrite (4) as a system of first order ODEs, together with the appropriate initial conditions. (b) Similarly, rewrite (5) as a system of first order IVPs.