mws_gen_ode_spe_finitedif

# mws_gen_ode_spe_finitedif - Chapter 08.07 Finite Difference...

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08.07.1 Chapter 08.07 Finite Difference Method for Ordinary Differential Equations After reading this chapter, you should be able to 1. Understand what the finite difference method is and how to use it to solve problems. What is the finite difference method? The finite difference method is used to solve ordinary differential equations that have conditions imposed on the boundary rather than at the initial point. These problems are called boundary-value problems. In this chapter, we solve second-order ordinary differential equations of the form b x a y y x f dx y d ), ' , , ( 2 2 , ( 1 ) with boundary conditions a y a y ) ( and b y b y ) ( ( 2 ) Many academics refer to boundary value problems as position-dependent and initial value problems as time-dependent. That is not necessarily the case as illustrated by the following examples. The differential equation that governs the deflection y of a simply supported beam under uniformly distributed load (Figure 1) is given by EI x L qx dx y d 2 ) ( 2 2 ( 3 ) where x location along the beam (in) E Young’s modulus of elasticity of the beam (psi) I second moment of area (in 4 ) q uniform loading intensity (lb/in) L length of beam (in) The conditions imposed to solve the differential equation are 0 ) 0 ( x y ( 4 ) 0 ) ( L x y Clearly, these are boundary values and hence the problem is considered a boundary-value problem.

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08.07.2 Chapter 08.07 Figure 1 Simply supported beam with uniform distributed load. Now consider the case of a cantilevered beam with a uniformly distributed load (Figure 2). The differential equation that governs the deflection y of the beam is given by EI x L q dx y d 2 ) ( 2 2 2 ( 5 ) where x location along the beam (in) E Young’s modulus of elasticity of the beam (psi) I second moment of area (in 4 ) q uniform loading intensity (lb/in) L length of beam (in) The conditions imposed to solve the differential equation are 0 ) 0 ( x y ( 6 ) 0 ) 0 ( x dx dy Clearly, these are initial values and hence the problem needs to be considered as an initial value problem. Figure 2 Cantilevered beam with a uniformly distributed load. q y L x q y L x
Finite Difference Method 08.07.3 Example 1 The deflection y in a simply supported beam with a uniform load q and a tensile axial load T is given by EI x L qx EI Ty dx y d 2 ) ( 2 2 (E1.1) where x location along the beam (in) T tension applied (lbs) E Young’s modulus of elasticity of the beam (psi) I second moment of area (in 4 ) q uniform loading intensity (lb/in) L length of beam (in) Figure 3 Simply supported beam for Example 1. Given,

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mws_gen_ode_spe_finitedif - Chapter 08.07 Finite Difference...

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