assignment2 - Instructor:Dr.EmreAlpman Assignment#2...

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ME 472 Computer-Aided Mechanical Engineering Instructor: Dr. Emre Alpman Assignment #2 Due Date: March 4th, 2010 SINGLE DEGREE-OF-FREEDOM SYSTEMS Summary of Previous Assignment The SDOF system in assignment #1, solved by forward Euler algorithm, which is known to have  inherent stability problems. The second order ODE I ¨ θ c ˙ θ mgl CG θ = 0 can be written as a system of linear first order ODE's as: [ ˙ y 1 ˙ y 2 ] = [ 0 1 mgl CG I c I ] [ y 1 y 2 ] Then forward Euler formulations looks like: y n 1 = I h A y n where A is the matrix given above. For stability the magnitude of the eigenvalues of the matrix ( I+hA must be less than one. Using the numbers given last week for the lab session one can find that for a  stable solution  h  must be less than approximately 0.065. The solutions obtained by  h  = 0.02,  h =  0.04  and  h  = 0.06 are given below. As you can see as  h  is increased numerical solution not only loses its  accuracy but also starts to have stability problems. Figure 1. Comparison of numerical solution ( red circles) with  h  = 0.02, with analytical solution.
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Figure 2. Comparison of numerical solution ( red circles) with  h  = 0.04, with analytical solution. Figure 3. Comparison of numerical solution ( red circles) with  h  = 0.06, with analytical solution. Lab Assignment In the previous section a clear example of loss of stability was illustrated, when the absolute stability  criterion posed by the forward Euler scheme was violated. One way to overcome this difficulty without  having to decrease the stepsize is to select an alternative numerical scheme, known to have better  stability properties.  It was shown in class that “backward Euler” scheme had much better stability characteristics compared  to forward Euler, by plotting their corresponding stability regions in the complex plane. As a matter of 
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This note was uploaded on 04/05/2012 for the course MECHANICAL ME372 CAME taught by Professor Emraalpman during the Spring '10 term at Yeditepe Üniversitesi.

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assignment2 - Instructor:Dr.EmreAlpman Assignment#2...

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