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# first order system

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### first order system

1. 1. MATLAB Tutorials Session V Mathematical Applications using MATLAB (Cont….) Rajeev Madazhy Email: rmadaz1@lsu.edu Dept of Mechanical Engineering LSUDepartment of Mechanical Engineering, LSU Session V
2. 2. Last Session….  Using fplot function  Minimization  Zero finding  Curve fitting  Interpolation  IntegrationDepartment of Mechanical Engineering, LSU Session V
3. 3. Session V Outline….  Solving Double Integrals  Ordinary Differential Equations  Examples of ODE using MATLAB….  Mention of DDE’sDepartment of Mechanical Engineering, LSU Session V
4. 4. Solving Double Integrals…. Consider the numerical solution of y max xmax ∫ ∫ f ( x, y )dxdy y min x min Let f ( x, y ) = y sin( x ) + x cos( y ) We write a function to calculate the double integral using the MATLAB Inline function dblquad.Department of Mechanical Engineering, LSU Session V
5. 5. MATLAB M-File…. Function out=integrnd(x,y) out= y*sin(x) + x*cos(y); To evaluate the double integral, use result = dblquad(‘integrnd’,xmin,xmax,ymin,ymax); at the command prompt.Department of Mechanical Engineering, LSU Session V
6. 6. In MATLAB….Department of Mechanical Engineering, LSU Session V
7. 7. Books on solving ODE’s using MATLAB.. Linear Algebra and Differential Equations Using MATLAB Martin Golubitsky, Michael Dellnitz Ordinary Differential Equations Using MATLAB John C. Polking & David ArnoldDepartment of Mechanical Engineering, LSU Session V
8. 8. Solving ODE’s….  MATLAB has the capability to solve the first order differential equations using numerical methods.  The functions used are ode23 and ode25  Both ode23 and ode25 work the same way except for the internal algorithm that is being used.  Let us use ode45 in solving the Differential equations.Department of Mechanical Engineering, LSU Session V
9. 9. Format…. The format is as follows:  [t,y] = ode45(‘function_name’,tspan,y0)  function_name is the name of a function type file where the differential equation is stored.  tspan is a vector specifying the initial and final values of independent variable  y0 is a column vector containing the initial conditions  Results for the command are stored in vector y.  t is the vector of independent variableDepartment of Mechanical Engineering, LSU Session V
10. 10. Example…. Solve a first order homogeneous differential equation with initial condition: . y+ = y 0 y (t = ) = 0 1 We rewrite it as : . y = y − y (0) =1Department of Mechanical Engineering, LSU Session V
11. 11. Function in MATLAB… Write the function in MATLAB and save it as ode1.mDepartment of Mechanical Engineering, LSU Session V
12. 12. Cont…. Write the following in another m-file:Department of Mechanical Engineering, LSU Session V
13. 13. Result….Department of Mechanical Engineering, LSU Session V
14. 14. Non-homogenous ODE’s…. If the differential equation is not homogenous then we do the following: . y + = −t y e 2 y (t = ) = 0 1 It is rewritten again as follows: . y = y + −t − e 2 y (0) =1Department of Mechanical Engineering, LSU Session V
15. 15. Solution…. All we need to do is change the function in the m-file ode1.m Rest remains the same as coded earlier.Department of Mechanical Engineering, LSU Session V
16. 16. Higher order differential Equations….  The higher order differential equations can be converted to a system of first order differential equations.  Next example shows how to solve second order differential equation using ode45 in MatlabDepartment of Mechanical Engineering, LSU Session V
17. 17. Problem….The following figure shows a spring-mass-damper system. Plot the response ofthe system when the initial displacement of mass m is 0.1 meters. c = 1 kg/s k = 100N/m m = 5 kg Department of Mechanical Engineering, LSU Session V
18. 18. Equation of motion…. The equation of motion is as follows: .. . m x +c x +kx =0 x (t =0) =0.1 Now we need to change it to first order equation to solve it. Rewriting the equation we get, c k  = − x − x x  m m x(t = 0) = 0.1Department of Mechanical Engineering, LSU Session V
19. 19. Cont…. . Consider the fact that x =vis just velocity. So we can rewrite the equation as two first order differential equations c k v =− v− x  m m x=v  v(t = 0) = 0 x(t = 0) = 0.1Department of Mechanical Engineering, LSU Session V
20. 20. Cont…. Or in general form we get: c k x1 = − x1 − x2  m m x2 = x1  x1 (t = 0) = 0 x2 (t = 0) = 0.1Department of Mechanical Engineering, LSU Session V
21. 21. Cont…. Write the function as follows in Matlab editor and save it as ode2.mDepartment of Mechanical Engineering, LSU Session V
22. 22. Cont…. Write the main program naming it as sorDiff.m Note that we have two initial conditions here.Department of Mechanical Engineering, LSU Session V
23. 23. Velocity Response….Department of Mechanical Engineering, LSU Session V
24. 24. Displacement Response….Department of Mechanical Engineering, LSU Session V
25. 25. Exercise…. Write a Matlab program to determine the time-temperature history of a sphere of radius r=5mm, initially at a uniform temperature of 400 0C. The sphere is exposed to 2000C air with a convection coefficient of h=10 W/m^2-K. The thermophysical properties of the sphere material are: ρ=Density=3000kg/m^3 k=Thermal conductivity=20 W/m-K c=specific heat=1000J/kg-KDepartment of Mechanical Engineering, LSU Session V
26. 26. Exercise cont…. The relation between the sphere temperature and time is given by an energy balance on the sphere, which results in the following differential equation dT − hA(T − T∞ ) = ρcV dt where H = convective heat transfer coefficient T = temperature of the sphere at any time A = surface area of the sphere = 4r2 V = volume of the sphere = 4/3 r3 T = timeDepartment of Mechanical Engineering, LSU Session V
27. 27. Exercise cont…. This differential equation has the following exact solution which can be used to check the accuracy of the numerical solution provided by Matlab T − T∞ − hA = exp( t) Ti − T∞ ρcVDepartment of Mechanical Engineering, LSU Session V
28. 28. Exact Temperature code….Department of Mechanical Engineering, LSU Session V
29. 29. Using as function….Department of Mechanical Engineering, LSU Session V
30. 30. MATLAB main code….Department of Mechanical Engineering, LSU Session V
31. 31. Result and plot….Department of Mechanical Engineering, LSU Session V
32. 32. Delay Differential Equations… Ordinary differential equations (ODEs) and delay differential equations (DDEs) are used to describe many phenomena of physical interest. While ODEs contain derivatives which depend on the solution at the present value of the independent variable (“time”), DDEs contain in addition, derivatives which depend on the solution at previous times. DDEs are a better approximation than ODEs to many physical systems.Department of Mechanical Engineering, LSU Session V
33. 33. Cont…. Consider a system of delay differential equations of the form: y(t) = f(t, y(t), y(t - τ1), y(t - τ2), . . . , y(t - τk)) that are solved on a ≤ t ≤ b with given history y(t) = S(t) for t ≤ a.Department of Mechanical Engineering, LSU Session V
34. 34. Function code….Department of Mechanical Engineering, LSU Session V
35. 35. Main program….Department of Mechanical Engineering, LSU Session V
36. 36. Output….Department of Mechanical Engineering, LSU Session V
37. 37. Website where you could obtain The dde23.m file….. http://www.radford.edu/~thompson/webddes/Department of Mechanical Engineering, LSU Session V
38. 38. Recap….  Solving Double Integrals  Ordinary Differential Equations  Examples of ODE using MATLAB….  Mention of DDE’sDepartment of Mechanical Engineering, LSU Session V
39. 39. Next Session…. Engineering Applications using MATLAB….  Solving non linear differential equations  Algorithm analysis  Common mechanical problems a) four bar linkage b) vibrations c) thermal and fluidsDepartment of Mechanical Engineering, LSU Session V
40. 40. Thank YouDepartment of Mechanical Engineering, LSU Session V