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Mathematical Modeling & Simulation HOME WORK # 5
Due Date: 10 days from date of distribution in the class.
Q.1 In following figure, we show a system of four tanks. Here, various tank have inlets and outlet for salt mixed water with given rates. First develop a mathematical model for coupled system. Then draw a patch diagram and simulate results using SIMULINK. Draw graphs for amount of salt in all tanks as a function of time respectively? Assume initial values as y 1 (0) = 150 (salt in Tank-1); y 2 (0) = y 3 (0) = y 4 (0) = 0. Find saturation values if any.
Q.2 Two masses are attached to two springs and a rigid body; then balancing the forces on each mass separately and taking into account the directions we obtain the following model equations:
Consider: B = 0.01, M 1 = M 2 = 1, k 1 = k 2 = 1. , Fext = 10. Both x 1 = x 2 = 0 at time t =
- Similarly, dx 1 / dt and dx 2 / dt are zero at t = 0. This system is based on coupled second ordinary differential equations. Draw a patch diagram and simulate it on SIMULINK to predict the behavior of the system. Plot graphs of x1(t) and x2(t) versus time. Then also draw phase portrait to predict the stability of the system.
2 2 2 1 1
kx dt
d x k x x M dt
dx dt
dx B
2 2 2 1 2
dt
d x k x x M dt
dx dt
dx Fext B
x 1 x 2
M 1 M 2
k (^1) K 2
B
fext
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Q.3 Use Euler’s method for the following predator – prey system and write a matlab program to solve this:
dx/dt = 3x – xy ; dy/dt = xy – 2y ;
The system is subjected to x(0) = 1, y(0) = 2 ; t(0) = 0. (a) Take Δt = 0.02 and consider time t in domain of [0 , 3]. Then plot y versus x. Why the trajectory do not comeback to the initial point (1, 2). Show the direction of time with arrow as solution progresses. (b) Reduce the step size to 0.003. Is it truly a periodic? Explain. (c) Reduce further the step size to 0.002 and comment on phase diagram. What is value of critical point and what is its type? (d) Use matlab solver ode45 to solve the same for step size of 0.03. Comment on results after comparison with part (a). (e) If you change initial conditions what happens. Show on a phase portrait for three different initial conditions.
Q.4 Use Euler’s method for the following van der pol system and write a matlab program to solve this:
y” - μ(1 – y^2 )y’ + y = 0.
The system is subjected to y(0) = -2, y’(0) = -2 ; t(0) = 0 and μ = 0. (a) Take Δt = 0.01 and consider time t in domain of [0 , 5]. Then plot y’ versus y. (b) Reduce the step size to 0.001. Explain what happened. (c) Reduce further the step size to 0.0001 and comment on phase diagram. What is value of critical point and what is its type? Show the direction of time with arrow as solution progresses. (d) Use matlab solver ode45 to solve the same for step size of 0.01. Comment on results after comparison with part (a). (e) Use matlab stiff solver to solve the same for step size of 0.01. Compare with part (a) and (d) respectively. (f) If you change initial conditions what happens. Show on a phase portrait for three different initial conditions.
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