Reference no: EM132158799

USE MATLAB TO SOLVE

Problems:

1. a) The motion of a given vehicle can be modeled by the ordinary differential equation y¨+4y?+6y=0. Analytically convert this ordinary differential equation into an equivalent system of coupled first order ordinary differential equations.

b) A particular spring-mass system can be modeled by the coupled ordinary differential equations

{m1x¨1+k1x1+k2(x1-x2)=0

m2x¨2+k2(x2-x1)=5sin(t).

Analytically convert this coupled ordinary differential equation system into an equivalent system of coupled first order ordinary differential equations.

2. Use ode45 and one other solver of your choice to numerically solve the systems in Problem 1, and plot your states vs. time.

Supply your own initial conditions, time steps, and time intervals. Experiment with how the results change based on varying these inputs, and comment on your observations in a statement that prints to the Command Window. In the case of Problem 1(a), since this is a second order ordinary differential equation, you will need to specify two initial conditions: y(0) and y?(0).

In the case of Problem 1(b), since this is a coupled system of second order ordinary differential equations, you will need to specify four initial conditions: x1(0), x?1(0), x2(0), and x?2(0). Use a value of 10 N/m for k1, 5 N/m for k2, 15 kg for m1, and 12 kg for m2.

3. Consider the integral ?e^x dx bounds (0,1). Analytically calculate the integral, and use that result to compare the accuracy against the:

a) Left Riemann Sum

b) Right Riemann Sum

c) Midpoint Riemann Sum

d) Trapezoid Rule. (Note that you could do this "by hand" by coding the algorithm, or by using a built-in technique.)

e) Simpson's Rule. (Remember that Simpson's rule requires an even number of subintervals; in the alternative of doing this by hand, you could use the integral command to similar effect.)

Experiment with the number of subintervals, making sure to study how this affects the solution accuracy. Comment on your observations in a statement that prints to the Command Window.