## Homework Assignment 7. Due March 19 1. Consider the differential equation: ?? ?? = − 1 2 ? sin? ? with initial condition given by ?(0) = 1 Solve this equation from t = 0 to t = 8π using the following methods: (a) Solve analytically by separating variables and integrating. (b) Solve using the 4th-order Runge-Kutta method (write your own code for this, do not use the MATLAB provided ODE solvers) for the following two step sizes: I. Maximum step size for stability (don’t try and do this analytically – try out your code for different step sizes to find the stability limit). II. Maximum step size for a time-accurate solution. “Good” accuracy can be defined in several ways, but use the definition that the numerical solution remains within 2% of the true solution a t = nπ. (c) Solve using the MATLAB function ode45. 2. A car and its suspension system traveling over a bumpy road can be modeled as a mass/spring/damper system. In this model, ?? is the vertical motion of the wheel center of mass, ?? is the vertical motion of the car chassis, and ?? represents the displacement of the bottom of the tire due to the variation in the road surface. Applying Newton’s law to the two masses yields a system of second-order equations: ???̈? + ??(?̇? − ?̇?) + ??(?? − ??) + ???? = ???? ???̈? − ??(?̇? − ?̇?) − ??(?? − ??) + ???? = 0 (a) Convert the two second-order ODE’s into a system of 4 first-order ODE’s. Write them in standard “state-space” form. (b) Assume the car hits a large pothole at t = 0 so that ??(?) = ?−0.2 m 0 ≤ ? < 0.2 s 0 ? > 0.2 s Create a MATLAB function that returns the right hand sides of the state-space equations for an input t and an input state vector. (c) Solve the system on the time interval [0 60] seconds using the MATLAB function ode45. Find the displacement and velocity of the chassis and the wheel as a function of time. Use the following data: ?? = 100 kg, ?? = 1900 kg, ?? = 145 N/m, ?? = 25 N/m, ?? = 150 N-s/m 3. Write a MATLAB program to simulate the dynamics of a helicopter lifting a survivor. When lifting the survivor into the helicopter with a constant speed winch, the resulting dynamics are non-linear, and stability is dependent upon the winch speed. Using polar coordinates, we can find the equations of motion to be: −?? sin ? = ????̈ + 2?̇?̇? ?̇ = constant (negative) Notice that the mass of the survivor factors out and thus the solution is independent of the mass of the person being lifted. In these equations, r is the instantaneous length of the winch cable, g, is the gravitational constant, and θ is the angle of the swing. You may choose to use either your Runge-Kutta solver from problem 1 or ode45 to integrate the equations of motion. This problem is of particular interest to the survivor since an unstable condition can cause the angle of the swing to exceed 90⁰, essentially placing him/her in danger of being beheaded by the rotor blades of the rescue helicopter. Also, it is desirable to retrieve the survivor as fast as possible to get away from the danger. Use your program to determine the maximum winch speed for which the survivor will not swing above the helicopter attach point for a lift from the initial conditions: ?? = 0.1 ??? ?? ̇ = 0 ?? = 34 ? And ending when ? = 0.5 ?. The maximum lifting speed of the winch is 5 m/s. Present your results for the above problems in an appropriate fashion. For problem 1, be sure to include a comparison of the numerical methods with each other and with the true solution. Be sure to discuss your findings with respect to the notions of stability and accuracy of the numerical methods. For problem 2, ensure that your results are easily interpreted by a reader. Students receiving a score of 70% or above on these two problems will receive credit for outcome #5. For problem 3, if you receive at least 70% of the points, you will receive credit for outcome #4.

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