Consider the inverted pendulum system shown in the figure. The cart has a mass M
ID: 3008410 • Letter: C
Question
Consider the inverted pendulum system shown in the figure. The cart has a mass M, the inverted pendulum has a length 1, and a point mass m. The distance from wall to the hinge point of the pendulum is x (this can be taken as the position of the cart), the angle of the pendulum with the vertical is theta, and u is an applied external force. Ideally, we would like to find a control force u(t) so that the pendulum remains in the upright position in the presence of disturbances. For small angles theta, the following system of equations approximates the dynamics of system. (M + m)x + mL theta = u (I + mL^2) theta + mLx = mgL theta Here I is the moment of inertia of the pendulum rod about its center of gravity, and g is the acceleration due to gravity. Assuming that I = 0, and after some algebra we get: ML theta = (M + m)g theta - u Mx = u- mg theta. Use the variables x_1 = theta, x_2 = theta, x_3 = x and x_4 = x rewrite system (2) in the form x = Ax + bu, where x = (x_1,x_2,x_3,x_4)^T, A is a 4 by 4 matrix, b is a 4 by 1 vector, and u is a scalar control force. Let u(t) = 0 and consider the initial conditions theta(0) = 0.01, theta(0) = -0.1, x(0) = 1 and x(0) = 2. Let M = 2, m = 0.1, L = 0.5, and g = 9.81. Use ode45 to solve the system for 0 lessthanorequalto t Greaterthanorequalto 5. Plot theta(t) versus t and x(t)versus t. Consider now u = -K^T x, with K = [-298.1504;-60.6972;-163.0989;-73.3945]. Use the above u, with the same initial conditions and parameter values as in part 2) to solve the system using ode45 for 0 lessthanorequalto t lessthanorequalto 5. Plot theta(t) versus t and x(t) versus t. Compare the results from parts 2) and 3). Did the control force work?Explanation / Answer
ANS-The free-body diagram above depicts four forces acting upon the object. Objects do not necessarily always have four forces acting upon them. There will be cases in which the number of forces depicted by a free-body diagram will be one, two, or three. There is no hard and fast rule about the number of forces that must be drawn in a free-body diagram. The only rule for drawing free-body diagrams is to depict all the forces that exist for that object in the given situation. Thus, to construct free-body diagrams, it is extremely important to know the various types of forces. If given a description of a physical situation, begin by using your understanding of the force types to identify which forces are present. Then determine the direction in which each force is acting. Finally, draw a box and add arrows for each existing force in the appropriate direction; label each force arrow according to its type. If necessary, refer to the list of forces and their description in order to understand the various force types and their appropriate symbols.