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Problem 7.1 : Derivation of the equation of the motion for a rolling half-disk
Half-disk is rolling without slipping on the plane surface.
i) Derive the equation of the motion. Keep all nonlinear terms and do not linearize.
ii) Linearize the nonlinear motion in case of small angle oscillation. Hint: use small angle approximations such
as sinθ≈θ
iii) Solve the linearized equation of motion obtained in ii) analytically with following initial conditions:
.

Solution:
Rolling half-disk has only one degree of freedom with the constraint of A v O = rq Generalized coordinate q1 is the
rotation angle of half-disk ( q1 = q ).
Frame A is attached to the ground, and frame B is attached to the center of disk, O. For kinetic energy of rolling half-
disk,
To obtain the speed of half-disk at the center of mass with respect to frame A, A |v CM | , the linear velocity for
rolling half-disk at the center of mass with respect to frame A, A v CM should be calculated first.

So, the speed of half-disk at the center of mass with respect to frame A is
In addition, the moment of inertia around the center of mass is
Therefore,
For the potential energy of rolling half-disk,
where A P is the position of center of mass
APCM can be calculated as below:

where L : Lagrangian, T : kinetic energy, V : potential energy, and q : generalized coordinate

(ii) For small θ, sinθ≈θ & cosθ≈1
In addition, the product of θ and higher order derivatives such as θ,θ
goes to zero: θ 2 ≈0
Therefore, the linearized equation of motion for rolling half-disk
(iii) Homogenous solution can be obtained as follows:
Therefore,

For the given initial conditions θ(0)= θ0, θ(0)= θ0
Therefore, the solution for the linearized equation of the motion is

Problem 7.2 : Generate simulation codes for motion for rolling half-disk
Generate functions to simulate the trajectory of θ for rolling half-disk based on following instructions. Simulation time
is 10 seconds. Set r =1m.
i) Use the nonlinearized equation of motion obtained in problem 7.1. i). Use “ode45” for simulation. Function
name (and m-file name) should be ‘RockerRK_your_kerberos_name’ and upload it to 2.003 MIT Server site. You
also submit the hardcopy of your code with appropriate comments. Function has following structure.
function [t,theta]= RockerRK_your_kerberos_name(theta0)
t: time matrix (N×1)
theta: angle matrix (N×1)
theta0: initial condition matrix (1×2)
ii) Use analytical solution obtained in 7.1. iii). Trajectory can be obtained by simply evaluating the analytical solution as
a function of time. Function name (and m-file name) should be ‘RockerAN_your_kerberos_name’ and upload it to
2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments.
Function has following structure.
function [t,theta]= RockerAN_your_kerberos_name(theta0)
t: time matrix (N×1)
theta: angle matrix (N×1)
theta0: initial condition matrix (1×2)

Solution:
i) The same method used in the homework #6 is used: Runge-Kutta. Most procedures are identical to the one used in
homework #6. Following is the m-code for the simulation of rolling half-disk.

ii) The analytic solution you obtained in P7.1 iv) is used to find the trajectory of rotation angle of rolling half-disk. First,
you make time vector which have numbers from 0 to 10 with enough step to describe motion well (I chose 0.01 sec.)
Then, some constants are given, and calculate solution with respect to time matrix. Matrix operation should be used.
The following is m-code for calculating the analytical solution for rolling half-disk.

Problem 7.3 : Trajectory of θ (t) for both small and large angle oscillations
For the initial conditions given below, simulate the nonlinear motion and the linearized motion for rolling half-disk
up to 10 seconds (use results of Problem 7.2). Compare these results by plotting both of them in the same figure
with the appropriate legends. Are they identical? Otherwise, explain why not. You should hand in hardcopy of the
plots.
i) Small angle oscillation: θ (0) = 5o and θ&(0) = 0 .
ii) Large angle oscillation: θ (0) = 30o and θ&(0) = 0 .
Solution:
i) As expected, the results with Runge-Kutta method and analytic approach are pretty close. The linearization
works for small angle rotation very well. Note that the unit of angle is radian, not degree, when you give the
initial conditions to function you made. Triangular function in MATLAB such as sin, cos, and tan accept for
only radian. The following codes describe how to generate the below plot where the result with different
simulation methods are compared.

ii) For the case of large angle rotation of half-disk, the linearized rotation motion is quite different from numerical
simulation of nonlinear rotation motion. Rotation obtained with nonlinear equation is a little slower than the one
with the linear equation since nonlinear terms in the differential equation is dominant when the rotation angle
becomes larger. The following codes describe how to generate the below plot where the result with different
simulation methods are compared.

Control System Assignment Help

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