We estimated the inertia of the motor armature through the use of a bifilar pendulum. This method involves hanging the ends of the arm by two long strings and measuring the time period of small angle oscillations in the horizontal plane. This gives the moment of inertia about the point halfway between the hanging strings, or the geometric center of the arm, through the relation:
Where m is the arm mass, g is gravity, r is half the distance between the hanging strings, h is the length of the strings, and T is the time period of a single oscillation. All of these values were easily measured with the time period taken as one tenth the value of the time required for 10 oscillations. As the resulting inertia is about the geometric center of the arm, two applications of the parallel-axis theorem are required to find the inertia about the axis of the rotation shaft. First, the inertia is translated from the geometric center to the center of mass and second from the center of mass to the shaft axis. This required knowing the distance between these three points. The geometric center and shaft axis were easily measured, while the center of mass position was found by balancing the arm on a fine edge. The net result for the inertia about the motor rotational axis was therefore found to be J = 5.64 x 10-5 kg•m2 or J = 564 g•cm2.
B
To find the damping coefficient of the motor, we ran an experiment where we attached a relatively large weight to the armature, held the motor vertically, and let it oscillate as a pendulum. Assuming small angle oscillations, the equation of motion describing the arm behavior can easily be derived from a standard 2nd order rotational mass-spring-damper system. Using the Hall effect sensor to estimate position, the results of our test appear below with the peaks of the oscillations describing a exponentially decreasing function:
H0 is the initial Hall effect reading when the arm is released and HC is the reading at the center when the arm is completely vertical and at equilibrium. Pulling out the data of just the oscillation peaks to measure the rate of decay, the values can be modified to fit the modified equation below, which is conducive to a linear fit.
The linear fit was then done on these data points, resulting in a value for T of 1.24s. From the equation of motion, T = J/b and compensating for the change in J from the mass added to arm, a value of b = 9.1 x 10-5 N•s/m can be calculated.
K_t
To find K_t, we mounted the motor sideways on the table and hung various weights off the end of the armature. The voltage at the power supply was then slowly increased until enough current was flowing to produce enough torque to suspend the weights. We then were able to collect the data points measuring the current with a multimeter and knowing the torque produced on the motor by the weights. As can be seen in the plot of our data, the data points do not tend towards the origin, but rather a point of no current and non-zero torque. We believe this the static friction inherent that has to be overcome before lifting the weight and so we ignore the offset and simply use the slope of the linear fit as our torque constant. This value was Kt = .01562 N•m/A.
R
R was found simply by using the multimeter to measure the resistance in the motor wire. We measured a value of R = 0.4 Ω.
I really like how you guys added weight to the motor armature to help record better data, and then compensated for the increased inertia in the equations later. This raises questions about how different the friction of the system is with the increased mass, but in this case with the unsealed cartridge ball bearings I'm sure the error could be shown to be much smaller than other groups with very noisy data and few oscillations.
ReplyDelete