Lab #8: The Unbalanced Wheel

Transcription

1 Lab #8: The Unbalanced Wheel OBJECTIVE Use observations to calculate the radius of gyration of a wheel. Apply energy models to the dynamics of an unbalanced rotating wheel. Learn new computing skills. PLEASE READ THE LAB BEFORE YOU START YOUR WORK. OVERVIEW Ideally, this lab would have a "point" mass attached to a wheel whose moment of inertia was known. Using your knowledge of energy methods, you could predict the motion of the assembly and compare it to measurements. Real life is not so simple. First, we don t have a wheel whose moment of inertia is known. Second, without some extra non-rotating hardware, we can t measure the rotation of the wheel. Therefore, before we can predict the motion of the unbalanced wheel, we must first do some work to calculate the moment of inertia (radius of gyration) of the balanced wheel. Walt Lund and Brad Dunkin from the ITLL staff have built an apparatus consisting of a bicycle wheel, supports (training wheels), and a "yellow" encoder that measures rotation and angular velocity. To this wheel, an extra mass may be attached. (See Figure 1.) Although the moment of inertia is not known, the following physical quantities are known: mass of bicycle wheel, M: 2460 g mass of training wheels, M 0 : 2120 g mass of point mass attachment, m: 545 g radius of the bicycle wheel, R: 32.2 cm radius to center of attachment, r: 24.1 cm radius of gyration of wheel: k (unknown) slope of ramp, β: 4.30 º g = 9.81 m/sec/sec Figure 1: The Unbalanced Wheel 1

2 EQUATIONS The Balanced Wheel The kinetic energy is the sum of the translational and rotational energies: KE = 1 2 (M + M o )V G I G 2 V G is the velocity of the center of mass of the wheel and I G is the moment of inertia, defined as Mk 2. If we assume that the bicycle wheel is rolling without slipping, we can make the usual assumptions about the relationship between V G and the angular velocity ω. The work done (equivalent to the change in potential energy) after the wheel has rotated though an angle θ is: work = (M+Mo)gR sin One can then show that k is defined as: k 2 work 1 = 2 (M + M o )R M 2 The Unbalanced Wheel Calculating the motions of the bicycle wheel with an extra mass, m, is more difficult. For this derivation we will use conservation of energy rather than work-energy. The kinetic energy equation now includes another term. KE = 1 2 (M + M o )V G I G mv b 2 We have assumed that the attached mass is a particle, and thus has no moment of inertia or rotational energy. The velocity of the attached mass, v b, is bω, where b is the distance between the contact point and the mass. By using the law of cosines, you can derive b in terms of r, R, and θ. We then define the kinetic energy: KE = 1 2 (M + M o )R Mk m(r2 + r 2 + 2rRcos ) 2 We can define potential energy by putting our x-y axis at the center of wheel at θ = 0. The potential energy at arbitrary θ is: 2

3 PE= (M + M o )gr sin +[ mgr sin + mgr cos( + )] You can then easily solve for ω as a function of θ using these expressions for kinetic and potential energy. DATA COLLECTION As a group, we will collect several trials of data for the wheel both with and without the extra mass. These files will be saved in the ASEN2003/Users directory in the ITLL. These data files have columns containing time (s), angle θ (revolutions), and angular velocity ω (rpm). As in previous labs, you ll be importing these data into MATLAB using the load command. The files contain data for 6 seconds and may include data after the wheel left the ramp. Use Matlab to restrict your analysis to the first three revolutions of data. ANALYSIS The Balanced Wheel One difficulty with the encoder is that since there are only 30 holes in it, it only outputs a value for θ every 12º. Depending on where the encoder is when we start recording data, the wheel could turn as much as 20º before the first θ value of 0º is recorded. This will produce a large error in our analysis. In homework and lecture we showed that the angular acceleration α of a wheel rolling down a ramp is constant. This means that ω should increase linearly with time and θ should increase as a quadratic function of time. We can use this fact to help us find the θ offset. By plotting θ (y-axis) against time (x-axis) you should see this quadratic dependence. Note that the parabolic shape does not reach its minimum. If we could extend the parabola back in time, the minimum of the parabola would occur just when we released the wheel. The value of θ at the minimum tells us how many degrees the wheel turned before the encoder read zero. For example, if the value of the parabola at its minimum is 15.3º, then we know that the θ=0 data point is really at 15.3º. Likewise, every data point should be corrected by this offset. How do we extend this parabola? Use the MATLAB command polyfit to fit a second order polynomial to the (θ,t) data. Then use the MATLAB command polyval to define an array of smooth (θ,t) values for each trial. The minimum θ value is the measurement offset described in the previous paragraph. The usage of polyfit and polyval is explained in the MATLAB help section. Once we have corrected the θ data for the measurement offset, we can calculate the radius of gyration, k. As discussed in the background section, k depends on both θ and ω. Which θ and ω should you use? For this lab you should find a k value for each (θ, ω) and then compute an average for each trial. 3

4 TURN IN: Plot θ (y-axis) vs time (x-axis) for all trials. Plot ω (y-axis) vs time (x-axis) for all trials. Include derived θ offsets on the figure. For trial 1, plot θ (y-axis) vs time (x-axis) data (symbol) and the polynomial fit (line). Plot ω (y-axis) vs θ (x-axis) for all trials. Include k values as added text (use num2str). Report the average k over all the trials. Use this average k value in the next section. The Unbalanced Wheel Unfortunately, there is not a simple polynomial that we can fit to the unbalanced data to find the θ offset. We could use more complicated methods to determine it, but instead, just assume that the offset is 15º. i.e. add 15º to all your unbalanced wheel measurements. TURN IN: Plot ω (y-axis) vs. θ (x-axis) for all four unbalanced wheel trials. Plot ω (y-axis) vs. θ (x-axis) for the theoretical model (line) and the average of the 4 trials (symbol). How well does the theoretical model agree with your measurements, qualitatively and quantitatively? Describe at least one way the theoretical model could be improved. Your commented MATLAB code for both the balanced and unbalanced wheel. GRADING COMMENTS FROM THE INSTRUCTOR All plots of θ should be in degrees. All plots of ω should be in radians/second. Your main program must call at least two functions - one to compute k and the other to read in and plot the unbalanced wheel trials. The constants of the experiment, listed on page 1, must be defined in your main code. If these values are needed in the functions, they need to be sent to the function as input. You are not limited to just two functions - you can use more if you like. While you could hardwire the value of g in your functions, I would like you to define it in the main code and send it to the function with your other code. Why? What if someone, i.e. Walt, gave you experimental data in English units? You would have to try and remember where you 4

5 hardwired g. This way you can just change the constants in the main code, and everything should work immediately. I've noticed that a lot of people let MATLAB choose which colors to use in their plots. This is fine if you plan to use a color printer. If you are going to use a black and white printer, then you need to use black dashed, dotted, dash-dotted, and regular lines. (Some) colors do not print well in black and white and make it more difficult to read your work. Also, some of you have been using color pencils to add the color back in after using a black and white printer. This detracts from the professional quality of your report. A small number of points will be deducted in these situations. No cover sheet, abstract, table of contents, description of the experiment, derivation of equations or conclusions are expected or required. The name of the lab, your lab section, and the lab group member names should be at the top of what you turn in. MATLAB The most significant new functions required for this lab are polyfit and polyval. Polyfit makes a polynomial least-squares fit to some data. It returns some statistics on how good the fit is and the coefficients of the polynomial that best-fits the data. Imagine you have two arrays of data, y and t. Where t is an array of times from 0 to 27 seconds. Let s imagine that y is supposed to equal t 3 +3t 2 +5, but the data are somewhat noisy. %do a third order polynomial fit on y based on t. [p,s]=polyfit(t,y,3); s will contain some stuff that we don t really care about right now. p will be a 1 x 4 array with the coefficients of the polynomial y=f(t). In our case, p will be [ ] if there is no observation error. Once you have these coefficients, it might be nice to use them to do something like plot a best-fit polynomial through the original data. To do that, you d use polyval. polyval takes the coefficients produced by polyfit and an array of independent variables and produces the polynomial curve. besty_array = polyval(p,t); 5