A wheel attached to a linear damper rolls without slipping as shown. Assume the wheel starts...

  

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A wheel attached to a linear damper rolls without slipping as shown. Assume the wheel starts from rest. 0 b R J, m a) Draw the FBD of the system b) Set up the differential equations that define the system above. Solve the differential equation to provide the solution of v(t). In Matlab, numerically integrate the differential equations to simulate the system and determine v(t) (There is a page on the website to help set this up - if you have questions come see me). What did you pick for At? Why? Assume the following values: m=50 kg J= 10 kg-m^2 R=0.25 m 0-10 degrees b=20 Ns/m c) Plot the analytical solution vs. the numerical integration on one plot (use solid and dashed lines with a legend, and label your axis - all in Matlab. There is a page on the website on how to do this as well.). d) Now lets assume instead of the damper the wheel is subjected to air drag (Fad-Dv²). Find the value of D to provide the same steady state velocity as before. What is the value of D? e) Now plot on the same plot the solution to this new differential equation and the solution to the differential equation from part (b). Bonus: The 1st order differential equation with non-linear air-drag can actually be solved analytically using separation of variables. Solve the non-linear differential equation and add the plot of the closed-form (analytical) solution to your plot. Note: Solve the problem symbolically. Place all of your equations in Matlab in symbol form. Then at the beginning of your script assign values for all of your symbols - this is part of the power of Matlab - you can change the value of "m" at the top without having to rewrite all of your equations! Make sure you set up the script correctly, since your m-files will be graded too. A wheel attached to a linear damper rolls without slipping as shown. Assume the wheel starts from rest. 0 b R J, m a) Draw the FBD of the system b) Set up the differential equations that define the system above. Solve the differential equation to provide the solution of v(t). In Matlab, numerically integrate the differential equations to simulate the system and determine v(t) (There is a page on the website to help set this up - if you have questions come see me). What did you pick for At? Why? Assume the following values: m=50 kg J= 10 kg-m^2 R=0.25 m 0-10 degrees b=20 Ns/m c) Plot the analytical solution vs. the numerical integration on one plot (use solid and dashed lines with a legend, and label your axis - all in Matlab. There is a page on the website on how to do this as well.). d) Now lets assume instead of the damper the wheel is subjected to air drag (Fad-Dv²). Find the value of D to provide the same steady state velocity as before. What is the value of D? e) Now plot on the same plot the solution to this new differential equation and the solution to the differential equation from part (b). Bonus: The 1st order differential equation with non-linear air-drag can actually be solved analytically using separation of variables. Solve the non-linear differential equation and add the plot of the closed-form (analytical) solution to your plot. Note: Solve the problem symbolically. Place all of your equations in Matlab in symbol form. Then at the beginning of your script assign values for all of your symbols - this is part of the power of Matlab - you can change the value of "m" at the top without having to rewrite all of your equations! Make sure you set up the script correctly, since your m-files will be graded too.

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