
#1
Apr1710, 07:50 AM

P: 6

I'm having a little trouble using the Latex thing so I've only used it for some of the equations.
1. The problem statement, all variables and given/known data Elevation control of a tracking antenna. Equation of motion for the system J[tex]\theta[/tex][tex]^{..}[/tex]+B[tex]\theta[/tex][tex]^{.}[/tex] = T_{c}+w ... Equation 1 (those dots are meant to be above the thetas, but I don't know how to do that... see attachment q1 for the equation) Antenna and drive mechanism have J = moment of inertia, B is the damping coefficient, T_{c} is the torque from the drive motor, w is a disturbance torque, and [tex]\theta[/tex] is elevation angle of the dish. Model the dish as a thin disc with diameter 2m, mass 40kg and rotates about central axis (J=MR^{2}/4) Damping coefficient = 20Nmsec Torque exerted by DC motor: T_{m}=K_{T}/R_{a}VaK_{T}K_{B}/R_{a}[tex]\theta[/tex]_{m}[tex]^{.}[/tex] ... equation 2 where [tex]\theta[/tex]_{m} is the angular position of the motor shaft which is connected to the load, via a 50:1 gearbox. Motor parameters J_{m}=0.01kgm^{2}, R_{a}=5ohms, K_{T}=0.2Nm/A, K_{B}=2Vsec a) Draw a block diagram of the motor itself(i.e. showing the relationship between the applied voltave V_{a}, the load torque T_{c} and motor speed [tex]\theta[/tex]_{m}[tex]^{.}[/tex] and position [tex]\theta[/tex]_{m}) b) Find the transfer function between the applied voltage V_{a} and the antenna angle theta assuming zero disturbance torque. c) Suppose the applied voltage is computed so that theta tracks a reference command theta_{r} according to the feedback law V_{A}=K(theta_{r}theta) where K is the feedback gain. Draw a block diagram of the resulting feedback system showing both theta, the reference position theta_{r} and disturbance torque w. Find the transfer function between theta_{r} and theta assuming zero disturbance torque. There are a few other parts to the question, but they need to be worked through sequentially and I'm a little more hopeful about having an idea of them once I know what to do here. 2. Relevant equations Laplace Transforms... The derivative one: Laplace{\ddot{f}}=s^{2}F(s)sf(0)s\dot{f}(0) Transfer function of a closed loop system T(s) = G(s)/(1+G(s)H(s)) 3. The attempt at a solution Taking equation 1 and laplace transforming, assuming zero initial conditions: Js^{2}theta(s) + Bstheta(s) = T_{c}(s) +w Rearrange: theta(s) = 1/(Js^{2}+Bs) * (T_{c}(s) +w) See attachment 1, after where I've written equations 1 and 2... I think I'm going ok up to where I try to draw the block diagram with T_{m} in attachment 2. Then I'm not sure what to do with the equation to make it into a block diagram. I don't think it's a closed loop (because that comes later in part c with the reference..?) but I seem to have two inputs, V_{a} and theta_{m}... And so I'm not sure how to go about putting it in a block diagram other than what I've done. Any help would be super. Thanks. 


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