Tracking Antenna Elevation Control: Equation to Block Diagram

[super sky]

I'm having a little trouble using the Latex thing so I've only used it for some of the equations.

Homework Statement

Elevation control of a tracking antenna.
Equation of motion for the system J$$\theta$$$$^{..}$$+B$$\theta$$$$^{.}$$ = 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 $$\theta$$ is elevation angle of the dish.

Model the dish as a thin disc with diameter 2m, mass 40kg and rotates about central axis (J=MR²/4)
Damping coefficient = 20Nmsec

Torque exerted by DC motor: T_m=K_T/R_aVa-K_TK_B/R_a$$\theta$$_m$$^{.}$$ ... equation 2
where $$\theta$$_m is the angular position of the motor shaft which is connected to the load, via a 50:1 gearbox.

Motor parameters J_m=0.01kgm², 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 $$\theta$$_m$$^{.}$$ and position $$\theta$$_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.

Homework Equations

Laplace Transforms... The derivative one: Laplace{\ddot{f}}=s²F(s)-sf(0)-s\dot{f}(0)

Transfer function of a closed loop system T(s) = G(s)/(1+G(s)H(s))

The Attempt at a Solution

Taking equation 1 and laplace transforming, assuming zero initial conditions:
Js²theta(s) + Bstheta(s) = T_c(s) +w

Rearrange:
theta(s) = 1/(Js²+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.

 

[KataKoniK]

Thank you for your post and for providing the necessary equations and information. I would be happy to assist you with your problem.

Firstly, to type the dots above the thetas in the equation, you can use the caret symbol "^" followed by the dot symbol "." in between the curly braces, like this: \theta^{..}. This will display the dots above the thetas.

Now, moving on to the problem, let's start with drawing the block diagram for the motor itself. The motor is a component in the overall system, so it is not a closed loop yet. We can start with the input voltage Va, which is the applied voltage to the motor. This voltage is responsible for driving the motor and producing the output torque Tc. The motor also has an angular position \thetam, which is connected to the load through a gearbox. This position is also an output of the motor. So, our block diagram for the motor would look something like this:

[Input] Va ---> [Motor] ---> [Output] Tc, \thetam

In this block diagram, the motor is represented as a box and the input and output are shown with arrows pointing towards the box. The output torque Tc and the angular position \thetam are the outputs of the motor and are affected by the input voltage Va.

Next, we can find the transfer function between the input voltage Va and the output angular position \thetam. Using equation 2, we can write the transfer function as:

G(s) = \thetam(s)/Va(s) = (KT/Ra)/(Js + B)

where G(s) is the transfer function, \thetam(s) is the Laplace transform of the angular position \thetam and Va(s) is the Laplace transform of the input voltage Va. This transfer function tells us how the input voltage affects the output angular position.

Moving on to part b), we can now find the transfer function between the applied voltage Va and the antenna angle \theta. Since the motor is connected to the antenna through the gearbox, we can use the transfer function we just found to find the overall transfer function between Va and \theta. Using the fact that the motor is connected to the antenna through a 50:1 gearbox, we can write the overall transfer function as:

H(s) = \theta(s)/Va(s) = G(s)/50 = (KT/R