System Dynamics and Control problem - finding the block diagram from given equations


by super sky
Tags: block, control, diagram, dynamics, equations
super sky
super sky is offline
#1
Apr17-10, 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] = Tc+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, Tc 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=MR2/4)
Damping coefficient = 20Nmsec

Torque exerted by DC motor: Tm=KT/RaVa-KTKB/Ra[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 Jm=0.01kgm2, Ra=5ohms, KT=0.2Nm/A, KB=2Vsec

a) Draw a block diagram of the motor itself(i.e. showing the relationship between the applied voltave Va, the load torque Tc and motor speed [tex]\theta[/tex]m[tex]^{.}[/tex] and position [tex]\theta[/tex]m)

b) Find the transfer function between the applied voltage Va and the antenna angle theta assuming zero disturbance torque.

c) Suppose the applied voltage is computed so that theta tracks a reference command thetar according to the feedback law VA=K(thetar-theta) where K is the feedback gain. Draw a block diagram of the resulting feedback system showing both theta, the reference position thetar and disturbance torque w. Find the transfer function between thetar 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}}=s2F(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:
Js2theta(s) + Bstheta(s) = Tc(s) +w

Rearrange:
theta(s) = 1/(Js2+Bs) * (Tc(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 Tm 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, Va and thetam... 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.
Attached Thumbnails
q1.JPG   q2.JPG  
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