Question

11. Fig.11 shows an alignment system being considered for directing a beam of microwave power onto a satellite in orbit. A rigid parabolic reflector unit has mass 30kg and its center of gravity is at point B. Its moment of inertia about B is 4 kgm². Beneath this, there is a rigid platform having mass 70kg. Its c.o.g. is at point E and its moment of inertia about E is 8 kgm². Points D and F on the rigid platform are each supported from ground by a spring-damper unit as shown where $k = 4000$ N/m and $c = 0.5Ns/m$. There are also springs of stiffness $k$ separating points A, D and points C, F. (a) Prepare an approximate STATE-SPACE model for the dynamics of this system. The inputs are to be the forces {$f_1, f_2$} (acting to separate A,D and C, F respectively). The outputs from this model are to be the vertical deflection (positive upwards), $u$, and angle (anticlockwise positive), $\theta$, of the reflector about its centre of gravity. Use the symbols $v$ and $\alpha$ to represent the vertical deflection and angle of the rigid platform. (b) A controller proposed for this system is intended to keep both $u$ and ($\theta_{ref} - \theta$) close to zero where $\theta_{ref}$ is a reference angle for the reflector determined by satellite position. The controller is described by: $f_1 = -200u - 10000(\theta - \theta_e)$ $f_2 = -200u + 10000(\theta_{ref} - \theta)$ Prepare a new STATE-SPACE model for the closed-loop system where $\theta_{ref}$ is the only input and the output is ($\theta_{ref} - \theta$). (c) Describe how you could determine whether the closed-loop system is stable and comment on what other information you could obtain from the same calculation. (Do not attempt to do the actual calculation) A 0.6m 0.6m $\theta$ B C $f_1$ k $f_2$ D E F k c k c Figure 11. A proposed beam-alignment mechanism.

          11.
Fig.11 shows an alignment system being considered for directing a beam of microwave power onto a
satellite in orbit. A rigid parabolic reflector unit has mass 30kg and its center of gravity is at point B. Its
moment of inertia about B is 4 kgm². Beneath this, there is a rigid platform having mass 70kg. Its c.o.g. is at
point E and its moment of inertia about E is 8 kgm². Points D and F on the rigid platform are each supported
from ground by a spring-damper unit as shown where $k = 4000$ N/m and $c = 0.5Ns/m$. There are also springs
of stiffness $k$ separating points A, D and points C, F.
(a) Prepare an approximate STATE-SPACE model for the dynamics of this system. The inputs are to be the
forces {$f_1, f_2$} (acting to separate A,D and C, F respectively). The outputs from this model are to be the
vertical deflection (positive upwards), $u$, and angle (anticlockwise positive), $\theta$, of the reflector about its
centre of gravity. Use the symbols $v$ and $\alpha$ to represent the vertical deflection and angle of the rigid
platform.
(b) A controller proposed for this system is intended to keep both $u$ and ($\theta_{ref} - \theta$) close to zero where $\theta_{ref}$ is
a reference angle for the reflector determined by satellite position. The controller is described by:
$f_1 = -200u - 10000(\theta - \theta_e)$
$f_2 = -200u + 10000(\theta_{ref} - \theta)$
Prepare a new STATE-SPACE model for the closed-loop system where $\theta_{ref}$ is the only input and the
output is ($\theta_{ref} - \theta$).
(c) Describe how you could determine whether the closed-loop system is stable and comment on what
other information you could obtain from the same calculation. (Do not attempt to do the actual
calculation)
A
0.6m
0.6m
$\theta$
B
C
$f_1$
k
$f_2$
D
E
F
k
c
k
c
Figure 11. A proposed beam-alignment mechanism.

11.
Fig.11 shows an alignment system being considered for directing a beam of microwave power onto a
satellite in orbit. A rigid parabolic reflector unit has mass 30kg and its center of gravity is at point B. Its
moment of inertia about B is 4 kgm². Beneath this, there is a rigid platform having mass 70kg. Its c.o.g. is at
point E and its moment of inertia about E is 8 kgm². Points D and F on the rigid platform are each supported
from ground by a spring-damper unit as shown where k = 4000 N/m and c = 0.5Ns/m. There are also springs
of stiffness k separating points A, D and points C, F.
(a) Prepare an approximate STATE-SPACE model for the dynamics of this system. The inputs are to be the
forces f1, f2 (acting to separate A,D and C, F respectively). The outputs from this model are to be the
vertical deflection (positive upwards), u, and angle (anticlockwise positive), θ, of the reflector about its
centre of gravity. Use the symbols v and α to represent the vertical deflection and angle of the rigid
platform.
(b) A controller proposed for this system is intended to keep both u and (θref - θ) close to zero where θref is
a reference angle for the reflector determined by satellite position. The controller is described by:
f1 = -200u - 10000(θ - )
f2 = -200u + 10000(θref - θ)
Prepare a new STATE-SPACE model for the closed-loop system where θref is the only input and the
output is (θref - θ).
(c) Describe how you could determine whether the closed-loop system is stable and comment on what
other information you could obtain from the same calculation. (Do not attempt to do the actual
calculation)
A
0.6m
0.6m
θ
B
C
f1
k
f2
D
E
F
k
c
k
c
Figure 11. A proposed beam-alignment mechanism.

Added by Soledad M.

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