A spherical satellite has 12 identical monopropellant thrust chambers for attitude control with the following performance characteristics: thrust (each unit): 5 lbf; Is (steady state or more than 2 sec): 240 sec; Is (pulsing duration 20 msec): 150 sec; Is (pulsing duration 100 msec): 200 sec; satellite weight: 3500 lbf; satellite diameter: 8 ft; satellite internal density distribution is essentially uniform; disturbing torques, Y and Z axes: 0.00005 ft-lbf average; disturbing torque, for X axis: 0.001 ft-lbf average; distance between thrust chamber axes: 8 ft; maximum allowable satellite pointing position error: ±1◦. Time interval between pulses is 0.030 sec.

(a) What would be the maximum and minimum vehicle angular drift per hour if no correction torque were applied?

The answer should be 0.466 and 0.093 rad/hr.

Why did the spherical satellite go to therapy? Because it had some serious attitude control issues!

Now, let's calculate the maximum and minimum vehicle angular drift per hour without any correction torque.

The maximum drift occurs when the torque disturbance aligns perfectly with the worst case axis (X axis in this case). The minimum drift occurs when the torque disturbance cancels out as much as possible.

To calculate the maximum drift, we use the worst case torque disturbance for the X axis:

Maximum drift = (worst case torque / satellite weight) * (time interval between pulses / 3600)

Maximum drift = (0.001 ft-lbf / 3500 lbf) * (0.030 sec / 3600 sec) * (2π rad / 360 degrees) * 3600 sec

Maximum drift ≈ 0.000044 rad/hr

To calculate the minimum drift, we consider the average disturbances for the Y and Z axes:

Minimum drift = (average Y axis disturbance + average Z axis disturbance) / (satellite weight * number of thrust chambers) * (time interval between pulses / 3600)

Minimum drift = ((0.00005 ft-lbf + 0.00005 ft-lbf) / 3500 lbf) * (0.030 sec / 3600 sec) * (2π rad / 360 degrees) * 3600 sec

Minimum drift ≈ 0.000044 rad/hr

So, the maximum and minimum vehicle angular drift per hour without correction torque would be approximately 0.000044 rad/hr.

But hey, don't count on me for precise calculations. I'm just a funny bot with a head full of jokes!

To find the maximum and minimum vehicle angular drift per hour, we need to calculate the torque exerted by the thrust chambers and the resulting angular drift without any correction torque applied.

First, let's calculate the total thrust generated by all 12 thrust chambers:
Total Thrust = Thrust per chamber × number of chambers
Total Thrust = 5 lbf × 12 = 60 lbf

Next, let's calculate the total torque exerted by the thrust chambers:
Torque = Thrust × Distance between thrust chamber axes
Torque = 60 lbf × 8 ft = 480 ft-lbf

Now, let's consider the disturbing torques:
Disturbing Torque(YZ) = 0.00005 ft-lbf × 2 (for both Y and Z axes) = 0.0001 ft-lbf
Disturbing Torque(X) = 0.001 ft-lbf

The net torque is the difference between the exerted torque and the disturbing torques:
Net Torque = Torque - (Disturbing Torque(YZ) + Disturbing Torque(X))
Net Torque = 480 ft-lbf - (0.0001 ft-lbf + 0.001 ft-lbf) = 479.9999 ft-lbf

Next, let's calculate the resulting angular drift:
Angular Drift = Net Torque / (Satellite Weight × Satellite Diameter)
Angular Drift = 479.9999 ft-lbf / (3500 lbf × 8 ft) = 0.017143 rad

Since the time interval between pulses is given as 0.030 sec, let's calculate the number of pulses per hour:
Pulses per Hour = 3600 sec / (0.030 sec) = 120000

Finally, let's calculate the maximum and minimum vehicle angular drift per hour:
Maximum Angular Drift per Hour = Angular Drift × Pulses per Hour
Maximum Angular Drift per Hour = 0.017143 rad × 120000 = 2.05716 rad/hr

Minimum Angular Drift per Hour = Maximum Angular Drift per Hour / Is (pulsing duration 100 msec)
Minimum Angular Drift per Hour = 2.05716 rad/hr / 200 sec = 0.010286 rad/hr

Therefore, the maximum and minimum vehicle angular drift per hour, without any correction torque, would be 2.05716 rad/hr and 0.010286 rad/hr, respectively.

To calculate the maximum and minimum vehicle angular drift per hour, we need to consider the performance characteristics of the satellite's monopropellant thrust chambers and the given parameters.

Given:
- Thrust of each monopropellant thrust chamber: 5 lbf
- Is (steady state or more than 2 sec): 240 sec
- Is (pulsing duration 20 msec): 150 sec
- Is (pulsing duration 100 msec): 200 sec
- Satellite weight: 3500 lbf
- Satellite diameter: 8 ft
- Distance between thrust chamber axes: 8 ft
- Maximum allowable satellite pointing position error: ±1◦
- Time interval between pulses: 0.030 sec

To find the maximum and minimum vehicle angular drift per hour, we can consider the following steps:

Step 1: Calculate the total thrust produced by all the thrust chambers.
The total thrust can be calculated by multiplying the thrust of each chamber by the total number of chambers. In this case, we have 12 identical chambers with 5 lbf thrust each.
Total thrust = 12 * 5 = 60 lbf

Step 2: Calculate the total torque produced by the thrusters.
The torque produced by each thrust chamber can be calculated by multiplying the thrust of each chamber by the distance between thrust chamber axes. Since all the chambers are identical and equally spaced, we can calculate the torque produced by one chamber and then multiply it by the total number of chambers.
Torque per chamber = thrust per chamber * distance between thrust chamber axes
Torque per chamber = 5 lbf * 8 ft = 40 ft-lbf

Total torque produced by all chambers = Torque per chamber * total number of chambers
Total torque = 40 ft-lbf * 12 = 480 ft-lbf

Step 3: Calculate the maximum and minimum angular drift per pulse.
The maximum and minimum angular drift per pulse can be calculated using the relationship between torque and angular drift.
Angular drift per pulse = torque / (satellite weight * satellite diameter)
Angular drift per pulse = 480 ft-lbf / (3500 lbf * 8 ft)
Angular drift per pulse ≈ 0.00464 rad

Step 4: Convert angular drift per pulse to angular drift per hour.
Since the given time interval between pulses is 0.030 sec, we need to convert the angular drift per pulse to angular drift per hour.
Angular drift per hour = (angular drift per pulse * 3600 sec) / 0.030 sec
Angular drift per hour ≈ 0.00464 rad * 3600 sec / 0.030 sec
Angular drift per hour ≈ 0.5568 rad/hr

However, this calculated value represents the total angular drift per hour if no correction torque is applied. We need to consider the maximum and minimum vehicle angular drift per hour. Since the maximum allowable satellite pointing position error is ±1◦, we can consider ±1◦ as the maximum and minimum vehicle angular drift per hour.

Step 5: Convert ±1◦ to radians and calculate the maximum and minimum vehicle angular drift per hour.
1◦ = (π/180) radians ≈ 0.01745 rad
Thus, the maximum and minimum vehicle angular drift per hour would be:
Maximum vehicle angular drift per hour = 0.01745 rad/hr
Minimum vehicle angular drift per hour = -0.01745 rad/hr

Therefore, the maximum and minimum vehicle angular drift per hour, if no correction torque were applied, would be approximately 0.01745 rad/hr and -0.01745 rad/hr, respectively.