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24 November 2024 16:15
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Welcome to the gyroscope forum. If you have a question about gyroscopes in general,
want to know how they work, or what they can be used for then you can leave your question here for others to answer.
You may also be able to help others by answering some of the questions on the site.
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Question |
| Asked by: |
Harvey Fiala |
| Subject: |
Precession Based Questions |
| Question: |
TAKE HOME QUIZ #2
For those who profess to fully understand the physics of a precessing gyroscope, here is take-home exam # 2. This series of questions cover basic material that is required in the design and development of mechanical precession-based inertial propulsion systems.
The basic equation governing gyroscopic motion is: T=Ir ωs ωp where T is the applied torque, Ir is the Mass Moment of Inertia of a rotor, ωs is the angular velocity of a rotor R about its spin axis, and ωp is the angular velocity of the rotor about the pivot point (Z-axis) of the resulting precession. The product (Ir ωs) is known as the Angular Momentum of the rotor. Assume a solid disk rotor. Assume natural precession where the torque T is provided by the force of gravity (g) at the surface of the earth and ωp is the angular velocity of the resulting natural precession in the horizontal plane. The torque T = WrLr = MrgLr , where Wr is the Weight of the rotor Rr and Lr is the Length of its lever arm or spin axis from the center of mass of the rotor to its pivot point. Mr is the Mass of the rotor and g is the acceleration of gravity at the surface of the earth. [s and p are supposed to be subscripts to ω. Likewise in the following paragraphs, 1 and 2 are supposed to be subscripts to Lr , r, ω and I and in several instances 2 is a superscript to r]
Assume a spin axis or lever arm of negligible mass and having a length of Lr from the pivot point to the center of mass of the rotor with a rotor R having a Mass Moment of Inertia I r about its spin axis, and an angular velocity about its spin axis of ωs . Neglect the effects of friction and air resistance. Calculate all answers to approximately three significant figures. Show the calculations and equations used to get the answers.
All answers count 2 points except 1.3, 1.6, 3.2, and 3.4 which each count 14 points. The highest correct score wins.
Use the following equations:
T=Ir ωs ωp, q = at2/2, T = Ir dw/dt = Ira,
Ir = Mr (3rr2 + hr2 )/12 = mass moment of inertia for a centroidal axis perpendicular to the spin axis for a solid disk rotor.
W = Mg, I = Ir + MrLa, w = a t , Work = I w2 /2, Ir = Mr R r2/2, Power = work/time
ωp2 = (W1 L1 + W2 L2 + W3 L3 + … Wn Ln ) / (Is1 ωs1 + Is2 ωs2 + Is3 ωs3 + …+ Isn ωsn )
Q1: What is the precessional angular velocity ωp in the horizontal plane for the 30 lb spinning rotor with a lever arm of negligible mass due to the earth’s gravity in radians/sec
A1: (answer: ωp = 1.37 radians/sec = 13.0 revolutions/minute)
Q1.1: Recall that the rotor weighs 30 lbs and the spin axle is of negligible weight. How long would it take for the spinning rotor to accelerate from zero rpm to the horizontal precessional angular velocity ωp arrived at in question 1.
A1.1:
Q1.2: How is it possible that a 30 lb object can be accelerated to the angular velocity ωp in zero time?
A1.2:
Q1.3: What is the rate of acceleration (a) in deg/sec2 for the 30 lb spinning and precessing rotor to get up to the angular velocity arrived at in question 1?
A1.3:
Q1.4: How is it possible for the acceleration rate to be that high?
A1.4:
Q1.5: How many radians and degrees did it travel before getting up to speed?
A1.5:
Q2: At what rate would a 30 ft-lb torque about the Z-axis accelerate the 30 lb non-spinning rotor with a lever arm of negligible mass of question 1?
Hint: Use the parallel transfer theorem for mass moments of inertia.
A2: (answer: a = 30.3 rad/sec2)
Q2.1: How long would it take to get up to the precessional angular velocity ωp arrived at in question 1?
A2.1:
A2.2: How much work was performed in accelerating the rotor and lever arm to the precessional angular velocity ωp?
A2.2
A2.3: What power level did it require to bring it up to the precessional angular velocity ωp?
A2.3:
Q2.4: How many radians and degrees did it travel before getting up to speed?
A2.4:
Q3: Calculate the precessional angular velocity ωp2 of the spinning 30 lb rotor with its spinning 3 lb lever arm.
Hint: Study the answer to Quiz #1A.
A3: (answer: ωp = 1.44 radians/sec)
Q3.1: How long would it take for the spinning rotor and lever arm to accelerate from zero rpm to the horizontal precessional angular velocity ωp arrived at in answer 3.
A3.1:
Q3.2: How is it possible that a 33 lb object can be accelerated to the angular velocity ωp in zero time?
A3.2:
Q3.3: What is the rate of acceleration (a) in deg/sec2 for the 33 lb spinning and precessing rotor and lever arm to get up to the angular velocity arrived at in answer 3?
A3.3:
Q3.4: How is it possible for the acceleration rate to be that high?
A3.4:
A3.5: How many radians and degrees did it travel before getting up to speed?
A3.5:
Q4: At what rate would a 30 ft-lb torque about the Z-axis accelerate a 30 lb non-spinning
Rotor with a 1.0 foot non-spinning lever arm weighing 3.0 lbs?
A4: (answer: a = 29.4 rad/sec2 )
Q4.1: How long would it take to get up to the precessional angular velocity ωp arrived at in question 3?
A4.1:
Now assume that the spin axis weighed 3.0 lbs and was not spinning (the rotor was on a bearing). Assume that the bearing was strong and of negligible size and mass.
Q5: Calculate the precessional angular velocity ωp2 of the spinning 30 lb rotor with its non-spinning 3 lb lever arm.
Hint: Study the answer to Quiz #1A.
A5: (answer: ωp = 1.44 radians/sec)
Q5.1: How come this answer is the same as for Question 3?
A5.1:
Q5.2: What is the rate of acceleration (a) in deg/sec2 for the 30 lb spinning rotor with its non-spinning to get up to the angular velocity arrived at in question 5?/
A5.2:
Q5.3: How long would it take to get up to the precessional angular velocity ωp2 arrived at in question 5?
A5.3:
Q5.4: How much work was performed in accelerating the rotor and the non-spinning lever arm to their precessional angular velocity arrived at in question 5?
A5.4:
Q5.5: What power level did it require to bring it up to its precessional angular velocity?
A5.5
Q5.6: How many radians and degrees did it travel before getting up to speed?
A5.6:
GOOD LUCK!
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| Date: |
13 October 2004
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