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23 November 2024 21:27
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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: |
Calculate Precessional Angular Velocity |
| Question: |
TAKE HOME QUIZ #1A
This quiz is intended to replace the one submitted earlier on September 9, 2004. A few of the terms were used incorrectly and have been corrected in this submission.
For those who profess to fully understand the physics of a precessing gyroscope, here is a short take-home exam:
The basic equation governing gyroscopic motion is: T=I ωs ωp where T is the applied torque, I is the Mass Moment of Inertia of a rotor, ωs is the angular velocity of a rotor R, and ωp is the angular velocity of the resulting precession. The product (I ωs) is known as the Angular Momentum of the rotor. Assume natural precession where the torque T is provided by the force of gravity at the surface of the earth and ωp is the resulting natural precession in the horizontal plane. [s and p are supposed to be subscripts to ω. Likewise in the following paragraphs, 1 and 2 are supposed to be subscripts to L, R, ω and I]
Assume a spin axis having a length of L1 from the pivot point with a rotor R1 having a Mass Moment of Inertia I1 and an angular velocity ωs1 . Now further assume that the spin axis is extended and is actually longer for a total length of L2 and that another rotor R2 at the length L2 has a Mass Moment of Inertia I2 and an angular velocity ωs2 .
It may be assumed that rotor R1 alone on a spin axis of length L1 would have a precessional angular velocity of ωp1 and that R2 alone on a spin axis of length L2 would have a precessional angular velocity of ωp2 .
Now we have a single spin axis or lever arm with two different rotors, each having a different Mass Moment of Inertia, angular velocity, and distance from the pivot point. Assume that both rotors are spinning in the same clockwise direction and are precessing in the horizontal plane and that the spin axis has negligible mass. What is the resulting precessional angular velocity ωp in the horizontal plane due to the force of gravity of the two spinning rotors on the one axis? _________ For example, the solution for the precessional angular velocity for a single rotor R1 is ωp1 = T /(I ωs1) . Do not plug in numbers, but keep the solution in algebraic form.
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| Date: |
22 September 2004
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Answers (Ordered by Date)
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| Answer: |
webmaster@gyroscopes.org - 24/09/2004 00:11:15
| | | I'll have a go at this.
precessional angular velocity = (T / ( I1 ws1)) + (T / (l2 ws2))
However I did wonder if T had to be spilt for each gyro. It doesn't seem right that its applied to both gyros.
Anyway, I'm sure you will tell me if I'm right.
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| Answer: |
Harvey Fiala - 25/09/2004 06:03:50
| | | This answer is in response to the answer submitted on 24/09/2004 00:11:15.
The answer is not correct. An easy way to determine this is to assume that both rotors have the same mass, the same mass moment of inertia (I), the same angular speed ωs , and the same lever arm length. This is exactly the same as simply saying that a single rotor has twice as much mass and has twice the mass moment of inertia on the same length of lever arm. Because the mass or weight is twice as much, the gravitational torque will also be twice as great.
Repeating, the basic equation for precession is: T=I ωs ωp where T is the Torque and ωp is the precessional angular velocity. Solving for ωp :
ωp1 = T /(I ωs1).
Substituting in two times the torque T and two times the mass moment of inertia (I), we get:
ωp1 = 2T /(2I ωs1) = T /(I ωs1) which is the same precessional angular rate as for a single rotor of the original mass. This is very interesting. It states that if you double or triple or quadruple only the width of a solid disk rotor while keeping the lever arm distance the same, the precessional angular velocity will not change.
However, your expression for the precessional angular velocity is: ωp = (T/(Is1 ωs1 ))+(T/(Is2 ωs2 ))
And letting (Is1 ωs1 ) = (Is2 ωs2 ), we get
ωp = (T / (Is1 ωs1 ))+(T/(Is1 ωs1 ) = 2T(1/(Is1 ωs1 ) which is twice what the precessional angular velocity should be.
The above test is a very easy test to help determine if a solution is correct. If the answer is not the same as for the basic equation, the solution is not correct. However, if the answer is the same, this does not guarantee that the solution is correct, although it may be. It is suggested that those submitting an answer apply this simple test first.
Only after someone provides an answer with the correct solution will its correctness be acknowledged unless three or four months go by. Otherwise those trying to solve the problem would have the answer given to them and that would defeat the purpose of this quiz.
This original answer was done in Microsoft Word and the s1, s2, s, and p are subscripts although they may not appear as subscripts on this web page.
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| Answer: |
Hsien-Lu Huang Elder, CM, PhD - 29/09/2004 16:12:21
| | | Answer: hsien-luhuang@sbcglobal.net- 29/09/2004 10:00:00
Answer to Mr. Harvey Fiala's Question: TAKE HOME QUIZ #1A,
Subject: Calculate Precessional Angular Velocity
Precessional angular velocity = g[(M1)(L1) + (M2)(L2)] / [( I1)(ws1) + (l2)(ws2)]
Where
g is the gravitational acceleration,
M1 and M2 are the mass of Rotors 1 and 2 respectively, and
I1 and I2, ws1 and ws2, ard L1 and L2 are the same as defined in the QUIZ #1A.
God bless you all.
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| Answer: |
Harvey Fiala - 04/10/2004 06:50:11
| | | Dr. Hsien-Lu Huang (CM, PhD) has the correct answer. The general answer is:
ωp = (W1 L1 + W2 L2 + W3 L3 + … Wn Ln ) / (Is1 ωs1 + Is2 ωs2 + Is3 ωs3 + …+ Isn ωsn ), where W is Weight, L is Length of the lever arm, I is the mass moment of inertia for a given rotor, ωs is the rotor speed for a given rotor, and ωp is the resulting precessional angular velocity,
The total torque on the shaft with multiple rotors is the sum of all weight times lever arm terms and the total angular momentum is the sum of the angular momentum of the individual rotors. 1 , 2 , 3 , n , s , and p are subscripts.
Note that this solution is very similar to the electrical engineering equation for the current through a number of resistors in series.
I = V/( R1 + R2 + R3 + … Rn ), where I is the current through all resistors, V is the voltage across the whole string of resistors, and R is for a resistor.
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