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20 May 2024 19:22
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Welcome to the gyroscope forum. If you have a question about gyroscopes in general,
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Question |
| Asked by: |
syafiah |
| Subject: |
report |
| Question: |
what is the relationship between :
1)precession and gyro-frequency and its dependence from torque
2)nutation frequency and gyro-frequency |
| Date: |
30 September 2004
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Answers (Ordered by Date)
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| Answer: |
Harvey Fiala - 04/10/2004 06:23:39
| | | Restating the question for clarity: What is the relationship between precession angular velocity and the gyro frequency (rotor speed) and the torque for a gyroscope? Also, What is the relationship between the nutation frequency and the gyro-frequency.
ANSWER: These are very good questions, insofar as the theory behind them is only assumed to be correct and is seldom challenged by established scientists.
There are two types of precession: Natural precession and forced precession. Natural precession is defined to be precession in the horizontal plane about a vertical axis at the pivot point. It is caused by the force of gravity pulling down on the spinning rotor while the support at the pivot point is pushing up on the lever arm or rotor shaft. These two forces form a torque on the spin axis, and the spinning rotor, instead of falling down, starts to move sideways, in an effort to conserve its angular momentum. Assume that the rotor is a solid disk with a mass (M) and a radius r, the length of the lever arm is L, the precessional angular velocity is ωp , the rotor speed is ωs , the mass moment of inertia is I, the angular momentum is the product of I and ωs , and the torque is T. For natural precession, the torque is supplied by the force of gravity acting at the center of mass of the rotor. Assume the rotor shaft or axle or lever arm has a negligible mass. The torque is the weight W times the lever arm length or WL. The equation governing the precessional angular velocity, the rotor speed, the mass moment of inertia of the rotor, the weight of the rotor, and the length of the lever arm is:
ωp = T/(Iωs) = WL/(/(Mr2 ωs /2) = MgL/(/(Mr2 ωs /2) = gL/(/(r2 ωs /2) = 2gL/(r2 ωs) rad/sec.
Note that the mass M of the rotor cancels out. Also ωp and ωs have an inverse relationship. That is, the higher the rotor speed, the slower will be the precessional angular velocity. The greater the torque, the greater will be the precessional angular velocity.
The simplest case of forced precession is where torque is applied about the vertical axis as if to speed up (a hurrying torque) or slow down (a slowing torque) the horizontal precessional angular velocity. The forcing torque is perpendicular to the force of gravity, and so the rotor will precess upward (or downward) in response to the forcing torque, depending on the direction of the forcing torque and the direction of spin of the rotor.
The relationship between nutation frequency and the rotor speed is more subtle and complex. Assume that the lever arm is non-spinning and is increased in length from L to a total length of h. Assume that a weight of mass m is added at the end of length h. Then the equation governing nutation is:
ωn = (Is ωs) / ((Is + mh2 m)/(m + M)) , where ωn is the nutation frequency and Is is the mass moment of inertial of only the spinning rotor.
A study of this equation will reveal that if the mass m is added with no vertical velocity there will be a U or W shaped nutation waveform superimposed on the circular path of the natural precession. If mass m has a small initial vertical velocity, then the nutation pattern will look like a prolate cycloid superimposed on the circular path of the natural precession. If mass m has a small horizontal velocity in the direction of the natural precession then the nutation pattern will look like a sine wave superimposed on the circular path of the natural precession.
The above equation does not give the amplitude of the nutation. However, as the rotor speed is increased, the amplitude of the nutation will decrease.
To understand precession and all of its variables it is highly recommended that units for all variables be clearly understood and carried along in any calculations to verify that the answer will be dimensionally correct.
This answer was originally typed in Microsoft Word and in the above equations n and p and s are supposed to be subscripts to ω and I and in some cases a superscript to h and r, although they may not show up as such in this text.
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