These are all very interesting results, in general the set of Lagrange variables most suitable to the problem should be chosen. There is some freedom of choice of Lagrange variables as you know, but also some rules. The very important advance is that the code is available for solution of the problem. this is code for the solution of complicated simultaneous differential equations. The angles are measured in the same way as in Problem 10.10 of Marion and Thornton, but the foot of the gyro is allowed to move. The reason for this is that the centre of mass of the gyro can move upwards, giving lift. For some reason I cannot see the graphics, can you resend them? I cannot see the protocol but your method seems to be OK. To check this, it should be equivalent to the simultaneous solution of Eqs. (17) to (19) and (32). Eq. (32) was set up deliberately to find an equation that involves both rotation and translation (i.e it contains both R and theta). As you infer, R can be interpreted as the height of the point of the gyro above a laboratory bench. The overall aim is to find whether the gyro can be elevated, i.e. to find whether its point can move upwards. Although I cannot see them, the graphics look to be very interesting, especially the space curve of the centre of mass. In order to get a complete solution can Eqs. (17) to (19) and (32) be solved simultaneously with R interpreted as the height above the laboratory bench? The nutations and rotations / precessions are also very interesting. This is probably the first time in 250 years that this problem has been completely solved, so it should be addressed in as many useful ways as are relevant, with different Lagrange variables and so on. This code is very useful and is the beginning of a new phase of research. I can suggest an experiment: fix the point of the gyro to a stand a distance R above the laboratory frame, so it is horizontal as in the Laithwaite experiment, and observe its motion. Does its centre of mass rise above the point at which its foot is fixed to the stand, R above the lab bench?
Sent: 28/01/2017 17:46:26 GMT Standard Time
Subj: Re: Discussion of 369(1): Complete Analytical Mechanics of the Gyroscope
If R in eq.(27) is the earth radius, it is in very good approximation:
h << R
so that R is practically not altered. Therefore R can better be measured from any point in the lab. To my understanding the angles are measured from the foot point of the gyro as before. We therefore obtain the additional equation
R dot dot = -g.
This is decoupled from the other gyro equations, i.e. the gyro moves in free fall.
I added this equation (in Hamilton form) to the set of equations:
r dot = v
v dot = -g
The results are in the protocol. First I re-computed the Lagrange equations directly, then I used the easier form with constants of motion. Due to a pdf printing problem, the formulas are not readable, but you can look at the graphics:
1. theta dot(t), theta(t) — nutation
2. phi(t), psi(t) — precession and rotation of body
3. v(t), R(t) — free fall solution
4. omega components in 1-2-3 frame and modulus omega (omega_3=const)
5. space curve of the centre of mass with nutation/precession
6. space curve of omega vector in 1-2-3 frame. This is motion in a plane (rosetta) since omega_3=const.
I will produce similar graphics for paper 368 in better quality.
Am 28.01.2017 um 10:12 schrieb EMyrone:
This will be the first time that this problem has been correctly solved, so your coding is full of interest, in particular the lab frame translational motion of the point of the gyro can be computed. If the upward Z axis force generated by the spin of the gyro is the same as the gravitational force downward, the point will float. So Laithwaite would feel no weight if he is holding a floating point. The reason why this two hundred and fifty year old problem has not been solved before is that it is exceedingly intricate, and code is needed to solve four simultaneous differential equations in four variables: r, theta, phi and chi.
Sent: 28/01/2017 08:54:57 GMT Standard Time
Subj: Re: 369(1): Complete Analytical Mechanics of the Gyroscope
This is a model with a free floating foot point of the gyro. It seems that the reference to the earth radius and gravitational earth potential is not necessary, it is easier to refer to any fixed point at the earth surface from where R is measured. This avoids appearance of G amd M in the Lagrange equations. I will set up this version today afternoon by computer.
Am 27.01.2017 um 14:39 schrieb EMyrone:
The geometry is defined in Figure(1), and in general the gyroscope’s point is allowed to move with respect to the centre of the earth. So the point can move up or down. The analytical problem reduces to the simultaneous solution of four differential equations, (17) to (19) and (32). The first three are the same as in UFT368, for the pure rotational motion of the gyro. They are supplemneted by Eq. (32) for the motion of R, where R is he distance between the point of hte gyro and the centre of the Earth. Eq. (32) is the necessary link between rotational and translation motions of the gyro. In the replicated Laithwaite experiment the point of the gyro (the common origins of (1, 2, 3) and (X, Y, Z)) is held at the height of Laithwaite’s arm.