2.1. Movement equations of a gyroscope

On the basis of an inertial system, the rate of change of components from angular L⃗ corresponds to the net torque τ⃗ exerted by external forces. However, based on a rotating system {e^1,e^2,e^3}, which is spinning with angular velocity ω⃗; the change on vector ω⃗×e^i is added to the rate of change of a vector’s components. Therefore, Newton’s equations take the form of Euler’s equations ¹⁰:

By aligning the base vectors from the rotating system with the main axes of a gyroscope, the Momentum of Inertia Tensor becomes diagonal, I = diag(I₁; I₂; I₃),, and components from the angular moment will depend only on the respective components of angular velocity, L_i = I_iω_i . Now, by aligning e^3 with the rotation axis, I₁ = I₂ is obtained, and the equation (1) can be written by components as follow ¹¹:

Gyroscope’s orientation is fully described by three Eulerian angles corresponding to the angles that relate the main axis of a gyroscope with the coordinate axis on a steady state system {e^x,e^y,e^z}; where by convention, z axis represents vertical direction. Each of these angles can be associated to each different movement on a gyroscope. The zenith angle θ corresponds to the angular separation between the symmetry axis e^3 and the vertical; and it determines the nutation movement. The azimuthal angle ϕ, formed between the x axis and the nodes line (intersection between horizontal plane and that other, perpendicular to the symmetry axis), it determines the precession. Finally, ψ angle which accounts for the body spinning around of symmetry axis, and determines the rotational movement. Figure 1 is a graphic representation of Euler’s angles and the angular velocity associated to them.

Figure 1 Euler’s angles and their associate angular velocities. 

The total angular velocity is a vector addition of angular velocities associated to each one of Euler’s angles. Figure 1 shows that the angular velocity components based on a rotating system are:

On the other hand, the torque exerted by gravitational force with respect to the fixed point, is given by a vector product between the center of mass position he^3, and the gyroscope weight -mge^z. This torque is a vector addressed to the nodes line and with components:

where μ ≡ mgh is the maximum torque. In the case where this torque was the only contribution to torsion momentum of system (neglecting friction), then the products of e^3⋅τ⃗ and e^z⋅τ⃗ are nulls, so the components from the angular momentum L₃ and L_z are constant of movement. These constants provide relations between variables, which in union with the energy conservation law, allows to determine values of the angular variables θ, ϕ and ψ as function of time. This procedure is equivalent to Lagrange’s method, it provides a solution in terms of elliptic functions ¹². Even, though in this way an exact solution for the values of frequencies can be obtained, the relation between the oscillation amplitudes of θ and ϕ, which define a typical gyroscope movement, are not direct.

Although approximately, the direct solution of Euler’s equations gives an intuitive method to correctly describe the movement of a gyroscope. To find out such a solution, is advisable to work in the complex plane, so that we add the first equation from (2), to the second one which was previously multiplied by the imaginary number i; all this to obtain the following complex equation:

with τ ≡ τ₁ + iτ₂ and ω ≡ ω₁ + iω₂. From (3) and (4) is obtained for these complex quantities that τ = μ sin θe^-iψ and ω = Ωe^-iψ , where Ω ≡ (̇θ + i̇ϕ sin θ).By using these values on (5), and factorizing the exponential e^-iψ , the following expression appears:

This is a differential equation on the real variables θ and ϕ, and can be expressed in terms of complex variable as

This variable corresponds to Riemann’s stereographic projection of a point on the sphere in the plane ¹³; here, the real part is directed to e^x, and the imaginary to e^y (see Fig. 2).

Figure 2 Variable a in the projective plane. 

By differentiating the expression (7) respect of time, we obtain

and

Replacing (8) and (9) on Eq. (6) the following equation shows up:

here, according with (7), cosθ = (1-|a|²)/(1+|a|²). Equation (100), together with ̇L₃ = τ₃ are equivalent to Euler’s equations (2). This equation is a generalization to θ large for the gyroscope’s equation used in external ballistic ¹⁴.

2.2. Approximate Solution to the gyroscope’s equation

Equation (10) is not linear, of second order on the complex variable a, and is not analytic. The exact solution for this equation is not easy to find. However, an approximate solution for small nutation can be easily obtained. In this case the value of θ can be replaced, on first approximation, by the mean value θ₀. After changing variables

Eq. (10) can be rewritten as:

The roots of the right hand polinomy on (12) are given by

which corresponds to the fast precession frequencies s₊ and slow s_-¹⁵. The differential Eq. (12) can be solved by separating variables followed by an integration using partial fractions, thus obtaining

where x = βt cosθ₀ + c, being c a complex constant of integration. The imaginary part of c is the relevant quantity because the real part may be eliminated through a temporal translation. Having on mind the definition of s given in (11), the value for a is obtained after integrate and exponentiate (14)

where A' is a constant resulting from the integration. The function sin x is proportional to e-iβtcosθ0(1-e2ix), so a can be expressed as

with A and B arbitrary constants. On the limit when B → 0 a slow precession solution for a is given, the case when B → ∞ (with AB finite) leads to a fast precession solution. Generally, precession turns out to be of lower frequency than nutation, therefore the solution for nutation of small amplitude corresponds to (16) with |B| ≪ 1. By making a binomial approximation of the term inside parenthesis in (16), we have

where ω_n and ω_p are frequencies of nutation and precession respectively, given by

Expression (17) implies that a is the sum of two rotating vectors of constant amplitude, |a_p | and |a_n |, which spin with frequencies ω_p and ω_p + ω_n respectively. Because of the fact that the amplitude of precession movement is greater than that of nutation, and its frequency is lower, the trajectory that vector a presents in a plane has a shape of rosette. On expression (18) ω_n is defined positive, and from (19) can be seen that ω_p has the same sign as μ. The rosette shape formed on the projection plane comes given by the relative sign between frequencies of precession and nutation, which means depends on the sign of μ. When the frequencies have the same sign (μ > 0) the projection created on the plane has the apices pointing inward, while if signs are opposite (μ < 0), the apices on the projection are going outward (See Fig. 3).

Figure 3 Graphic representation of the gyroscope solution, (a) for μ > 0 and (b) for μ < 0. ϕ_n y ϕ_p are the arguments of a_n and a_p , respectively. 

The precession frequency does not diverge for θ ₀ approaching = π/2, despite first appearances. In this limit is obtained ω_p = μ/L ₃, like the simple theoretical expression for precession frequency found in most elementary physics textbooks, and ω_n = L ₃/I ₁. They also correspond to the expressions of frequency values on the limit of fast rotation L32≫μI1. If θ ₀ is less than π/2 and μ > 0 then the precession frequency will be greater than μ/L₃, while nutation smaller than L ₃/I₁. These relations are inverted for angles higher than π/2 or μ < 0. All of this can be sumarized in an expression coming from (19) at first order in r≡μI1/L32 given by

From Eqs. (19) and (18), can be found relations between precession and nutation frequencies which are independent of the state of movement, given by:

Given that ω3=⟨ψ̇⟩+ωpcosθ0, from expression (22) is obtained that the number of rotations per nutation Δn=⟨ψ̇⟩/ωn, in function of precession and nutation frequencies, and the azimuthal angle, is given by

This relation predicts that the number of rotations by nutation tends to the value I₁/I₃ for θ = π/2 or L₃ large, but in general (23) has a linear behavior with respect to the parameter (ω_p /ω_n )cosθ₀, where the slope and the axis intersection are determined by I₁/I₃.

Finally, to obtain the expressions of these angles as a function of time, solution (17) and definition (7) are equating, and considering |a_n /a_p | ≪ 1 we got:

and except a constant

where δ is the argument of a_n /a_p . From this last expression is evident that the mean value of the angular velocity ϕ is the precession velocity ω_p , while for tanθ/2 is |a_p |, what we make to coincide with tanθ₀/2, having an relative uncertainty of order (|a_p |/|a_n |)².

Results aforementioned coincide with the Lagrangian description in the approximation of harmonic potential to first order in the parameter μI1/L32 ¹⁵.

3.1. Experimental Assembly

With an experimental set up we looked for a corroboration of the gyroscope solution, Eq. (17), with its corresponding parameter values, fits in an exactly way to an experimental situation. For this experience was used a Pasco ME8960 gyroscope, it consist of a disk of 1.755 ± 0.001 kg mass with a radius of 12.7 ± 0.1 cm and a thickness of 2.0 ± 0.1 cm, which can spin around an axis with little friction, at the same time this axis may be orientated in any azimuthal direction but inside a range of 30° to 140° respect to the zenith. The orientable axis has a fixed point at 12.7 ± 0.1 cm from the disk, which is pivoted at 30.7 ± 0.1 cm from the floor over a vertical bar. This bar is nailed in the center of an A-shaped base of molten iron, and it can rotates around its own axis. At the opposite end of orientable axis, opposite to the disk, there is a counterweight that works to vary the position of the center of mass.

To visualize the a vector, a simple laser pointer device is set up in such a way that it remains on the same vertical plane than orientable axis, but presenting an angle with respect to the vertical equal to half of zenithal angle of the axis. This can be achieved by setting the pointer on the base of an isosceles triangle where one of its congruent sides aligns with the vertical bar and the other with the orientable axis, as showed in Fig. 4.

Figure 4 Laser assembly on the gyroscope. 

To have information about the angular velocity of rotation from the disk, an electric switch was installed to the pointer device so, it turns on during half of revolution cycle and then turns off on the other half. The switch was fixed by sticking a copper semi-disk to one of the faces of the spinning disk. The laser’s feeder circuit becomes closed only when two small brushes, in contact with the disk, also get in touch with the cooper semi-disk, thus the circuit is open every half revolution. (see Fig. 4).

In our case the laser light is projected upward over a horizontal screen 70 cm away from floor, which is formed by a framed fabric in a way that the luminous dot can be observed over the screen. A mirror just above the screen works as a guide of images from screen to the digital camera; the camera is installed in a tripod, on side of the mirror and pointing out to it, as can be seen in Fig. 5.

Figure 5 Experimental Assembly. 

By taking a photography of luminous dot in a time interval large enough is obtained a discontinuous light beam, each segment of the bright line along with the following dark interval, correspond to a disk rotation. Initially counterweights were set on the disk axis in a way that the center of mass will stay just on the pivoted point from orientable axis. Later, an additional counterweight, with a mass of 151.48 ± 0.01 g, was set on one of the orientable axis ends. Experiments with the counterweight on the same disk side at 21.0 ± 0.1 cm from pivoted point (forward) were made, and then with the counterweight on opposite side (backward) to 34.5 ± 0.1 since pivoted point. The counterweight mass value multiplied by the pivoted point distance and by the constant of gravity, accounts for the maximum torque μ to which the gyroscope is subjected. Given the laser pointer position, with an additional counterweight ahead, the rosette shape formed by the movement of the dot of light shows its apices outward; reciprocally, by adding the counterweight back, rosette’s apices point inward, as show Fig. 3.

3.2. Results

With the mass values of both, counterweights and disk, and taking into account their position, the calculation of values for the main inertia momentum and maximum torque, related to each situation of additional counterweight, is direct. These values are given by Table I.

The disk was set up to spin around the orientable axis at a specific velocity ψ and certain value of θ₀. The system was left to evolve and the trajectory of luminous dot was photographed, with a digital camera which has an aperture time control, inside a dark room. The camera was located above screen and by over the gyroscope, an exposure time of 10 seconds for each photo was set, also a green laser was used which is why the result of each photography is a dotted green line with a dark background. After inverting colors using an editor of images, pictures are seen as a red trace with white background.

From photographic records is obtained: an angular interval of a nutation Δϕ, the number of rotations per nutation Δn, the total number of rotations n, and the circumference diameters D₁ and D₂, intern and extern, tangent to the trajectory of luminous dot. The values of these diameters must be given using units of a reference diameter D_r, which is defined as the trajectory diameter of luminous dot when gyroscope precesses uniformly with θ = π/2. These measurements, along with the exposition time t, allow us to find the frequencies of rotation, precession and nutation, as well as the average value of zenithal angle, through next relations:

Using these values, and according to (21) and (22), relations given by μ/I₁ and I₁/I₃ can be calculated. 60 pictures of each counterweight, back and front, were analyzed. Figure 6 shows one example of each case, where the way how every measurement was obtained is described. The average values of these quantities are called experimental values; the associated uncertainty is the standard deviation of data, divided by square root of the amount of data. Theoretical data correspond to those obtained from Table I. The average values for μ/I₁ and I₁/I₃, experimental and theoretical, are shown in Table II.

Figure 6 Pictures of the projection of gyroscope movement with inverted colors. Counterweight ahead and back respectively. 

According to (23) the graphs of ω_n in function of (ω_p cosθ + ω_n )^-1, present a linear form with a slope of μ/I₁.

Figure 7 shows the experimental results using counterweight back and then ahead. Solid lines represent theoretical values.

Figure 7 Experimental value of μ/I₁ for experiments with counterweight ahead (light circles), and with it on back (dark circles). 

A kind of dispersion in values can be noticed; it possibly is because of a high value of the relative uncertainties on Δn. If it is only considered one nutation cycle, it shall an uncertainty of order 1/Δn, which, in our case is around 12%. In order to diminish the uncertainty, N cycles of nutation are taken, reducing the error in one factor of 1/N. However, for low velocities, N cannot be too large because the Δn variation per cycle of nutation will be significantly affected by friction, therefore on the beginning exists an insuperable limit on precision of measurements, determined by the disk friction.

Figure 8 shows experimental results for the number of rotation per nutation as a function of the relation between precession and nutation frequencies, multiplied by cosine of θ₀. Solid lines represent theoretical values.

Figure 8 Relation between the number of revolutions per nutation cycle for the experiments with the counterweight ahead (light circles) and back (dark circles). 

Figure 9 shows the ratio between experimental precession speed ω_p and the expected in the simplest model μ/L₃, as a function of the parameter r=μI1/L32. The dotted line indicates the predicted value by the elementary model and the continuous line shows the predicted by the model for this work, according to the Eq. (20).

Figure 9 Relationship between experimental precession speed !p and its expected value μ/L₃ in the simplest model, in function on the parameter r=μI1/L32, for both counterweight ahead and counterweight back 

For the data reported in Fig. 7, 8 and 9 the rotation frecuency was between 3 to 13 revolutions per second.