Deactivation of gyroscopic inertial forces

Usubamatov, R.

doi:10.1063/1.5047103

An inertial gyroscope in engineering manifests several unexplainable properties which physical nature is still unknown in classical mechanics. The new study demonstrates that the origin of the gyroscopic effects is more complex than presented in known publications. The gyroscopic effects manifest the inertial resistance and precession torques acting around different axes. The first torque is generated by the centrifugal and Coriolis forces and the second one by the common inertial forces and the change in the angular momentum. Blocking of the gyroscope motion around one axis deactivates the resistance torques but the precession torques are acting and the gyroscope turns under the action only of the gravity force. These phenomena are the manifestation of the unknown property of a physical matter that represents the new challenge for the researchers of classical mechanics. Newton’s laws are justified for the simple action, but for complex one should be formulated, validated and written new physical laws. For the example, the rotation of the mass around two axes demonstrates the disappearance of the inertial forces that contradicts the principles of physics. This work presents the mathematical model and practical validation of the deactivation of the inertial forces for the spinning rotor around two axes.

I. INTRODUCTION

Since the Industrial Revolution, famous, outstanding and ordinary scientists studied the gyroscopic effects and published probably tons of the manuscripts and several dozens of theories.^1–4 However, the known publications in the area of gyroscope theory do not match practical results. The gyroscopic forces and motions do not describe well by mathematical models.^5,6 This is the reason that researchers created the artificial terms like gyroscopic forces, effects, resistance, etc. Today, for practical applications of the gyroscopic properties developed numerical modelling with the software that is relayed in many engineering calculations of rotating parts.⁷ However, the nature of the gyroscope effects is more complex than presented in known manuscripts.^8–11 Recent investigations in the area of the inertial gyroscopes demonstrate that the load torque applied to the gyroscope generates the four fundamental torques based on the action of the centrifugal, Coriolis and common inertial forces of the spinning rotor and the change in the angular momentum. The centrifugal and Coriolis forces generate the resistance torque in change the rotor’s location. The common inertial force and the change in the angular momentum of the spinning rotor generate the precession torque. The simultaneous action of these torques and their interrelation represent the fundamental principles of the of inertial gyroscope theory. New inertial torques are represented by the new mathematical equations in Table I.¹²

TABLE I.

Equations of the gyroscope’s inertial torques.

Type of torque generated by	Equation, (N.m)
Centrifugal forces, T_ct.i	$T_{c t . i} = T_{i n . i} = 2 {(\frac{π}{3})}^{2} J ω ω_{i}$
Common inertial forces, T_in.i
Coriolis forces, T_cr.i	$T_{c r} = (8 / 9) J ω ω_{i}$
Change in angular momentum, T_am.i	$T_{a m . i} = J ω ω_{i}$
Resistance torque T_r.i = T_ct.i + T_cr.i	$T_{r . i} = [2 {(\frac{π}{3})}^{2} + \frac{8}{9}] J ω ω_{i}$
Precession torque T_p.i = T_in.i + T_am.i	$T_{p . i} = [2 {(\frac{π}{3})}^{2} + 1] J ω ω_{i}$

Type of torque generated by	Equation, (N.m)
Centrifugal forces, T_ct.i	$T_{c t . i} = T_{i n . i} = 2 {(\frac{π}{3})}^{2} J ω ω_{i}$
Common inertial forces, T_in.i
Coriolis forces, T_cr.i	$T_{c r} = (8 / 9) J ω ω_{i}$
Change in angular momentum, T_am.i	$T_{a m . i} = J ω ω_{i}$
Resistance torque T_r.i = T_ct.i + T_cr.i	$T_{r . i} = [2 {(\frac{π}{3})}^{2} + \frac{8}{9}] J ω ω_{i}$
Precession torque T_p.i = T_in.i + T_am.i	$T_{p . i} = [2 {(\frac{π}{3})}^{2} + 1] J ω ω_{i}$

Where J is the rotor’s mass moment of inertia around the spinning axle; ω_i is the angular velocity of the precession of a gyroscope around axis i and ω is the angular velocity of the spinning rotor. The following analysis of the actions of several torques and motions around the two axes has used the system of subscripts signs. All components of the equations are marked by subscript signs that indicating the axis of action. For example, T_r.x is the resistance torque acting around axis ox, ω_y is the angular velocity of precession around axis oy, etc.

The derived equations of the inertial torques are used for mathematical modelling of the gyroscope motions^12,14 that are well-matched with practical results.¹³ Nevertheless, practical and analytical studies of the action of the inertial torques and gyroscope motions manifest the new unknown property of the material object. This property is manifested in the form of the deactivation of the inertial forces generated by rotating objects at the conditions of the complex motions. The disappearance of the inertial forces of the running object contradicts the principles of physics. This work represents unknown interrelations of the gyroscope torques and motions, deactivation of the centrifugal and Coriolis forces that are new properties, which practically tested and validated. The physics of the new gyroscope properties is extremely unclear and present the challenge to the science of the classical mechanics.

II. METHODOLOGY

The action of the external and several inertial torques on the gyroscope is validated by practical tests^1–4 of the mathematical models for the motions of the gyroscope suspended from the flexible cord.¹³ The phenomena of deactivation of gyroscope’s inertial forces are manifested in case of the gyroscope motion around one axis. The mathematical model for gyroscope motion around one axis derived for the test stand of the Super Precision Gyroscope “Brightfusion LTD” with one side support (Figs. 1 and 2). The stand contains the gyroscope connected with the counter-weight G by the axle s. The mass of the counter-weight is selected with the aim to have the slow turn of the gyroscope around one axis and with the ability to record the time of motions. The axle s is fixed on the centre beam b that has the ability of the free rotation around axis ox on the supports B and D. The centre beam located on the vertical arms of the frame that assembled on the horizontal bar b_s. The bar has the ability to rotate about the fixed pivot C (vertical axis oy) of the platform. Schematic of the test stand with the sizes of gyroscope components and the angle of gyroscope turn are represented in Fig. 2. The technical data of the stand is represented in Table II.

FIG. 1.

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The gyroscopic stand with the counter-weight.

FIG. 2.

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Schematic of the gyroscope’s stand with the counter-weight.

TABLE II.

Technical data of the test stand with Super Precision Gyroscope, “Brightfusion LTD”. Location of the centre mass is l_m = 0.0036 m, difference of masses left and right sides is m_r = 0.043 kg.

Parameters	Mass, kg	Mass moment of inertia, J kg.m²	Around axis
Spinning rotor	0.1159	J = mR_c²/2 = 0.5726674×10^-4	oz
Gyroscope with rotor, W	0.146	J_iW = (2/3)m₅r₅² + m₅l² + (mR²_c/4) +	ox, oy
		ml² = 2.284736*10^-4
Counter-weight, G	0.098	J_G = m_Gr²/2 + m_G a² = 2.45784×10^-4	ox
Axle, s	0.005	J_s = m_sl²_s/3 = 0.0352666×10^-4	ox, oy
Centre beam with journals and screw, b	0.028	J_b = m_br²_b/2 = 0.00224×10^-4	ox
		J_b = m_bl²_b/12 = 0.2105833×10^-4	oy
Components, A = W+s+b+ G	0.277	J_xA = 4.780082×10^-4	ox

Parameters	Mass, kg	Mass moment of inertia, J kg.m²	Around axis
Spinning rotor	0.1159	J = mR_c²/2 = 0.5726674×10^-4	oz
Gyroscope with rotor, W	0.146	J_iW = (2/3)m₅r₅² + m₅l² + (mR²_c/4) +	ox, oy
		ml² = 2.284736*10^-4
Counter-weight, G	0.098	J_G = m_Gr²/2 + m_G a² = 2.45784×10^-4	ox
Axle, s	0.005	J_s = m_sl²_s/3 = 0.0352666×10^-4	ox, oy
Centre beam with journals and screw, b	0.028	J_b = m_br²_b/2 = 0.00224×10^-4	ox
		J_b = m_bl²_b/12 = 0.2105833×10^-4	oy
Components, A = W+s+b+ G	0.277	J_xA = 4.780082×10^-4	ox

The tests of the gyroscope conducted for conditions of the gyroscope turn around horizontal axis ox and blocking of the rotation around vertical axis oy. The aim of the tests is the demonstration and validation of the mathematical model for the gyroscope motion under the action of the external and inertial torques. Recorded results are represented by the time motion of the gyroscope. The measurement of the data for the tests were conducted by the laboratory instrumentation. The angular velocity of the spinning rotor was measured by the Optical Multimeter Tachoprobe Model 2108/LSR Compact Instrument Ltd. with the range of measurement 0 – 60,000.00 rpm. The time spent on the turn of the gyroscope around axis ox measured by the stopwatch of Model SKU SW01 with resolution 1/100 s.

Table II contains the following symbols: J_x.W = J_y.W = (MR²_c/4) + Ml² is the mass moment of the movable gyroscope component inertia around axes ox and oy respectively; l is the overhang of the centre of mass of the gyroscope from the centre beam; M is the mass of the movable gyroscope component; J_r.x = J_r.y = (m_rR²_c/4) is rotor’s mass moment of inertia around axis ox and oy; R_c is the conditional radius of the rotor; J = (m_rR²_c/2) is rotor’s mass moment of inertia around axis oz.^8–10 The computed mass moments of the gyroscope’s components inertia around axes ox and oy is presented in publication.¹³

The mathematical model for the gyroscope motion around axis ox is similar to the model for the gyroscope suspended from the flexible cord.¹³ The difference of the equation for the gyroscope motion around one axis is the absence of the precession torques that originated around axis oy and acing around axis ox (Fig. 2). Figure 3 represents the action of the external and inertial torques on the gyroscope stand. Defined information and data enable for hypothetical formulating the mathematical model for motions of the gyroscope under the action of the external and inertial torques that presented by the following Euler’s differential equation:

J_{x} \frac{d ω_{x}}{d t} = T - T_{f x} - T_{c t . x} - T_{c r . x}

(1)

where ω_x is the angular velocity of the gyroscope around axes ox; T is the resulting torque generated by the weight of the gyroscope components, T_ct.x, T_cr.x, are the inertial torques generated by the by the centrifugal and Coriolis forces acting around axis ox (Table I); T_f.x is the frictional torques acting on the gyroscope’s supports and pivot respectively; other components are as specified above.

FIG. 3.

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Schematic of the forces and torques acting on the gyroscope with the counter-weight.

The components of Eq. (1) are represented the torques that described by the correspondent equations. The load torque T around axis ox is generated by the weight of gyroscope components and represented by the following expression:

T = (W l - s a / 2 - G a) g \cos γ

(2)

where W is the gyroscope mass; s is the axle mass; G is the mass of the counter-weight; g is the gravity acceleration; l is the distance of location the centre mass of the gyroscope; a is the length of the axle and the distance of location the center mass of the counter-weight; γ is the angle of the axle inclination, m_r is resulting load mass, l_m is location of resulting load mass (Table II, Fig. 3).

Equation (2) shows that the acting torque T is variable and depends on the angle γ (Fig. 3 and Table I). The several external and inertial forces acting on the gyroscope generate the frictional forces of the supports. The weight of the gyroscope components is acting on the supports B and D, and produce the frictional torque that is represented by the following equation:

T_{f A} = (W + s + b + G) g f \frac{d}{2 \cos δ} = A g \frac{f d}{2 \cos δ}

(3)

where b is the mass of the center beam; f is the frictional coefficient of supports, d is the diameter of the supports B and D; δ = 45° is the angle of the beveled sliding bearing, other parameters are as specified above.

The angular velocity of the gyroscope around axis ox generates the centrifugal force produced by the centre mass and hence the frictional torque acting on the supports B and D that is expressed by the following equation:

T_{f m} = m_{r} l_{m} \frac{f d}{2 \cos δ} ω_{x}^{2}

(4)

where all parameters are as specified above.

The load torque T produces the resistance and precession torques of the spinning rotor. The first torque generated by of the centrifugal T_ct and Coriolis T_cr forces acting around axis ox. Second one generated by the inertial forces T_in and the change in the angular momentum T_am acting around axis oy. The precession torques are causing the additional load on the supports B and D that increases the value of the frictional force of the supports. Hence, the frictional torque generated by the precession torques are represented by the following equation:

T_{f p x} = T_{i n . x} + T_{a m . x} = [2 {(\frac{π}{3})}^{2} + 1] \frac{f d \cos (δ - τ)}{2 h \cos δ} J ω ω_{x}

(5)

where T_px is the precession torque (Table I); h is the distance from the centre mass of the gyroscope to contact points of the centre beam journals with supports B and D; τ is the constructive angle (Fig. 3); other parameters are as specified above.

The total frictional torque acting on supports of the gyroscope stand is represented by the following equation:

T_{f x} = T_{f A} + T_{f m} + T_{f p x} = A g \frac{f d}{2 \cos δ} + m_{r} l_{m} \frac{f d}{2 \cos δ} ω_{x}^{2} + \{2 {(\frac{π}{3})}^{2} + 1\} \frac{f d \cos (δ - τ)}{2 h \cos δ} J ω ω_{x}

(6)

where all parameters are as specified above.

Substituting defined equations of the external and inertial torques of the gyroscope (Table I), Eqs. (3)–(6) into Eqs. (1) yields the following differential equation:

\begin{matrix} J_{x} \frac{d ω_{x}}{d t} & = & (W l - s a / 2 - G a) g \cos γ - A g \frac{f d}{2 \cos δ} - m_{r} l_{m} \frac{f d}{2 \cos δ} ω_{x}^{2} \\ - \{2 {(\frac{π}{3})}^{2} + 1\} \frac{f d \cos (δ - τ)}{2 h \cos δ} J ω ω_{x} - [2 {(\frac{π}{3})}^{2} J ω ω_{x} + \frac{8}{9} J ω ω_{x}] \end{matrix}

(7)

where all parameters are as specified above.

Practical observation of the gyroscope motion around axis ox demonstrates the following properties.^5–7 The blocking motion of the gyroscope around axis oy leads to the fast turn down of the gyroscope around axis ox under action only the gravity and frictional forces. In such condition is obviously that the resistance torques T_ct and T_cr are deactivated but the precession torques T_in and T_am acting around axis oy, that demonstrated by the following tests and computing. Then, the equation of the motion around axis ox of the gyroscope with the spinning rotor is represented by the following equation:

\begin{matrix} J_{x 1} \frac{d ω_{x}}{d t} & = & (W l - s \frac{a}{2} - G a) g \cos γ \\ - [A g \frac{f d}{2 \cos δ} + m_{r} l_{m} \frac{f d}{2 \cos δ} ω_{x}^{2} + \{2 {(\frac{π}{3})}^{2} + 1\} \frac{f d \cos (δ - τ)}{2 h \cos δ} J ω ω_{x}] \end{matrix}

(8)

If the rotor of the gyroscope does not spin, the precession torques T_in and T_am around axis oy are not generated. Hence, the frictional torques T_fin and T_fam are removed from Eq. (8). Then, the equation of the motion around axis ox of the gyroscope with the stopped rotor is represented by the following equation:

J_{x 1} \frac{d ω_{x}}{d t} = (W l - s \frac{a}{2} - G a) g \cos γ - [A g \frac{f d}{2 \cos δ} + m_{r} l_{m} \frac{f d}{2 \cos δ} ω_{x}^{2}]

(9)

where all parameters are as specified above.

The results of computing the time of the gyroscope motion by Eqs. (9) and (8) are compared and yields the interesting information.

III. CASE STUDY AND PRACTICAL TESTS

The experimental part of the gyroscope test is presented by the measurement of its time of the turn from the upper side to down one at the condition of the motion around one axis. The angle of the turn of the gyroscope on the defined angle was calculated on a base of the geometrical parameters of the stand (Fig. 2). At the starting condition of the practical test, the counter-weight of the gyroscope has contacted with the surface of the platform. At the end of the turn, the spherical frame of the gyroscope has contacted the platform (Fig. 2). The total turn of the gyroscope constituted the angle of γ = 91.57°. The time of the gyroscope motion down from the upper side until the contact with the platform was measured several times by the stopwatch. The mean time of the gyroscope motion was accepted for the following analysis. The angular velocity of the spinning rotor was maintained for each experimental test around 10000 rpm. The frictional forces acting on the bearings of the spinning rotor leaded to its drop of speed on 67 rps. Following parameters of the gyroscope stand were computed: the coefficient of friction in sliding supports of the beam is accepted as f = 0.1 that defined empirically; the geometrical parameters h = 56.925 mm and τ = 38.581° are computed from the stand’s geometry (Fig. 2). Defined parameters and presented in Table II were used for the experimental tests and analytical computing of the angular velocity and the time of the turn for the gyroscope.

Substituting defined data and parameters that represented in Fig. 2 into Eq. (8) and transformation yield the following equation:

\begin{matrix} 4.780082 \cdot 1 0^{- 4} \frac{d ω_{x}}{d t} & = & (0.146 \cdot 0.0355 - 0.005 \cdot 0.025 - 0.098 \cdot 0.05) \cdot (9.81) \cdot \cos γ \\ - \frac{(0.277) \cdot (9.81) \cdot (0.1) \cdot (0.00424)}{2 \cos 4 5^{°}} - (0.043) \cdot (0.03674418) \\ \times (\frac{0.1 \times 0.00424}{2 \cos 4 5^{o}}) ω_{x}^{2} - [2 {(\frac{π}{3})}^{2} + 1] \cdot [\frac{0.1 \times 0.00424 \times \cos (4 5^{o} - 38.58126 6^{o})}{2 \times 0.0569254 \cos 4 5^{o}}] \\ \times (0.5726674) \cdot (1 0^{- 4}) \cdot (10000) \cdot (\frac{2 π}{60}) ω_{x} \end{matrix}

(10)

where component 4.737048969·10^-7 $ω_{x}^{2}$ have the small value of high order that can be neglected for the following computing.

Simplification of Eq. (10) yields the following result:

4.780082 \times 1 0^{- 4} \times \frac{d ω_{x}}{d t} = 1.54998 \times 1 0^{- 3} \cos γ - 8.147035996 \times 1 0^{- 4} - 6.771139167 \times 1 0^{- 7} ω ω_{x}

(11)

Equation (11) enables for defining the angular velocity of gyroscope velocity around axis ox. Separating variables and transformation for differential Eq. (11) gives the following equation:

\frac{d ω_{x}}{2289.097833868 \cos γ - 1203.200199414 ω - ω_{x}} = 1.416532010 \times 1 0^{- 3} ω d t

(12)

Representing Eq. (12) in integrals from at definite limits, yields the following expression:

\int_{0}^{ω_{x}} \frac{d ω_{x}}{\frac{2289.097833868 \cos γ - 1203.200199414}{ω} - ω_{x}} = 1.416532010 \times 1 0^{- 3} ω \int_{0}^{t} d t

(13)

The left integral of Eq. (13) is tabulated and represented the integral $\int \frac{d x}{a - x} = - \ln |a - x| + C$ ⁠. The right integral is simple and integrals have the following solution:

- \ln (\frac{2289.097833868 \cos γ - 1203.200199414}{ω} - ω_{x}) |_{0}^{ω_{x}}) = 1.416532010 \times 1 0^{- 3} ω t |_{0}^{t})

giving rise to the following

1 - \frac{ω_{x}}{\frac{2290.2860039 \cos γ - 1203.8247284}{ω}} = e^{- 1.416532010 \times 10^{- 3} ω t}

(14)

Solving Eq. (14) gives the expression of the angular velocity for the running gyroscope around axis ox as the result of the action of the gyroscope weight and frictional torques:

ω_{x} = (\frac{2289.097833868 \cos γ - 1203.200199414}{ω}) (1 - e^{- 1.416532010 \times 10^{- 3} ω t})

(15)

Expression $e^{- 1.416532010 \times 10^{- 3} ω t}$ of Eq. (15) contains the angular velocity ω of the rotor, which value is around n = 10000 …. 30000 rpm. Hence, this expression has the small value of the high order and can be neglected. Then, the equation of the angular velocity for the gyroscope around axis ox is represented by the following expression:

ω_{x} = \frac{2289.097833868 \cos γ - 1203.200199414}{ω}

(16)

where all parameters are as specified above.

The angular velocity of the gyroscope around axis ox is variable and depends on the angle γ of location the gyroscope. The time of the turn of the gyroscope on the angle γ is derived from the following equation:

t = \frac{γ}{ω_{x}}

(17)

where t is the time of the gyroscope turn, γ is the angle of the turn.

The change of the time of the gyroscope motion with the change of its angle of location is represented by the following differential equation:

d t = \frac{d γ}{ω_{x}}

(18)

Substituting Eq. (17) into Eq. (18) and transformation enables for representing the integral form of Eq. (18) is as follows:

\int_{0}^{t} d t = \frac{ω}{2289.097833868} \int_{0}^{γ} \frac{d γ}{\cos γ - 0.525622007}

(19)

The right integral is converted to the integral of the rational function using the versatile trigonometric substitution $x = \tan (γ / 2)$ ⁠, which the derivative is $d x = \frac{d γ}{2 \cos^{2} (γ / 2)}$ ⁠. Using the trigonometric identities $\cos γ = \frac{1 - \tan^{2} (γ / 2)}{1 + \tan^{2} (γ / 2)}$ ⁠, $\cos^{2} \frac{γ}{2} = \frac{1 + \cos γ}{2}$ ⁠, and substituting into Eq. (19), and transformation yield the following integral equation:

\int_{0}^{t} d t = - 5.726888089 \times 1 0^{- 4} ω \int_{0}^{x} \frac{d x}{x^{2} - {0.557620579}^{2}}

(20)

The right integral of Eq. (20) is tabulated and represented by the following equation: $\int \frac{d t}{x^{2} - a^{2}} = \frac{1}{2 a} \ln |\frac{x - a}{x + a}|$ ⁠. Solving the integral Eq. (22) yields the following equation:

t |_{0}^{t}) = - 5.726888089 \times 1 0^{- 4} ω \times \frac{1}{2 \times 0.557620579} \ln |\frac{x - 0.557620579}{x + 0.557620579}| |_{0}^{x})

(21)

Substituting the expression of x into Eq. (21), changing the limits and transformation yield the following equation:

t |_{0}^{t}) = - 5.726888089 \times 1 0^{- 4} ω \times \frac{1}{2 \times 0.557620579} \ln |\frac{\tan (γ / 2) - 0.557620579}{\tan (γ / 2) + 0.557620579}| |_{- {29.11}^{0}}^{{62.46}^{0}})

(22)

Substituting the given speed of the gyroscope rotor ω = 10,000 rpm into Eq. (22), transformation and calculation yield the time of the gyroscope turn on the angle γ = 91.47° = 62.46° + 29.11° that is as follows:

\begin{matrix} t & = & - 5.726888089 \times 1 0^{- 4} \times \frac{10000 \times 2 π}{60} \times \frac{1}{2 \times 0.557620579} \\ \times [\ln |\frac{\tan ({62.46}^{0} / 2) - 0.557620579}{\tan ({62.46}^{0} / 2) + 0.557620579}| - \ln |\frac{\tan (- {29.11}^{0} / 2) - 0.557620579}{\tan (- {29.11}^{0} / 2) + 0.557620579}|] = 2.24 s \end{matrix}

(23)

For the gyroscope, which rotor does not spin, the time of the gyroscope turn is calculated by Eq. (9) that is the typical example from the textbooks of the engineering mechanics. Definitely, the solution of Eq. (9) yields the angular velocity of the gyroscope turn, which value will be less than the value of the angular velocity of the gyroscope turn with the spinning rotor.

The left component of Eq. (9) dω_x/dt = ε_x is presented as the acceleration of the gyroscope turn. Substituting defined parameters into Eq. (9) and transformation yields the following result:

4.780082 \times 1 0^{- 4} \times ε_{x} = 1.54998 \times 1 0^{- 3} \cos γ - 8.147035996 \times 1 0^{- 4}

(24)

Simplification and transformation of Eq. (24) yields the following expression:

ε_{x} = 3.242580357 \cos γ - 1.704371597

(25)

The acceleration is variable due to the change of the angle γ. The change of the acceleration is defined by the first derivative of Eq. (25)

d ε_{x} = - 3.242580357 \sin γ d γ

(26)

Equation (26) is presented by the integral form with defined limits: $\int_{0}^{ε_{x}} d ε_{x} = - 3.242580357$ $\int_{{62.46}^{o}}^{- {29.11}^{o}} \sin γ d γ$ giving rise to the following

ε_{x} = 3.242580357 \times [\cos (- 29.1 1^{o}) - \cos 62.4 6^{o}] = 1.333736781 r a d / s^{2}

(27)

The time of the gyroscope turn on the angle γ = 91.57° is defined by the following formula

γ = \frac{ε_{x} t^{2}}{2}

(28)

Substituting defined parameters into Eq. (28) and solving yield the following result

t = \sqrt{\frac{2 γ}{ε_{x}}} = \sqrt{\frac{2 \times {91.57}^{0} \times π}{180^{0} \times 1.333736781}} = 1.54 s

(29)

The final computing is conducted for Eq. (7) with the aim to define the time of the gyroscope motion around axis ox with the hypothetical action of the resistance torques for the comparative analysis of the results of practical tests and the analytical model. Substituting defined parameters that represented in Table II and Fig. 2 into Eq. (7) and transformation yield the following equation:

\begin{matrix} 4.780082 \times 1 0^{- 4} \times \frac{d ω_{x}}{d t} & = & (0.146 \times 0.0355 - 0.005 \times 0.025 - 0.098 \times 0.05) \times 9.81 \\ \times \cos γ - 0.277 \times 9.81 \times 0.1 \times \frac{0.00424}{2 \cos 45^{o}} - [2 {(\frac{π}{3})}^{2} + 1] \frac{0.1 \times 0.00424}{2 \times 0.056925} \\ \times \cos (4 5^{o} - 38.58 1^{o}) \times 0.5726674 \times 1 0^{- 4} ω ω_{x} \\ - [2 {(\frac{π}{3})}^{2} + \frac{8}{9}] \times 0.5726674 \times 1 0^{- 4} ω ω_{x} \end{matrix}

(30)

Simplification of Eq. (30) yields the following result:

\begin{matrix} 4.780082 \times 1 0^{- 4} \times \frac{d ω_{x}}{d t} & = & 1.54998 \times 1 0^{- 3} \cos γ - 8.147035996 \times 1 0^{- 4} \\ - 1.7718089817 \times 1 0^{- 4} ω ω_{x} \end{matrix}

(31)

Following the steps of computing are the same as represented in Eqs. (10)–(23)

\frac{d ω_{x}}{8.751560678190 \cos γ - 4.598145782161 ω - ω_{x}} = 0.370664976395 ω d t

(32)

\int_{0}^{ω_{x}} \frac{d ω_{x}}{\frac{8.751560678190 \cos γ - 4.598145782161}{ω} - ω_{x}} = 0.370664976395 ω \int_{0}^{t} d t

(33)

- \ln (\frac{8.751560678190 \cos γ - 4.598145782161}{ω} - ω_{x}) |_{0}^{ω_{x}}) = 0.370664976395 ω t |_{0}^{t})

1 - \frac{ω_{x}}{8.751560678190 \cos γ - 4.598145782161 ω} = e^{- 0.370664976395 ω t}

(34)

ω_{x} = (\frac{8.751560678190 \cos γ - 4.598145782161}{ω}) (1 - e^{- 0.370664976395 ω t})

(35)

ω_{x} = \frac{8.751560678190 \cos γ - 4.598145782161}{ω}

(36)

t = \frac{γ}{ω_{x}} a n d d t = \frac{d γ}{ω_{x}}

(37)

\int_{0}^{t} d t = \frac{ω}{8.751560678190} \int_{0}^{γ} \frac{d γ}{\cos γ - 0.525408661522}

(38)

\int_{0}^{t} d t = - 0.149816028235 ω \int_{0}^{γ} \frac{d x}{x^{2} - {0.557784960059}^{2}}

(39)

t |_{0}^{t}) = - 0.149816028235 ω \times \frac{1}{2 \times 0.557784960059} \ln |\frac{x - 0.557784960059}{x + 0.557784960059}| |_{0}^{x})

(40)

t |_{0}^{t}) = - 0.149816028235 ω \times \frac{1}{2 \times 0.557784960059} \ln |\frac{\tan (γ / 2) - 0.557784960059}{\tan (γ / 2) + 0.557784960059}| |_{- {29.11}^{0}}^{{62.46}^{0}})

(41)

\begin{matrix} t & = & - 0.149816028235 \times \frac{10000 \times 2 π}{60} \times \frac{1}{2 \times 0.557784960059} \\ \times [l n |\frac{t a n (62.4 6^{0} / 2) - 0.557784960059}{t a n (62.4 6^{0} / 2) + 0.557784960059}| - \ln |\frac{\tan (- {29.11}^{0} / 2) - 0.557784960059}{\tan (- {29.11}^{0} / 2) + 0.557784960059}|] = 588.646 s \end{matrix}

(42)

Obtained result t = 588.646 s is definitely being not appropriate for the solution and proves that the inertial resistance torques acting around axis ox are deactivated. Results of practical tests and theoretical calculation of the time motion of gyroscope with the counter-weight recorded and represented in Table III.

TABLE III.

Experimental and theoretical results of the gyroscope turn time.

	Time of the gyroscope turn on 91.57^o, (s)
Parameters	Test	Theoretical	Difference, %
The rotor is spun, Eq. (10)	2.10	2.24	6.25
The rotor is stopped, Eq. (11)	1.42	1.54	7.79
The rotor is spun, Eq. (9)		588.64

	Time of the gyroscope turn on 91.57^o, (s)
Parameters	Test	Theoretical	Difference, %
The rotor is spun, Eq. (10)	2.10	2.24	6.25
The rotor is stopped, Eq. (11)	1.42	1.54	7.79
The rotor is spun, Eq. (9)		588.64

Analysis of the results of the theoretical calculations and practical tests for the gyroscope precessions demonstrate discrepancies between them. This divergence is explained by the several factors that are the accuracy of calculations for the gyroscope technical data, the accuracy of tests measurement, the drop of the spinning rotor velocity, and variable values of the frictional coefficient in supports. Nevertheless, the results in tolerance of 10% that accepted in engineering practice for divergence in theoretical and practical results.^15,16 These data enable for stating that results are well matched and theoretical model for the forces acting on the gyroscope and its motions are satisfied for engineering practice.

Conducted test for the gyroscope with the spinning rotor and recorded time of its motion is the validation of the analytical statement that blocking of the rotation of the gyroscope around axis oy leads to deactivation of the resistance torques generated by the gyroscope inertial forces acting around axis ox. At the same time, the precession torques are acting around axis oy¹¹ and generating the frictional torques on the sliding supports that increase the time of gyroscope turn (t = 2.10 s). Hence, the time of the turn of the gyroscope with the stopped rotor around axis ox is shorter (t =1.42 s) < (t = 2.10 s), Table III. This result is the validation of the action the precession torques on the gyroscope. The hypothetical time of the gyroscope turn with the action of the inertial resistance torques is t = 588.64 s that is not validated by the practical tests.

Obtained results demonstrate the deactivation of the inertial resistance torques and the action of the inertial precession torques for the gyroscope motion around one axis. These physical phenomena contradict known principles of the classical mechanics and need deep analytical validation.

IV. RESULTS AND DISCUSSION

The load torque applied to the gyroscope produces the torques generated by the action the centrifugal, inertial, Coriolis forces and the change in the angular momentum of the spinning rotor. The mathematical models for the resistant and precession torques acting on the gyroscope are formulated on the basis of inertial forces. The motion around axis ox of the gyroscope with one side support and blocked motion around axis oy studied at the accepted system of coordinates. Experimental tests and the mathematical model for the gyroscope motion around axis ox demonstrate that resistance inertial torques are deactivated but at the same time, the precession inertial torques are acted around axis oy. The obtained result presents the time of the gyroscope motion around one axis that computed by the mathematical models and measured by practical tests. The analytical and practical results are well matched. Deactivation of the centrifugal and Coriolis forces of the running gyroscope is new unknown and unexplainable property that contradict to the principal of physics and need detailed investigation.

V. CONCLUSION

The gyroscopic effects in classical mechanics are the most complex and intricate in terms of analytical solutions. The recent investigation demonstrated that the origin of the gyroscopic effects is more complex than presented in the known publications. On the gyroscope are acting the several inertial forces generated by the rotating mass of the spinning rotor. New mathematical models for the gyroscope’s inertial torques and motions around one axis, as well as practice tests, discovered the new unknown effects. The rotation of the spinning rotor around two axes demonstrates the deactivation of the inertial forces generated by the rotor’s mass. This fact contradicts the principles of classical mechanics. These phenomena are the manifestation of the unknown property of a physical matter that represents the new challenge for the researchers of classical mechanics. Newton’s laws are justified for the simple action, but for complex one should be formulated, validated and written new physical laws. The new property of the physical matter needs very deep investigations and define the reasonable links of the manifestation.

ACKNOWLEDGMENTS

The work supported by the Kyrgyz State Technical University named after I. Razzakov.

NOMENCLATURE

a, d, h, r_i: geometrical sizes of the gyroscope components
b: mass of the center beam
G: mass of the counter-weight
g: gravity acceleration
e: base of natural logarithm
f: frictional coefficient
i: index for axis ox or oy
J: mass moment of inertia of the rotor’s disc
J_i: mass moment of inertia of the gyroscope around axis i
l: distance between the gyroscope centre mass and axis of the support
l_m: distance between a gyroscope component centre mass and axis of the support
M: mass of the movable gyroscope component
m_r: resulting mass of the gyroscope components
m_i: mass of element i of the gyroscope component
R_c: conventional radius of the rotor
s: mass of the axle
T: load torque
T_am.i: torque generated by the change in the angular momentum acting around axis i
T_cti, T_cr.i, T_in.i: torque generated by centrifugal, Coriolis and common inertial forces respectively, and acting around axis i
T_p.i: precession torque acting around axis i
T_r.i: resistance torque acting around axis i
t: time
W: mass of the gyroscope
α, δ, τ: constructional angles of the gyroscope stand
γ: angle of inclination of the rotor’s axle
ω: angular velocity of the rotor
ω_i: angular velocity of precession around axis i

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2018

Author(s)

Deactivation of gyroscopic inertial forces

I. INTRODUCTION

II. METHODOLOGY

III. CASE STUDY AND PRACTICAL TESTS

IV. RESULTS AND DISCUSSION

V. CONCLUSION

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

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Deactivation of gyroscopic inertial forces

I. INTRODUCTION

II. METHODOLOGY

III. CASE STUDY AND PRACTICAL TESTS

IV. RESULTS AND DISCUSSION

V. CONCLUSION

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

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