Class 31- 32 Modeling Of A Gyroscope
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System Modeling Coursework
Class 31-32: Modeling of Gyroscope
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of Gyroscope • Principles and Types of gyroscope • Modeling of gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
Principle of Gyroscope • A spinning wheel in the absence of external torque, maintains the direction of its spin in space. Such a spinning wheel is known as a gyro. • When a torque is applied to an axis inclined to the spin axis of a wheel, the wheel rotates in the plane of the two axes in a direction tending to align the spin axis with the input torque axis. This rotation is termed as precession. • The basic principle behind this spinning gyro wheel which has an angular momentum of Hs remains fixed in space even though the frame in which it is fixed is turning. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
Principle • The gyro wheel can be used as reference in space to measure the angular displacement of the frame. This is true only when there is no friction in the bearings of the gyro wheel. • There will be gradual drift in the gyro wheel due to the friction in the bearings. Therefore the measurement is limited to short duration and smaller angles (approximately does not exceed 100). • The gradual drift velocity in the gyro wheel is proportional to the frictional torque Tf in the bearing and inversely proportional to the angular momentum Hs of the gyro wheel. T ω = • That is the velocity of the angular drift ωd is Therefore H high angular momentum Hs is required to reduce the drift f
d
s
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Gyroscope representation
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Schematic of Gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
Types of Gyroscope • Two types – Free and restrained gyro
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Free type gyroscope • Free gyros there are no restraining forces in any direction of the gyro and the gyro wheel remains in fixed space. • The angular motion of supporting frame is not transmitted to gyro wheel. Hence the gyro measures the angular motions of the vehicle with respect to the gyro as a reference. • This type of gyros are used in automatic plotting, inertial navigational equipment, artificial horizons, stable platform etc. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Restrained gyro • Restrained gyro – rate gyro or integrating gyro. – The restrained gyros are employed to provide either derivative or integral feedback from roll, pitch or yaw to damp out vehicle oscillations about these axes.
• In rate gyro, a helical spring is connected between outer gimbal and the frame and a synchro is fitted on the outer frame such that the shaft of the synchro transmitter is coupled to the z-axis of the outer gimbal. Now, if the gyro wheel turns by an angle Φ about z axis then the signal developed in the synchro mounted on the gyro will be proportional to dΦ/dt. Hence this called rate gyro. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
Integrating gyro • In integrating gyro, only one axis is free to move and the gyro has a single degree of freedom with respect to the frame. • The outer gimbal of a gyro is fixed to the frame to convert it into an integrating. • In integrating gyros it can be proved that the angular motion about y-axis is proportional to integral of the angle through which gyro wheel turns about the zaxis. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
General applications of gyroscope • Gyroscope instruments are used in ships and aircrafts either as gyroscopes or/and turn and bank indicators. • It is also used to measure drilling direction in oil well drilling. In this case it is attached to the drilling string.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Representation of gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
Analysis of gyroscope
July – December 2008
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14
Analysis of Gyroscope • The analysis of gyroscope is too complex but simple relations can be obtained for smaller angular displacements. • The previous Figure shows a gyro wheel whose gimbals (inner and outer gimbals) displaced small angles θ and Φ with respect to XZ plane and Z axis respectively. Newton’s law states that the sum of torque is equal to the rate of change of angular momentum. • The x axis and y axis components of angular momentum are considered separately. For the x axis components, the angular velocity dΦ/dt (due to motion of entire system attached to outer gimbals or outer frame) and moment of inertia Ix of the entire system attached to outer gimbals are responsible for angular momentum in addition to the momentum of spinning wheel about x-axis. On applying Newton’s Law we have Tx = July – December 2008
d dΦ ( H s sin θ + I x ) dt dt
prv/System Modeling Coursework/MIT-Manipal
15
Analysis of Gyroscope • Similarly for y axis components, the angular velocity dθ/dt (due to the motion of all subsystems attached to inner gimbals or inner frame) and moment of inertia Iy of the subsystems attached to the inner gimbals are responsible for angular momentum in addition to the angular momentum of spinning wheel. dθ d dθ Ty − B
dt
=
dt
(− H s cos θ sin Φ + I y
dt
)
• Where B is the frictional coefficient in the bearing. When θ and Φ are very small then cosθ =1, cosΦ=1, sinθ = θ and sinΦ=Φ. Therefore the equation becomes
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
16
Analysis of Gyroscope • Similarly, Ty becomes
• Taking laplace transform on both sides, the above equations yield,
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
17
Analysis of Gyroscope • Eliminating θ(s) from the above equations we have
• Considering Tx and Ty separately we have
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
18
Analysis of Gyroscope • Similarly eliminating Φ(s) from the equation, we get.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
19
Block diagram representation of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
20
Analysis of Gyroscope • In practical situation there is no outer gimbal. The body of the ship or aircraft acts as outer gimbal. Therefore the two axes (Tx and Ty) gyros are replaced by single axis gyros. The torque Tx is input, the angle Φ is the displacement due to Tx. • θ is the angle proportional to Φ. There is no torque is applied along Ty. Therefore we can neglect Ty. Considering the equations, we have
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
21
Analysis of Gyroscope • It is a first order system. In this case input is Φ(s) (angular displacement of outer gimbal or body of ship or aircraft) and the output is θ(s). In other words, θ(s) is measure of Φ(s). To measure the position of an aircraft or ship we need three angular measurements with respect to x-axis, y-axis and zaxis.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
22
References • Control Systems – Anantha Natarajan and Ramesh Babu • www.gyroscopes.org (for videos)
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
23
And, before we break… • Hence, that general is skillful in attack whose opponent does not know what to defend; and he is skillful in defence whose opponent does not know what to attack – Sun Tzu
Thanks for listening… July – December 2008
prv/System Modeling Coursework/MIT-Manipal
24
Class 31-32: Modeling of Gyroscope
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of Gyroscope • Principles and Types of gyroscope • Modeling of gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
Principle of Gyroscope • A spinning wheel in the absence of external torque, maintains the direction of its spin in space. Such a spinning wheel is known as a gyro. • When a torque is applied to an axis inclined to the spin axis of a wheel, the wheel rotates in the plane of the two axes in a direction tending to align the spin axis with the input torque axis. This rotation is termed as precession. • The basic principle behind this spinning gyro wheel which has an angular momentum of Hs remains fixed in space even though the frame in which it is fixed is turning. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
Principle • The gyro wheel can be used as reference in space to measure the angular displacement of the frame. This is true only when there is no friction in the bearings of the gyro wheel. • There will be gradual drift in the gyro wheel due to the friction in the bearings. Therefore the measurement is limited to short duration and smaller angles (approximately does not exceed 100). • The gradual drift velocity in the gyro wheel is proportional to the frictional torque Tf in the bearing and inversely proportional to the angular momentum Hs of the gyro wheel. T ω = • That is the velocity of the angular drift ωd is Therefore H high angular momentum Hs is required to reduce the drift f
d
s
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Gyroscope representation
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Schematic of Gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
Types of Gyroscope • Two types – Free and restrained gyro
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Free type gyroscope • Free gyros there are no restraining forces in any direction of the gyro and the gyro wheel remains in fixed space. • The angular motion of supporting frame is not transmitted to gyro wheel. Hence the gyro measures the angular motions of the vehicle with respect to the gyro as a reference. • This type of gyros are used in automatic plotting, inertial navigational equipment, artificial horizons, stable platform etc. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Restrained gyro • Restrained gyro – rate gyro or integrating gyro. – The restrained gyros are employed to provide either derivative or integral feedback from roll, pitch or yaw to damp out vehicle oscillations about these axes.
• In rate gyro, a helical spring is connected between outer gimbal and the frame and a synchro is fitted on the outer frame such that the shaft of the synchro transmitter is coupled to the z-axis of the outer gimbal. Now, if the gyro wheel turns by an angle Φ about z axis then the signal developed in the synchro mounted on the gyro will be proportional to dΦ/dt. Hence this called rate gyro. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
Integrating gyro • In integrating gyro, only one axis is free to move and the gyro has a single degree of freedom with respect to the frame. • The outer gimbal of a gyro is fixed to the frame to convert it into an integrating. • In integrating gyros it can be proved that the angular motion about y-axis is proportional to integral of the angle through which gyro wheel turns about the zaxis. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
General applications of gyroscope • Gyroscope instruments are used in ships and aircrafts either as gyroscopes or/and turn and bank indicators. • It is also used to measure drilling direction in oil well drilling. In this case it is attached to the drilling string.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Representation of gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
Analysis of gyroscope
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
14
Analysis of Gyroscope • The analysis of gyroscope is too complex but simple relations can be obtained for smaller angular displacements. • The previous Figure shows a gyro wheel whose gimbals (inner and outer gimbals) displaced small angles θ and Φ with respect to XZ plane and Z axis respectively. Newton’s law states that the sum of torque is equal to the rate of change of angular momentum. • The x axis and y axis components of angular momentum are considered separately. For the x axis components, the angular velocity dΦ/dt (due to motion of entire system attached to outer gimbals or outer frame) and moment of inertia Ix of the entire system attached to outer gimbals are responsible for angular momentum in addition to the momentum of spinning wheel about x-axis. On applying Newton’s Law we have Tx = July – December 2008
d dΦ ( H s sin θ + I x ) dt dt
prv/System Modeling Coursework/MIT-Manipal
15
Analysis of Gyroscope • Similarly for y axis components, the angular velocity dθ/dt (due to the motion of all subsystems attached to inner gimbals or inner frame) and moment of inertia Iy of the subsystems attached to the inner gimbals are responsible for angular momentum in addition to the angular momentum of spinning wheel. dθ d dθ Ty − B
dt
=
dt
(− H s cos θ sin Φ + I y
dt
)
• Where B is the frictional coefficient in the bearing. When θ and Φ are very small then cosθ =1, cosΦ=1, sinθ = θ and sinΦ=Φ. Therefore the equation becomes
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
16
Analysis of Gyroscope • Similarly, Ty becomes
• Taking laplace transform on both sides, the above equations yield,
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
17
Analysis of Gyroscope • Eliminating θ(s) from the above equations we have
• Considering Tx and Ty separately we have
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
18
Analysis of Gyroscope • Similarly eliminating Φ(s) from the equation, we get.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
19
Block diagram representation of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
20
Analysis of Gyroscope • In practical situation there is no outer gimbal. The body of the ship or aircraft acts as outer gimbal. Therefore the two axes (Tx and Ty) gyros are replaced by single axis gyros. The torque Tx is input, the angle Φ is the displacement due to Tx. • θ is the angle proportional to Φ. There is no torque is applied along Ty. Therefore we can neglect Ty. Considering the equations, we have
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
21
Analysis of Gyroscope • It is a first order system. In this case input is Φ(s) (angular displacement of outer gimbal or body of ship or aircraft) and the output is θ(s). In other words, θ(s) is measure of Φ(s). To measure the position of an aircraft or ship we need three angular measurements with respect to x-axis, y-axis and zaxis.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
22
References • Control Systems – Anantha Natarajan and Ramesh Babu • www.gyroscopes.org (for videos)
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
23
And, before we break… • Hence, that general is skillful in attack whose opponent does not know what to defend; and he is skillful in defence whose opponent does not know what to attack – Sun Tzu
Thanks for listening… July – December 2008
prv/System Modeling Coursework/MIT-Manipal
24
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