Class 31- 32 Modeling of a Gyroscope

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 31-32: Modeling of Gyroscope

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mailto:[email protected]

http://www.esnips.com/web/SystemModelingClassNotes

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

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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.

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For best results, it is always suggested you read the source material.


Contents

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Description of Gyroscope•

Principles and Types of gyroscope

•

Modeling of gyroscope


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.

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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.


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. •

That is the velocity of the angular drift ωd

is Therefore high angular momentum Hs

is required to reduce the drift s

fd H

T=ω


Gyroscope representation


Schematic of Gyroscope


Types of Gyroscope

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Two types–

Free and restrained gyro


Free type gyroscope

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Free gyros there are no restraining forces in any direction of the gyro and the gyro wheel remains in fixed space.

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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.


Restrained gyro

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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.


Integrating gyro

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In integrating gyro, only one axis is free to move and the gyro has a single degree of freedom with respect to the frame.

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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 z-

axis.


General applications of gyroscope

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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.


Representation of gyroscope


Analysis of gyroscope


Analysis of Gyroscope

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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

)sin(dtdIH

dtdT xsx

Φ+= θ



•

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.

•

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

)sincos(dtdIH

dtd

dtdBT ysy

θθθ+Φ−=−



•

Similarly, Ty

becomes

•

Taking laplace

transform on both sides, the above equations yield,



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Eliminating θ(s) from the above equations we have

•

Considering Tx

and Ty

separately we have



•

Similarly eliminating Φ(s) from the equation, we get.


Block diagram representation of the system



•

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



•

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 z-

axis.


References

•

Control Systems –

Anantha

Natarajan

and Ramesh Babu

•

www.gyroscopes.org

(for videos)

http://www.gyroscopes.org/


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…

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