Explanation Given information: The rate of spin of the rotor is ψ ˙ about an axis mounted in a single gimbal, the axis of the rotation of the gimbal is A B , the angle between the rotor and the plane of the meridian is θ , the latitude of the position on the earth is λ , O C is parallel to the axis of the earth, and angular velocity of the earth is ω e . The figure-(1) shown below represents the components of angular velocity vector of the gyrocompass Figure-(1) Write the expression for the angular velocity of point O in the reference frame of x y z . Ω = ω e ( − i ^ cos λ sin θ + j ^ sin λ + k ^ cos λ cos θ ) + j ^ θ ˙ = − i ^ ω e cos λ sin θ + j ^ ( ω e sin λ + θ ˙ ) + k ^ ω e cos λ cos θ ...... (I) Here, the rate of rotation of the frame of reference is θ ˙ . Write the expression for K . K = i ^ cos λ sin θ + j ^ sin λ + k ^ cos λ cos θ The K is represented in figure (1). Write the expression for the angular velocity of the rotor. ω = − i ^ ω e cos λ sin θ + j ^ ( ω e sin λ + θ ˙ ) + k ^ ω e cos λ cos θ + k ^ ψ ˙ = − i ^ ω e cos λ sin θ + j ^ ( ω e sin λ + θ ˙ ) + k ^ ( ω e cos λ cos θ + ψ ˙ ) ...... (II) Write the expression for the angular velocity of the rotor in x direction. ω x = − ω e cos λ sin θ ...... (III) Write the expression for the angular velocity of the rotor in y direction. ω y = ω e sin λ + θ ˙ ...... (IV) Write the expression for the angular velocity of the rotor in z direction. ω z = ω e cos λ cos θ + ψ ˙ ...... (V) Substitute ω e cos λ cos θ + ψ ˙ for ω z in Equation (II). ω = − i ^ ω e cos λ sin θ + j ^ ( ω e sin λ + θ ˙ ) + k ^ ω z Write the expression for the angular momentum of the rotor. H 0 = i ^ I x ω x + j ^ I y ω y + k ^ I z ω z ...... (VI) Here, the moment of inertia in x direction is I x , the moment of inertia in y direction is I y , and the moment of inertia in z direction is I z . Since the rotor is symmetric about x axis and y axis, substitute I ′ for I x , and I y , I for I z , ω e cos λ sin θ for ω x , ω e sin λ + θ ˙ , ω y , and ω e cos λ cos θ + ψ ˙ for ω z in Equation (VI). H 0 = − i ^ I ′ ω e cos λ sin θ + j ^ I ′ ( θ ˙ + ω e sin λ ) + k ^ I ω z Write the expression for the angular momentum in the reference frame of x y z . ( H 0 ) xyz = i ^ I x ω ˙ x + j ^ I y ω ˙ y + k ^ I z ω ˙ z ...... (VII) Here the angular velocity in the direction of x is ω ˙ x , the angular velocity in the direction of y is ω ˙ y the angular velocity in the direction of z is ω ˙ z in the reference frame of x y z . Write the expression for ω ˙ x . ω ˙ x = − ω e cos λ sin θ θ ˙ ...... (VIII) Write the expression for ω ˙ y . ω ˙ y = θ ¨ Here, the angular velocity of frame of the reference about y axis is θ ¨ . Substitute − ω e cos λ sin θ θ ˙ for ω ˙ x , θ ¨ for ω ˙ y , I ′ for I x , and I y , and I for I z in Equation (VII). ( H ˙ 0 ) x y z = i ^ I ′ − ω e cos λ sin θ θ ˙ + j ^ I ′ θ ¨ + k ^ I ω ˙ z ...... (IX) Write the expression for the rate of change of the angular momentum. H ˙ 0 = ( H ˙ 0 ) xyz + Ω × H 0 ...... (X) Substitute i ^ I ′ − ω e cos λ sin θ θ ˙ + j ^ I ′ θ ¨ + k ^ I ω ˙ z for ( H ˙ 0 ) xyz , − i ^ ω e cos λ sin θ + j ^ ( ω e sin λ + θ ˙ ) + k ^ ω e cos λ cos θ for Ω , and − i ^ I ′ ω e cos λ sin θ + j ^ I ′ ( θ ˙ + ω e sin λ ) + k ^ I ω z for H 0 in Equation (X) To determine (b) The equation of gyrocompass for small value of θ , and neglecting the term containing ω e 2 .

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter 18, Problem 18.158RP

To determine

(a)

The equation of motion of the gyrocompass.

To determine

(b)

The equation of gyrocompass for small value of θ, and neglecting the term containing ωe2.

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Chapter 18, Problem 18.157RP

Chapter 19.1, Problem 19.1P

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(a) The equation of motion of the gyrocompass.

Solution Summary: The author explains the equation of motion of the gyrocompass.