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Inertial Navigation/Positioning Analysis – GS7745 Spring Semester 2016 Date Distributed: 3 February 2016 Date Due: 22 February 2016 Lab Problem Set No.2 1. Consider the following numerical values for the SDF mechanical H s = 80, 000 gm cm 2 /s , I1g = 75 gm cm 2 , I 2g = 120 gm cm 2 , I 3g = 117 gm cm 2 . gyro: a) If the third moment of inertia of the gimbal, alone, is 95 gm cm 2 , what is the nominal rotation rate of the rotor? b) Derive the mass un-balance error (p.64) with appropriate discussion. Compute the angular rate error ( δω1 ) due to mass-unbalance if ∆ℓ = 10 −6 m , m = 10 gm , and a = 10 m/s 2 . c) Considering only the output axis rotation error entering the output axis equation (3.26), and assuming the steady state ( η ≈ 0 ), provide an expression for the measurement ω1c and the corresponding error ( δω1 ) (mentioned also on p.63). Compute this error ( δω1 ) if ωɺ 2 = 0.2 rad/s 2 (where ω2 is the angular rate about the output axis). d) Determine the optimal placement of three gyros on a typical aircraft so that all its three mutually orthogonal angular rates are sensed, but the errors (computed in 1.b and 1.c) are avoided as much as possible. Consider that the largest sensed acceleration is in the vertical (due to the reaction of gravity – i.e., the lift on the wings of the aircraft) and that the largest angular rate is the roll rate (usually). Draw a figure of the three gyros, each represented as boxes, with all coordinate axes labeled for each one, and within a larger frame representing the aircraft. In your discussion show how and why the errors mentioned in 1.b and 1.c are minimized as much as possible. 2. a) Assume an ideal circular RLG with λ = 0.6 × 10 −6 m/cy . Determine the diameter that is needed to sense rotation rates of 10−11 rad/s (at which point you would begin to see variations in Earth’s rotation rate); i.e., sense an angular increment of 10−11 rad over one second. b) To what size could the gyro be reduced if the integration time is increased to 10 sec? c) For a circular RLG with radius equal to 5 cm, determine the number of quarter-fringes (quarter cycles) that pass by the detector per second if it senses an inertial rotation rate of 1 rad/s. d) What is the count if its input axis is tangent to the local level and is pointing due north, when the gyro is located at latitude 40° and is stationary with respect to the Earth? e) For the two input rotations of parts c) and d), calculate the phase shift (in cycles) of a FOG with the same circular dimension and 1000 windings of optical fiber.