MECH2210 - Intermediate Mechanical and Space Dynamics - Engineering Assignment Help

Assignment Task

Topic: Angular Momentum, Kinetic Energy/3D Equations of Motion

1. A wheel shown of outer radius 340mm is found to cause excessive vibrations on a car due to unbalance. The wheel is then found to be perfectly balanced using two corrective masses; one of 50g placed at B and one of 150g placed at D.

(a) Prior to balancing the wheel it is rolled down a hill of height h=15m from rest and gains a speed of 50km/hr. Determine the radius of gyration of the wheel about the spin (z) axis and the ratio of the rotational kinetic energy to the translational kinetic energy. (246mm, 0.525)

(b) Determine angular momentum components with respect to the body fixed axes x and y of the wheel at a speed of 50km/hr, both before and after the corrective masses are attached. (0,0.112,0,0 Nms)

(c) Determine the amplitude of the unbalanced vibratory force and moment components with respect to the body fixed axes x,y and z of the wheel at a constant speed of 50km/hr both before and after the corrective masses are attached.

Topic: Angular Momentum, Kinetic Energy/3D Equations of Motion

2. The helicopter model shown consists of four blades A of total mass mA and diameter d, thin rotor of mass mR = mA /5 ,and a body B with 24 times the spin mass moment of inertia of A. The axis of rotor rotation passes through the centre of mass G of the body B.

Assume no friction between the helicopters body and rotor and the helicopter rotor, blades A may be modeled as uniform thin rods. The rotor is spunup to a speed of p (in rad/s) relative to the body B using a motor torque T acting between the rotor and body. The helicopter body B is fixed to the ground during spinup and then is released and hovers in a stationary position.

(a) Explain why it is important to hold the body of the helicopter fixed to the ground while it is being spunup. [2 mks]

(b) Calculate the total angular momentum components and kinetic energy of the helicopter after it is spunup to p rad/s and hovers in a stationary position (as functions of mA, d and p). [4 mks]

(c) Repeat (b) for the case where the helicopter body is not fixed to the ground during spinup and explain any differences. [5 mks]

(d) After an accident, an imbalance caused by a bent blade is fixed by placing a balance mass of 2% of mA at the blade tip A and offset 1% of d in the vertical direction. Calculate the shaking forces and moments on the helicopter under the conditions in (b) after the accident but before and after balancing. [5 mks]

(e) After balancing the helicopter, during hovering under the conditions in (b) it suddenly pitches downwards due to turbulence at a steady rate of ?x. Determine gyroscopic moment on the rotor bearing and explain what control action needs to be made to regain level flying.

Topic: 3D Equations of Motion of Rigid Body

3. An aircraft uses a rate gyroscope as a turn indicator as shown. Assume:

The aircraft has a total mass of, M, radius of gyration, K, and a speed of, V, during a horizontal (XZ plane) turn of radius, R.

The turn indicator has a thin uniform disk of mass m, radius r, spinning at a rate of, on a shaft AB of length, L. Each spring (AC & BD) has linear stiffness, k.

Neglect aircraft engine rotor inertia and shaft AB inertia.

Topic: Angular Momentum, Kinetic Energy/3D Equations of Motion/Precession

4. A fidget spinner of total spinning mass M=30g has 3 balance masses of m=4g positioned at the radius of gyration about its centre of mass O and spin axis z of k=25mm. After the fidget spinner is dropped it is found to shake when spunup to a speed of p=20Hz due to one of the balance masses being pushed out by ?=3mm. Approximate the balance masses as point masses, assume the spinner is thin ie Io=½I and neglect the mass of non-spinning components.

(a) Calculate the angular momentum components and kinetic energy of the fidget spinner when spunup to p both before and after it is dropped. Explain any differences. (0,0,0.00236 Nms, 0.148J, 3.77x10-5 ,0,0.00236 Nms, 0.148J, unbalanced mass in xz plane=>hxu?0)

(b) Determine the amplitude of the unbalanced vibratory force and moment components with respect to the body fixed axes x,y and z both before and after it is dropped. Explain any differences.

(c) Before it is dropped, the fidget spinner is made to perform a precessional motion when spunup to p by tilting it on a central pivot (ie pencil) on Oz at a distance c=6mm. Approximate the precession rate and predict how it changes as the speed is reduced.

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