ENME352-16B Dynamics Assignment 1 Submit by: 8 August 2016, 5.00 pm You may use MATLAB or EXCEL. Present all calculations and including intermediate steps. Include units in your answers. It is not necessary to submit your codes or spreadsheet but please retain these to show if required. Question 1. A child sitting on a swing is given a horizontal impulse H by a parent. Using the solution to the equation of motion of a pendulum, determine the required magnitude of H if the mass of the child and the seat is 15 kg and the length of the swing is 2m for the following values of peak angle of swing: 5°, 10°, 15°, 20°, 25°, 30°. Take the mass of the child and seat as being concentrated at a point 2 m below the axis of the swing. Neglect the mass of the chain and the effect of any friction. Repeat the calculations by using the conservation of energy. Explain the reasons for any discrepancy. The acceleration due to gravity is 9.81 ms-2. Comment on possible effects of neglecting the mass of the chain. Question 2. This question is designed to help you understand the concepts of damped and undamped natural frequency of a single dof system, critical damping, and transient response. The door of a Hobbit house is a circular wooden panel of 1.0 m diameter and has a mass of 8 kg. Neglecting the gap between the axis of the hinges and the door, and any unevenness in the distribution of mass, determine the moment of inertia of the door about the axis of rotation. The moment of inertia of a thin circular plate about a diameter is given by can be determined from the formula I=md2/16. If the hinges are critically damped, and the door opens 60° when given an initial velocity of 1.2 m/s at its centre, determine the rotational stiffness of the door control mechanism, modelling it as a coil spring and rotational damper at the axis of the hinges. Plot the displacement of the centre of the door as a function of time from the time the application of the impulse until the door returns to the original closed position. Use your judgement to decide on the scales and limits. Repeat the plots for the same initial velocity, using the stiffness value obtained above, but for the following special cases: (a) When the damping ratio is 0.05 (b) When the damping ratio is 1.1 (c) If the hinge is undamped d=1 m Model of the door H �Question 3. A trailer is being towed on a rough road at a speed of 100 km/h. The road surface can be approximated by a regular sinusoidal form with a distance of 5.2 m between the peaks. A simplified model of the trailer is shown in the Figure (neglecting the flexibility of the tyres and considering the flexibility to be entirely due to the suspension system with a stiffness of 320 kN/m. When it is empty, its mass is 200 kg, and the damping ratio is 0.25. Determine the amplitude ratio Y2/ Y1. If the trailer were to carry a load of 600 kg load, what will be the damping raio? Recalculate the amplitude ratio for the fully loaded trailer. Hint: Derive the equation of motion in terms of k, m, c and the vertical displacements of the wheel and the trailer y1 and y2 where y1= Y1 sin(t). Find  from the speed of the trailer and the gap between the peaks. Then solve the equation of motion to show that �2 �1 = √1+(2��)2 √1(1−�2)2+(2��)2 where � = Ω � ; � =damping ratio. k c 5.2 m m