# statics and Dynamics

So, 6. 2. The number of revolutions made by each wheel during the take-off run. Revolutions 6. 3. The torque exerted on each wheel if the mass is keg and the effective radius of gyration is CACM. , where is the moment of inertia. So 7. A solid wheel starts from rest at the top of a slope. The slope is mm long and at an angle of 200 to the horizontal. If the radius of gyration of the wheel (k) is given by: , where r is the radius of the wheel, calculate the linear velocity of the wheel at the bottom of the slope, stating any assumptions you have made.

Let: mass of wheel moment of inertia potential energy at the top of the slope kinetic energy at the bottom of the slope acceleration due to gravity Assumptions made are that there isn’t any friction, air resistance acting on the wheel and that it doesn’t skid. 8. A 2. Egg mass, when attached to the lower end of a vertical, helical spring, causes it to extend by mm. Determine the period of vertical oscillation of the system. Restoring force Spring stiffness Displacement The motion is simple harmonic because is proportional to .

The magnitude of the force per unit displaced is and thus, using the equation: Therefore: So the period of vertical oscillation is: 9. Describe an experiment using a simple pendulum to determine the value of acceleration due to gravity (g), deriving any formulae that will be required. The simplest of experiments using a pendulum to determine the value of acceleration due to gravity, would be to tie a weight to the end of a piece of string creating a pendulum. The time of the back and forth motion the pendulum shows is called the period. It does not depend on the mass or the size of the arc, only the length and acceleration due to gravity.

The formula for finding the period off simple pendulum is: Where Period Engel of pendulum Transpose the simple pendulum formula to find g: To solve the equations for any pendulum, time the pendulum through say 20 back and forth motions. Then record the time and divide it by 20 to find : Once has been found, measure the length of the pendulum, to the centre of the weight and input these values into the equations for . Now the acceleration due to gravity can be found. 10. Discuss forced mechanical vibration, resonance and damping in engineering, egg. Aircraft, bridges, ships, cars, etc.

Include the sequence of events and a description of the contribution of each to the final outcome. You are encouraged to draw on your own experience where you have been involved in a vibration issue on aircraft. Vibration can be described as the movement on a body, back and forth from its resting place when acted upon by an external force. There are three main parameters that can be measured from vibration. The first being amplitude, measuring how much vibration, frequency, measuring how many times it occurs in relation to time, and phase, which describes how it is vibrating.

Forced mechanical vibration is when an external force from a mechanical imbalance causes oscillations through the system. For example when there is an imbalance on the rotors on a helicopter, the resulting vibrations travel wrought the aircraft. If the vibration matches the natural frequency of the aircraft, this can cause resonance. Resonance is a potentially destructive vibration as the oscillations will continue to grow in amplitude until the initial forced vibration ceases or failure occurs.

For example the well-known ground resonance test on a Chinook aircraft, where a vibration matches the natural frequency of the fuselage and rips itself apart. The likelihood of resonance can be minimizes by the use of damping. Damping is the use of systems or components to reduce the amplitude of any oscillations to limit the damage vibrations can cause. This can be done in various ways; springs are used on cars suspension, viscous fluid is used In aircraft landing gear and on the Apache aircraft, rubber lead/lag dampers are used on the rotor head to minimize the vibration from the blades.

An example where forced mechanical vibration leading to resonance has resulted in failure is the collapse Of the Tacoma Narrows Bridge, Washington State, USA in 1940. Problems began to arise when on particularly windy days, construction workers on the bridge noticed that the deck oscillated vertically giving the bridge the nickname ‘Galloping Grittier’, nevertheless the bridge was penned to traffic on 1st July 1940. The ‘Galloping’ motion continued and various attempts to correct it proved ineffective. These included extra strengthening cables and hydraulic dampers.

Fig 1 On the day of the collapse, 7th November 1940, the wind speed was MPH which resulted in, at first small oscillations of the deck. The wind caused a phenomenon known as rare elastic fluttering (fig 1 where the Centre Of the deck remains still and either side of the bridge twists in opposite directions. This then escalated into a resonance effect as the oscillations increased periodically. Once the vibration had moved past the bridges damping mechanisms and matched the natural frequency the result was unavoidable as resonance took hold (fig 2).