EG260 Modelling & Energy Methods

EG260 Modelling & Energy Methods - A.K. Slone...

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Unformatted text preview: A.K. Slone EG-260 Dynamics (1) a.k.slone 2010 1 of 26 EG-260 DYNAMICS I Modelling and Energy Methods 1. Kinetic Energy .................................................................... 3 2. Potential Energy ........................................................................................ 4 2.1. Potential Energy in Elastic springs ............................. 4 2.2. Gravitational PE .......................................................... 5 3. Relating Potential and Kinetic Energy ............................. 5 4. A vertical spring-mass system .......................................... 7 5. A Simple Pendulum ........................................................... 9 5.1. Method of forces ........................................................... 9 5.2. Energy Method for Simple Pendulum ...................... 11 6. Modelling the mass of a spring ....................................... 13 7. The Compound Pendulum .............................................. 15 7.1. The Non uniform Compound Pendulum ................. 15 7.2. The Uniform Compound Pendulum ......................... 19 8. Fluid systems .................................................................... 22 9. The Shaft and Disc ........................................................... 25 A.K. Slone EG-260 Dynamics (1) a.k.slone 2010 2 of 26 Introduction Modelling is the process of deriving a system of equations to model the motion of a physical device. So far only force summation methods have been used, for instance in the previous SHM notes. There a spring mass system was modelled by considering all the forces acting on the system and using Newtons Laws of Motion to derive the equations of motion. These notes explain energy methods as distinct from force summation methods in the context of modelling dynamic systems and the two methods compared to show their equivalence. Consider a body free to rotate about a fixed axis. The sum of torques about the rotation axis through the centre of mass of the object must equal the rate of change of angular momentum of the mass. Thus: & & I M i i = (1) where i M are the torques acting through point 0, I is the moment of inertia about the rotation axis and & & is the angle of rotation. So that the force summation method implies a knowledge of the torque of the system. If the torques are difficult to determine it may be more efficient to use an energy approach, where use is made of the principle of conservation of energy. Consider the energy of a system; there are two forms of energy, Kinetic and Potential energy. The notation used for potential energy is U and T for kinetic energy. A.K. Slone EG-260 Dynamics (1) a.k.slone 2010 3 of 26 1. Kinetic Energy Kinetic energy is defined as the work needed to accelerate a body from rest to a velocity v. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work would also be required to return the body to a state...
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This note was uploaded on 05/18/2010 for the course ENGINEERIN EG 260 taught by Professor Stone during the Spring '10 term at Swansea UK.

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EG260 Modelling & Energy Methods - A.K. Slone...

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