notes%20update

notes%20update - Chapter 1: Replace the first paragraph of...

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Unformatted text preview: Chapter 1: Replace the first paragraph of modelling example 3 (page 21) with the following. In the system shown below, a motor is connected to a drum by a gearbox. As the drum turns, a rope wound around it draws up a weight. To keep things simple, it is assumed that the rope has no mass and that the effective radius of the drum does not change as rope is wound onto it. The moment of inertia of motor/shaft/gear assembly is J 1 , and that of gear/shaft/drum assembly is J 2 . The system has one output (the velocity of the weight) and two inputs. One input is the voltage applied to motor and the other is gravity (note that, if the value of gravity is changed, the behaviour of the system will also change). Chapter 1: Replace last part of modelling example 3 (page 23) with the following. This may look complicated (and, as always, checking to see whether the units are consistent is definitely a good idea), but the complexity it largely illusionary. If we ignore the complex looking coefficients, we are left with an equation of the form dg cv bv v a IN + = + & where a, b, c and d are constants Apart from the fact that both inputs show up on the right hand side (note that, as d is negative, increasing the voltage and increasing gravity have opposite effects, as one might expect), this equation is just like those we derived in the previous two examples and, assuming constant gravity, this system will in fact behave just like a charging capacitor. If the mass is initially stationary and the input voltage is increased from zero to some value at time t = 0 (i.e. if the system is subject to a step input), the velocity of the mass will conform to an equation of the form ) 1 ( / t MAX e v v = where v MAX is a limiting velocity is the time constant of the system Exercise: what will happen if the input voltage applied is such that gm n r n k R v t IN ) / )( / ( 2 1 < Chapter 3: Replace page 3 and the first page of page 4 (down to "Let us now turn") with the following. Consider the simple pendulum shown below. It assumed that the mass of the pendulum is concentrated in the bob. Gravity acting on the bob generates a restoring torque equal to - ) sin( mgl , where is the deflection of the pendulum from the vertical, and the pendulum&s moment of inertia is ml 2 . The defining equation is therefore ) sin( ) / ( = + l g & & This is not linear because of the presence of...
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notes%20update - Chapter 1: Replace the first paragraph of...

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