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04_SEMD3_Ch3_LaplaceText

04_SEMD3_Ch3_LaplaceText - 1 Standard notation in dynamics...

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L l T f Laplace Transforms 1. Standard notation in dynamics and control (shorthand notation) = how you communicate with other engineers r 3 with other engineers. 2. Converts differential calculus mathematics to hapte algebraic operations, very useful for: Initial Value Problems (IVPs). Ch 3. Advantageous for block diagram analysis = a “systems” way to analyze more complex systems. (study subsystems then combine to large system) (study subsystems, then combine to large system) r 3 hapte For “simple” Ch For simple Y(s) -- maybe a short cut ! Laplace Transform Definition - 0 [ ]= ( ) ( ) ( ) st e F t dt f s f t = L Two derivations (“a” is a constant, f ’ is derivative (“a” is a constant, f ’ is derivative wrt wrt time): time): -st st a a a [ ]= e dt e 0 a a = − = − − = L 3 0 0 -st -(b+s)t (b s)t -bt -bt [ ] s s s 1 1 [ ]= e dt e dt -e b b e e + = = = L hapte 0 0 0 -st b+s s+b df df and a theorem ... [f ] e dt ( ) ( 0) dt dt sF s f t = = = = L L Ch 0 Very convenient for zero initial condition f(0) = 0 An example is when f = the “error” in a process control IVP An example is when = the error in a process control IVP, i.e at t = 0 the process is at a “nominal design” steady-state. Other Transforms 2 where we assign ( ) d f dg df g f t L L [ ] 2 where we assign ( ) - (0) dt dt dt sG s g = = = = 2 ( ) - (0) (0) ( ) (0) (0) s sF s f f s F s sf f = = n r 3 etc. for higher orders - [ ( )] sin 2 j t j t e e t j ω ω ω + = L L n d f dt - j t j t e e ω ω + + hapte 1 1 1 + 2 2 s ω ω = + cos [ ( )] 2 t ω = L L Ch Note: 2 2 2 2 2 1 2 s j s j s j s j ω ω ω ω = +
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