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second_order - Second Order Systems Do not distribute these...

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Second Order Systems Do not distribute these notes 41 Prof. R.T. M’Closkey, UCLA
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Second Order Linear Differential Equations The second order linear time invariant ODE is given by: ¨ y + a 1 ˙ y + a 2 y = b 1 ˙ u + b 2 u, (1) where y = dependent variable t = independent variable u = “forcing" function specified on interval I a 1 , a 2 , b 1 , b 2 = constant coefficients (may be real or complex) One block diagram that corresponds to this ODE is u y a 1 a 2 b 1 b 2 There are other “topologies" that also implement (1) but their study will be deferred. Do not distribute these notes 42 Prof. R.T. M’Closkey, UCLA
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Homogeneous and particular solutions are also associated with this ODE: 1. Homogeneous solution. The homogeneous differential equation is the ODE in which the forcing function is zero: ¨ y + a 1 ˙ y + a 2 y = 0 , A non-zero solution to this ODE, denoted y h , is called a homogeneous solution . A basis for all homogeneous solutions can be generated by computing the roots of the associated characteristic polynomial, λ 2 + a 1 λ + a 2 = ( λ - λ 1 )( λ - λ 2 ) = λ 2 - ( λ 1 + λ 2 ) | {z } a 1 λ + λ 1 λ 2 | {z } a 2 , where the roots have been denoted λ 1 and λ 2 . The roots may be real or complex. Note that if the coefficients a 1 and a 2 are real, then the roots, if complex, appear as conjugate pairs. There are two cases to consider: (a) Distinct roots. The roots are distinct when λ 1 6 = λ 2 . In this case any homogeneous solution can be expressed as y h ( t ) = α 1 e λ 1 t + α 2 e λ 2 t , where α 1 and α 2 are constants. (b) Repeated roots. If the roots are repeated, that is λ 1 = λ 2 , then any homogeneous solution can be expressed as y h ( t ) = α 1 e λ 1 t + α 2 te λ 1 t , where α 1 and α 2 are constants. 2. Particular solution. For the non-homogeneous ODE, a particular solution, denoted y p , is any function that satisfies ¨ y p ( t ) + a 1 ˙ y p ( t ) + a 2 y p ( t ) = b 1 ˙ u ( t ) + b 2 u ( t ) , t ∈ I . Particular solutions are not unique because any homogeneous solution can be added to a particular solution to yield another particular solution. Do not distribute these notes 43 Prof. R.T. M’Closkey, UCLA
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Initial Value Problem (IVP). Let the initial conditions associated with the ODE by denoted y 0 and ˙ y 0 . These ICs are specified at time t 0 (in other words, y ( t 0 ) = y 0 and ˙ y ( t 0 ) = ˙ y 0 are specified) and the forcing function u is known for time in the interval t ∈ I (note that the beginning of the time interval coincides with the time when the initial condition is specified). Then, find y over the same time interval. Solution of the IVP. The solution of the IVP is unique and may be found by several different methods. 1. Ad hoc approach. The ad hoc approach relies on finding any particular solution for the IVP and then adding to it a homogeneous solution with the two free parameters that are used to match the initial conditions at t 0 . In the case of distinct roots, y ( t ) = y p ( t ) + α 1 e λ 1 t + α 2 e λ 2 t .
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