lec15 (1)

lec15 (1) - Introduction to Simulation - Lecture 15 Methods...

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Introduction to Simulation - Lecture 15 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Methods for Computing Periodic Steady-State Jacob White
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Outline • Periodic Steady-state problems – Application examples and simple cases • Finite-difference methods – Formulating large matrices • Shooting Methods – State transition function – Sensitivity matrix • Matrix Free Approach
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SMA-HPC ©2003 MIT Basic Definition Periodic Steady-State Basics • Suppose the system has a periodic input • Many Systems eventually respond periodically () { { input state dx t Fx t u t dt   =+  ( ) 0 xt T f o rt += > > t T2 T 3 T
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SMA-HPC ©2003 MIT Basic Definition Periodic Steady-State Basics • If x satisfies a differential equation which has a unique solution for any initial condition • Then if u is periodic with period T and ( ) () dx t Fx t u t dt =+ ( ) 00 0 xt T f o r s o m e t += Interesting Property ( ) ( ) 0 x t T x t for all t t ⇒+ = >
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SMA-HPC ©2003 MIT Application Examples Periodic Steady-State Basics • Periodic Input –W ind • Response – Oscillating Platform •D e s i r e d I n f o – Oscillation Amplitude Swaying Bridge
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SMA-HPC ©2003 MIT Application Examples Periodic Steady-State Basics Communication Integrated Circuit • Periodic Input – Received Signal at 900Mhz • Response – filtered demodulated signal •D e s i r e d I n f o – Distortion
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SMA-HPC ©2003 MIT Application Examples Periodic Steady-State Basics Automobile Vibration • Periodic Input – Regularly Spaced Road Bumps • Response – Car Shakes •D e s i r e d I n f o – Shake amplitude
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SMA-HPC ©2003 MIT Simple Example Periodic Steady-State Basics RLC Filter, Spring+Mass+Dashpot Spring-Mass-Dashpot RLC Circuit • Both Described by Second-Order ODE Force { 2 2 () input dx d x M Dx u t dt dt ++ =
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SMA-HPC ©2003 MIT Simple Example Periodic Steady-State Basics RLC Filter, Spring+Mass+Dashpot Cont. • Both Described by Second-Order ODE • u(t) = 0 lightly damped (D<<M) Response 2 2 () dx d x M Dx u t dt dt ++ = 2 cos D M t xt K e M φ  ≈+  
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