chapter_4_solutions_TFrev2

chapter_4_solutions_TFrev2 - ME 375 System Modeling and...

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1 ME 375 System Modeling and Analysis Section 4 – Solution Methods and Transfer Function Analysis System (w/ I.C.s) Input Output Spring 2009 School of Mechanical Engineering Douglas E. Adams Associate Professor
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2 Motivational Example Differential equations vs. algebra – choose wisely 4.1 Consider the vibration of the INCHES POWERED TETRASILICIDE AND BOROSILICATE GLASS WITH LIQUID CARRIER .025 to .065 following ceramic-coated silica tile during launch and reentry of the orbiter. TILE DENSITY HIGH PURITY SILICA 99.8% AMORPHOUS FIBERS 1 .008 SIP -- NEEDLED STRUCTURE . 016 to .018 .09/.115/.16 LUDOX AMMONIA-STABILIZER BINDER TO PREVENT STRESS CONCENTRATIONS CAUSED BY THE SIP NEEDLE FIBERS AND RTV BOND .125 NOMEX FELT INSULATION – RESISTANT UP TO 800F RTV BOND – CURES AT RM TEMP & VACUUM BAG PRESS Which of the following problems would you RTV BOND -- CURES AT ROOM TEMP UNDER VACUUM BAG .008 VEHICLE SURFACE TILES NOMINAL SIZE IS 1”(to 5”) x 6” x 6” rather solve? 2 0 as bs c ++ = OR We can find solutions using algebra if we transform into the frequency domain Find s © 2009 D. E. Adams ME 375 – Solutions and Transfer Function Analysis 0 ay by cy ++= && & OR We can find solutions using differential equations if we stay in the time domain Find y
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3 Frequency-Domain Solutions Partial fraction expansions When the Laplace transform, This limit of integration is 4.2 [] +∞ = = 0 ) ( ) ( L ) ( dt t y e t y s Y st This limit of integration is associated with associated with steady-state response is applied to a differential equation, each term is transformed into the frequency domain where we can use algebra to solve simultaneously for both responses transient response Initial Condition Ys =+ j j j j i i i i dt u d b dt y d a = ( ) Terms Forcing Function Terms © 2009 D. E. Adams ME 375 – Solutions and Transfer Function Analysis
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4 Important Examples of Transforms Finding Laplace transforms of common functions Some simple examples Steps dt t y e s Y st ) ( ) ( = +∞ 4.3 Exponentials Ramps Trigonometric Impulses t y e s dt e st st 1 1 0 0 0 = = = + y s t y α = +∞ dt e e s Y t st ) ( 0 t ) ( tdt e s Y st = +∞ + = + = + s e s t s 1 1 0 ) ( 2 0 1 s = © 2009 D. E. Adams ME 375 – Solutions and Transfer Function Analysis
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5 Some simple examples Trigonometric +∞ Important Examples of Transforms Finding Laplace transforms of common functions 4.4 Impulses 2 2 0 sin ) ( ω + = = s tdt e s Y st t y () ) ( 0 = +∞ st dt t e s Y δ t y 0 22 co s st Ys e t d t s s +∞ = = + 1 0 = = st e Impulses transform into constants with spectral energy at all frequencies © 2009 D. E. Adams ME 375 – Solutions and Transfer Function Analysis
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6 An impulse in the time domain produces a constant in the frequency domain; in other words short events translate What Is the Frequency Domain? The special case of an impulse 4.5 frequency domain; in other words, short events translate into a “broadband” of energies that excite systems throughout their frequency ranges (e.g., TPS impacts) Typical Example of Original Input Time History vs Estimated 5 10 15 20 25 Force - lb f Original Estimated History vs. Estimated 0.008
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chapter_4_solutions_TFrev2 - ME 375 System Modeling and...

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