5 351 2nd order systems

# 5 351 2nd order systems - 2nd Order Systems Nice web site...

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1 2 nd Order Systems Nice web site on this topic: http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/SysDyn/SysDyn2.html 2nd-Order Systems Second-order systems contain 2 independent energy storage elements that exchange stored energy. They typically also contain dissipative elements. Examples?

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2 2nd-Order Systems 2 12 nn yy y K F ( t ) ς ωω ++ = ±± ± For systems with storage/dissipative elements plus inertia , the model used to describe them is: where ω n is the natural frequency of the system (rad/sec) ζ is the damping ratio of the system (unitless ratio) Discussion Compare this to our model for a 1 st -order system. yyK F ( t ) τ += ± units of seconds units of m/sec units of m units of seconds 2 2nd-Order Systems: Mass on a Spring Figure from R. Resnick & D. Halliday Physics , 3 rd Ed., John Wiley & Sons, New York, 1977 kx mx = F = ma 0 k xx m + = Obeyed by elastic bodies generally (below the elastic limit). Analogous equations occur in optics, electrical circuits. The spring force acting on the mass is a restoring force, acting to return the mass to its equilibrium position. The total mechanical energy for the mass is the sum of its kinetic and potential energies. simple harmonic oscillator k x x m =−
3 2nd-Order Systems: Mass on a Spring k m ω = k x x m =− ±± The solution of this equation must be a function whose second derivative is the negative of itself (except for a constant k/m). This is true for sines and cosines.

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## This note was uploaded on 11/28/2010 for the course ENES enes100 taught by Professor Staff during the Spring '10 term at Maryland.

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5 351 2nd order systems - 2nd Order Systems Nice web site...

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