Question

The following problems should be turned in for credit. Show all of your work for full credit. If you use MATLAB for calculations, attach your code. 1. A linear damped harmonic oscillator is defined by the equation drz+ 25W + w x = 0 where w, is the undamped frequency of the system and is the damping coefficient. The damping coefficient has a strong effect on the overall behavior of the oscillator. a) For several (positive) values of wo, simulate (using appropriate initial conditions) the oscillator in MATLAB under the following conditions: i. (>1 iii. iv. V. 0<5<1 (=0 (<0 b) Based on your results, what is the effect of the parameters and woon system behavior? c) Calculate the eigenvalues of the Jacobian for this system. Based on these results, discuss the expected system stability properties expected over various ranges of -- do these results match your previous results? d) Plot your simulation results in state space - that is, plot yi versus y, where yı and y2 are the two state variables in your canonical form model - showing one example for each of the (What kind of plot will a purely oscillatory system make in state-space?) Explain the relationship between these results and your stability predictions.

Answer 1

Solo Transcher Senesion can be written as 8² + 2qwo t ot was natural forcequency & za demping soon (2) 371 , this i case of overdamped syte (1) z21 , this is case of critical damped system Cililazzel , this is case of underdanped system (iv) 0 No darping () 340 , Unstable system (Poles in R. H.p.) guerdapet critical unstable unstable darped Underdarpy No dauphift