MAE 340 Lecture Notes - Lecture 4: Damping Ratio, Settling Time

• School: University at Buffalo
• Department: Mechanical & Aerospace Engr
• Course: MAE 340
• Professor: J.Mook

Document Summary

De ne the characteristic equation as a2 2 + a1 + a0 = 0 = 2 + 2 n + 2 n. For an underdamped system, the two roots form a complex conjugate pair, which may be written. Using these roots, we know that the homogeneous solution has the form xh(t) = ae tsin( t + ) = ae ntsin(cid:0)( np1 2)t + )(cid:1) N and the damped natural frequency of the system is. D = = np1 2. The characteristic equation can be written in terms of and as ( i )( + i ) = 2 2 + ( 2 + 2) Thus, the relationship between the actual roots and and n are. In general, it is easier to obtain and from a plot than and n. Therefore, the preferred method for obtaining these parameters from a plot is to rst obtain and from the plot, and then calculate and n using.