MAE 340 Lecture 3: 3-Complex Plane representation of roots

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

Document Summary

Mae 340 i y r a n g a m. All roots are either real or complex conjugate pairs. Gure to the left shows a complex plane with a total of 6 di erent types of roots. 4: a complex conjugate pair of roots with (0) real part. Alternatively, the right-half plane (rhp) is associated with unstable systems. Since the system settling time is based on the slowest root, we may obtain it directly from the plot using the root that is farthest to the right in a stable system. All damped natural frequencies in a system are easily read from the same complex plane. If all roots appear on the real axis, the homogeneous solution has no oscillation: the damping ratio for any pair of complex conjugate roots is equal to the cosine of the angle . I np1 2 i , with = + i shown in the gure.