Final answer:
The first-order pole is at s = -2.5 with a time constant of τ = 1/2.5. The second-order poles are a complex conjugate pair with a damping ratio ξ = -1 / (2*ω_n) and undamped natural frequency ω_n = √(4).
Step-by-step explanation:
The student is looking to determine the pole locations, time constant, damping ratio, and undamped natural frequency for a third-order system with a given transfer function G(s) = 8/(s+2.5)(s²+2s+4). To find the poles, we factor the denominator of the transfer function. The first-order pole is at s = -2.5, and the second-order poles come from the quadratic term s²+2s+4, which does not factor into real roots and therefore represents a complex conjugate pair. To determine the time constant for the first-order pole, we use τ = 1/|Real part of the pole|, giving us τ = 1/2.5. For the second-order poles, we find the damping ratio ξ = -Real part / ω_n, where ω_n is the undamped natural frequency, which can be found by the square root of the constant term in the quadratic expression, ω_n = √(4). The damping ratio can then be determined by ξ = -1 / (2*ω_n).