Mech 215 Flashcards
Newton’s 2nd law in a non-inertial frame
∑Fext + Ffict = ma, where Ffict is the fictitious force, and a is the acceleration of the mass within the moving frame.
Ffict
-ma, where a is the acceleration of the moving frame.
Damping ratio > 1
Amplification
Damping ratio < 1
Mitigation
Phase shift < 0
Output delayed
Phase shift > 0
Output in advance
Harmonic forced response
Output is always delayed and mitigated with respect to the input.
When does resonance occur?
When the system is excited at the resonance frequency.
Requirement of frequency analysis
All initial conditions are set to zero.
Output delay of a first order system
Phase shift of the FRF/ input frequency
Output amplitude of a first order system
alphamodulus(FRF), where alpha is the coefficient of the harmonic input.
Bode plot positive gain
Amplification of the harmonic forced response.
Bode plot negative gain
Mitigation of the harmonic forced response.
If T(s) = T1(s)T2(s), then then gain and phase are (respectively)…
gain1(w) + gain2(w) and phiH1 + phiH2
If T(s) = T1(s)/T2(s), then then gain and phase are (respectively)…
gain1(w) - gain2(w) and phiH1 - phiH2
Bode stability criterion
A system is stable if the open loop gain is negative at the critical frequency, wc.
Critical frequency, wc
The frequency when the phase is -π.
Gain margin
GM = -Gain(wc). This corresponds to the gain that can be added to the system whilst preserving its stability.
Phase margin
PM = π + phase(wgo), where wgo is the frequency at zero gain. This corresponds to the phase delay that can be added to a system whilst preserving its stability.
Steady state error, ess
lim(s->0)(SU(s)/1 + T1(s)T2(s))
What does gamma represent in the open-loop transfer function equation?
The system type number.
First order rise time, Tr
2.2 tau
First order 2% settling time, Ts
4 tau
Second order 2% settling time, Ts
4/xiwn, where xi is the damping ratio and wn is the natural frequency.
Second order peak time, Tp
π/wd
Second order percentage overshoot, D%
100e^(-πxi/√(1-xi^2))
Three effects of increasing the damping ratio
The settling time and overshoot decrease, which is good. The peak time increases, which is bad.
Optimal damping ratio
xi = 0.7, which gives an overshoot of D% = 5%.
Proportional controller, Kp
Improves rapidity, but can introduce a steady state error. This error can be reduced by increasing Kp, but this reduces stability margins.
Proportional integral controller, Kp + Ki/s
The integral action can suppress the steady state error, but it also reduces the stability margins. It increases the order of the system, so it can introduce undesirable oscillations.
Proportional derivative controller, Kp + Kds
Increases the stability margins, but introduces a steady state error. This can be reduced by increasing Kp and Kd, while still keeping good stability margins. The main drawback is that it introduces high frequency noise.
Proportional integral-derivative controller
Kp + Ki/s + Kds
