ME 363 Exam #2

characteristic roots (poles)

values of p

can solve using quadratic formula or by factoring into parts

the poles correspond to two distinct solutions to the ODE

the damping ratio is equal to 1 when there are 2 distinct poles (T/F)

false: it is not equal to one

the damping ratio is equal to 1 when there are two similar poles

plotting poles

poles can be represented as vectors and put on a graph

ex) 𝑥_1,2 = 𝑎 ± 𝑏𝑗 ⟹ 𝑥₁ = 𝑎 + 𝑏𝑗, 𝑥₂ = 𝑎 − 𝑏𝑗

can be represented as (a, b) and (a, -b) and then plotted

they are plotted on a real (x-axis) and imaginary (y-axis) plane

what is the damping ratio

a measure of how quickly a solution decays (damped out)

what happens to oscillations when 𝜁 = 0

oscillations do not decay

what happens to oscillations when they have a low 𝜁 < 1

oscillations decay slowly

what happens to oscillations when they have a high 𝜁 < 1

oscillations decay quickly

what happens to oscillations when 𝜁 ≥ 1

what is undamped natural frequency

a measure of the speed of a system’s response

what does a small natural frequency mean for oscillations

slow oscillations

slow decay

what does a large natural frequency mean for oscillations

fast oscillations

fast decay

what is damped natural frequency

frequency of oscillations for an underdamped system

free response

when input = 0

f(t) = 0

forced response

response resulting from an external force f(t)

what is the stability when the response decays to zero

stable/critically damped

happens when all of the terms in y(t) contain e^pt where p has a negative real part

what is the stability if the response diverges to infinity

unstable

happens if any of the terms in y(t) contain e^pt where p has a negative real part

what is the stability if the response diverges to neither infinity or nor zero

neutrally stable/critically unstable

y(t) oscillates forever because there are no exponential terms

what happens with stability if there are repeated sets of purely imaginary poles

marginal stability

increasing amplitude of oscillations with time

what is the system response for a pole located to the right of the origin

exponential increase

what is the system response for a pole located to the left of the origin

exponential decrease

what is the system response for a pole located further from the real axis

more oscillatory

what is the system response for a pole located further from the origin

faster response

what is the system response for a pole located closer to the origin

slower response

time constant

1/real part of pole

a system with multiple poles can have multiple time constants if they have different real parts

represents time for system to exhibit 63% of total change

dominant root/pole

for systems that have more and one time constant, the dominant root is the one that exhibits the slowest response

larger time constant = slower response

logarithmic decrement

natural log of the ratio of the amplitudes of any peaks in an underdamped response

represents the rate at which the amplitude of oscillation decreases

laplace transform

allows us to convert differential equations into algebraic equations to solve them more easily

step/impulse responses

represent how systems react to sudden changes