Problem Set #5 Out: 10/17/14

Due: 10/24/14

Problem 1.

An engineer determines (from the differential equation) that a system's transfer function is

given by:

( )( ) ( )100102

12000

)(

)(2 +⋅+⋅+⋅

+⋅=

ssss

s

sU

sY

(i) Draw the Bode Plot for this O.L. system by hand using the four guides presented in class,

and label all axes. Is the phase lag at the cross-over frequency greater than -180°?

(ii) Draw the polar plot of the open loop transfer function by hand. Does the plot cross the

negative real axis with a magnitude greater than 1?

(iii) Determine if the closed loop (negative unity feedback) system which uses the previous OLTF

is stable.

Problem 2.

An open-loop plant has a transfer function of ( ) ( )31

1

+⋅− ss. In an attempt to stabilize this

system, an engineer designs a controller (used in a negative unity feedback system) of the

form s

asK

+⋅ so that there is zero steady-state error to a step input.

(i) Determine "a" such that as ∞→K , the closed-loop poles go to ∞⋅±− j2

1 . Draw the root

locus, and determine the values of K for which the closed loop (negative unity feedback)

system is stable.

(ii) For K = 10, draw the closed-loop Bode diagram.

Problem 3.

An engineer wants to determine the differential equation which describes the unknown

system G(s)

G(s) = ? F.F.T. G(j )ωg(t)

(t)δ

Since G(jω) is simply the Fourier transform of the unit impulse response, the engineer uses a

Fast Fourier Transform (spectral analyzer) to generate a Bode diagram (shown on the next page).

Determine the system differential equation, and draw any pole(××××) or zero(����) locations on the s-

plane, (straight line approximations for |G(jω)| are given).

Problem 4.

k c

m

F

x

k = 1 m = 0.0025 c = 0.07071

An engineer wants to experimentally determine the frequency response G(jω) of the dynamic system shown above for F (force) as the input and x (displacement) as the output. The mass “m” is struck with an instrumented hammer (accelerometer mounted on a hammer of known mass) which gives an input of R•δ(t) = F(t) (i.e., hammer mass times hammer acceleration). Another accelerometer mounted on the mass “m” outputs x&& .

k c

m

x

x • •

R • (t) = mh⋅x δ ••

These signals (acceleration of the input and output) go into a spectral analyzer (fast Fourier

transform, or FFT). The acceleration signal of the mass is integrated twice to get displacement,

and it is divided by the input multiplied by the mass of the hammer to represent the input force.

The Fourier transform of this result can be plotted on the FFT display as a Bode plot. Another

way of looking at this is if we divide both the input and output by R (scale each by the same

amount) and integrate the output twice, then we see the displacement response of the system to an

impulse input.

a) Draw the output that is displayed on the spectral analyzer. Have you seen this plot before

in your dynamics classes?

b) Draw the poles or zeros of the system on the s-plane.

c) Draw the polar plot of the analyzer output.

d) How would the Bode diagram and the polar plot change if the damping "c" got very small

(explain and show)?

e) Sketch the Bode diagram of the input F(t). What significance on the experiment do you

draw from the shape of this Bode plot?

( ) hammerhammer xmtR &&⋅=⋅δ

k=1N/m

m=2.5 grams

c=0.07071 N·sec/m