MAE 2381

Measurement System Behavior 2

Assigned reading: remainder of 3.3, 3.4, 3.6, 3.8

Second Order Systems

• A measurement system that contains storage and inertia

elements

• Applicable to many measurement instruments

• Modeled by a 2nd order differential equation that can be

compared with a mass–spring system (storage and inertia)

• Piezoelectric sensors are good examples of a 2nd order

measurement system

• piezoelectric effect mechanical stress on some solids

can create charge

• Materials crystals, ceramics

• Inertia from flexing of material

• Amplifier is needed

• Dynamic measurements

Examples of piezoelectric measurement equipment (by PCB)

Force in three directions

Microphone

(low pressure)

High pressure

(shocks, blasts)

Acceleration

Physical Parameter Definitions

• The differential equation coefficients are arranged into

important physical parameters:

Natural frequency

Damping ratio

So then…

• Some mathematics...

z

2z

• The damping ratio determines the form of the DE roots

(positive distinct, equal values, complex conjugates)

• The damping ratio describes the ability of the system to

internally dissipate energy. Four solutions to the equation

exist based on ζ since the quadratic equation has two roots:

2z

2z z 2 z

General Solutions

• No damping (z = 0):

• Example: hit a mass–spring system once and it oscillates

for forever at constant amplitude since there is no internal

energy dissipation

tC

tCtCtjCtjCKA

y

j

n

nnnnh

nn

cos

sincosexpexp

1

2121

General Solutions

• Overdamped (z > 1):

• Two real roots of the equation

• Example: System with sluggish response, never gets to the

steady state output

General Solutions

• Critically damped (z = 1):

• Example: System gets to the steady-state output, but the

response is still slow. None of these solutions seem to

correlate with good response behavior

n

th neCtCKA

y 21

General Solutions

• Underdamped (0 < z < 1):

• Example: Good for measurement devices as it reaches the

steady state value quickly. However, a ringing frequency

exists (ωd) since there is oscillatory behavior.

2

21

1,

sincos

zz

ndn

dd

th tCtCeKA

y

Step Function Input

• We will look at the behavior of this system with a simple

step function input of y(0) to KA

• The NH solution to the DE may be found using a Laplace

transform, three different solutions are obtained based on

the damping ratio

z

z

z 2

z 2

z 2 z 2

Step Function Input

• Using F(t) = AU(t), the graph below shows typical

solutions:

nt

0 2 4 6 8 10 12 14

0.0

0.5

1.0

1.5

2.0z = 0

0.1

0.2

0.65

0.7071

2

KA

y

Characteristic Times

• Time constant: transient response time before the system

settles to equilibrium

The system settles to equilibrium more quickly when the time

constant is smaller.

• Rise time: time to reach 0.9(KA – y0)

• Settling time: measure of time to achieve steady response

where oscillations are inside ± 10% of KA

Characteristic Times - Example

Td

±10%

Characteristic Times

• A good measurement device has a low rise and settling

time

• However, devices with a low rise time typically have a

long settling time and vice versa since ζ depends on

multiple variables

• Usually the damping ratio is 0.6–0.8 leads to a tradeoff

where the rise and settling times are adequate

Realistic Example

yi(t) = 5 U (t)

Step Function and Calibration

• A step function is often created in a lab environment in

order to test a device to find the ringing frequency and

natural frequency and damping ratio

• These values are often reported in calibration certificates

dd

df

T12

21 z nd

2

2

1ln/21

1

yy

z

Sine Function Input

• For F(t) = Asin(ωt), the following general output is

achieved (yh form is ζ dependent; see Eq. 3.15):

• The phase shift is damping & frequency ratio dependent

this time:

Sine Function Input

• We are usually interested in the steady-state response of

the measurement system

• Again, the response is characterized by parameters

Magnitude Ratio

• The second order magnitude ratio M(ω) = B(ω)/KA is:

• This is frequency & damping dependent

• M(ω) is no longer limited to 0–1. What happens if ζ = 0?

Magnitude Ratio

• Based on ω/ωn and ζ (divided into resonance,

transmission, and filter bands based on decibels)

/n

0.01 0.1 1 10

0.01

0.1

1

Decib

els

10

3

-3

-10

-20

-30

-40

M

z = 00.10.2

z = 0.650.707

5

2

10

1

0.3

resonance

transmission

filter

Resonance Band

• Resonance is based on ω/ωn and ζ (the resonance band is

defined as a magnitude ratio above +3 dB)

• Maximum M(ω) occurs around ω/ωn ≈ 1, but changes

slightly based on ζ

/n

0.01 0.1 1 10

0.01

0.1

1

Decib

els

10

3

-3

-10

-20

-30

-40

M

z = 00.10.2

z = 0.650.707

5

2

10

1

0.3

Resonance

• We do not want measurement devices to operate in the

resonance band

• Resonance frequency: related to natural frequency and

damping (not the ringing frequency equation)

• Systems in resonance can be destroyed by excessive

energy addition

Resonance Example

• Video example (University of Salford)

http://www.acoustics.salford.ac.uk/feschools/waves/wine3

video.htm

• A thin, cheaply manufactured glass has low damping and a

natural frequency that can be matched to a high pitch noise

from a speaker

• When affixed to the floor, the forced oscillations and high

magnitude ratio response cause it to break (even though the

sound power is low)

Resonance Example

• Tacoma Narrows Bridge (Wikimedia video) or B&W

http://www.youtube.com/watch?v=xox9BVSu7Ok

• Collapsed on November 7, 1940 from resonance caused by

high winds

• Wind velocity was not ‘periodic’ but wind flowing over

objects can create periodic forces due to vortex shedding

the resonance was due to several factors

Transmission Band

• Good dynamic measurements are made by systems in the

transmission band where -3 dB < M(ω) < +3 dB

• Ideally, ω/ωn << 1, but this is not always possible so ζ is

important and can only take on a certain range of values if

ω/ωn ≈ 1

/n

0.01 0.1 1 10

0.01

0.1

1

Decib

els

10

3

-3

-10

-20

-30

-40

M

z = 00.10.2

z = 0.650.707

5

2

10

1

0.3

Magnitude Ratio

• Systems where ζ > 0.707 do not resonate (fast response?)

• Below – 3 dB, the frequency range is in the filter band

/n

0.01 0.1 1 10

0.01

0.1

1

Decib

els

10

3

-3

-10

-20

-30

-40

M

z = 00.10.2

z = 0.650.707

5

2

10

1

0.3

Phase Lag

• A phase lag diagram indicates the response of the

measurement system based on ζ and ω/ωn.

• Overdamped systems have high phase lag

/n

0.01 0.1 1 10

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

deg

z = 00.10.2

0.650.707

5 2z = 10

1 0.3

Phase Lag

• Large phase shifts occur when damping is low and ω/ωn ≈

1; not a good situation for making measurements

/n

0.01 0.1 1 10

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

deg

z = 00.10.2

0.650.707

5 2z = 10

1 0.3

Multiple Function Inputs

• All of the examples covered so far considered a forcing

function with only one frequency however many

forcing functions have more than one

• When the model equation and input function are both

linear (means output is proportional to input), a

simplification can be used

• Superposition: input signals simply add together to

produce an output signal

Superposition

• If a forcing function can be written as:

• A combined steady response will be:

• B(ωk) and Φ(ωk) are calculated for each component

Superposition Example

• Input function:

• Assume: K = 1, ζ = 2, ωn = 500 rad/s, 2nd order system,

(neglect transient effects, just look at steady state output)

Superposition Example

• Output graph: minor magnitude reduction, phase lag