Acceleration Measurement and
Applications
Prof. R.G. Longoria
Updated for Fall 2011
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Lab: choose a question to answer
1. How does material damping depend on length
of a cantilevered beam?
2. How do bungee cord characteristics influence
the peak acceleration at the extreme end of a
bungee drop?
3. How do you interpret acceleration signals from
the impact of a mass dropped into a sand target?
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Mass-spring-damper concept is key
Mass-spring-damper systems are ubiquitous in engineering, and understanding
their natural (unforced) and forced response lends insight into system dynamics
and provides tools to aid design of physical experiments and sensors.
Practical problems arise involving two different configurations:
Fixed-base – study
response x to
forces on mass
This models many
simple vibration
problems
m
k
m
x
b
k
x
b
y (t )
This case forms the basis for the simple
experiments we will conduct in lab.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Base-excited –response of
x subject to forces
induced by motion of
base
This is similar to a
vehicle suspension, and
also models seismic
sensors
This case forms the basis for understanding the
frequency response of seismic sensors,
particularly accelerometers.
Department of Mechanical Engineering
The University of Texas at Austin
Fixed-base mass-spring-damper
m
k
x
b
A discussion of all cases of the mass-spring-damper system response
can be found in pre-lab readings (Ogata handout). From that
reading, you should review :
1. How undamped natural frequency (ωn) and damping ratio (ζ)
are defined from the basic 2nd order differential equation,
2. How the mass, stiffness and damping influence ωn and ζ,
3. How the response in each case depends on ωn , ζ, and the initial
conditions (initial position and initial velocity), and that these
are closed-form solutions you can use for basic design and
predictive calculations.
mxɺɺ + bxɺ + kx = 0
Unforced case, F(t) = 0
b
k 
ɺɺ
x +   xɺ +   x = 0
m
m


≜2ζωn
≜ωn2
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
ζ = 0 ⇒ undamped
0 < ζ < 1 = 0 ⇒ under-damped
ζ = 1 ⇒ critically-damped
ζ > 1 ⇒ over-damped
Department of Mechanical Engineering
The University of Texas at Austin
No damping – harmonic motion
Useful measures:
xo = peak displacement
xoωn = peak velocity
+x
xoωn2 = peak acceleration
Basic model:
mxɺɺ + kx = 0
Initial conditions: x(0) = xo
Response: x = xo cos(ωn t )
ωn = k m = 'natural frequency'
2π
= 'natural period'
Tn =
ωn
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
xɺ (0) = 0
Vibration relations:
Velocity
v = xɺ = − xoωn sin(ωnt )
Acceleration
a = ɺɺ
x = − xoωn2 cos(ωn t )
For additional information, see Ogata
handout.
Department of Mechanical Engineering
The University of Texas at Austin
Now, introduce damping
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Measuring the damping ratio
The logarithmic decrement refers to the relationship between the
amplitude of the peaks in the response of an under-damped
system versus the cycle of oscillation.
This is a specific analysis of the response for the case where an
under-damped system is given an initial condition set: x(0) = x
o
xɺ (0) = 0
The response data allows you to determine the damping ratio, ζ,
without any other information about the system.
See the pre-lab readings for more details.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Note on logarithmic decrement
• The logarithmic decrement helps you find the
damping ratio by measuring the slope of a line
formed by the natural log of the amplitude
ratios plotted against cycle number.
• If you plot this data, and it does not form a
straight line, we usually interpret this to mean
that the decay is NOT exponential. This means
that the assumption that the damping in the
system is linear is NOT valid – i.e., damping
must be nonlinear.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Using logarithmic decrement to identify
dominant damping
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
The following slides review the response of the baseexcited mass-spring-damper system to a sinusoidal
input.
This response is referred to as the harmonic or
frequency response.
This solution helps explain how the system responds
to a wide range of inputs with different frequency
components.
These concepts help define key sensor
specifications.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Motion sensor dynamics
• Sensors used to measure motion rely on base-excited
mkb system configuration.
• A seismic mass is used and a displacement sensing
mechanism monitors the relative position between
the seismic mass and the housing.
Seismic
mass
m
k
z = relative position
z
b
y (t )
Most sensing mechanisms
either detect or respond to z.
Sensing mechanisms are
discussed in the Appendix.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Mathematical model
Forcing function
z = x− y
mzɺɺ + bzɺ + kz = mω 2Y sin ωt = mω 2Y (t )
z (t ) = Z sin(ωt + φ )
Amplitude response
Z
=
Y
ω 
 
 ωn 
2
2
2
 ω   
ω
1 −    +  2ζ

ω
ω
  n   
n 
2
Appendix A shows how these functions are derived.
Mostly it is important to see how the amplitude z(t) changes
depending on frequency. This helps understand how
different types of seismic sensors are defined (next slide).
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
If Y(t) is a sinusoid,
the response is also
a sinusoid.
Need to find
amplitude and
phase, which
depend on ω
phase response
2ζ
tan φ =
ω
ωn
ω 
1−  
 ωn 
2
Department of Mechanical Engineering
The University of Texas at Austin
Frequency response function
Magnitude: |Z/Y|
Seismometers
operate in this
region
Accelerometers
operate in this
region
From Thomson (1993)
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
‘Seismometer’
Frequency response of Z to Y (displacement) input
Z (ω )
=
Y (ω )
ω 
 
 ωn 
 ω 
1 −  
  ωn 
2
2
2
2
 
ω
 +  2ζ

ω
 
n 
This ratio is the ‘sensitivity’ – basically,
how much does the spring element
compress for a given displacement input.
Remember, the spring element represents a
sensing element of some type.
ω
Z → Y for
≫1
ωn
i.e., mass does not move!
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
‘Accelerometer’
Frequency response of Z to Y (acceleration) input
2
ω 
First, note:
 
Y (t ) = Y sin(ωt )
Then from the Z (ω ) =
 ωn 
2
Y (ω )
2
frequency
Yɺ (t ) = ωY cos(ωt )
  ω 2  
ω 


1
−
+
2
ζ
response:
 


ω
ω
Yɺɺ(t ) = −ω 2Y sin(ωt )
  n   
n 
For frequencies well below natural frequency of the sensor:
2
ωY
ω
Z → 2 for
≪1
ωn
ωn
This indicates that for this frequency range, Z (which is the sensed
variable) is proportional to acceleration of Y.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Accelerometer – alternative function
Z (ω )  1 
= 
Ay (ω )  ωn 
2
1
2
2
 ω   
ω
1 −    +  2ζ

ω
ω
  n   
n 
2
This is now a magnitude ratio
defining how Z depends on the
acceleration. This ratio is the
sensitivity!
Bode Diagrams
From: U(1)
10
That is, we want to use the sensor in
a region of frequencies where the
ratio is essentially constant.
-10
-20
-30
The ‘flat region’ of the
response is where we
want to operate.
Bandwidth
-40
0
-50
To: Y(1)
useful frequency range = bandwidth.
Magnitude response
0
Phase (deg); Magnitude (dB)
A magnitude plot helps us to
understand a critical specification
for any sensor :
-100
Phase response
-150
-200
10-1
100
101
Frequency (rad/sec)
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
SENSITIVITY
Calibration sheet for a Sensotec (Honeywell) JFT flat pack accelerometer
This is a piezoresistive-type accelerometer
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
(Good) Sensors avoid the dynamics
• A good sensor should be designed so that the
forcing is in a frequency range well away from
the natural frequency.
• If we force it close to the natural frequency, we
induce ‘dynamics’ in the sensor. This is
generally not a good thing. You want to
‘operate in the flat region’.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Summary
• Mass-spring-damper system models help understand a
wide range of problems and basic sensors.
• You can understand the underlying design of many
types of sensors such as accelerometers by
understanding 2nd order dynamics.
• The frequency response function for a sensor basically
shows you how the sensitivity is a function of the
input (forcing) frequency. It is essentially constant
within the bandwidth.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Appendix A: How
the frequency
response is
derived
The frequency response
function can be
derived by:
1. Converting ODE to
s-domain
2. Letting s = jω
3. Deriving the
magnitude and phase
functions*
*These are functions of
frequency, ω
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Appendix B
• Types of accelerometers and specifications
• Discussion of some sensing mechanism:
– Capacitive
– Piezoresistive
– Piezoelectric
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Types of accelerometers
• A short note on accelerometers is provided in the
laboratory web documents.
• There are several types of accelerometers
distinguished by the type of sensing element used to
monitor displacement of the seismic mass.
• The type used in this lab will either be a capacitive or
piezoresistive accelerometer.
• These types give reasonably good low frequency
response, and both are made using microelectromechanical devices (MEMS).
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Capacitive sensing mechanism
• The measurand directly or indirectly causes a change in the capacitance.
• The easiest conceptualization is to imagine parallel plates.
C=
εA
d
where ε is the permittivity, A is the
area, and d is the distance between
the plates.
d
q
x•
Energy is stored by virtue of
changes in q and x.
v
F
•
q
C
•
x
v
•Typical scenarios leading to change in C:
–changing the distance between capacitor plates
–changes in the dielectric constant (e.g., due to humidity)
–changes in the area (e.g., a variable capacitor)
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Some C sensors
Pressure
dielectric
“fixed plate”
Level
insulating material
1
2
Humidity
chromium layer
fluid level
pressure
h
Polymer
dielectric
Tantulum layer
H
glass
substrate
deflected diaphragm
Acceleration
flexible/support beam
ho
mass
insulating material
dielectric and
damping
“fixed plate”
motion of
case
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
ADXL05 (capacitive) accelerometer
Note: the construction is
basically a mass-spring-damper
system, where the beam and
spring elements deflect
horizontally, and their position is
sensed by the capacitor plates.
However, it is not a simple
‘passive’ system, because there
is feedback in the operation.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
ADXL05 operation
You actively ‘null’ the output, then measure the
voltage or current required. Contrast with how
a Wheatstone bridge works.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Commonly used with other types of
sensing/actuation
Department of Mechanical Engineering
The University of Texas at Austin
ADXL05 accelerometer
This accelerometer has the frequency response shown below.
This region defines the bandwidth of
this accelerometer. Strictly speaking,
the bandwidth is defined by the
frequency range for which the deviation
is 3 decibels from 0 dB.
This would dictate that you can use this
accelerometer to measure signals with
frequencies out to about 1000 Hz.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Piezoresistive accelerometer
These devices rely on strain gauges that are
typically solid-state and directly
manufactured into the deflecting beam.
The basic design still relies on a seismic
mass (here labeled inertial mass).
The gauges monitor strain induced by
deflection during acceleration.
The calibration sheet for a piezoresistive
accelerometer from Honeywell (Sensotec)
is shown on the next slide.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Calibration sheet for a Sensotec (Honeywell) JFT flat pack accelerometer
This is a piezoresistive-type accelerometer
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
On sensitivity of accelerometers
We saw that the amplitude function for an accelerometer relates the
displacement response (Z) to the input.
If the displacement response represents the deflection of capacitor
plates or the bending of a beam with strain gauges, you can see
how the amplitude response is related to the sensor output,
typically in voltage. Hence, sensitivity is usually specified as the
ratio voltage/acceleration. Typical units are mV/g.
Further, the frequency response curve should give you a ‘picture’
of how this sensitivity varies with frequency, and as such helps
define the bandwidth by some appropriate measure (e.g., the 3 dB
point).
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Piezoelectric accelerometers
Many high grade accelerometers use piezoelectric material in shear
(left) and the other uses it in compression to form the sensing
element. (Diagram from Bruel & Kjaer). Can you see how
these are basic seismic devices in accelerometer form?
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Bruel & Kjaer PZT accelerometer
This particular specification is for a B&K
accelerometer used for structural response
studies.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
“Home-made” solutions
Courtesy of F. Mims, “Sensor Projects” Mini-Notebook
Using a piezo-electric buzzer element, you can build your own vibration sensor.
Since the PZ material is self-generating you
will get “some” signal to drive the diode.
Mims claims that this setup
detected a train that was 1
mile away.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
http://www.sparkfun.com/datasheets/Sensors/Flex/MiniSense_100.pdf
See technical manual at sparkfun.com
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin