Frequency Response of Sensors
By Albert E. Brendel
For most force measurement
applications, the strain gage based
transducer can be relied upon to
faithfully convert a mechanical input
into a proportional electrical analog
output signal. However, when the
mechanical input is either applied or
changes at a rapid rate, the sensor
may distort the conversion and the
user should be aware of the possible
consequences of this distortion. This
article will explore the sources of
these distortions, the possible
problems created, and the normally
applied corrections.
A force sensor (which also includes
torque sensors), consists of a
structure which deforms upon the
application of force and a system
which senses this deformation,
producing a proportional output
signal. While these articles concentrate on strain gage based sensors, it
should be noted that the above
definitions are extremely broad and
define many other force sensing
systems as well. A spring is also a
structure which deforms under an
applied load, and the structures of
transducers may be considered as
stiff springs for analysis purposes.
When a mass is connected to one
end of a spring, and the mass is
deflected and then released suddenly,
the mass will be observed to oscillate
and eventually return to its undisturbed position. The frequency at
which the mass will oscillate is
determined by the magnitude of the
mass and the stiffness (or spring
constant) of the spring. The equation
which describes this NATURAL
FREQUENCY is given below:
Where K is the spring constant in
Ibs/inch and M is the mass of the
body in Ibs/in/sec*. On earth, we
often relate the mass of a body to its
Reproduced with permission from Rice Lake Weighing Systems.
weight and M may be replaced by the
weight of the object in Ibs. divided by
the acceleration due to gravity of 386
in/set*.
If the spring was perfect and there
were no frictional losses as the body
passed through our atmosphere, the
system would not halt, but would
continue to oscillate forever. The fact
that the oscillations do eventually stop
shows that there is some form of
retarding action which in a springmass system is referred to as
DAMPING. The existence of damping
does affect the natural frequency,
lowering it somewhat, but for the
small amount of damping normally
found in force sensors the above
equation is still accurate enough for
most purposes. Since the natural
frequency of a spring-mass system
depends upon not only the stiffness of
the structural element but also upon
the mass, the natural frequency of a
force sensor is not constant, but
varies over a certain range of possible
natural frequencies. When a force
sensor is unloaded, its structure still
supports a small mass consisting of
its input mounting structure and some
of the mass of the spring element
itself. If this unloaded transducer is
lightly struck, it will oscillate at its
highest possible natural frequency
often referred to as its RINGING
FREQUENCY Ringing frequencies
typically range from 50 Hz for low
capacity sensors to 10 KHz for high
capacity sensors. On the other hand,
when these same sensors are
deadweight loaded up to their rated
force capacities, their natural frequencies drop because of the increased
mass term in the equation. In this
condition, it will be found that all force
sensors, regardless of capacity, will
have about the same natural frequency. For strain gage based
sensors, this loaded natural frequency will always be approximately
60 Hz. For lesser mass loads, the
natural resonant frequency will be
higher than 60 Hz but always less
than the unloaded ringing frequency.
The importance of these natural
resonant frequencies will become
clear later on.
Another way of considering the
oscillation characteristics of a force
sensor, is to consider it as an energy
converter. For example, an electronic
scale normally contains a strain gage
based sensor and when an object is
placed upon the platform, the platform
starts to accelerate toward a new
equilibrium position. When the
platform reaches this position,
however, it now has gained energy in
the form of momentum and the
platform continues past its equilibrium
point. Now, the spring element of the
sensor starts applying more and more
restraining force, causing the platform
to stop its motion and start returning
to the equilibrium point. But again on
its return trip, the platform gains
momentum and again passes by. If
no damping is present in the system,
the platform would never come to rest
but instead would continue to oscillate
about the static equilibrium point. The
method used to account for this
phenomena in electronic scales is to
either add some form of damping and/
or recognize that the oscillations
occur equally about the equilibrium
point and use some form of electronic
signal conditioning to extract the
correct weight reading from the scale.
Figure 1 below shows the expected
motion of both a damped and an
undamped system.
W-damped
Figure 1
The response of a force transducer
to a varying force input can best be
explained by an example:
Figure 2 shows a simplified
mechanism used to convert rotary to
reciprocating motion. The mechanism
is commonly found in hand tools such
as saber saws and reciprocating
sanders. The bearings in such a
mechanism are subjected to radial
forces that are sinusoidal in nature
and occur at the rotational frequency
of the drive motor. From physics, the
dynamic force produced is relatively
easy to predict, being proportional to
the mass of the reciprocating part, its
maximum displacement and the
square of the angular velocity of the
drive motor.
One method of measuring this
oscillating force would be to mount the
tool on a force sensor with the
reciprocating axis aligned with the
force sensitive axis of the sensor and
then turn on the motor. If we examine
the output signal of the transducer
and compare its peak value to the
predicted value at various motor
speeds, as shown plotted in Figure 3,
we would find that they do not agree.
Measured
F
0
As the rotational speed increases,
we find that the error also increases
and eventually reaches a maximum
value. The point at which this
maximum error occurs will coincide
with the Natural Frequency of the
transducer measured with the mass
of the supported tool assembly.
As the rotational speed is further
increased, we note that the error
starts to diminish until the predicted
value and the measured value again
agree. This point will also be found to
be dependent upon the Natural
Frequency of the system, occurring at
1,414 times the natural frequency.
Unfortunately, these three rotational
velocities 0, Fn and 1.414Fn), are the
only velocities where we can be
certain about the data obtained from
a force sensor. And, even worse, the
data obtained is only known to be
correct as far as magnitude at 0 and
1.414Fn.
If we closely examine the direction
of the force input and compare it with
the direction of the output signal from
the sensor, we would see that they
agree in direction only at values close
to 0 RPM. In fact, at speeds above
the natural frequency, the sensor
signal indicates a reversed signal,
which is often interpreted as a wiring
error rather than an
acknowledgement of a misunderstood
physical phenomena. If this PHASE
information relating the time/direction
information is plotted, we can better
see the relationship between the input
force and the output signal typically
produced by a force sensor. This is
shown in Figure 4.
180
R
C
E
Expected
z
Motor Speed
Figure 3
At low speeds, the values closely
agree but the force transducer always
indicates a higher force than is
predicted.
A
:
cJo’.------------
------------.
e
’
--------T
I
I
0
F/Fn
f
0
At this point, it is tempting to simply
“give up” the measurement and
relegate all force sensors to the local
super-market where they can
accurately weigh non-moving objects
such as meat or apples. If, however,
we can accept some small errors, we
can still measure dynamic forces.
Figure 5
x/x0
1
0 k
F/Fn
1
Figure 5 shows the correction factor
which can be applied to a typical
strain gage based transducer to
correct its output signal at various
frequencies expressed in relation to
the sensor’s natural frequency. The
curve is identical to that found in all
textbooks of mechanical vibration and
is known as the transmissibility of a
spring/mass system. The shape of
the curve and the magnitude of the
error will also be found to be dependent upon the amount of DAMPING
present in the system which is
generally unknown but can normally
be estimated as having a DAMPING
FACTOR of less than 0.1.
The equation which describes this
curve (in BASIC notation) is:
+&AL
(9’)‘+(2*d*p)q
n
n
Where X/X0 is the magnification
factor created by the sensor at
frequency F related to its natural
frequency Fn with a damping factor d.
The phase relationship, plotted in
Figure 4, can be described with the
following equation:
Angle = tan - ’ (*)
Where angle is in radians and
TAN -’ is the arc tangent function.
These equations show that force input
frequencies up to l/l 0 the natural
frequency of the sensor will produce
errors of less than 1%. Force inputs
of II4 the natural frequency produce
errors of 10% and a 30% error can be
expected at forcing frequencies equal
to i/2 the natural frequency.
With this in mind, the general
solution to choosing a force transducer for a dynamic force
measurement is to choose one that,
when installed, has a natural frequency at least 10 times that of the
highest frequency to be measured. In
this way, measurement errors can
always be assumed to be less than
1% which is generally accurate
enough for most engineering purposes. This value of l/i 0 the natural
frequency can be considered as a
good “rule of thumb”.
Normally, a force sensor is chosen
by selecting a full scale capacity
slightly above the highest force level
anticipated for the measurement. In
this way, maximum signal levels and
maximum resolutions are obtained.
However, for dynamic force measurements, it should now be apparent that
the spring constant of the sensor is
an equally important consideration.
Since higher spring constants are
normally associated with higher
capacity sensors, oftentimes it is
necessary to lose some resolution to
gain higher natural frequencies.
The above example and discussion
is related to the effects of a forcing
function of a single frequency.
A common application for sensors
is that of determining the magnitude
of forces acting in various structures.
For example, it might be desirable to
know the forces acting upon the
engine mount of an automobile as it is
driven over various road surfaces. If
such a sensor is fabricated, installed,
and its output recorded, the resulting
signal might appear as shown in
Figure 6.
‘A
m
P
I
i
t
:
e
v Time
frequency domain representative as
shown in Figure 7. This type of signal
transformation is commonly referred
to as a Fourier Transformation.
Another way of viewing this
transformation process is to consider
a method in which the frequency
domain plot may be determined
directly from the sensor output. Figure
8 shows a series of electronic notch
filters which are set up at various
pass frequencies.
V
Figure 6
How accurately does this signal
correspond to the actual force input to
the sensor? The sensor will distort the
signal by various degrees depending
upon the ratio of the force input
frequency to that of the natural
frequency of the sensor. Therefore, if
we know the frequency of the input
force, we could correct the output
data. But looking at the recorded
signal...what frequency do we use?
Over a century ago, a French
mathematician by the name of JeanBaptiste Fourier determined that any
time varying phenomenon could be
described by a series of sine waves at
various frequencies and amplitudes,
which when added together would
duplicate the original phenomenon.
Because of the equivalence, it is
possible to convert a typical sensor
signal recorded in the time domain
such as shown in Figure 6 to a
(fl)
(f2)
(f3)
(f4)
(f5)
(f6)
(J
-
(fn)
-
A signal entering the input to this
series of filters will be separated into
components of different frequencies,
much like sand being sifted through
meshes of different sizes. If we add
these individual frequencies back
together again (perhaps using a
summing amplifier), we could
reconstruct the original signal.
Since sensor inaccuracies are
related to input frequencies, the
frequency domain representation of
the output is very helpful. For example, Figure 9 (below) again shows
the recorded signal as previously
shown in Figure 7, along with the
sensor’s frequency related correction
curve. Examining the amplitude of
P I
Figure 7
w
Figure 8
‘A
ml
Frequency
VI/
Q/L
“wv\rl
wwwwi
^sycwr
“MW
J&gJ
Frequency
Figure 9
each frequency in turn, we can
determine the amount that the sensor
amplified (or attenuated) each
individual component and construct a
new frequency domain curve representing what the input signal must
have been like to produce the
recorded sensor output. This corrected frequency domain plot is
shown in Figure 10.
Frequency
Figure 10
We can even go further, if necessary, and construct a time domain plot
from the frequency domain plot to see
what the actual force input had to look
like.
The above discussion serves only
to produce a good mental picture of
how to analyze frequency related
sensor distortions. As a practical
matter, keeping sensor/mass natural
frequencies high relative to the forcing
frequencies eliminates most correction requirements. Another technique
used is to simply eliminate high
frequency signal components which
may be distorted by using low pass
electronic filtering. Most signal
conditioning systems already contain
this type of filtering and the user
should be cautioned to check to
ensure electronic response characteristics compatible with the
measurement objectives.
Elimination of high frequency
components is not generally damaging to most measurement situations.
For example, if sensor output data is
simply displayed on a digital or analog
display for real-time interpretation by
an operator, frequencies above
approximately 4 Hz will do nothing
more than blur the display and,
therefore, should be eliminated in the
first place. Recording typically roll off
at frequencies below 100 Hz for
mechanical pen writers and slightly
higher for light beam recorders. Very
high speed tape systems might get to
1000 Hz and direct computer analog
to digital converters run anywhere
from 1 Hz to 100 KHz sampling data
and assume the signal samples
(Aliasing errors). Also, when real
mechanical components are involved,
they possess mass and in themselves
tend to restrict rapid force changes.
Virtually everything you ever want to
know about a mechanical system
occurs at frequencies below 100 Hz,
with probably +90% of the action
occuring below 30 Hz.
As a final note, instrumentation
systems (spectrum analyzers) are
available which accept time domain
signals and convert them into
frequency domain plots in real time
(i.e., as they occur). These instruments perform the same function as
the filter sieve and present measurement data in a different format for
analysis.
Reproduced with permission from Rice Lake Weighing Systems.