Post on 23-Dec-2015
Dynamic Characteristics of
Instruments
P M V SubbaraoProfessor
Mechanical Engineering Department
Capability to carry out Transient Measurements….
Cyclic Input: Hysteresis and Backlash
• Careful observation of the output/input relationship of an instrument will sometimes reveal different results as the signals vary in direction of the movement.
• Mechanical systems will often show a small difference in length as the direction of the applied force is reversed.
• The same effect arises as a magnetic field is reversed in a magnetic material.
• This characteristic is called hysteresis• Where this is caused by a mechanism that gives a sharp
change, such as caused by the looseness of a joint in a mechanical joint, it is easy to detect and is known as backlash.
Hysteresis Loop
Dynamic Characteristics of Instrument Systems
• To properly appreciate instrumentation design and its use,
• it is necessary to develop insight into the most commonly encountered types of dynamic loading &
• to develop the mathematical modeling basis that allows us to make concise statements about responses.
• The response at the output of an instrument Gresult is obtained by multiplying the mathematical expression for the input signal Ginput by the transfer function of the instrument under investigation Gresponse
responseinputresult GGG
Standard Forcing Functions
Standard Forcing Functions
Standard Forcing Functions
Characteristic Equation Development
• The behavior of a block that exhibits linear behavior is mathematically represented in the general form of expression given as
)(........... 012
2
2 txyadt
dya
dt
yda
Here, the coefficients a2, a1, and a0 are constants dependent on the particular instrument of interest. The left hand side of the equation is known as the characteristic equation. It is specific to the internal properties of the block and is not altered by the way the insturment is used.
• The specific combination of forcing function input and instrument characteristic equation collectively decides the combined output response.
• Solution of the combined behavior is obtained using Laplace transform methods to obtain the output responses in the time or the complex frequency domain.
Behaviour of the Instrument
)(0 txya
)(01 txyadt
dya
)(012
2
2 txyadt
dya
dt
yda
)(........... 012
2
21
1
1 txyadt
dya
dt
yda
dt
yda
dt
yda
n
n
nn
n
n
Zero order
First order
Second order
nth order
Behaviour of the Block
• Note that specific names have been given to each order. • The zero-order situation is not usually dealt because it has no
time-dependent term and is thus seen to be trivial. • It is an amplifier (or attenuator) of the forcing function with
gain of a0. • It has infinite bandwidth without change in the amplification
constant.• The highest order usually necessary to consider in first-cut
instrument analysis is the second-order class. • Higher-order systems do occur in.• Computer-aided tools for systems analysis are used to study the
responses of higher order systems.
Solution of ODE
• Define D operator as
dt
dyD
The nth order system model:
)(........... 012
21
1 txyaDyayDayDayDa nn
nn
)(........... 012
21
1 txyaDaDaDaDa nn
nn
The solution of equations of this type has been put on a systematic basis by using either the classical method of D operators orLaplace Transforms method.
Laplace Transforms: Solution of ODE
• The Laplace transform, is an elegant way for fast and schematic solving of linear differential equations with constant coefficients.
• Instead of solving the differential equation with the initial conditions directly in the original domain, the detour via a mapping into the frequency domain is taken, where only an algebraic equation has to be solved.
• Thus solving differential equations is performed in the following three steps:
• Transformation of the differential equation into the mapped space,
• Solving the algebraic equation in the mapped space, • Back transformation of the solution into the original space.
Schema for solving differential equations using the Laplace transformation
Laplace Transforms; nth Order Equation
The nth order system model:
)(........... 012
21
1 txyaDyayDayDayDa nn
nn
Laplace Transform:
sX
sYassYasYassYassYas nn
nn
........... 0122
11
sX
sYasaasasas nn
nn
........... 0122
11
012
21
1 ...........
asaasasas
sXsY
nn
nn
0122
11
1
...........
asaasasas
sXty
nn
nn
Laplace Transformations for Sensors
sXsYasaasasas nn
nn
012
21
1 ...........
012
21
1 ...........
1
asaasasassX
sY
nn
nn
0
1 :
asX
sYOrderZero
01
1 :
asasX
sYOrderFirst
012
2
1 :
asaassX
sYOrderSecond
Generalized Instrument System : A combination of Blocks
The response analysis can be carried out to each independent block.
Response of the Different Blocks
• Zero-Order Blocks• To investigate the response of a block, multiply its frequency
domain forms of equation for the characteristic equation with that of the chosen forcing function equation.
• This is an interesting case because Equation shows that the zero-order block has no frequency dependent term, so the output for all given inputs can only be of the same time form as the input.
• What can be changed is the amplitude given as the coefficient a0. • A shift in time (phase shift) of the output waveform with the
input also does not occur.• This is the response often desired in instruments because it
means that the block does not alter the time response. • However, this is not always so because, in systems, design
blocks are often chosen for their ability to change the time shape of signals in a known manner.
Zero Order Instrument: Wire Strain Gauge
:
ll
RR
KFactorGauge
This is the response often desired in instruments because it means that the block does not alter the time response. All instruments behave as zero order instruments when they give a static output in response to a static input.
Wire Strain Gauge
Strain Gauge
• A strain gauge's conductors are very thin: • if made of round wire, about 1/1000 inch in diameter. • Alternatively, strain gauge conductors may be thin strips of
metallic film deposited on a nonconducting substrate material called the carrier.
• The name "bonded gauge" is given to strain gauges that are glued to a larger structure under stress (called the test specimen).
• The task of bonding strain gauges to test specimens may appear to be very simple, but it is not.
• "Gauging" is a craft in its own right, absolutely essential for obtaining accurate, stable strain measurements.
• It is also possible to use an unmounted gauge wire stretched between two mechanical points to measure tension, but this technique has its limitations.
Wire Strain Gauge Pressure Transducers
In comparison with other types of pressure transducers, the strain gage type pressure transducer is of higher accuraciy, higher stability and of higher responsibility.The strain gage type pressure transducers are widely used as the high accuracy force detection means in the hydraulic testing machines.
TypeFeatures and
ApplicationsCapacity
RangeNonlinearity(%
RO)
Rated Output(m
V/V)
Compensated Temp.Range
( )℃
HVSHigh Accuracy
type0.5,..50 MPa 0.2,0.3 1.0,1.5±1 % - 10 to 60
HVUGeneral Purpose
type1,..50 MPa 0.3,0.5 1.5,2.0±1 % - 10 to 60
HVJSSmall & High
Temperature type
1,..50 MPa 0.5 1.0,1.5±20 % - 10 to 150
HVJS-JS
Small & High Temperature type,Vibratio
n-proof
1,..10 MPa 0.5 1.0,1.5±20 % - 10 to 150
Micro Sensor Technology Tokyo
First Order Instruments
• A first order linear instrument has an output which is given by a non-homogeneous first order linear differential equation
)(01 txyadt
dya
01
asa
sXsY
01
1
asa
sXty
• In these instruments there is a time delay in their response to changes of input.
• The time constant is a measure of the time delay.
• Thermometers for measuring temperature are first-order instruments.
• The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.
• A cup anemometer for measuring wind speed is also a first order instrument.
• The time constant depends on the anemometer's moment of inertia.
First‐order instrument step response
b0
0)( 0 tbtx
s
bsX 0)(
01
0
01
asas
b
asa
sXsY
The complex function F(s) must be decomposed into partial fractions in order to use the tables of correspondences. This gives
01
1
0
0
0
0 1
asa
a
a
b
sa
bsY
1
00
0
0
0 11
aa
sa
b
sa
bsY
1
00
0 11
aa
ssa
bsY
1
00
01 11
aa
ssa
bty
1
0
0
0 exp1 a
ta
a
bty
constant time:0
1 a
a
Factor Gauge:0
0 Ka
b
t
Kty exp1
K
ty
Dynamic Response of Liquid–in –Glass Thermometer
Liquid in Glass Thermometer
TVV bulbT 1
bulbbulbbulb lrV 2
materialvolume
(10−6 K−1)alcohol, ethyl 1120gasoline 950jet fuel, kerosene 990mercury 181water, liquid (1 )℃ −50water, liquid (4 )℃ 0water, liquid (10 )℃ 88water, liquid (20 )℃ 207water, liquid (30 )℃ 303water, liquid (40 )℃ 385water, liquid (50 )℃ 457water, liquid (60 )℃ 522water, liquid (70 )℃ 582water, liquid (80 )℃ 640water, liquid (90 )℃ 695
Thermometer: A First Order Instrument
Conservation of Energy during a time dt
Heat in – heat out = Change in energy of thermometer
Assume no losses from the stem.
Heat in = Change in energy of thermometer
System theof eTemperatur ousInstantanetTs
tTs tTtf
fluid ric thermometof eTemperatur ousInstantanetTtf
Rs Rcond Rtf
Ts(t) Ttf(t)
tfconds RRRU
1
dtTTUAinheat tfsbulb
Change in energy of thermometer: tftfbulb dTCV
tftfbulbtfsbulb TCVdtTTUA
sbulbtfbulbtf
tfbulb TUATUAdt
dTCV
stftf
bulb
tfbulb TTdt
dT
UA
CV
bulb
tfbulb
UA
CV
Time constant
stftf TT
dt
dT
s
Tsss s
tftf
s
Tss s
tf 1 1
ss
Ts s
tf
Step Response of Thermometers
1
s
T
s
Ts ss
tf
1
s
T
s
Ts ss
tf
1
11
ss
Ts stf
1
111
ss
TtT stf
t
TtT stf exp1
t
TtT stf exp1
bulb
tfbulb
UA
CV
Response of Thermometers: Periodic Loading
• If the input is a sine-wave, the output response is quite different;
• but again, it will be found that there is a general solution for all situations of this kind.
stftf
bulb
tfbulb TTdt
dT
UA
CV
22
max,max
s
stSinTT ssss
22
max,
ssss s
tftf
22
max,1
sss s
tf
122
max,
ss
s stf
122
max,1
sstT s
tf
t
tTe
TtT s
t
stf
1
22
max,
22max,
tan
sin11
Ts,
max
- T
tf,m
ax