Measurement Systems. Lecture 7- Combination of Component Errors in Overall System-Accuracy Calculations

Measurement Systems Lecture 7- Combination of Component Errors in Overall System-Accuracy Calculations Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Combination of Component Errors in Overall System-Accuracy Calculations When planning an experiment, we must decide how accurate the final results must be to meet the goals of the study. The method of equal effects: it seems reasonable to, at least initially, force all the instruments to contribute equally to the overall error.

An example: Dynamometer Consider an experiment for measuring, by means of a dynamometer, the average power transmitted by a rotating shaft.

Uncertainties for each item Our preference, is to treat uncertainty at the 95 percent level of confidence. It is assumed the counter does not miss any counts, the maximum error in R is ±1, because of the digital nature of the device. In assigning an error to t, the synchronization error is considered which involves human factors; a total starting and stopping error is taken as ±0.50 s

Uncertainties for each item The measurement of the torque arm length L with 95 percent uncertainty is assigned ±0.05 in. The scales used to measure the force F is calibrated with deadweights, yielding an uncertainty ±0.040

Uncertainties for each item The calibration uncertainty must be translated into a corresponding measurement uncertainty, the scales is subject to vibration (not present at calibration), which may reduce frictional effects and decrease the error (uncertainty). the pointer on the scale will not stand perfectly still when the dynamometer is running; thus may introduce a new error not present at calibration. Suppose we assume the two mentioned effects cancel and thus take the force measurement uncertainty as ± 0.040 lbf.

Most sensitive parameters Absolute error The total uncertainty

An example: Dynamometer Suppose we wish to measure hp to 0.5 percent accuracy; what accuracies are needed in the individual measurements? Most sensitive parameters Where possible one or more of the quantities, R, F, L, and t must be measured more accurately.

Theory Validation by Experimental Testing An important application of experimental work is testing of new theoretical relations Such confirmation is usually of a statistical nature; both the theoretical prediction and the experimental measurement have some uncertainty attached to them. An "overlap graph" of the theoretical and the measured results, each with its own uncertainty band reviles correlations between two sets. When there is some overlap, then the theory gets more and more likely to be correct as the overlap region increases.

Theory Validation by Experimental Testing: An Example To check the thin beam theory we might build a beam with certain properties (L, b, t, E). We then apply a force F and measure the resulting δ. Suppose our interest is in the "spring constant" F/δ. Measuring F, δ and estimates of their 95 percent confidence interval lead to computation of the uncertainty in the ratio F/δ.

Theory Validation by Experimental Testing: An Example Every "theoretical" result relies on one or more parameter values that can only be found by "experiment", we need values for the dimensions and the material property E. these can only be found by experiment! we can as usual attach an uncertainty to each. We calculate the uncertainty in the theoretical value, and plot the "overlap graph". Inaccurate as shearing effects are neglected!!!

Effect of Measurement Error on Quality- Control Decisions in Manufacturing All measurement systems exhibit statistical scatter; resulting partially defeat of our qualitycontrol goals. some good product may be measured as bad and some bad product may be measured as good. If both the process and measurement variabilities are assumed to be Gaussian, statistical analysis can be used to quantify these effects.

Effect of Measurement Error on Quality- Control Decisions in Manufacturing The "gage" (dashed curves) is set to reject product which measures below 4 and above 16. Due to measurement errors, some percentage of bad product will be measured as good and vice versa, as shown by the shaded squares. We can bias our gage limits, that is, we choose to reject units which measure as large as say, 6, and as small as 14. Then the chance of accepting bad product is nearly zero. Of course the price we pay is that now we will be rejecting more good product. target acceptable range

Effect of Measurement Error on Quality- Control Decisions in Manufacturing Quality-control managers have to decide where to strike a balance between these conflicting goals. The effect of measurement errors can usually be ignored if the measurement system standard deviation is less than about 4 percent of the product acceptance range, Here the percentage is 0.5/(16-4) = 4.2%, so this example almost meets the criterion target acceptable range

Measurement Systems Lecture 8- Static Characteristics (Cont.) Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Static Characteristics (Cont.) Static Sensitivity Computer-Aided Calibration and Measurement: Multiple Regression Linearity Threshold, Noise Floor, Resolution, Hysteresis, and Dead Space Scale Readability Span Generalized Static Stiffness and Input Impedance: Loading Effects

Static Sensitivity The static sensitivity of the instrument can be defined as the slope of the calibration curve. If the curve is not nominally a straight line, the sensitivity will vary with the input value. To get a meaningful definition of sensitivity, the output quantity must be taken as the actual physical output, not the meaning attached to the scale numbers, In this form the sensitivity allows comparison of different instruments as regards their ability to detect measurand changes.

Static Sensitivity Sensitivity to interfering and/or modifying inputs is also of interest. Temperature can cause a relative expansion/contraction in a pressure gage (an interfering input). Also, temperature can alter the modulus of elasticity of the pressure-gage spring, thereby affecting the pressure sensitivity (a modifying input). The first effect is often called a zero drift while the second is a sensitivity drift or scale-factor drift. The superposition of these two effects determines the total error due to temperature. numerical knowledge of zero drift and sensitivity drift allows correction of the readings If such corrections are not feasible, then knowledge of the drifts is used mainly to estimate overall system errors due to temperature

Static Sensitivity To evaluate zero drift, the pressure is held at zero while the temperature is varied over a range and the output reading recorded. For reasonably small temperature ranges, the effect is often nearly linear Sensitivity drift may be found by fixing the temperature and running a pressure calibration to determine pressure sensitivity. Repeating this for various temperatures should show the effect of temperature on pressure sensitivity.

Computer-Aided Calibration and Measurement: Multiple Regression A more cost/effective approach uses multipleregression statistical techniques rather than the classical "one variable at a time. Consider an example where we want to cover a temperature range of 40 to 100 F (the design temperature is 70 F) and the range of the pressure transducer is from 0 to 100 psig. In laying out the calibration plan, we must choose the total number of runs and also the specific combinations of pressure and temperature to actually use.

Computer-Aided Calibration and Measurement: Multiple Regression Assume that the pressure transducer output voltage e 0 is given in terms of pressure p and temperature T by If our model and data were perfect, it would require only four sets of p, T, and e o measurements to find the parameter values (four equations in four unknowns)

Computer-Aided Calibration and Measurement: Multiple Regression Exact Model Identified Models (in presence of noise effects)

Computer-Aided Calibration and Measurement: Multiple Regression the 9 runs we have decided to try validation experiment

Linearity If an instrument's calibration curve for desired input is not a straight line, the instrument may still be highly accurate. Specifications relating to the degree of conformity to straight-line behavior are common. The conversion from a scale reading to the corresponding measured value of input quantity is most convenient if we merely have to multiply by a fixed constant. When the instrument is part of a larger data or control system, linear behavior of the parts often simplifies design and analysis of the whole.

Linearity Several definitions of linearity are possible. However, independent nonlinearity seems to be preferable in many cases, A measure of the maximum deviation of any calibration points from this straight line.

Hysteresis, and Dead Space Overall hysteresis effect Internal friction / hysteretic damping External sliding friction

Generalized Static Stiffness and Input Impedance: Loading Effects The introduction of any measuring instrument into a measured medium always results in the extraction of some energy from the medium, thereby changing the value of the measured quantity from its undisturbed state and thus making perfect measurements theoretically impossible. Should be minimized by impedance matching of source with measuring instrument.

Generalized Static Stiffness and Input Impedance: Loading Effects P E 2 / Z2 Input Impedance Measured source voltage potential The more Z 2 / Z 1 is, the less the loading error

Example: 3.1A force-measuring transducer has an opencircuit output voltage of 95 mv and an output impedance of 500 Ω. To amplify the signal voltage, it is connected to an amplifier with a gain of 10. Estimate the input loading error if the amplifier has an input impedance of: (a) 4 kω or (b) 1 MΩ.

Solution: a) Solving for the current yields The voltage across the amplifier input resistor is then The loading error is thus 10.6 m V or 11 % of the transducer unloaded output. b) Repeating the analysis replacing the 4-k Ω resistor with a 1-M Ω resistor, the error becomes 0.047 mv or 0.05%.

Measurement Systems Lecture 9- DYNAMIC CHARACTERISTICS Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Dynamic Characteristics Dynamic characteristics tell us about how well a sensor responds to changes in its input. For dynamic signals, the sensor or the measurement system must be able to respond fast enough to keep up with the input signals. In many situations, we must use q o (t) to infer q i (t), A qualitative understanding of the operation that the sensor or measurement system performs is imperative to understanding the input signal correctly.

Generalized Mathematical Model of Measurement System The most widely useful mathematical model for the study of measurement-system dynamic response is the ordinary linear differential equation with constant coefficients. The relation between any particular input (desired, interfering, or modifying) and the output can be put in the form:

Operational Transfer Function In the analysis, design, and application of measurement systems, the concept of the operational transfer function is very useful: Their utility for graphic symbolic depiction of system dynamic characteristics by means of block diagrams, It is helpful in determining the overall characteristics of a system made up of components whose individual transfer functions are known

Forcing functions The typical ones for dynamic response analysis :

Characteristic Equation Development Nature of characteristic equation block behavior Highest order usually necessary to consider in first-cut instrument analysis : 2nd-order class

Zero-order blocks No frequency dependent term Not even for phase shift Just For amplification ( ) a 0 7

First-Order Systems A system where energy is stored (or dissipated) A change at the input is NOT seen immediately at the output. Many instruments exhibit a 1 st order response. Thermal Systems Digital filters Analog filters Capacitances

First Order Systems: Provides a parameter, called the time constant, for choosing an instrument. This parameter let us know if the instrument will respond quick enough to capture changes in the system. a 1 dy a dt 0 y bx

()First-Order Systems: Step Input Conservation of Energy Rate of energy added Rate of energy removed = Rate of energy change within system (CV) E s SyE out E stemhatts dedtcv d dsystemeout dtdcvcvdthaddhatt ( )Tdt T s T A s TTT 1 st order ODE hs T = T(t) T(0) = T o >T cvae cvt

TTo T 1 0.9 T e R = Thermal resistance C = Thermal capacity c v has RC Response for Varying Values of Tau 1 R ha s C c v 0.8 0.7 T T o T T 0.6 0.5 0.4 0.3 0.2 0.1 Tau = 1 Tau = 0.5 Tau = 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time 11 K. Alt

12 1 M T T T T M M T o o 10,RatioMagnitude01,inErorDynamic5 )(tettttto 0.632 or 63.2% 0.368 90% of the final value Rise Time

Time Constant Is the time it takes a first order system to reach 63.2% (0.632) of its final value in response to a step change in the system 1.0 0.632 Time

First-order blocks Time dependent terms Output response to step forcing function y( t) AK(1 e t / ) Static gain of the block Step amplitude 14

Example A balloon is equipped with temperature and altitude measuring instruments. The balloon is initially anchored to the ground with the instrument output readings in steady state. The altitude-measuring instrument is approximately zero order and the temperature transducer first order with a time constant of 15 seconds. The temperature on the ground, T 0, is 10 C and the temperature T x at an altitude of x meters is given by the relation: T x = T 0-0.01x

Example a) If the balloon is released at time zero, and thereafter rises upwards at a velocity of 5 m/sec, draw a table showing the temperature and altitude measurements at intervals of 10 seconds over the first 50 seconds of travel. Show also in the table the error in each temperature reading. b) What temperature does the balloon report at an altitude of 5000 meters?

Solution of part a) Tx T0 0.01x 10 0.01x Tr 1 D 1 D 1 15D t/15 T Ce 10 0.05( t 15) r @ t 0., T 10 C 0.75 r

Solution of Part b) @5000 mt, 1000sec T T r r 1000/15 10 0.75e 0.05 1000 15 10 0.05 985 39.25 For large values of t, the transducer reading lags the true temperature value by a period of time equal to the time constant of 15 seconds. o C

First-order blocks Output response to sine-wave forcing function amplitude of the steady state response phase shift Dynamic error: a measure of an inability of a system to adequately reconstruct the amplitude of the input for a particular frequency 19

First-Order Systems: Frequency Response Example Suppose we want to measure the following input signal with a first-order instrument whose τ is 0.2 s and static sensitivity K, Solution: By superposition, one can write,

Dynamic Characteristics

Dynamic Characteristics Example: The approximate time constant of a thermometer is determined by immersing it in a bath and noting the time it takes to reach 63% of the final reading. If the result is 28 s, determine the delay when measuring the temperature of a bath that is periodically changing 2 times per minute.

Measurement Systems Lecture 10- Dynamic Characteristics (cont.) Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Second-Order Systems In general, a second-order measurement system subjected to arbitrary input, x(t): The essential parameters:

Second-Order Systems The characteristic equation: Roots of quadratic equation: Complementary solutions:

Second-order Systems: Example: The force-measuring spring Consider a spring with spring constant Ks under applied force fi and the total mass M. At start, the scale is adjusted so that xo =0 when fi =0;

Second-order Systems: Step Response With zero initial conditions:

Second-order Systems: Step Response Non-dimensional step response of second-order instrument

Second-order Systems: Step Response For over-damped (ζ >1) or critical damped (ζ = 1), there is neither overshoot nor steady state dynamic error in the response. In an under-dameped system (ζ < 1) the steady-state dynamic error is zero, but the speed and overshoot in the transient are related.

Second-order Systems: Step Response

Second-order Systems: Ramp Response For a ramp input: With zero initial conditions the solutions are:

Second-order Systems: Ramp Response Typical ramp response of second-order instrument

Second-order Systems: Frequency Response The response of a second-order to a sinusoidal input

Example: A pressure transducer has a natural frequency of 30 rad/s, damping ratio of 0.1 and static sensitivity of 1.0 μv/pa. A step pressure input of 8*10 5 N/m 2 is applied. Determine the output of a transducer. Solution:

Example: A second order instrument is subjected to a sinusoidal input. Undamped natural frequency is 3 Hz and damping ratio is 0.5. Calculate the amplitude ratio and phase angle for an input frequency of 2 Hz. Solution:

Example: An Accelerometer is to selected to measure a timedependent motion. In particular, input signal frequencies below 100 Hz are of prime interest. Select a set of acceptable parameter specifications for the instrument, assuming a dynamic error of ±5% and damping ratio ζ =0.7

Response of a General Form of System to a Periodic Input The steady state response of any linear system to the complex periodic signal can be determined using the frequency response technique and principle of superposition.

Response of a General Form of System of a Periodic Input

Response of a General Form of System to a Periodic Input

Measurement Systems Lecture 11- Measuring Devices Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Motion and Dimensional Measurement Considering measuring devices with motion and dimensional measurements as, they are based on two of the fundamental quantities in nature (length and time), so many other quantities (such as force, pressure, temperature, etc.) are often measured by transducing them to motion and then measuring this resulting motion. Mainly concerned with electromechanical transducers which convert motion quantities into electrical quantities, To provide sufficient detail for practical application of the relatively small number of transducer types which form the basis of the majority of practical measurements.

Motion and Dimensional Measurement Fundamental Standards Relative Displacement: Translational and Rotational Calibration Resistive Potentiometers Resistance Strain Gage Differential Transformers Synchros and Resolvers Variable-Inductance and Variable-Reluctance Pickups Eddy-Current Non-contacting Transducers Capacitance Pickups Piezoelectric Transducers Electro-Optical Devices Photographic and Electronic- Imaging Techniques Photoelastic, Brittle-Coating, and Moire Fringe Stress-Analysis Techniques Displacement-to-Pressure (Nozzle- Flapper) Transducer Digital Displacement Transducers (Translational and Rotary Encoders) Ultrasonic Transducers

FUNDAMENTAL STANDARDS In the SI system, there are seven basic measurement units from which all other units are derived,

Derived Standards All of the other units are derived from the seven basic units,

RELATIVE DISPLACEMENT: TRANSLATIONAL AND ROTATIONAL We consider here devices for measuring, the translation along a line of one point relative to another and the plane rotation about a single axis of one line relative to another. They form the basis of many transducers for measuring pressure, force, acceleration, temperature, etc.

Calibration Static calibration of translational devices often can be satisfactorily accomplished by using ordinary micrometers as the standard A precision dividing head may be used to both produce the angular motion and measure it.

Resistive Potentiometers A resistive potentiometer consists of a resistance element provided with a movable contact. The contact motion can be translation, rotation, or a combination of the two (helical motion in a multi turn rotational device), Allowing measurement of rotary and translatory displacements. The resistance element is excited with either de or ac voltage, and the output voltage is (ideally) a linear function of the input displacement.

Resistive Potentiometers Ex.: It is necessary to measure the position of a panel. It moves 0.8 m. Its position must be know within 0.1 cm. Part of the mechanism which moves the panel is shaft that rotate 250 o when Panel is moved from one extreme to the other. A control potential has been found which is rated at 300 o full scale movement. It has been 1000 turns of wire. Can this be used? Solution: Potentiometer resolution is, The shaft provides a conversion, The required resolution translates into, So the available potentiometer will work.

Resistive Potentiometers Self-heating occurs because of the power dissipation in sensor. The increase in temperature from self-heating ΔT due to P D =I 2 R T is: δ is heat dissipation factor (mw/k) θ is thermal resistance (K/mW) To minimize self-heating effect, the power dissipation must be limited.

Resistive Potentiometers Ex: A control potentiometer is rated as 150 Ω 1 W (derate at 10 mw/ o Cabove 65 o C) 30 o C/W thermal resistance Can it be used with 10 V supply at 80 o C ambient temperature? Solution: The power dissipated by the potentiometer is, The actual temperature of the potentiometer The allowable power dissipation P allowed < P

Resistive Potentiometers The potentiometer output voltage is the input to a meter or recorder that draws some current from the potentiometer. Potentiometer loading effect causes:

Resistive Potentiometers Ex.: Plot the transfer curve and determine endpoint linearity of a 1 kω potential driving a 5 kω load, powered from a 10 V source.

Resistive Potentiometers

Resistance Strain Gage Consider a conductor of area A and length L, made of a material with resistivity ρ. The resistance R is given by: If this conductor is now stretched or compressed, its resistance will change Neglecting higher orders

Resistance Strain Gage Strain gages, in general, are applied in two types of tasks: in experimental stress analysis of machines and structures and in construction of force, torque, pressure, flow, and acceleration transducers.

Resistance Strain Gage

Wheatstone Bridge: Deflection Method Wheatstone bridges are often used in the deflection mode: This method measures the voltage difference between both dividers or the current through a detector bridging them.

Temperature Compensation

Strain gage arrangements

Effect of Lead Wire Resistance

Effect of Lead Wire Resistance: Defection method

Load Cell

Differential Transformers Schematic and circuit diagrams for translational and rotational linear variable-differentialtransformer (LVDT) displacement pickups

Differential Transformers When the secondaries are connected in series opposition, a null position exists e 0 is zero. Motion of the core from null then causes a larger mutual inductance (coupling) for one coil and a smaller mutual inductance for the other, the amplitude of e 0 becomes a nearly linear function of core position for a considerable range either side of null.

Differential Transformers The output e 0 is generally out of phase with the excitation e ex inductances no voltage-measuring device attached For each differential transformer there exists a particular frequency (numerical value supplied by the manufacturer) at which this phase shift is zero.

Measurement Systems Lecture 12- Measuring Devices (Contd.) Hamid Ahmadian School of Mechanical Engineering Iran University of Science and Technology ahmadain@iust.ac.ir

Eddy current proximity probe Measures Displacement Dynamic range : 500:1 Frequency range : DC-10 KHz (Theoretical) DC-2000Hz (Practical) How it works? Driver Probe Extension cable

Eddy current proximity probe How it works? (cont.) Produces 2 signals : AC proportional to vibration DC proportional to the gap size

Eddy current proximity probe Application Relative motion Shaft eccentricity Oil film thickness & etc. Generally Smooth running rotor is critical (Turbines & Compressors) High speed or very low speed rotors Advantages Non-contacting No moving parts, no wear Works to DC

Eddy current proximity probe Limitations Shaft magnetic properties Variations Shaft geometric irregularities erroneous signal components Local calibration necessary Limited practical frequency range as displacement relatively small at high frequencies

Piezoelectric Accelerometers How it works? Sensing element put under load by a mass Crystal squeezed or released as stack vibrates Charge output proportional to force

Piezoelectric Accelerometers Principle of Operation Polarization principle Generator action Motor action Mass in direct contact with piezoelectric Proportional electric charge

Piezoelectric Accelerometers Measures acceleration Dynamic range Contacting 10 8 :1(160dB) Measures absolute casing motion Advantages Self generating Rugged No moving parts, no wear Very large dynamic range Wide frequency & amplitude range Compact & often low weight Orientation not important Velocity or displacement output available

Piezoelectric Accelerometers Limitations High impedance output No true DC response Types Compression type design Traditional simple construction Very stable but high environmental influence Typically used for high shock levels P :piezoelectric element B :Base M :Seismic mass S :Spring

Piezoelectric Accelerometers Types (cont.) Shear type design Piezoelectric arranged subjected to shear forces from seismic mass Rather insensitive to environmental parameters like temperatures DeltaShear Design 3 piezoelectric elements & 3 masses arranged in triangular configuration Excellent overall specifications Very low sensitivity to environmental influences P :piezoelectric element B :Base M :Seismic mass R :Clamp ring

Piezoelectric Accelerometers Types (cont.) Planar-Shear Design Simplified DeltaShear Design with 2 elements Annular-Shear Design Theta-Shear Design Ortho-Shear Design P :piezoelectric element B :Base E :Built-in Electronics M :Seismic mass R :Clamp ring

Accelerometer Dynamics Seismic Accelerometer A Deflection type accelerometer Considering only the mass-spring system Adding the motion of the base... A 2 nd order System

Piezoelectric Accelerometers Typical frequency response for a PZT Determining resonant frequency Mathematically a 3 rd order system

Cross-Axis sensitivity A Vibrating structure may be subjected to Torsional vibration Compressional vibration Transverse vibration Cross-Axis sensitivity Response to a plane perpendicular to the main axis Expressed in percent of the main

Accelerometer Selection First glance categorization General purpose various sensitivities, frequencies, full scale, & overload ranges Special types Characteristics to be considered Transient response Cross-axis sensitivity Frequency range Sensitivity Mass & dynamic range Environmental conditions

Accelerometer Selection : Some Hints Frequency Range Upper limit Rule of thumb Upper limit : one-third of resonance vibrations measured less than 1 db in linearity Applications with lower linearity (e.g., 3 db) As for internal conditions of machines (reputability more important) ½ or 2/3 of natural freq. Lower limit 2 factors Amplifier s cut-off Ambient temperature fluctuations

Accelerometer Selection : Some Hints Sensitivity, Mass & Dynamic Range Sure better if higher is the sensitivity... BUT compromises may have to be made versus Frequency Range Overload capacity Size Mass for small & light test objects Should not load the test object Rule of thumb Dynamic range Should match high or low acceleration levels General purpose linear up to 5000g or 10,000g

Accelerometer Selection : Some Hints Transients Releases of energy in short-duration pulses : various shapes and rise times Overall linearity limited to Zero Shift Phase nonlinearity in preamplifiers The accelerometer not returning to steady-state conditions after subjected to high shocks Ringing High frequency components of excitation near resonance Environmental effects Base strain Reduced by Thick base Delta shear type Humidity For the connector Use silicon rubber sealants

Accelerometer mounting Main requirement Close mechanical contact Bad mounting reduces the usable frequency range Stud mounting Cementing stud With Beeswax (limited by temp.) Isolated mounting

Accelerometer mounting Isolating the accelerometer Electrical Preventation of ground loops Mechanical Protection against high shocks

Calibration Back-to-back method Using a Reference Standard Accelerometer Accelerometers with very high accuracy (1%) At a reference frequency (normally 160 or 80 Hz) Over wider frequency ranges with slightly less accuracy Details in ISO 5347-3 ISO 5347 Methods for Calibration and Characterization of Vibration and Shock Transducers