MECH 373 Instrumentation and Measurement Lecture 3 (Course Website: Access from your “My Concordia” portal) Contents of today’s lecture: • Dynamic measurements – Zero order, first order, second order systems – Time constant, response time, rise time, settling time – Frequency response • Experimental design

Dimensions and Units

Calibration • Calibration: A test in which known values of the input are applied to a measurement system (or sensor) for the purpose of observing the system (or sensor) output. • Static calibration: A calibration procedure in which the values of the variable involved remain constant (do not change with time). • Dynamic calibration: When the variables of interest are time dependent and time-based information is need. The dynamic calibration determines the relationship between an input of known dynamic behavior and the measurement system output.

Static Calibration Curve Sensitivity is defined as the ratio of the change in the magnitude of the output to the change in the magnitude of measurand. That is: In other words, the slope of a calibration curve provides the sensitivity of the measurement system Static Sensitivity: The slope of a static calibration curve. It relates changes in the indicated output to changes in the input

Least-Squares Regression Given a set of N data points of the form (x,y), the least-squares regression yields the mth order polynomial that minimizes the sum of the squares of the deviations For example, the best straight line fit in the least squares sense is obtained for the case m=1, for which

Least-Squares Regression To find the values of that minimize we compute the derivatives and set them equal to zero

Dynamic Calibration We have seen so far how to obtain the input-output relationship of an instrument when the input is held constant, neglecting the measurement system dynamics. This process is called static calibration However, when the input variables of interest are time dependent and time-based information is sought, we need to consider the system dynamics A dynamic calibration determines the relationship between an input of known dynamic behavior and the measurement system output Usually such calibrations involve either a sinusoidal signal or a step change as the known input signal To perform a “static calibration” we use a piecewise-constant function as input. The measurements can only be performed when the system reaches steady state

Static vs. Dynamic Calibration Example: What is the difference between static and dynamic calibration? What type of calibration would you recommend for (1) an oral thermometer (2) a pressure gage used in a water line, and (3) a car speedometer? Explain your reasoning.

Dynamic Measurements If a measurand is unchanging in time and if the measurement system instantaneously shows an equilibrium response to the measurand, the measurement system is siad to be static. However, in the general case, when the measurand is changing in time and the measuring system does not show instantaneous response, the measurement process is said to be dynamic. In making dynamic measurements, we must acount for the dynamic charactristics of the measuring system, the dynamic interaction between the measuring system and the test system, and the dynamic charateristics of the test system.

Dynamic Measurements The dynamic respnse of a measurment system can usually be placed into one of three categories: zero order, first order and second order. In the following slides, the dynamic chracteristics of different measurement systems are discussed.

General Model For A Measurement System nth Order ordinary linear differential equation with constant coefficient F(t) = forcing function Where m ≤ n y(t) = output from the system x(t) = input to the system t = time a’s and b’s = system physical parameters, assumed constant Measurement system x(t) y(t) y(0) The solution Where yocf = complementary-function part of solution yopi = particular-integral part of solution

Complementary-Function Solution The solution yocf is obtained by calculating the n roots of the algebraic characteristic equation Characteristic equation Roots of the characteristic equation: Complementary-function solution: 1. Real roots, unrepeated: 2. Real roots, repeated: each root s which appear p times 3. Complex roots, unrepeated: the complex form: a  ib 4. Complex roots, repeated: each pair of complex root which appear p times

-- undetermined coefficients, variation of parameters etc. Particular Solution -- undetermined coefficients, variation of parameters etc. Method of undetermined coefficients: Where f(t) = the function that describes input quantity A, B, C = constant which can be found by substituting yopi into ODEs Important Notes  After a certain-order derivative, all higher derivatives are zero. After a certain-order derivative, all higher derivatives have the same functional form as some lower-order derivatives. Upon repeated differentiation, new functional forms continue to arise.  

Zero-Order Systems All the a’s and b’s other than a0 and b0 are zero. where K = static sensitivity = b0/a0 The behavior is characterized by its static sensitivity, K and remains constant regardless of input frequency (ideal dynamic characteristic). xm Vr + Where 0  x  xm and Vr is a reference voltage y = V x = 0 - A linear potentiometer used as position sensor is a zero-order sensor.

First-Order Systems All the a’s and b’s other than a1, a0 and b0 are zero. where K = b0/a0 is the static sensitivity  = a1/a0 is the system’s time constant

First-Order Systems: Step Response Assume for t < 0, y = y0 , at time = 0 the input quantity, x increases instantly by an amount A. Therefore t > 0 The complete solution: yocf yopi Transient response Steady-state response Applying the initial condition, we get C = y0-KA, thus gives

First-Order Systems: Step Response Here, we define the term error fraction as Non-dimensional step response of first-order instrument

Determination of Time Constant 0.368 Slope = -1/

First-Order Systems: Frequency Response From the response of first-order system to sinusoidal inputs, we have The complete solution: Transient response Steady state response = Frequency response If we do interest in only steady-state response of the system, we can write the equation in general form Where B() = amplitude of the steady-state response and () = phase shift

First-Order Instrument: Frequency Response The amplitude ratio The phase angle is Dynamic error -3 dB 0.707 Cutoff frequency Frequency response of the first-order system Dynamic error, () = M(): a measure of an inability of a system to adequately reconstruct the amplitude of the input for a particular frequency

Dynamic Characteristics Frequency Response describe how the ratio of output and input changes with the input frequency. (sinusoidal input) Dynamic error, () = 1- M() a measure of the inability of a system or sensor to adequately reconstruct the amplitude of the input for a particular frequency Bandwidth the frequency band over which M()  0.707 (-3 dB in decibel unit) Cutoff frequency: the frequency at which the system response has fallen to 0.707 (-3 dB) of the stable low frequency.

Dynamic Characteristics Example: A first order instrument is to measure signals with frequency content up to 100 Hz with an inaccuracy of 5%. What is the maximum allowable time constant? What will be the phase shift at 50 and 100 Hz? Solution: Define From the condition |Dynamic error| < 5%, it implies that But for the first order system, the term can not be greater than 1 so that the constrain becomes Solve this inequality give the range The largest allowable time constant for the input frequency 100 Hz is The phase shift at 50 and 100 Hz can be found from This give  = -9.33o and = -18.19o at 50 and 100 Hz respectively

Second-Order Systems = the static sensitivity In general, a second-order measurement system subjected to arbitrary input, x(t) The essential parameters = the static sensitivity = the damping ratio, dimensionless = the natural frequency

Second-Order Systems Consider the characteristic equation This quadratic equation has two roots: Depending on the value of , three forms of complementary solutions are possible Overdamped ( > 1): Critically damped ( = 1): Underdamped (< 1): :

Second-Order Systems Case I Underdamped (< 1): Case 2 Overdamped ( > 1): Case 3 Critically damped ( = 1):

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; the second-order model:

Second-order Systems: Step Response For a step input x(t) With the initial conditions: y = 0 at t = 0+, dy/dt = 0 at t = 0+ The complete solution: Overdamped ( > 1): Critically damped ( = 1): Underdamped (< 1): :

Second-order Instrument: Step Response Ringing frequency:  = 0 Ringing frequency: 0.25 Rise time decreases  with but increases ringing 0.5 Optimum settling time can be obtained from  ~ 0.7 1.0 Practical systems use 0.6<  <0.8 2.0 Non-dimensional step response of second-order instrument

Dynamic Characteristics overshoot 100%  5% settling time rise time Typical response of the 2nd order system

Second-order Instrument: Frequency Response The response of a second-order to a sinusoidal input of the form x(t) = Asint where The steady state response of a second-order to a sinusoidal input Where B() = amplitude of the steady state response and () = phase shift

Second-order Instrument: Frequency Response The amplitude ratio The phase angle  = 0.1 0.3 0.5 1.0 2.0 0 = 0.1 0.3 0.5 1.0 2.0 Magnitude and Phase plot of second-order Instrument

Second-order Systems For overdamped ( >1) or critical damped ( = 1), there is neither overshoot nor steady-state dynamic error in the response. In an underdameped system ( < 1) the steady-state dynamic error is zero, but the speed and overshoot in the transient are related. Rise time: overshoot Td Maximum overshoot: Peak time: peak time Resonance frequency: settling time Resonance amplitude: rise time

Dynamic Characteristics Speed of response: indicates how fast the sensor (measurement system) reacts to changes in the input variable. (Step input) Rise time: the length of time it takes the output to reach 10 to 90% of full response when a step is applied to the input Time constant: (1st order system) the time for the output to change by 63.2% of its maximum possible change. Settling time: the time it takes from the application of the input step until the output has settled within a specific band of the final value. Dead time: the length of time from the application of a step change at the input of the sensor until the output begins to change

Experimental Design Experimental design is the first step in any measurement experiment. It involves developing a measurement test plan following three steps: Parameter Design Plan – test objective and identification of process variables and parameters and a means for their control. You should ask: What question am I trying to answer ? What variables to be measured ? What variables will affect my results ? System and Tolerance Design Plan – selection of measurement technique, equipment and test procedure based on some preconceived tolerance limits for error. You should ask: How will I do the measurement and how good do the results have to be ?

Experimental Design (cont’d) 3. Data Reduction Design Plan – Plan ahead on how to analyze, present, and use the anticipated data. You should ask: How will I interpret the resulting data ? How will I use the data to answer my question ? Step 1 needs the following important concepts: Variables Parameters Noise and Interference Random Tests Replication and Repetition Concomitant Methods

Variables One variable in the measurement system is obviously the targeted measured variable, but there might be other variables that affect the outcome All known process variables should be listed and evaluated for any possible cause and effect relationships A variable that can be changed independently of other variables is known as an independent variable A variable that is affected by changes in one or more other variables is known as a dependent variable Variables that cannot be controlled during measurements but that affect the value of the measured variables are called extraneous variables. If the values of a variable can be enumerated it is called discrete. Otherwise it is called continuous

Variables Example: For the following calibration system. Identify the independent, dependent, and possible extraneous variables. Answer: Independent variables: piston displacement x, temperature T Dependent variables: gas pressure p Extraneous variables: noise effects due to room temperature; line voltage variations, connecting wires

Parameters A parameter is a function relationship between variables A parameter that has an effect on the behavior of the measured variable is called a control parameter A control parameter is completely controlled if it can be set and held at a constant value during a set of measurements Example (Pendulum): period Ratio of l and g is a completely controlled parameter because l is fixed and g is known

Noise and Interference The way of extraneous variables affects measured data can be classified into noise and interference Noise is a random variation of the value of the measured signal as a consequence of the variation of the extraneous variables Interference produces undesirable deterministic trends on the measured value because of extraneous variables Example:

Random Tests It is important to minimize the effects of extraneous variables in a measurement using random tests A random test is defined by a measurement matrix that sets a random order in the value of the independent variable applied to measure the dependent variable Example: Extraneous Variable Pressure Transducer + Voltmeter Independent Varables Dependent Variable Apply Random Sequence (holding temperature constant)

Random Tests (cont’d) Assuming that we can hold the temperature fixed, applying a random sequence of volume values rather than a sequential sequence will allow us to average out the effects of extraneous variables We will see later that when the measurement system is subject to hysteresis, applying an increasing sequence of values for the independent variable produces a different result as compared to the case of applying the same set of values in a decreasing order A random sequence enables us to deal with this effect by providing a unique value of the independent variable for each value of the dependent variable in the random sequence After applying several random sequencies the results are averaged out to find a best-fit curve

Random Tests (cont’d) We have seen that random tests are effective for the local control of extraneous variables that change in a continuous manner Discrete extraneous variables can also be dealt with by performing a random test The use of different instruments and different test operators are examples of discrete extraneous variables that can affect the outcome of a measurement These effects are usually reduced by randomizing a test matrix by using random blocks A block consists of a data set of the measured variable in which the control variable is varied but the extraneous variable is fixed The extraneous variable is then varied between blocks

Example: Randomized Matrix The manufacture of a particular composite material requires mixing a percentage by weight of binder with resin to produce a gel. The gel is used in a lay-up procedure to impregnate the fiber. The strength will depend on both the percent binder in the gel and the test operator performing the lay-up. Formulate a test matrix to find the percent binder influence on strength Solution: Pick three different bider-gel ratios A, B, C and three typical operators to produce N separate composite test samples for each of the 3 ratios Create the following test pattern:

Replication and Repetition In general the estimated value of a variable improves with the number of measurements. The mean over measurements is taken as the estimated value Repeated measurements made during any single test run or on a single batch are called repetitions. It allows for quantifying the variation in a measured variable as it occurs during any one test or batch while the operating conditions are held under nominal control An independent duplication of a set of measurements using similar operating conditions is referred to as a replication. It allows for quantifying the variation in a measured variable as it occurs between different tests, each having the same nominal values of operating conditions

Concomitant Methods A good strategy is to incorporate the use of concomitant methods in a measurement plan. The goal is to obtain two or more estimates for the result, each based on a different method, which can be compared as a check for agreement Example: Establish the volume of a cylindrical rod of known material. Method 1: Measure the diameter and length Method 2: Measure the weight and compute volume based on specific weight of material