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

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

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. (4) Accelerometers. (5) Strain gauges.

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 account 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 response 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.

Zero order systems Zero order: Systems response instantly to measurands, no instrument is trully zero order. e.g. Linear potentiometer used as position sensor. 10

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 and second order systems 12

First order systems Mechanical analogs to capacitance – springs or devices that stored thermal energy – e.g. Common thermometer. First order equation : (Response A) [y(t) – y(0)]/[(ye) – y(0)] = 1 – e(-t/) where y(t) – response at time t, y(0) – initial response. ye - steady state response.  - time constant (s) – determine the curve – Find by t =  , y(t)/ye = 0.632. 13

First-Order Systems: Step Response Here, we define the term error fraction as: (KA = ye) Non-dimensional step response of first-order instrument 14 14

Determination of Time Constant – KA = ye 0.368 Slope = -1/ τ t 15 15

Second order systems Sping/mass systems – Load cells, pressure transducers, accelerometers. Have damping – dissipate energy. Inf only : Low damping – underdamped – oscillation. Overdamped – No oscillation. Critical damping – No oscillation. 16

1st and 2nd order systems’ representation Response time – time to achieve a fraction of steady response from initail condition – 95% response time = 0.95 of y/ye. (Ensure overshoot passed for 2nd order system.) Rise time – time requied y/ye to increase from 0.1 to 0.9 . Settling time – time until the oscillation is less than some fraction of steady state amplitude, say 5% = 95% of steady state amplitude. Minimal dynamic error – small settling time (response time). 17

1st and 2nd order systems 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. Speed of response - indicates how fast the sensor (measurement system) reacts to changes in the input variable. (Step input). 18

2nd orders - Dynamic Characteristics overshoot 100%  5% settling time rise time Typical response of the 2nd order system 19 19

First order system example To calculate rise time (10% y/ye to 90% ) and 90% response time. Initial thermometer temperature =20C. Thermometer immersed into water, its temperature = 80C. Thermometer time constant = 4 seconds. Solution: y/ye = (T – Ti)/(T – Ti) = 1 – et/ y/ye = 0.1 = 1 – et1/4 ; t1 = 0.42s. y/ye = 0.9 = 1 – et2/4 ; t2 = 9.21s. Rise ∆T = t2 – t1 = 8.79s. 90% Response time = 9.21s. 20

First order system example 21

Frequency response Useful dynamic response. Output determined by pure sine wave input. Repeat for a range of frequencies. See figure. Bandwidth – constant ratio of output and input amplitude – good area. Outside – large systematic errors. Ensure dynamic response frequency much less than the device natural frequency (transducer), say 0.2 to 0.4n, but n influence by mounting. 22

Frequency response 23

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 (output/input) (20log 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.