Lecture3 Mech SU

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Lecture 3

Measurement System Behavior

Goals of the lecture• Understand how the system responds to a variety of

input signals

• Understand how the instrument being used modifiesthe measurand

• Characterize the behavior of 0th, 1st and 2nd ordersystems

• Use the response of first and second order systems to

predict the behavior of simple instruments andtransducers

• Use transfer functions to describe system behavior



Example

Your boss asks you to find the temperature of theliquid in a tank which is part of a manufacturing

process. Using a mercury thermometer, you record

the temperatures over several minutes – what value

do you report to her?



The Problem

For every sensor (as well as every part of the sensor system),

we need to describe how that portion of the system respondsto the imposed input



General Model for a Measurement System

Most measurement system dynamic behavior can be characterized by a linear

ordinary differential equation of order n:

where:



• System response is determined by mechanical elements (mass, stiffness, and damping)

and electrical elements (resistance, inductance, and capacitance) that form the components

of most measurement system.

– Sensors may use mechanical elements (e.g., a spring of known stiffness can be used to

determine static force by measuring spring deflection) or electrical elements (change of

resistance to measure strain as in a strain gage), or most commonly, a combination of both.

– Electrical filters, amplifiers, and other electronics use electrical elements to eliminate

noise, remove “DC” signals, or to boost signal level.

• No element is “pure” - our spring also has mass that will restrict its ability to determine

dynamic forces (i.e., the spring’s motion creates an additional force due to its

acceleration).

• Elements interact - for example, the spring’s stiffness and mass create its natural

frequency

•All systems possess damping (from friction, viscosity, inductance, etc.) that is usually

helpful



Zero-Order System

The simplest model of a system is the zero-order system, for which all the derivatives

drop out:y = K F(t)

K is measured by static calibration.

Used for static measurements only - cannot be used dynamically

A tire gauge determines unknown pressure by measuring the deflection of a spring.



First-Order Systems

Example

Suppose a bulb thermometer originally indicating 20ºC is suddenly exposed to a fluidtemperature of 37 ºC. Develop a simple model to simulate the thermometer output

response.The rate at which energy is exchanged between the sensor and the

environment through convection, , must be balanced by the

storage of energy within the thermometer, dE/dt.

For a constant mass temperature sensor,

This can be written in the form

dividing by hAs

This equation can be written in the

form first order differential equation:or



First-Order Systems (Cont.)

The solution to the first ordinary differential equation:

The output response of the system to unit step input

τ is the time required for the output to achieve 63% of the difference between the initial

and final values

or

Rise time, settling time – these are instrumentation specs (e.g., rise time = 15 ms, 5%

settling time = 30 ms)



Harmonically Excited First –Order System


The angular frequency ω( Ω) of the input and output are the same, but the amplitude

(and phase lag) of the output depend on ωτ.

We can therefore describe the entire frequency response characteristics in terms of a

magnitude ratio and a phase shift

Note: τ is the only system characteristic which affects the frequency response



The amplitude is usually expressed

in terms of the decibel dB = 20 log10M(ω).


The frequency bandwidth of aninstrument is defined as the frequency

below which M(ω)=0.707, or dB = -3 ("3

dB down").

First order systems act as low pass

filters, in other words theyattenuate high frequencies.

A useful measure of the phase shift is thetime delay of the signal:

Zero phase lag is ideal, but a linear phase

lag is often acceptable



Second-Order Systems

Example of a second order system

where ωn is the natural frequency and ζ (zeta)

is the damping ratio.

or



Homogeneous Solution

The form of the homogeneous solution depends on the roots of the characteristic

equation

The particular solution will depend on the forcing

function F(t).







Frequency Response

If F(t) = A sin ωt , the solution is given by

The first term is a transient which willeventually die out

the steady-state

response

where

and

Resonance Frequency

Under-damped second order systems may

resonate or oscillate at a greater magnitude

than the input, M(ω) > 1.

Systems with a damping ratio greater than ζ >

0.707 do not resonate.

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Lecture3 Mech SU