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Dynamic Analysis of First Order Instruments. P M V Subbarao Professor Mechanical Engineering Department. Capability to Reach Exact Equilibrium ……. First Order Instruments.

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dynamic analysis of first order instruments

Dynamic Analysis of First Order Instruments

P M V Subbarao

Professor

Mechanical Engineering Department

Capability to Reach Exact Equilibrium ……

first order instruments
First Order Instruments

A first order linear instrument has an output which is given by a non-homogeneous first order linear differential equation

  • In these instruments there is a time delay in their response to changes of input.
  • The time constant t is a measure of the time delay.
  • Thermometers for measuring temperature are first-order instruments.
slide3
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
First‐order Instrument Step Response

b0

The complex function F(s) must be decomposed into partial fractions in order to use the tables of correspondences. This gives

thermometer a first order instrument
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

slide8

Ts(t)

Ttf(t)

Rs

Rcond

Rtf

Change in energy of thermometer:

response of thermometers periodic loading
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.

u tube manometer a second order system
U-tube Manometer : A Second Order System
  • The pressure to be measured is that of a system that involves a fluid (liquid or a gas) different from the manometer liquid.
  • Let the density of the fluid whose pressure being measured be ρf and that of the manometer liquid be ρm.
  • Equilibrium of the manometer liquid requires that there be the same force in the two limbs across the plane AA.
  • We then have

p

patm

This may be rearranged to read

dynamic response of a u tube manometer
Dynamic response of a U tube manometer

h

  • The manometer liquid is assumed to be incompressible the total length of the liquid column remains fixed at L.
  • Assume that the manometer is initially in the equilibrium position.
  • The pressure difference Δp is suddenly applied across it.
  • The liquid column will move during time t > 0.
slide18

The forces that are acting on the length L of the manometer liquid are:

Force disturbing the equilibrium

Inertial Force

Forces opposing the change:

a. Weight of column of liquid

b. Fluid friction due to viscosity of the liquid :

  • The velocity of the liquid column is expected to be small and the laminar assumption is thus valid.
  • The viscous force opposing the motion is calculated based on the assumption of fully developed Hagen-Poiseuelle flow.

The fricitional pressure drop

second order system
Second Order System

The essential parameters

The static sensitivity:

The dimensionless damping ratio:

The Natural Frequency:

slide23

The transfer function is parameterized in terms of ζ and ωn.

  • The value of ωn doesn’t qualitatively change the system response.
  • There are three important cases—withqualitatively different system behavior—as ζ varies.
  • The three cases are called:
  • Over Damped System (ζ >1)
  • Critically Damped System (ζ =1)
  • Under Damped System (ζ <1)
slide27

z=0.707

z=1.0

t

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