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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

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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

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.

• 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

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

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

Ts(t)

Ttf(t)

Rs

Rcond

Rtf

Change in energy of thermometer:

### Step Response of Thermometers

Time constant

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.

Ts,max- Ttf,max

f

### 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

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.

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

Newton’s Law of Motion

### Second Order System

The essential parameters

The static sensitivity:

The dimensionless damping ratio:

The Natural Frequency:

Transfer Function of a second order system:

• 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)

• Over Damped System (ζ >1)

y(t)

t

t

z=0

y(t)

t

z=0.5

z=0.707

z=1.0

t