Dynamic Characteristics of Instruments P M V Subbarao Professor Mechanical Engineering Department Capability to carry out Transient Measurements….

Cyclic Input: Hysteresis and Backlash Careful observation of the output/input relationship of an instrument will sometimes reveal different results as the signals vary in direction of the movement. Mechanical systems will often show a small difference in length as the direction of the applied force is reversed. The same effect arises as a magnetic field is reversed in a magnetic material. This characteristic is called hysteresis Where this is caused by a mechanism that gives a sharp change, such as caused by the looseness of a joint in a mechanical joint, it is easy to detect and is known as backlash.

Hysteresis Loop

Dynamic Characteristics of Instrument Systems To properly appreciate instrumentation design and its use, it is necessary to develop insight into the most commonly encountered types of dynamic loading & to develop the mathematical modeling basis that allows us to make concise statements about responses. The response at the output of an instrument G result is obtained by multiplying the mathematical expression for the input signal G input by the transfer function of the instrument under investigation G response

Standard Forcing Functions

Characteristic Equation Development The behavior of a block that exhibits linear behavior is mathematically represented in the general form of expression given as Here, the coefficients a 2, a 1, and a 0 are constants dependent on the particular instrument of interest. The left hand side of the equation is known as the characteristic equation. It is specific to the internal properties of the block and is not altered by the way the insturment is used.

The specific combination of forcing function input and instrument characteristic equation collectively decides the combined output response. Solution of the combined behavior is obtained using Laplace transform methods to obtain the output responses in the time or the complex frequency domain.

Behaviour of the Instrument Zero order First order Second order nth order

Behaviour of the Block Note that specific names have been given to each order. The zero-order situation is not usually dealt because it has no time-dependent term and is thus seen to be trivial. It is an amplifier (or attenuator) of the forcing function with gain of a 0. It has infinite bandwidth without change in the amplification constant. The highest order usually necessary to consider in first-cut instrument analysis is the second-order class. Higher-order systems do occur in. Computer-aided tools for systems analysis are used to study the responses of higher order systems.

Solution of ODE Define D operator as The n th order system model: The solution of equations of this type has been put on a systematic basis by using either the classical method of D operators or Laplace Transforms method.

Laplace Transforms: Solution of ODE The Laplace transform, is an elegant way for fast and schematic solving of linear differential equations with constant coefficients. Instead of solving the differential equation with the initial conditions directly in the original domain, the detour via a mapping into the frequency domain is taken, where only an algebraic equation has to be solved. Thus solving differential equations is performed in the following three steps: Transformation of the differential equation into the mapped space, Solving the algebraic equation in the mapped space, Back transformation of the solution into the original space.

Schema for solving differential equations using the Laplace transformation

Laplace Transforms; n th Order Equation The n th order system model: Laplace Transform:

Laplace Transformations for Sensors

Generalized Instrument System : A combination of Blocks The response analysis can be carried out to each independent block.

Response of the Different Blocks Zero-Order Blocks To investigate the response of a block, multiply its frequency domain forms of equation for the characteristic equation with that of the chosen forcing function equation. This is an interesting case because Equation shows that the zero- order block has no frequency dependent term, so the output for all given inputs can only be of the same time form as the input. What can be changed is the amplitude given as the coefficient a 0. A shift in time (phase shift) of the output waveform with the input also does not occur. This is the response often desired in instruments because it means that the block does not alter the time response. However, this is not always so because, in systems, design blocks are often chosen for their ability to change the time shape of signals in a known manner.

Zero Order Instrument: Wire Strain Gauge This is the response often desired in instruments because it means that the block does not alter the time response. All instruments behave as zero order instruments when they give a static output in response to a static input.

Wire Strain Gauge

Strain Gauge A strain gauge's conductors are very thin: if made of round wire, about 1/1000 inch in diameter. Alternatively, strain gauge conductors may be thin strips of metallic film deposited on a nonconducting substrate material called the carrier. The name "bonded gauge" is given to strain gauges that are glued to a larger structure under stress (called the test specimen). The task of bonding strain gauges to test specimens may appear to be very simple, but it is not. "Gauging" is a craft in its own right, absolutely essential for obtaining accurate, stable strain measurements. It is also possible to use an unmounted gauge wire stretched between two mechanical points to measure tension, but this technique has its limitations.

Wire Strain Gauge Pressure Transducers In comparison with other types of pressure transducers, the strain gage type pressure transducer is of higher accuraciy, higher stability and of higher responsibility. The strain gage type pressure transducers are widely used as the high accuracy force detection means in the hydraulic testing machines.

Type Features and Applications Capacity Range Nonlinearity(% RO) Rated Output(m V/V) Compensated Temp.Range ( ℃ ) HVS High Accuracy type 0.5,..50 MPa0.2,0.31.0,1.5±1 % － 10 to 60 HVU General Purpose type 1,..50 MPa0.3,0.51.5,2.0±1 % － 10 to 60 HVJS Small & High Temperature type 1,..50 MPa0.51.0,1.5±20 % － 10 to 150 HVJS- JS Small & High Temperature type,Vibratio n-proof 1,..10 MPa0.51.0,1.5±20 % － 10 to 150 Micro Sensor Technology Tokyo

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  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 b0b0 The complex function F(s) must be decomposed into partial fractions in order to use the tables of correspondences. This gives

Dynamic Response of Liquid–in –Glass Thermometer

Liquid in Glass Thermometer material volume  (10 −6 K −1 ) alcohol, ethyl1120 gasoline950 jet fuel, kerosene990 mercury181 water, liquid (1 ℃ ) −50 water, liquid (4 ℃ ) 0 water, liquid (10 ℃ ) 88 water, liquid (20 ℃ ) 207 water, liquid (30 ℃ ) 303 water, liquid (40 ℃ ) 385 water, liquid (50 ℃ ) 457 water, liquid (60 ℃ ) 522 water, liquid (70 ℃ ) 582 water, liquid (80 ℃ ) 640 water, liquid (90 ℃ ) 695

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

RsRs R cond R tf T s (t) T tf (t) Change in energy of thermometer:

Time constant Step Response of Thermometers

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.

T s,max - T tf,max 