MECH 373 Instrumentation and Measurements Lecture 19 Measurement of Solid-Mechanical Quantities (Chapter 8) • Measuring Strain • Measuring Displacement • Measuring Linear Velocity • Measuring Accepleration and Vibaration • Measuring Force

Measuring Strain (Strain Gages) • If the temperature is held constant, the change in resistivity is proportional to the strain. • The strain gage factor is approximately constant, although it is sensitive to the temperature change. • In summary, we have following equations:

Measurement Systems Displacement is one of the fundamental measurements in engineering, and the base for velocity, acceleration, strain & force measurements. The displacement can be either translational (linear) or rotational. Many displacement sensors are commercially available.

Measuring Displacement • Potentiometers are very common devices used to measure displacement. A linear potentiometer is used for linear measurements and an angular potentiometer is used for angular measurements. • The linear potentiometer is a device in which the resistance varies as a function of the position of a slider, shown below: • With the supply voltage (Vs), the output voltage (Vo) will vary between zero and the supply voltage. • For linear potentiometer, the output is a simple linear function of the slider position. That is

Measuring Displacement • It should be noted that the device measuring Vo must have a high impedance to maintain a linear response and to avoid loading error. • Linear potentiometers can be used to measure displacements as small as 0.1 to 0.2 in. (2.5 to 5 mm) up to displacements of more than 1 ft.

Measuring Displacement • Angular potentiometers can measure angular displacements in multiple rotations. • Angular potentiometers are used in common devices such as radios and televisions as volume and tone controls. • Potentiometers do have limitations. • Because of sliding contact, they are subject to wear. • Furthermore, the output tends to be somewhat electrically noisy since the slider-resistor contact point has some resistance, and this can affect the output in a somewhat random manner. • This effect often becomes worse with the age of the device due to contamination of the contact surface. • The wire-wound potentiometers display output as the slider contacts successive turns of the wire winding. • Conductive plastic potentiometers were developed to eliminate this stepwise output and now widely used in mechatronic systems.

Measuring Linear Velocity Linear Velocity Transducer • The linear velocity transducer (LVT) is an inductive device suitable for measuring the velocity of components in machines. • As shown below, the velocity is measured by driving a permanent magnet core past a fixed coil. • As the north pole of the magnet approaches the coil, the magnetic flux cuts across the coils and generate a voltage that is proportional to the velocity. • However, the number of flux lines cutting the coil varies with position, and hence, the output voltage is a function of both velocity and position. • When the magnet is centered with respect to the coil, the flux lines due to the negative pole also induce voltage in the coil of opposite polarity to that induced by the north pole, and the net output is zero.

Measuring Linear Velocity • To avoid this problem, two coils are used as shown below: • The south pole induces a voltage primarily in winding 1 and the north pole induces a voltage primarily in winding 2. • By connecting the coils with opposing polarity, a voltage proportional to velocity is generated and this voltage is relatively independent of position over a limited range. • For a typical LVT 7 in. long, the output is proportional to the velocity and independent of position for displacements that are within ±1 in. of the centered position.

Measuring Acceleration and Vibration • The measurement of acceleration is required in a variety of purposes, ranging from machine design to guidance systems. • A wide variety of transducers and measurement techniques exists for acceleration and vibration measurements, each associated with a particular application. • Displacement, velocity and acceleration measurements are also referred to as shock or vibration measurements, depending on the waveform of the forcing function that causes the acceleration. Acceleration is a derivative of velocity, second derivative of displacement. • A forcing function that is periodic in nature generally results in acceleration that are analyzed as vibration. Piezoelectric Accelerometers • Certain materials when deformed are capable of generating an electric charge. For example, quartz crystal. • This property of generating an electric charge when deformed makes piezoelectric materials useful sensors in several common types of transducers. • The following figure shows a piezoelectric material under compression.

Measuring Acceleration and Vibration • The faces have been coated with a conducting material such as silver. • When the load is applied, electrons move to one of the conducting faces and away from the other, which results in a charge being stored by the inherent capacitance of the piezoelectric material itself. • The arrangement shown in the above figure is called longitudinal effect. For this arrangement, the charge generated is given by Q = F×d where, F is the applied force and d is the piezoelectric coefficient of the material.

Measuring Acceleration and Vibration • The piezoelectric coefficient, d, depends on the piezoelectric material and its crystal orientation relative to the force, F. • For a typical quartz element, d has a value of 2.3×10^(-12) C/N. • The above equation shows that the charge is proportional to the applied force which can also be viewed as being proportional to the displacement. • The piezoelectric element is slightly flexible, and the imposition of a force produces a small, proportional displacement. • Another configuration of the piezoelectric sensor is called transverse effect which is shown below: • The charge generated in this configuration is given by • If the ratio of the dimensions, b/a, is greater than one (the usual case), the transverse effect produces a greater charge than the longitudinal effect.

Measuring Acceleration and Vibration

Measuring Acceleration and Vibration • With either loading, the charge is proportional to the applied force. • This charge results in a voltage. However, this voltage depends not only on the capacitance of the piezoelectric element but also on the capacitances of the lead wires and the signal-conditioner input. • Since the piezoelectric element generates a charge when loaded, this charge must be sensed in a manner that does not dissipate the charge. • This is normally performed with a device called a charge amplifier. • A typical configuration of the charge amplifier is shown below:

Measuring Acceleration and Vibration • In this circuit, the resistance R1 is set very high, so the circuit draws very low current and produces a voltage output that is proportional to the charge. • It should be noted that the capacitance of the lead wire from the transducer to the amplifier is important and affects calibration. • Normally the charged amplifier is located close to the transducer. • The charge amplifiers are normally supplied by the manufacturer of the piezoelectric transducer, and in some cases, the amplifier is incorporated into the body of the transducer. • Although piezoelectric sensors are capable of measuring quasi-static forces, they cannot measure completely static forces since the charge decays with time. • Consequently, in many common applications, transducers using piezoelectric sensors have a lower limit on frequency response and are not suitable for steady or quasi-steady measurands. • The primary advantage of piezoelectric sensors is their higher ability to respond to high-frequency measurands. • Most piezoelectric materials are very stiff and have a high modulus of elasticity. This leads to high natural frequencies of piezoelectric transducers.

Measuring Acceleration and Vibration Accelerometers using Piezoelectric Sensing Elements • An accelerometer using a piezoelectric material as the sensing element is shown below: • It consists of a housing, a mass called the seismic mass, and a piezoelectric sensing element, which typically uses the longitudinal piezoelectric effect. • An initial force between the mass and sensor is obtained with a preloading spring sleeve.

Measuring Acceleration and Vibration • As the housing for the accelerometer is subject to an acceleration, the force exerted by the mass on the quartz crystal is altered. • This generates a charge on the crystal, which can be sensed with a charge amplifier. • Piezoelectric accelerometers are available in many ranges up to ±1000g, where g is the acceleration due to gravity. • Quartz crystal accelerometers can have very high values of natural frequency up to 125 kHz. • This allows them to measure frequencies as high as 25 kHz.

Measuring Acceleration and Vibration Vibrometer • An instrument that is used to measure the ground motion in earthquakes and sometimes to measure vibration in machines is called the vibrometer. • Although the basic components are the same as the piezoelectric or strain-gage accelerometers, the mode of operation is different. • In the vibrometer, the spring is quite soft and as the housing moves, the mass remains approximately stationary. The relative motion, y, is large and sensed with a potentiometer.

Measuring Acceleration and Vibration • These devices are used to measure vibrations with frequencies that are high relative to the natural frequency of the spring-mass system, which is often less than 1 Hz. • The vibrometer effectively measures the displacement of the base rather than the acceleration. • Thus, these devices are most sensitive to vibrations with moderate frequencies and fairly large displacement amplitudes. • High frequency vibrations usually have small values of displacement amplitude and are better measured with accelerometers.

Measuring Force Load Cell • “Load cell” is a term used to describe a transducer that generates a voltage signal as a result of an applied force, usually along a particular direction. • Virtually any simple metal structure deforms when subjected to a force, and as long as the resulting stresses are below the material yield stress, the deflection (δ) and resulting strain (ε) are linear functions of the applied force: where C1 and C2 are constants determined from analysis or calibration. • The most common force-measuring devices are strain-gage load cells. They are often constructed of a metal and have a shape such that the range of forces to be measured results in a measurable output voltage over the desired operating range. • Figure 8.34(a) shows a cantilever beam instrumented with four strain gages, two on the top and two on the bottom, to measure normal or bending stresses. These four gages form the Wheatstone bridge and offer effective temperature compensation. The output of the bridge is four times the output of an individual gage.

Measuring Force

Measuring Force • Figure 8.34(b) shows a hollow-cylinder load cell. It also uses four strain gages and is also temperature compensated. As the cylinder is compressed, it becomes slightly shorter, while the diameter becomes slightly larger. As a result, two gages measure the axial compression. The other two, mounted transversely, measure the tensile diametral strain. Since the transverse strain is only Poisson's ratio times the axial strain, the output is less than four times the output of a single axial strain gage. For a Poisson's ratio of 0.3, the output will be about 2.6 times the output of a single axial strain gage. • Due to their simple design, any range can be readily manufactured. Commercial load cells are available with ranges from ounces up to several hundred thousand puns. • Unlike accelerometers, it is not useful to specify the frequency response of commercial load cells because the mass and flexibility of the instrumented system control the dynamic response. • Furthermore, an installed load cell will add flexibility to the system and also affect the dynamic response. • If the flexibility of strain-gage load cells is too high, load cells using piezoelectric sensors, which are much stiffer, are commercially available.

Measurement Systems