DSC Lab 2: Force and Displacement Measurement Page 1 Overview of Laboratory on Force and Displacement Measurement This lab course introduces concepts in force and motion measurement using strain-gauge based sensors as well as displacement measurement using an LVDT. A cantilevered beam instrumented with foil strain-gauges wired in a full-bridge configuration forms the focus of the study into measurement of force and motion. It will be shown how both force and motion (displacement) can be inferred from the signals generated by this beam sensor. Typically, a first exercise focuses on familiarization with how strain-gauges are used in a typical sensor arrangement. The objectives usually include: 1. To learn principles underlying strain measurement using strain-gauges. 2. To study signal conditioning concepts: bridge circuits with application to strain gauges. 3. To study a typical strain-gauge bridge circuit and associated signal conditioning equipment. 4. To review methods for how strain-gauges can be used to measure several physical quantities (e.g., force, pressure, acceleration, or displacement). 5. To learn about displacement measurement using an LVDT. Before beginning this lab, it is expected that pre-laboratory review has provided: 1. familiarity with basic strain-gauge concepts 2. an understanding for how strain gauges are configured in a bridge circuit in order to amplify the effect of small changes in resistance induced by strain. 3. understanding of how the gauge resistance values relate to bridge outputs (in case the circuit must be tested) 4. an understanding of the force-deflection relations for a simple cantilever beam under tip force loading. Strain Gauge Beam Sensor In this laboratory, a cantilevered beam will be used to illustrate how measurement of force can be made using a foil strain-gauge. The investigations in this lab are focused on learning about strain-gauges, the use of strain-gauges and their use in beam-sensor configurations, bridge circuits, signal conditioning, and calibration for force. The displacement of the beam tip will be monitored using a Linear Variable Differential Transformer (LVDT), and the basic principles governing the operation of this variable reluctance device should be understood (from pre-lab study).
DSC Lab 2: Force and Displacement Measurement Page 2 Figure 1: Photo of laboratory configuration for studying beam-sensor. Figure 2: Schematic showing strain-gauged cantilever beam and LVDT.
DSC Lab 2: Force and Displacement Measurement Page 3 Table 1: List of equipment typically used for this lab strain-gauge beam digital multimeter strain gauge amplifier digital oscilloscope (e.g., OMEGA DMD-465WB) beam support and base plate circuit breadboard calibration weights and apparatus LVDT or DCDT digital weighing scale power supply (if needed) As shown in the schematic in Figure 2, the beam will deflect under tip loading. It is expected that this relation between tip load (force), deflection, and beam geometry and material properties are familiar concepts. This concept will be used to relate measured force and displacement data. Basic Equipment The equipment used in lab may vary depending on the particular exercise(s) to be completed from one semester to another. The TA will update the requirements as needed. Usually, it is helpful to have most of the items listed in Table 1, but not all of these are needed in every case. The schematic of the cantilever beam configuration shown in Figure 2 features an aluminum beam with a rectangular cross-section. The beam has been instrumented with a full-bridge strain-gauge circuit, and it is cantilevered on a support stand. Deflection of the beam will be measured using a linear variable differential transformer (LVDT) that has been pre-calibrated (by the manufacturer). The sensitivity of the LVDT should be recorded (e.g, in units of mm/mv). The LVDT is mounted on a plate that can slide on two metal rods, allowing the LVDT to be positioned at any point along the length of the beam. The LVDT plunger will rest on the beam at the desired location (see schematic above). Full-Bridge Circuit The circuit diagram for the bridge circuit is shown in Figure 3. These diagrams describe the fullbridge stain-gauge circuit for the type of beam sensor used in this lab. Side A refers to the side that has the electrical terminal connections attached, as shown in the figure. The physical location of each gauge on the beam and within the bridge circuit is specified by these drawings and as shown in beam diagram. Because any amplifier may be modified or damaged by previous use, it is good practice to check for any faults in the circuit before calibrating the sensor. If time permits, the TA may introduce methods for quickly testing the setup. Strain Gauge Amplifier The output from the bridge is connected to a strain-gauge amplifier as shown in Figure 5 (e.g., the Omega DMD-465WB). A description of the particular amplifier to be used in this lab can be found in the manufacturer s specification sheet (attached as appendix).
DSC Lab 2: Force and Displacement Measurement Page 4 Figure 3: (a) Layout of strain gauges on beam. (b) Wheatstone configuration with full-bridge. Figure 4: Cantilever beam with full-bridge strain-gauge configuration. Figure 5: Diagram showing full-bridge strain-gage circuit connected to amplifier.
DSC Lab 2: Force and Displacement Measurement Page 5 LVDTs Our study of the strain-gauged cantilevered beam also involves measurement of beam deflection using a Linear Variable Differential Transformer (LVDT). Differential transformers (DTs) use a variable-reluctance (inductance) principle to infer displacement. A linear variable differential transformer (LVDT) is the most popular type of DT, and a sectional diagram is shown in part (a) of Figure 6. Figure 6: A sectional view of a linear variable differential transformer (LVDT). (b) Circuit diagram for an LVDT. An LVDT consists of three symmetrically spaced coils wound onto an insulated bobbin. The magnetic core can move within the bobbin and provides a path for the magnetic flux linkage between the primary and the secondary coils. The motion (from a mechanical input) changes the characteristics of the flux path and the changes can be detected in the circuit. The primary coil is excited by an a.c. signal and voltages are induced in the two secondary coils. The induced voltages depend on the position of the core inside the bobbin. The circuit is shown in part (b) of the figure. The secondary coils are wired in a series-opposing circuit so that when the core is centered between them the voltages induced are opposite but equal. When the core is centered between the two secondary coils, the induced voltages v 1 and v 2 are equal but out of phase by 180 degrees, and they cancel to give a zero output voltage. When the core moves from the center position, an output voltage v 0 = v 1 v 2 is developed, and as long as the displacement is within the working range of the LVDT, the voltage and displacement will be linearly related. Modern commercially available LVDTs usually come with built-in signal conditioners that provide a very easy to use package. Most require a dc excitation voltage. The actual excitation of the circuit is a voltage with a frequency that can range from 50 to 20,000 Hz and an amplitude of 5 to 15 volts. Frequencies between 1 and 5 khz typically give the best sensitivity (output voltage to displacement). The signal conditioner must provide this ac signal as well as circuitry to perform
DSC Lab 2: Force and Displacement Measurement Page 6 a demodulation and amplification of the output voltage to give a useful dc voltage output. The modern LVDTs that have built in signal conditioning are referred to as DCDTs or direct current differential transformers. Another type of DT is rotary variable differential transformer (RVDT). RVDTs measure angular displacements using the same principle of variable-reluctance. Useful references for both strain gauges and LVDTs 1. R.S. Figliola and D.E. Beasley, Theory of Mechanical Measurements, John Wiley and Sons, New York, NY (any edition; e.g., for LVDTs see Ch. 12) 2. T.G. Beckwith, R.D. Marangoni, J.H. Lienhard V, Mechanical Measurement, Prentice-Hall, Upper Saddle River, NJ (any edition)