laser) for optimum quality. A schematic diagram is shown to the right, and the fundamentals of interferometry are analyzed here.

Transcription

1 Measurement f Mechanical Quantities, Page 1 Measurement f Mechanical Quantities Authr: Jhn M Cimbala, Penn State University Latest revisin: 0 December 009 Intrductin Sme eamples f mechanical quantities that need t be measured include psitin (r displacement), angular velcity (rtatin rate f a ), frce, trque (mment), and pwer Instruments f varius kinds have been invented t measure each f these quantities In this learning mdule, several f these instruments are discussed; the list here is by n means ehaustive The purpse here is t give yu a feel fr hw mechanical instruments wrk, and t make yu aware f the variety f ways t measure things Psitin and Displacement Measurement The length unit in the SI system is the meter (m) Fr many years, the standard meter was defined as 1,650,76373 wavelengths f kryptn 86 (an range-clred istpe f the element kryptn) in a vacuum The standard meter is nw defined as the distance that light in a vacuum travels in 1/(99,79,458) s Varius kinds f instruments have been invented t measure distance, length, r displacement Sme f these (especially the mechanical devices) are simple and straightfrward, and are discussed first Sme mre sphisticated measurement devices als eist fr measuring displacement, and these are discussed later Mechanical devices Principle f peratin: A simple cmparisn is made between a displacement (r an bject s length) and that f a pre-measured displacement r length The length r displacement is inferred t be the same as that f the (knwn) pre-measured length Eamples include: Lengths f varius magnitudes are measured with a standard ruler r tape measure Small lengths are measured with a mre precise device called a micrmeter (see abve picture) Cylindrical r spherical diameters are measured with vernier calipers Varius gaps and the inner diameters f small tubes are measured with gage blcks (a set f precisin grund hardened steel bjects f knwn size) Interfermeter Principle f peratin: Interacting light waves prduce interference patterns Displacement is inferred frm the interference patterns prduced by light reflecting ff a surface An interfermeter uses single wavelength light (like that frm a Optical flat laser) fr ptimum quality A schematic diagram is shwn t the right, and the fundamentals f interfermetry are analyzed here Surface being analyzed Defect The light is passed thrugh a precisin grund quartz disk called an ptical flat The ptical flat is tilted at sme small angle, as shwn Incming light passes thrugh the ptical flat, and is reflected as sketched Interference patterns are visible when viewing the reflected rays f light, since the ptical path f individual rays differs Each dark and light band represents a distance equal t half f the wavelength f the light Optical flat When the ptical flat is viewed frm the tp, the user sees smething Interference fringes Defect like the image sketched t the right On a perfectly flat prtin f the surface, as n the left side f the sketch, the interference fringes are straight and parallel Where there is a defect, as n the right side f the sketch, the fringe lines get distrted The elevatin f the defect can be inferred since each dark band represents a half wavelength f light

2 Ptentimeter Principle f peratin: A material s electrical resistance increases with length The displacement is inferred by measuring the change in resistance as a slider is displaced alng the ptentimeter Measurement f Mechanical Quantities, Page V s L Resistr Slider V In a practical applicatin, a linear ptentimeter is simply a vltage divider, with L being the ttal length f the resistr, and being the displacement t be measured, as sketched As displacement changes, the slider mves alng the resistr The equatin fr utput vltage is simply V = Vs L Ptentimeters are cheap and fairly accurate, but wear ut eventually due t the physical cntact at the slider The cntact pint itself can be electrically nisy, which is als undesirable There are als rtary ptentimeters, which wrk under the same principle, but measure angular displacement rather than linear displacement Rtary ptentimeters are als used in electrnics as variable resistrs r pts Linear variable displacement transducer Principle f peratin: A linear variable displacement transducer (LVDT) wrks n the same principle as electric mtrs, electrmagnets, etc, namely the link between electricity and magnetism as fund by H A Lrentz If a magnetic field mves near an electrical wire, current flws thrugh the wire Or (vice-versa), if the current flwing thrugh an electrical wire changes, it changes the magnetic field Sme authrs call this device a linear variable differential transfrmer (als LVDT) Cre An LVDT cnsists f a rd called the cre (a ferrmagnetic Cil B Cil A Cil C material like irn), that slides inside a hllw cylindrical tube called a bbbin, that is surrunded by three electrical cils: Cil A is the primary cil, wrapped arund the center f the bbbin (see sketch t the right) Cils B and C are the secndary cils, als wrapped arund V s the bbbin as shwn Bbbin V V s is the supply vltage, an AC supply alternating current V is the utput vltage Nte that ne lead t V cmes frm the utside f cil B, but the ther lead cmes frm the inside f cil C is the displacement f the cre frm the center f the bbbin Hw des the LVDT wrk? AC current (nt DC current) is supplied t the primary cil (cil A, in the center f the bbbin) This causes an alternating magnetic field in the cre The magnetic field induces an alternating current in cils B and C as well The electrical cnnectins in cils B and C are ppsite, as in the sketch, where we ntice that cil B is wund frm left t right electrically, while cil C is wund frm right t left electrically If the cre is eactly at the center f the bbbin, then V = 0, since the tw currents (r vltages) frm cils B and C are equal and ppsite Hwever, if the cre is displaced clser t ne f the secndary cils, V is nt zer since that cil will have a strnger current (In the sketch, the cre is clser t cil C; it is displaced t the right) Nte that the utput vltage V is als an alternating (AC) vltage Thus, the rms value f V (V,rms ) is f interest rather than V itself In the sketch t the right, V,rms is pltted as a functin f displacement

3 Measurement f Mechanical Quantities, Page 3 It turns ut that the rms utput vltage is linear ver a certain range f, and is symmetric with respect t psitive and negative V,rms Mst LVDTs are designed t perate nly within the linear regin Yu can distinguish between psitive and negative displacements by lking at the phase f the utput signal: If the cre is displaced t the left, twards cil B, the utput vltage is in Linear phase with the input vltage If the cre is displaced t the right, twards cil C, the utput vltage is 180 ut f phase with the input vltage One clear advantage f the LVDT ver the ptentimeter is that there is n slider cntact t wear ut LVDTs are als typically mre accurate and mre precise A disadvantage is that they are mre epensive than linear ptentimeters, and require mre sphisticated electrnics and signal cnditining Snic and ultrasnic transducers Bat Principle f peratin: If the speed f sund thrugh a material is knwn, it s thickness can be Reflected measured by timing hw lng it takes fr the sund t pass thrugh Thickness r displacement is inferred by measuring the time it takes fr the sund wave t travel thrugh the material An age-ld eample used by mariners is water depth sunding r snar (eg, Acts 7:8 in the Bible arund 000 years ag!) Transmitted Ocean flr Suppse the speed f sund a in water is knwn The snar device n the bat can transmit sund waves and can receive r detect reflecting sund waves The snar device transmits a sund wave, and detects its reflectin frm the bttm, as sketched The time fr the transmitted signal t travel t the bttm, reflect, and return t the ship is measured aδt The distance (in this case water depth) is calculated as = Pulse-ech transducer The factr f tw appears because the sund wave must travel twice the water depth befre being detected by the receiver This same principle is used in the labratry t measure the thickness f materials t and r r displacement (thrugh air r sme ther medium), but we typically use sund at frequencies abve that f human hearing (ultrasnic) Object There are tw types f ultrasnic transducers, as sketched t the right: A pulse-ech ultrasnic transducer has the transmitter (t) and receiver (r) n the same side ( as in the bat eample abve) The gray vertical line represents sme hard surface ff f which the sund aδt waves reflect Fr the pulse-ech case, = A piezelectric transducer is used in pulse-ech ultrasnic transducers, as shwn t the right In the Pulse-ech transducer transmit mde, the piezelectric element cnverts electrical signals int mechanical vibratins t generate the pulse In the receive mde, mechanical vibratins frm the ech are cnverted int electrical signals by the piezelectric element A thrugh-transmissin ultrasnic transducer has the Thrugh transducer transmitter (t) and receiver (r) n ppsite sides N ther surface is needed, but access t bth sides f the bject is required, and this is nt always pssible Fr the thrugh-transmissin case, t r = aδ t Withut the factr f tw, the sensitivity f the thrugh transmissin transducer is half that f the pulse-ech transducer, all else being equal

4 Measurement f Mechanical Quantities, Page 4 Capacitance displacement sensr Principle f peratin: The capacitance between tw metal plates is a functin f the distance between the plates and the verlapping area f the plates d A A C By measuring the change in capacitance, we infer the displacement The ne shwn here is designed fr ultra-high precisin psitining systems (they claim ±5 nm) A schematic diagram is sketched t the right The equatin fr the capacitance C is C = Kε 0 A/ d, where d is the distance between the plates, as sketched 1 C ε 0 is a cnstant called the permittivity f free space, ε 0 = N m K is the dielectric cefficient f the material between the plates (K = 1 fr air) A is the verlapping plate area If the tw plates are aligned right n tp f each ther, as in the sketch, A is the entire surface area f ne side f a plate There are tw ways t mve the plates t cause a change in capacitance: Nrmal displacement is when the plates mve perpendicularly away frm each ther, ie, d changes linearly with, but A remains cnstant In this case, is nrmal t the plates As can be seen frm the abve equatin, C changes inversely with d (C falls ff as 1/d); thus C changes nnlinearly with Parallel displacement is when the plates mve laterally away frm each ther, ie, area A changes linearly with, but d remains cnstant In this case, is d A Δ C parallel t the plates As can be seen frm the abve equatin, C changes linearly with A; thus C changes linearly with, at least fr sme range f d A Δ C Laser displacement meter Principle f peratin: The angle f reflected light frm a laser beam changes as the reflecting bject is displaced By measuring the reflected angle, the distance t the bject is inferred The diagram t the right illustrates hw a laser beam is transmitted t an bject and reflects in all directins One f the reflected waves ges t a phtdetectr in the instrument [Ntice that since 1 <, θ 1 > θ ] Sphisticated ptics and electrnics in the unit measure angle θ f the reflected beam, and cnvert t a digital Δ readut f displacement, r mre cmmnly Δ 1 [Displacement Δ is inferred by measuring Δθ] Laser emitter Transmitted wave The main advantage f this measurement technique is θ θ 1 bvius n physical cntact with the bject is required This is especially attractive in the electrnics Phtdetectr Reflected wave industry where physical cntact can lead t impurities Laser displacement meters are quite accurate and precise; the ne in the M E 345 labratry can detect displacement changes f ±01 mm (±0004 inch) Hwever, the range is limited; the laser displacement meter in the M E 345 labratry has a nminal perating pint f = 800 mm, with a measurement range f ±0 mm (±078 inch) frm this perating pint Angular velcity r rtatin rate measurement First sme nmenclature and units: The mst cmmn unit fr rtatin rate is rtatins per minute, r revlutins per minute, r rpm Rtatin rate is given the symbl N rpm here t emphasize that its units are rtatins per minute (rpm) The mre mathematically useful measure f rtatin rate is angular velcity ω, in units f radians per secnd (ω is nt as ppular amng engineers, but is required in mst equatins) The cnversin between these tw units is rtatin radian rtatin 60 s N rpm ω = min s π radian min

5 Measurement f Mechanical Quantities, Page 5 60 Sme authrs write this mre cmpactly as Nrpm = ω, where it is understd that N rpm is in units f π rpm and ω is in units f radians per secnd Hwever, this can lead t unit errrs if we are nt careful 60 rpm s T avid cnfusin, it is best t always define a unity cnversin factr, namely, = 1 π radian π radian radian Fr eample, if N rpm = 3600 rpm, the angular velcity is ω = ( 3600 rpm) = rpm s s In mst engineering applicatins, a rtatin speed (in either rpm r radians/s) is t be measured The instruments that are used t measure rtatin speed are cmmnly called tachmeters There are tw main categries f tachmeters, cntacting and nncntacting Cntacting tachmeter Principle f peratin: The instrument is physically attached t the end f a spinning, s that part f the instrument rtates at the same rpm as the Shaft rpm is measured directly and internally by the instrument The actual measurement f rpm internally in the instrument can be perfrmed by a number f methds including: Mechanical: The spinning ges thrugh a gear b that rtates a springladed dial Electric generatr: A small DC mtr is used backwards as a generatr Output vltage is prprtinal t rpm Chrnmeter: An electrnic clck measures the time between rtatins and calculates rpm (like the ne shwn t the right) Nncntacting tachmeter Principle f peratin: There is n direct physical cntact between the instrument and the rtating Instead, the rpm f the rtating is inferred by timing pulses remtely, either frm magnetic r ptical signals There are several types f nncntacting tachmeters: Permanent magnet Magnetic pickup tachmeter: This device, als called a variable reluctance pickup tachmeter, wrks n the same principle as the LVDT, ie, distrtin f a magnetic field when ferrmagnetic material passes by A schematic diagram is shwn t the right As ferrmagnetic material in a gear tth passes in frnt f the device, as shwn, the magnetic field is distrted, which causes a pulse in the utput vltage A plt f utput vltage as a functin f time may lk smething like that sketched t the right, where T is the perid f ne pulse (each gear tth causes ne pulse f time duratin T) An electrnic circuit then either measures T, r cunts the number f pulses per secnd, P It then cnverts the time signal int frequency, and then int rpm The equatin t cnvert frm measured number f P pulses per secnd t rpm is Nrpm =, where n teeth V T Gear Cil Gear teeth N rpm = rtatin rate (typically in rpm) P = measured pulse rate (typically in pulses per secnd) n teeth = number f teeth n the gear (n teeth als equals the number f pulses per rtatin) The number f gear teeth must be knwn in rder t calculate the rpm crrectly, and units must be kept straight t avid errrs, as the fllwing eample illustrates V t

6 Measurement f Mechanical Quantities, Page 6 Eample: Given: A magnetic pickup tachmeter is used t measure the rtatin speed f a gear Suppse n teeth = 30 The number f pulses per secnd P is cunted by the electrnic circuit T d: Calculate the rtatin speed f the gear (in rpm) as a functin f the measured value f P Slutin: pulse P P s 60 s rtatin We use the equatin given abve, Nrpm = = P n pulse = r Nrpm = P rpm teeth min min 30 rtatin Alternatively, in terms f the pulse perid, T = 1/P, and thus Nrpm = / T rpm Fr eample, if P = 750 pulses per secnd, N rpm = 1500 rpm Discussin: Since 30 gear teeth is half f 60, the relatinship invlves a simple factr f If instead we use a gear with 60 teeth, the relatinship wuld be ne-t-ne, ie, N rpm = P, where P is the number f pulses per secnd and N rpm is the number f rtatins per minute Strbscpic tachmeter: A strbe light is used t illuminate a spinning, gear, r pulley at precisely timed intervals The strbe light frequency is adjusted (by a knb) until the spinning bject appears t stp Smetimes yu need t paint a dt r ther mark n the surface f the spinning bject in rder t see it clearly, as sketched here Yu are prbably familiar with strbe lights at dance halls and frm physics class, where it is ften used t illustrate the acceleratin f a falling bject (see yutube vide) Strbes are als useful t measure rpm The digital readut f mst strbscpic tachmeters is in rpm Cautin: Strbscpic tachmeters can suffer frm a type f Rtating, pulley, etc Painted dt r mark Strbe light Flash rate adjust knb Display in rpm aliasing errr the appears t be rtating at sme incrrect rpm (We can be fled by this!) Furthermre, if the flashing rpm is an integer fractin f the true rpm, it will still lk like the is statinary, and the wrng rpm will be inferred! Fr eample, suppse the true speed is 300 rpm, and the strbe is adjusted t 100 rpm, where the mark n the appears t stand still (a frzen dt) Althugh it lks like the crrect speed has been determined, the rtatin rate wuld be wrng by a factr f 3! The actually rtates three times arund between each flash f the strbe T avid this prblem, the user shuld keep increasing the flashing frequency until the crrect rpm is determined until the can n lnger be made t appear t stand still Althugh strbscpic tachmeters are widely used, their main disadvantage is that the adjustment knb must be turned manually there is human interactin Rtating invlved Reflective tape Phtelectric tachmeter: This instrument is smewhat similar in principle t the strbscpic tachmeter, ecept electrnics remve the manual cmpnent A steady light surce, rather than a strbscpic light surce, is transmitted frm the device A phtdetectr (r phtcell) prduces a pulse each time light is reflected frm a piece f reflective tape n the rtating Steady light surce Pwer supply and display Cable Phtdetectr A schematic diagram is shwn t the right Since the reflected light prduces ne pulse per revlutin, the rtatin rate is easily determined by either cunting pulses r measuring the time between pulses Mst mdern phtelectric tachmeters have a digital readut directly in rpm, like the unit shwn t the left

7 Measurement f Mechanical Quantities, Page 7 Trque and Pwer Measurement (Dynammeters) A dynammeter measures the mment r trque T f a rtating [D nt cnfuse trque T with temperature T] The wrd dynammeter cmes frm dynamic mment meter The angular velcity ω f the is als measured s that pwer W can be calculated, since fr a rtating, pwer = angular velcity trque, W = ωt, r W π = NrpmT 60 Nte that if the angular velcity is given in rpm rather than radians per secnd, it must first be cnverted in rder t calculate the pwer, as in the secnd equatin abve Althugh the SI system f units is becming mre ppular, many engineers still use hrsepwer (hp) as the unit fr pwer Belw are sme useful cnversins when dealing with pwer; first the unit descriptin, and then the unity cnversin factr(s): 1 hp s 1 hp = 550 ft lbf/s = ft lbf 1 hp = 7457 watts 1 hp = 1 r 7457 W 1 hp min 1 hp = 44 Btu/min = 1 r 44 Btu kw = 1 1 hp 546 Btu = 1 1 hp hr 1 W s 1000 N m 1 watt = 1 N m/s = 1 r = 1 1 N m 1 kw s The dynammeter pictured t the right is frm DyneSystems, Inc, and is used t measure wind turbine pwer up t 6000 hp and 31,500 ft-lbf f trque There are several types f dynammeters Only three are discussed here the three that are used in the labratry prtin f this curse Prny brake dynammeter: Tensin adjustment r Principle f peratin: Mechanical frictin in a braking Bearings Band F mechanism absrbs the unit s pwer Trque and rpm are measured t calculate Unit under F the pwer test Typically, either a band brake r a disk brake is used A prny brake dynammeter with a band brake is shwn in the schematic diagram t the right Cradle supprts Cable The unit under test can be anything with a rtating eg, a mtr r engine The tensin n the brake is adjusted t cntrl the trque (due t frictin n the band) As trque is applied, the prny brake mechanism tries t rtate (cunterclckwise in the sketch abve) Hwever, a cable keeps the mechanism frm rtating The tensin frce F in the cable is measured by a frce scale, as shwn Frce F is measured at a knwn mment arm called the trque arm r t calculate the trque, T = Fr Shaft rtatin rate is measured simultaneusly Shaft pwer is then calculated: W = ωt = ωfr, r W π π = N T = N Fr 60 rpm 60 rpm The advantages f a prny brake dynammeter are that it is simple, small in size, and inepensive The disadvantages are that limited pwer can be dissipated with a prny brake, the mechanical brake is smetimes nt very stable, and the unit cannt supply pwer it can nly absrb pwer The band can als get very ht, and this can be dangerus Cradled DC mtr dynammeter (als called an electric mtring dynammeter) Principle f peratin: An electric mtr, attached directly t the f the unit under test, absrbs the unit s pwer when it functins as a generatr Shaft Scale

8 Measurement f Mechanical Quantities, Page 8 The lad n the mtr is varied t cntrl the trque Specifically, the generated pwer is dissipated (as heat) in a resistr bank as shwn in the schematic diagram t the right The resistance in the resistr bank is adjusted t cntrl the trque Pwer supply Resistr bank Just as with the prny brake dynammeter, the mtr/generatr Bearings r tries t rtate (cunterclckwise in the sketch here) Hwever, the cable Unit under Mtr / keeps the mechanism frm rtating test generatr F The trque arm r the frce scale, etc Shaft are identical t thse f the prny Scale brake dynammeter Shaft rtatin rate is measured Cradle supprts Cable simultaneusly, and pwer is calculated with the same equatins as fr the prny brake, W = ωt, r W π = NrpmT 60 The pwer dissipated by the resistr bank can als be calculated easily Namely, if either the current I r the vltage V acrss the resistr bank is measured, the dissipated pwer is W = V dissipated V But since V = IR frm Ohm s law, W dissipated = VI = I R= R The advantages f a cradled DC mtr dynammeter are that it is versatile, and it can supply pwer as well as absrb pwer, which is smetimes a useful feature The disadvantages are that it is large in size and epensive Eddy current dynammeter Principle f peratin: A fluctuating magnetic field absrbs the unit s pwer, and dissipates it as heat The magnetic field strength is varied t cntrl the trque A schematic diagram is shwn here The internal structure f the eddy current dynammeter cnsists f: A statr (the statinary part), which has Unit under test Bearings Shaft Pwer supply Statr Rtr Cradle supprts Cable a cil arund a ferrmagnetic material, pwered by a DC supply The supplied pwer is adjusted t cntrl the lad A rtr (the rtating part), which has radial lbes As the rtr turns, the lbes disrupt the magnetic field (same principle as the magnetic pickup device), and cause internal currents knwn as eddy currents The energy f the eddy currents dissipates internally due t electrical resistance Unlike the cradled DC mtr dynammeter, there is n resistr bank the unit itself heats up, and this heat must be dissipated On smaller units, the heat is dissipated int the surrunding air On larger units, the dissipated energy is carried away by cling water The trque arm, the frce scale, etc are identical t thse f the cradled DC mtr dynammeter Shaft pwer is calculated with the same equatins as previusly, W = ωt, r W π = NrpmT 60 The main advantage f an eddy current dynammeter is it is gd fr very high pwer applicatins, like autmbile engines The main disadvantage is that it cannt supply pwer as can the cradled DC mtr dynammeter An eddy current dynammeter can nly absrb pwer r F Scale