Cantilever Beam Experiment

Transcription

1 Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio of the disk drive i extreme eviromets (drop loads). Typical disk drive armatures are show i Figure 1 alog with sapshots of drop load usig high-speed photography; these have revealed sigificat armature/head displacemets ad acceleratios resultig i udesirable separatio of the head mechaism from the armature. Typical Armature Arragemet Armature ad Head Mechaism Separatio - 130g/1msec Simulated Pulse Figure 1 - Several Armatures ad Sapshots usig High-Speed Photography As part of the redesig process, the armature s dyamic characteristics must be idetified. This is eeded to determie the atural frequecy, dampig ad tip displacemet/acceleratio at the read/write head of the armature for overall performace evaluatios. While the first atural frequecy is expected to cause the largest displacemets, the secod atural frequecy is also of cocer due to the high acceleratios aticipated. A redesig of the armature alog with both active ad passive dampig treatmets are aticipated to provide the best overall solutio to the problem. I order to evaluate this problem, several measuremet scearios are evisioed. Due to the high cost of prototype developmet, a simple catilever structure will be used to evaluate measuremets usig strai gage, eddy curret probe, accelerometer ad LVDT measurig devices; three of these devices must be used to assess the performace of the disk drive armature i the productio vibratio screeig that will be performed. Vibratio screeig tests will be routiely performed to cofirm the adequacy of productio hardware due to vibratio eviromets Fial Project - Catilever Beam Experimet 1 Rev

2 Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Assigmet - Develop a mass, sprig, dashpot model to characterize the catilever beam A aalytical ad experimetal model eeds to be developed for evaluatio of the various desigs. Certai aspects of the model are to be developed usig aalytical approaches ad others usig experimetal approaches. However, where possible, both aalytical ad experimetal approaches are required together to assure that a accurate dyamic model is obtaied. It is required that the tip displacemet of the catilever be evaluated usig both displacemet ad acceleratio approaches usig three differet measurig type devices. Correlatio of the measured displacemet ad acceleratio is required to assure that a accurate dyamic model is obtaied. The atural frequecy ad dampig of the system is to be evaluated. While the secod atural frequecy is of cocer, the first steps i the developmet of the aalytical ad experimetal models will oly be performed for the first atural frequecy of the system. Theory A mathematical model ca be used to represet the characteristics of the system. Ay vibratio textbook cotais the material ecessary; Referece 1 was used as the referece for the material preseted herei. However, all aalytical models iheretly have assumptios relatig to the material properties, elastic properties, boudary coditios ad dampig. As such, these aalytical represetatios must always be verified through the use of accurately measured dyamic data. Th simplest model is the sigle degree of freedom lumped mass model defied by secod order differetial equatio with costat coefficiets. This model is show i Figure 2. x(t) m f(t) k c Figure 2 Sigle Degree of Freedom Model The equatio of motio describig this is system ca easily be show to be 2 d x dx m c kx f (t) dt = or m & x + cx& + kx = f (t) (1) dt where m is the mass, c is the dampig ad k is the stiffess with the displacemet, velocity ad acceleratio ad the forcig fuctio as show. Several pertiet relatioships are show i Referece 2. These iclude the atural frequecy, damped atural frequecy, dampig factor, critical dampig, ad techiques for estimatig dampig; other otes o sigal processig ad experimetal modal aalysis are also cotaied i that documet Fial Project - Catilever Beam Experimet 2 Rev

3 Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell A cotiuous solutio ca be obtaied for the catilever beam or a aalytical model ca be developed usig the fiite elemet techique i a software package such as MATLAB [3]. (Referece 4 cotais some brief otes cocerig the fiite elemet modelig process as well as some rudimetary MATLAB script files for the geeratio of a simple catilever beam model). Usig some basic stregth of materials approximatios alog with the cotiuous beam vibratio equatio, a equivalet model ca be developed for aalysis purposes. The mode shapes for a cotiuous catilever beam are give i [1] as f (x) = A {(si β + (cosβ L sih β L coshβ L)(siβ L)(cosβ where = 1,2,3, K ad x sih β x coshβ β x) x)} L = π (2) ad the atural frequecies are give as 2 E I ω = α where α = 1.875, 4.694, (2) 4 m L The frequecies ad mode shapes are show i Figure 3. Figure 3 Frequecies ad Mode Shapes for Catilever Beam Fial Project - Catilever Beam Experimet 3 Rev

4 Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell From Stregth of Materials, the deflectio, x, at the tip of a catilever beam is give by where 3 x = P L / 3 E I (3) x tip displacemet P applied load (also referred to as F) L legth of the catilever beam E Youg s Modulus of Elasticity I bedig momet of iertia The deflectio at the ed of the catilever beam ca be expressed as F = k x (4) ad therefore, the stiffess of the catilever beam ca be expressed as 3 k = 3 E I / L (5) Realizig that the atural frequecy is k ω = (i rad/sec) (6) m ad the atural frequecy for the first mode of the catilever from the cotiuous solutio is 2 E I g ω = (1.875) (where w - weight ad g gravitatioal costat) (7) 4 w L allows the effective mass at the tip of the catilever beam to be determied. This approximatio allows the catilever beam to be modeled as a sigle degree of freedom system sice the mass ad stiffess are kow Fial Project - Catilever Beam Experimet 4 Rev

5 Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Equipmet The Mechaical Egieerig Laboratory has a assortmet of equipmet to evaluate this beam system. Measuremets o the beam must be made with a miimum of three differet measurig devices simultaeously to determie the beam tip displacemet durig vibratio as well as the atural frequecy, damped atural frequecy ad dampig i the system. Time domai ad/or frequecy domai techiques ca be employed i the solutio of this problem. Post Aalysis ad Report The report should address (but is ot limited to) the test setup, calibratio, test procedure, digital data acquisitio system, uderlyig priciple of operatio for the trasducers ad acquisitio system, measuremets made, umerical processig of the data, problems associated with the data collectio ad/or reductio, digital sigal processig cosideratios, accuracy, aalog measurig devices used ad ay other related iformatio to substatiate the results preseted. The results ad coclusios should address ay problems that were observed. Recommedatios to improve the measuremet system should be discussed. Recommedatios for the productio testig of these disk drive armatures must be provided so that suitable equipmet ca be procured for the product tests. Refereces 1) Dyamics of Structures, Humar, J.L., Pretice-Hall, 1990, ISBN ) Overview of Digital Sigal Processig ad Experimetal Modal Aalysis, Peter Avitabile, excerpts of otes preseted at IMAC 19 semiar o Modal Aalysis, Orlado, Florida 3) MATLAB, Versio 5.3, Math Works, Ic., Natick, Massachusetts 4) Some Brief Fiite Elemet Modelig Notes (excerpted from Fiite Elemet Modelig Notes) ad MATLAB scripts for Catilever Beam Evaluatio, Peter Avitabile, October Fial Project - Catilever Beam Experimet 5 Rev