ME 498CA1 · Abaqus Project 1 Due November 3 Tuning Fork Frequencies Tuning orks are steel bars #$ic$ acoustica""% resonate at a &'e( )itc$* They are requent"% use( to tune musica" instruments (ue to t$eir consistenc% an( ease-o +use. The standard tuning ork soun(s A (44- ./01 #$ic$ is t$e stan(ard concert )itc$* 2n genera"1 t$e ollo#ing ormula can be use( to determine t$e frequenc% o a given tuning fork* 1.8345 a5 E f = 5π l 5 √ 15ρ #$ere f is t$e frequenc% t$e fork vibrates (H/06 1.834 is t$e smallest positive solution o cos x cos$ x = –16 l is t$e lengt$ o t$e prongs (m) (t%)ica""% 8-89- mm06 E is t$e Young's mo(ulus o t$e material (Pa06 a is t$e edge o t$e square area A o t$e prong cross+section (m) (t%)ica""% 484 mm06 an( ρ is t$e densit% o t$e materia" (kg·m830* 9ou #ill create a tuning ork model in t#o an( t$ree dimensions #$ic$ a($ere to t$e s)eci&cations given belo#1 an( t$en carr% out severa" simulations* 2n eac$ o t$ese, %ou #ill determine t$e initia" vibrationa" requenc% o t$e model an( t$e &rst nine overtones )lus stresses* (Create a linear )erturbation; requenc% ana"%sis step #it$ ten modes using t$e <Lanc/os> eigensolver.) I recommen( using a = 4 mm an( starting wit$ L from 8-89- mm* 1* Calculate t$e c$aracteristics o a tuning fork designe( to sound 44- H/ as t$e dominant )itc$. Then simulate t$is tuning fork as? 1* A t#o+dimensiona" )lanar simulation using beam elements #it$ t$ic!ness t$e same as t$e prong widt$ (solid square cross+section o prong ■0* 5* A t$ree-dimensiona" simulation using solid $exa$edra" elements , u"" integration0* This ma% require )artitioning t$e mes$ as in t$e bottom+u) mes$ing exam)le. Do not use tetra$edra if at a"" possible. @eport t$e modes o %our defau"t model, an( t$en tune it suc$ t$at t$e dominant mode alls #it$in 4A o t$e correct tuning requenc% (if necessar%0* @eport t$e actua" lengt$ o t$is ca"cu"ated tuning fork* 1/2 ME 498CA1 · Abaqus Project 1 Due November 3 5* Bin( dimensions or a middle C ,5C1*5C ./0 tuning ork rom t$e ormula above. Construct and model t$is fork (in t$ree dimensions0* @eport t$e modes o %our defau"t model, an( t$en tune it suc$ t$at t$e dominant mode alls #it$in 4A o t$e correct tuning requenc% (if necessar%0* @eport t$e actua" lengt$ o t$is ca"cu"ated tuning fork* Assume t$e tuning ork is made o aluminum1 E 7 C9×1-9 Pa1 ν 7 -.331 an( ρ 7 5*3×1-83 g·mm83* 9ou s$ou"( verif% #$et$er t$is unit s%stem is consistent an( ma!e certain t$at %ou carr% out an% unit trans ormations necessar% before using t$e formula above. 9ou are res)onsible to s$o# t$at %our mes$ converges reasonab"% an( ma!e a case or its suEcient accurac%* (A mes$ densit% o 4A o t$e overa"" dimensions is usua""% a goo( starting )oint*0 2 t$e $andle a))ears to be causing trouble, %ou can delete t$e $andle an( use a F+ s$a)e( geometr% or t$e model; in an% case, you nee( to constrain t$e base o t$e 9 or F at t$e junction as it is a requenc% node. 2 oun( t$at constraining t$e lines or )oints along its edge #orke( #ell. 9ou s$ou"( c$ec! t$e requenc% range 5-83-1--- ./ (a bit beyon( t$e limits o $uman hearing). ,2ntriguingly1 tuning orks can be use( to detect bone ractures noninvasively* A "o# C (158 ./0 tuning ork is tone( on t$e sus)ecte( bone, causing t$e )eriosteum (bone membrane) to vibrate. This in turn causes nociceptors (pain receptors0 to fire, resulting in a local s$ar) pain*0 9ou will document t$e simu"ations in a 881- page report (#it$ figures) containing t$e sections? ■ Problem descri)tion (tuning fork s$a)e, grid, etc*0 ■ Numerica" va"ues (element )arameters6 number o nodes an( t$eir requencies6 boundar% conditions6 etc*0 ■ Com)utational times (CPF time to solve) ■ Gbservations o numerica" behavior (boundar% conditions6 mes$ be$avior, including convergence; etc*0 ■ Discussion o t$e )$%sics (modes1 etc*His t$e difference bet#een t$e ormu"a an( t$e actua" value a constant ratioJ0 2nclude t$e follo#ing plots in your report1 wit$ data from eac$ case you will stu(%? ■ Mes$ o model ■ Contour plot o dis)"acement in x and y directions ■ Ktills o animation o mo(al vibration6 you ma% also wis$ to pro(uce movie files The report s$ould be ormatte( wit$ 1*4 line s)acing1 1 inc$ margins on all sides1 an( set in 11 )oint seri font* Figures and tables s$ould be numbere( wit$ labels and ca)tions* 2/2.