. MATLAB tutorial Course Outlie. Motio of systes that ca be idealized as particles Descriptio of otio; Newto s laws; Calculatig forces required to iduce prescribed otio; Derivig ad solvig equatios of otio 3. Coservatio laws for systes of particles Wor, power ad eergy; Liear ipulse ad oetu Agular oetu 4. Vibratios Exa topics Characteristics of vibratios; vibratio of free DOF systes Vibratio of daped DOF systes Forced Vibratios 5. Motio of systes that ca be idealized as rigid bodies Descriptio of rotatioal otio; ieatics forulas Dyaics forulas for rigid bodies; calculatig oets of iertia Motio of systes of rigid bodies Eergy ad oetu for rigid bodies
Particle Dyaics: Cocept Checlist Uderstad the cocept of a iertial frae Be able to idealize a egieerig desig as a set of particles, ad ow whe this idealizatio will give accurate results Describe the otio of a syste of particles (eg copoets i a fixed coordiate syste; copoets i a polar coordiate syste, etc) Be able to differetiate positio vectors (with proper use of the chai rule!) to deterie velocity ad acceleratio; ad be able to itegrate acceleratio or velocity to deterie positio vector. Be able to describe otio i oral-tagetial ad polar coordiates (eg be able to write dow vector copoets of velocity ad acceleratio i ters of speed, radius of curvature of path, or coordiates i the cylidrical-polar syste). Be able to covert betwee Cartesia to oral-tagetial or polar coordiate descriptios of otio Be able to draw a correct free body diagra showig forces actig o syste idealized as particles Be able to write dow Newto s laws of otio i rectagular, oral-tagetial, ad polar coordiate systes Be able to obtai a additioal oet balace equatio for a rigid body ovig without rotatio or rotatig about a fixed axis at costat rate. Be able to use Newto s laws of otio to solve for uow acceleratios or forces i a syste of particles Use Newto s laws of otio to derive differetial equatios goverig the otio of a syste of particles Be able to re-write secod order differetial equatios as a pair of first-order differetial equatios i a for that MATLAB ca solve
Particle Kieatics Iertial frae o acceleratig, o rotatig referece frae Particle poit ass at soe positio i space Positio Vector Velocity Vector r( t) = x( t) i + y( t) j + z( t) v() t = v () t i+ v () t j+ v () t x y z d dx dy dz = ( xi+ yj+ z) = i+ j+ dx dy dz vx() t = vy() t = vz() t = Directio of velocity vector is parallel to path Magitude of velocity vector is distace traveled / tie Acceleratio Vector j O r(t) r(t+) d dv dvy () () () () x dv a t = a z x t i+ ay t j+ az t = ( vxi+ vyj+ vz) = i+ j+ dv dv () x y dv () () z x y z d x d y d z a t = = a t = = a t = = v(t) i path of particle
Particle Kieatics Straight lie otio with costat acceleratio r = X + Vt + at = ( V + at) = a i v i a i Tie/velocity/positio depedet acceleratio use calculus t t r = X + v() t = V + a() t i v i vt () xt () V vdv = v dv g() t a = = f ( v) dv = g() t f () v V xt () dx g() t v = = f ( x) dv = v() t f () v X a( x) dx t t
Geeral circular otio ( cosθ siθ ) R( si cos ) r = R i+ j v= ω θi+ θj = Vt Particle Kieatics Circular Motio at cost speed θ = ωt s= Rθ V = ωr ( cosθ siθ ) R( si cos ) r = R i+ j v= ω θi+ θj = Vt V a= ω R(cosθi+ si θj) = ω R= R a= Rα( siθi+ cos θj) Rω (cosθi+ si θj) dv V = αrt+ ω R= t+ R ω = dθ / α = dω / = d θ / s = Rθ V = ds / = Rω j siθ t cosθ R θ Rsi θ i Rcosθ
Particle Kieatics Arbitrary path v= Vt dv V a= t + R r = x( λ) i+ y( λ) j dx d y dy d x dλ dλ dλ dλ = R dx dy + dλ dλ 3/ t t R Polar Coordiates e θ e r dr dθ v= er + r e θ d r dθ d θ dr dθ r r r a= e + + e θ j i r θ
Newto s laws For a particle F = a For a rigid body i otio without rotatio, or a particle o a assless frae j i M c = You MUST tae oets about ceter of ass N A W T B N B
Calculatig forces required to cause prescribed otio of a particle Idealize syste Free body diagra Kieatics F=a for each particle. M (for rigid bodies or fraes oly) c = Solve for uow forces or acceleratios
Derivig Equatios of Motio for particles. Idealize syste. Itroduce variables to describe otio (ofte x,y coords, but we will see other exaples) 3. Write dow r, differetiate to get a 4. Draw FBD 5. F= a 6. If ecessary, eliiate reactio forces 7. Result will be differetial equatios for coords defied i (), e.g. d x dx + λ + x = Y siωt 8. Idetify iitial coditios, ad solve ODE
Motio of a projectile r i j dr = X + Y + Z t = Vxi Vyj Vz = + + X V i j ( ) ( ) r = X + Vxt i+ Y + Vyt j+ Z + Vzt gt ( V ) ( V ) ( V gt) v= i+ j+ a= g x y z
Rearragig differetial equatios for MATLAB Exaple d y dy + ζω + ωy = Itroduce v = dy / The d y v v = ζωv ωy This has for dw y = f(, t w) w = v
Coservatio laws for particles: Cocept Checlist Kow the defiitios of power (or rate of wor) of a force, ad wor doe by a force Kow the defiitio of ietic eergy of a particle Uderstad power-wor-ietic eergy relatios for a particle Be able to use wor/power/ietic eergy to solve probles ivolvig particle otio Be able to distiguish betwee coservative ad o-coservative forces Be able to calculate the potetial eergy of a coservative force Be able to calculate the force associated with a potetial eergy fuctio Kow the wor-eergy relatio for a syste of particles; (eergy coservatio for a closed syste) Use eergy coservatio to aalyze otio of coservative systes of particles Kow the defiitio of the liear ipulse of a force Kow the defiitio of liear oetu of a particle Uderstad the ipulse-oetu (ad force-oetu) relatios for a particle Uderstad ipulse-oetu relatios for a syste of particles (oetu coservatio for a closed syste) Be able to use ipulse-oetu to aalyze otio of particles ad systes of particles Kow the defiitio of restitutio coefficiet for a collisio Predict chages i velocity of two collidig particles i D ad 3D usig oetu ad the restitutio forula Kow the defiitio of agular ipulse of a force Kow the defiitio of agular oetu of a particle Uderstad the agular ipulse-oetu relatio Be able to use agular oetu to solve cetral force probles
Wor ad Eergy relatios for particles Rate of wor doe by a force (power developed by force) P = F v i O F P j v Total wor doe by a force W t = F v W = F dr r r i r O F(t) j r P Kietic eergy T= v = v + v + v ( ) x y z v Power-ietic eergy relatio Wor-ietic eergy relatio dt P = r W = F dr = T T r i r O P j r
Potetial eergy Potetial eergy of a coservative force (pair) Type of force Gravity actig o a particle ear earths surface Potetial eergy V = gy j i F y r r V( r) = F dr+ costat F = grad( V ) Gravitatioal force exerted o ass by ass M at the origi GM V = r r F r i r O F j r P Force exerted by a sprig with stiffess ad ustretched legth L Force actig betwee two charged particles V = ( r L ) F j F QQ +Q j V = +Q i 4πε r r i r Force exerted by oe olecule of a oble gas (e.g. He, Ar, etc) o aother (Leard Joes potetial). a is the equilibriu spacig betwee olecules, ad E is the eergy of the bod. 6 a a E r r j i r F
Eergy relatios for coservative systes subjected to exteral forces Iteral Forces: (forces exerted by oe part of the syste o aother) Exteral Forces: (ay other forces) R ij ext F i Syste is coservative if all iteral forces are coservative forces (or costrait forces) ext F 4 R R 3 R 3 R ext F R 3 R 3 F 3 ext 3 t Eergy relatio for a coservative syste ext 4 F ext F 3 3 ext F = t t = t Total KE T Total PE V TOT TOT Exteral Power P ext () t t ext Exteral wor W P() t = t ext F ext F Total KE T Total PE V 4 ext F 3 3 TOT TOT ext ( ) TOT TOT TOT TOT W = T + V T + V Special case zero exteral wor: TOT + TOT = TOT + TOT T V T V KE+PE = costat
Ipulse-oetu for a sigle particle Defiitios Liear Ipulse of a force Liear oetu of a particle t I= F() t t p= v i O F(t) j v Ipulse-Moetu relatios d F = p I= p p t=t p=p i O F(t) j t=t v p=p
Ipulse-oetu for a syste of particles R ij ext Fi vi Force exerted o particle i by particle j Exteral force o particle i Velocity of particle i ext F 4 R R 3 R 3 R ext F R 3 R 3 F 3 ext 3 Total Exteral Force Total Exteral Ipulse ext 4 F ext F 3 F I TOT TOT () t t = F t TOT () t ext F 4 ext F 3 Ipulse-oetu for the syste: F TOT d = p TOT I = p p TOT TOT TOT t = t 3 F ext Total oetu p TOT F ext t = t 3 Total oetu p TOT Special case zero exteral ipulse: p = p TOT TOT (Liear oetu coserved)
Collisios v x A A v x A A * v x B v x B B B Moetu Restitutio forula v + v = v + v A B A B A x B x A x B x ( ) v v = ev v B A B A v = v ( + e) v v + ( ) B B A B A A v = v + ( + e) v v + B ( ) A A B B A A B A v A v A A B v B B B v Moetu Restitutio forula v + v = v + v B A B A B A B A B A B A B A ( ) ( ) ( e) ( ) v v = v v + v v + + ( ) ( ) v B B A B A = v + e v v B A ( ) ( ) v A A B B A = v + + e v v B A
Agular Ipulse-Moetu Equatios for a Particle Agular Ipulse Agular Moetu Ipulse-Moetu relatios t A= r() t F() t t h= r p= r v r d F= h i x O y r(t) F(t) z A= h h j Special Case A= h = h Agular oetu coserved Useful for cetral force probles (whe forces o a particle always act through a sigle poit, eg plaetary gravity)
Free Vibratios cocept checlist You should be able to:. Uderstad siple haroic otio (aplitude, period, frequecy, phase). Idetify # DOF (ad hece # vibratio odes) for a syste 3. Uderstad (qualitatively) eaig of atural frequecy ad Vibratio ode of a syste 4. Calculate atural frequecy of a DOF syste (liear ad oliear) 5. Write the EOM for siple sprig-ass-daper systes by ispectio 6. Uderstad atural frequecy, daped atural frequecy, ad Dapig factor for a dissipative DOF vibratig syste 7. Kow forulas for at freq, daped at freq ad dapig factor for sprig-ass syste i ters of,,c 8. Uderstad uderdaped, critically daped, ad overdaped otio of a dissipative DOF vibratig syste 9. Be able to deterie dapig factor ad atural frequecy fro a easured free vibratio respose. Be able to predict otio of a freely vibratig DOF syste give its iitial velocity ad positio, ad apply this to desig-type probles
Vibratios ad siple haroic otio Typical vibratio respose Period, frequecy, agular frequecy aplitude Displaceet or Acceleratio y(t) Period, T Pea to Pea Aplitude A tie Siple Haroic Motio ( ω φ) xt ( ) = X + Xsi t+ vt ( ) = Vcos( ωt+ φ) at ( ) = Asi ( ωt+ φ) V = ω X A= ω V
Vibratio of DOF coservative systes Haroic Oscillator Derive EOM (F=a) d s + s= L Copare with stadard differetial equatio x= s C = L x = s ω = Solutio st ( ) = L + ( s L) + v / ω si( ω t+ φ) Natural Frequecy ω =
Vibratio odes ad atural frequecies Vibratio odes: special iitial deflectios that cause etire syste to vibrate haroically Natural Frequecies are the correspodig vibratio frequecies x x
Nuber of DOF (ad vibratio odes) I D: # DOF = *# particles + 3*# rigid bodies - # costraits I 3D: # DOF = 3*# particles + 6*# rigid bodies - # costraits Expected # vibratio odes = # DOF - # rigid body odes A rigid body ode is steady rotatio or traslatio of the etire syste at costat speed. The axiu uber of rigid body odes (i 3D) is 6; i D it is 3. Usually oly thigs lie a vehicle or a olecule, which ca ove aroud freely, have rigid body odes. x x
Calculatig at freqs for DOF systes the basics y,l EOM for sall vibratio of ay DOF udaped syste has for ω ω d y + y = C is the atural frequecy. Get EOM (F=a or eergy). Liearize (soeties) 3. Arrage i stadard for 4. Read off at freq.
Trics for calculatig atural frequecies of DOF udaped systes Usig eergy coservatio to fid EOM ds ( KE + PE = + s L ) = cost d ds d s ds KE PE s L ds ( + ) = + ( ) = + s = L s, d Nat freq is related to static deflectio ω = g δ,l L +δ
Liearizig EOM Soeties EOM has for d y + f( y) = C We cat solve this i geeral Istead, assue y is sall d y df + f() + y+... = C dy y= y= d y df + y = C f dy () There are short-cuts to doig the Taylor expasio
Writig dow EOM for sprig-ass-daper systes s=l +x, L c Coit this to eory! (or be able to derive it ) d x c dx F= a + + x= d x dx c + ζω + ωx = ω = ζ = x(t) is the dyaic variable (deflectio fro static equilibriu) Parallel: stiffess = + Series: stiffess = + c c c Parallel: coefficiet c= c + c c Parallel: coefficiet = + c c c
Caoical daped vibratio proble EOM d s ds + c + s = L Stadard For s x d x ς dx + + x= C ω ω c ω = ς = C = L with s= s = v t = x dx x= x = v t = s ds s=l +x, L c ω = ω ζ d Overdaped ς > Critically Daped ς = Uderdaped ς < Overdaped ς > Critically Daped Uderdaped ς < v + ( ςω )( ) ( )( ) ( ) exp( ) + ωd x C v + ςω exp( ) ωd x C xt = C+ ςωt ω exp( ω) ωd ωd { } ς = [ ω ] x( t) = C+ ( x C) + v + ( x C) t exp( ω t) v + ςω ( ) ( ) exp( ) ( )cos x C xt = C+ ςωt x C ω+ siω ω d
Calculatig atural frequecy ad dapig factor fro a easured vibratio respose Displaceet x(t ) x(t ) x(t ) x(t 3 ) tie t t t t 3 t 4 T Measure log decreet: xt ( ) δ = log xt ( ) Measure period: T The δ 4 π + ς = ω δ = 4π + δ T
Forced Vibratios cocept checlist You should be able to:. Be able to derive equatios of otio for sprig-ass systes subjected to exteral forcig (several types) ad solve EOM usig coplex vars, or by coparig to solutio tables. Uderstad (qualitatively) eaig of trasiet ad steady-state respose of a forced vibratio syste (see Java siulatio o web) 3. Uderstad the eaig of Aplitude ad phase of steady-state respose of a forced vibratio syste 4. Uderstad aplitude-v-frequecy forulas (or graphs), resoace, high ad low frequecy respose for 3 systes 5. Deterie the aplitude of steady-state vibratio of forced sprig-ass systes. 6. Deduce dapig coefficiet ad atural frequecy fro easured forced respose of a vibratig syste 7. Use forced vibratio cocepts to desig egieerig systes
EOM for forced vibratig systes L x(t), L λ F(t)=F si ωt Exteral forcig d x ς dx + + x = KF siω t ω ω λ ω =, ς =, K = L, L y(t)=y siωt L, L λ x(t) λ x(t) y(t)=y siωt ω Base Excitatio Rotor Excitatio d x ς dx ς dy + + x= K y+ ω ω ω λ ω =, ς =, K = d x dx K d y Y + ς ω ω + = = x K siωt ω ω ω λ ω = ς = K = M = + M M M
Equatio Steady-state ad Trasiet solutio to EOM d x ς dx + + x = C + KF si( ωt) ω ω Iitial Coditios dx x= x = v t = Full Solutio xt () = C+ x () t + x () t Steady state part (particular itegral) h p ( ω φ) xp ( t ) = X si t+ X KF = φ = ta / ω / ω ( ω / ω) + ( ςω / ω) ςω / ω Trasiet part (copleetary itegral) h h h h v + ( ςω ) ( ) Overdaped ( ) exp( ) + ωd x v + ςω exp( ) ωd x ς > xh t = C+ ςωt ω exp( ω) ωd ωd Critically Daped Uderdaped ς < h h h ς = { ω } ω = ω ς d xh( t) = C+ x + v + x t exp( ωt) h h h v ( ) exp( ) + ςω cos x xh t = C+ ςωt x ω+ siω ω d h h dxp x = x C xp () = x C Xsi v = v = v X cos t= φ ω φ
d x ς dx Steady state solutio to + + x = C + KF si( ωt) ω = π /T ω ω c ω = ζ = K = ( ω / ω) + ( ςω / ω) p ( ) si ( ω φ) x t = X t+ ςω / ω = ( ω / ω, ) ta ζ = φ = / ω / ω X KF M M Caoical exterally forced syste (steady state solutio) M ax s=l +xft ( ) = F siωt, L c ζ
Caoical base excited syste (steady state solutio) Steady state solutio to c ω = ζ = K = Magificatio X /KY ζ =.6 d x ς dx ς dy + + x= C+ K y+ ω ω ω ζ =. ζ =. ζ =.5 ζ =. ( ω φ) xp ( t ) = X si t+ / { + ( ςω / ω ) } 3 3 ςω / ω = ( ωω,, ) ta ζ = φ= / ( 4 ς ) ω / ω X KY M M M ax ( ω / ω) + ( ςω / ω) ζ ζ =. ζ =.4 -.5.5.5 3 Frequecy Ratio ω/ω
Steady state solutio to ω Caoical rotor excited syste (steady state solutio) c c = ζ = K = + ( + ) + d x ς dx K d y + + x= C ω ω ω p ( ) si ( ω φ) x t = X t+ y = Y siωt X = KY M ( ωω,, ζ) M ω / ω ςω / ω = = ( ω / ω) + ( ςω / ω) φ ta / ω / ω M ax ζ
Dyaics of Rigid Bodies cocept checlist. Uderstad agular velocity ad acceleratio vectors; be able to itegrate / differetiate agular velocities / acceleratios for plaar otio.. Uderstad forulas relatig velocity/acceleratio of two poits o a rigid body 3. Uderstad costraits at joits ad cotacts betwee rigid bodies 4. Be able to relate velocities, acceleratios, or agular velocities/acceleratios of two ebers i a syste of lis or rigid bodies 5. Be able to aalyze otio i systes of gears 6. Uderstad forulas relatig velocity/agular velocity ad acceleratio/agular acceleratio of a rollig wheel 7. Be able to calculate ass oets of iertia of siple shapes; use parallel axis theore to shift axis of iertia or calculate ass oets of iertia for a set of rigid bodies coected together 8. Uderstad M = I G Gα for plaar otio of a rigid body 9. Uderstad ad ow whe you ca use M = Iα. Be able to calculate acceleratios / forces i a syste of plaar rigid bodies subjected to forces usig dyaics equatios ad ieatics equatios. Uderstad power/wor/potetial eergy of a rigid body; use eergy ethods to aalyze otio i a syste of rigid bodies. Uderstad agular oetu of a rigid body; use agular oetu to aalyze otio of rigid bodies
Describig rotatioal otio of a rigid body Agular velocity vector:. Directio parallel to rotatio axis (RH screw rule). Magitude agle (radias) tured per sec dθ ω= = ω Agular acceleratio vector: α = dω Axis of rotatio For plaar otio: d d d ω = α = = θ ω θ dθ d θ ω = α = Pure Moets (torques): M = M A pure oet is a geeralized force that iduces rotatioal otio without traslatio of ceter of ass A otor shaft is a exaple of a object that exerts a oet the shaft is parallel to the directio of the oet θ
Rigid body ieatics Velocities of two poits o a rigid body are related by v = v + ω r A B AB / Acceleratios of two poits o a rigid body are related by a = a + α r + ω ( ω r ) A B AB / AB / Cotiuity coditios A B va = vb A B a A = a B A B v = v aa = ab A B No slip v A = v Slip va = vb B Tagetial accels equal Accels arbitrary
Kieatics of a Rollig Wheel Wheel has agular velocity Wheel has agular acceleratio ω = ω α Wheel rolls without slip This eas that velocity of A is zero (wheel has sae velocity as the groud, see aiatio) α = ωα, B D A C * A E j i Poit A also has zero acceleratio i the i directio (tagetial acceleratios are equal at the cotact A has a ozero upwards acceleratio, however) The rigid body forula tells us that v = v + ω r C A C/ A = + ω Rj v C = ω Ri The differetiate wrt tie to see a C = α Ri To fid velocity or accel at A, B, D, E use the stadard rigid body forulas.
Dyaics of rigid bodies Preliiary defiitios: ass oets of iertia used i plaar otio (geeral 3D ore coplex) Mass desity ρ Total Mass : M = ρdv COM positio : rg = ρ dv M r V Iertia about a axis through origi I r dv Parallel Axis Theore : I = I + d O G : = V ρ V G G d r ρdv O Equatios of Motio Traslatioal otio Rotatioal otio F= a (ust use acceleratio of COM) G M = r F+ M = I α G F/ G G Forces Pure Moets a G F This rotatioal otio equatio is valid ONLY for plaar otio 3D otio has aother ter α Μ For rotatio about a fixed axis oly MO = rf/o F+ M = IOα Forces Pure Moets G ω O
Free body diagras with frictio Rollig without slip v a T C C = ω = α < µ N Ri Ri T Both FBDs below are correct N N T ωα, B D A C * A E j i Rollig with slidig: Frictio force ust oppose slidig vcx + ωr> A oves to right wrt A * T vcx + ωr< A oves to left wrt A * N T = µ N T N
Aalyzig otio of systes of rigid bodies. Idetify each particle/rigid body i the syste. Draw a FBD for each particle / rigid body separately F= a 3. Write dow for each rigid body ad particle 4. Write dow M = I G Gα for each rigid body (for rotatio about a fixed poit ca also use = I α M 5. Loo for poits i syste where acceleratio is ow or related (eg cotacts, joits, etc) a = a + α r + ω ( ω r ) O O 6. Use G A G/ A G/ A to relate acceleratios ad agular acceleratios of rigid bodies 7. Solve syste of equatios fro 3, 4, 6 to calculate uow reactios ad acceleratios / agular acceleratios
Eergy ethods for rigid bodies Power (rate of wor doe) by forces ad oets actig o a rigid body P = F v + M ω Forces F Pure Moets F v F G ω Μ Total wor doe t t W = P() t = F vf + M ω t t Forces Pure Moets Gravitatioal potetial eergy of a rigid body use positio of COM V = gh G Potetial eergy of a costat oet (plaar otio oly) V = Mθ h Potetial eergy of a torsioal sprig V = κθ G θ M=M Kietic eergy of a rigid body T = vg + IG ω T = I ω O Power-KE relatio Geeral ca always use this Rotatio about a fixed axis oly (use parallel axis theore to fid Io) dt P = Wor-KE relatio W = T T Wor- eergy relatio for a coservative syste ext W = T+ V ( T + V) If o exteral wor is doe o a coservative syste T+ V = ( T + V) ω G G v G ω O
Agular oetu for rigid bodies Agular ipulse about COM (ote that COM eed ot be fixed) t ΑG = ( r rg) F+ M t Forces Pure Moets F j r r G G ω Μ Agular ipulse about a fixed poit t Α = r F+ M t Forces Pure Moets O i Agular oetu about COM h G = I G ω Agular oetu about a fixed poit ho = rg vg + IGω Special case: rotatio about a fixed poit h = I ω O O G O ω dhg Ipulse-oetu relatios (COM) ( r rg ) F+ M = AG = hg hg Forces Pure Moets Moetu is coserved if A = G Ipulse-oetu relatios (Fixed poit) dh A = h h O r F+ M = O O O Forces Pure Moets Moetu is coserved if A = O