Course Outline. 2. Motion of systems that can be idealized as particles

Transcription

1 . 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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)

18 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

19 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)

20 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

21 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

22 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 =

23 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

24 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

25 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.

26 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

27 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

28 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

29 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

30 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

31 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

32 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

33 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

34 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

35 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 = Frequecy Ratio ω/ω

36 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

37 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

38 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

39 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

40 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.

41 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

42 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

43 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

44 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

45 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