Euler s Formula. Complex Numbers - Example. Complex Numbers - Example. Complex Numbers - Review. Complex Numbers - Review.

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1 Chaper ahemaical ehods Slides o accompay lecures i Vibro-Acousic Desig i echaical Sysems by D. W. Herri Deparme of echaical Egieerig Lexigo, KY el: dherri@egr.uky.edu Euler s Formula he Power of Complex Numbers θ e j cosθ j siθ aylor s Series Proof 4 x x x x f 4 4 x f a xf a f f f f 4 6 θ θ θ cosθ θ θ θ siθ θ e j θ θ θ θ θ θj j j 4 5 Complex Numbers - Example z j Addiio ad Subracio z z 7 j z z 4 j z 5 j Complex Numbers - Example uliplicaio ad Divisio Raioalizaio z j z 5 j z z 5j j j j 5 j 7 5 j 5 j 6 * z zz 7 j * z z z 4 Close ad Frederick, Complex Numbers - Review Close ad Frederick, Complex Numbers - Review 5 6

2 Close ad Frederick, Polar Form - Review x z cosθ y z siθ z x y y θ arca x jθ z z e Close ad Frederick, Complex Number Noaios - Review Recagular Form z x jy cosθ jsiθ z z Polar Form z z θ jθ z z e 7 8 Polar Form - Review z e z e π j π π j π cos j si j j π cos π j si z e π π cos π j si j j π cos π j si z e π z z Polar Form - Examples z j z 5 j z e.98 j π j z e j 6.97 z z 8e 4 j *.98 j zz e. 8 j 7 7 j e *.97 j zz 6e 6 9 Polar Form - Examples z j z 5 j z z z z 8 45 z z z z Polar Form Complex Cojugae z j * z j z e z e.98 j *.98 j *.98 j.97 j zz e 6e. 8 j 7 7 j e *.97 j.97 j zz 6e 6e 6

3 ALAB Commads Example Wrie he harmoically varyig force as a complex force. r absz hea aglez x realz y imagz ˆF cos ω ˆFe jω ˆF cos ω jsi ω F F F Re jω ˆFe 4 Example Wrie he harmoically varyig force as a complex force. F ˆF cos ω ˆF si ω e jω ˆF j ˆF F ˆF j ˆF F ˆF cos ω ˆF si ω cos ω jsi ω j ˆF si ω ˆF cos ω Example Wrie he harmoically varyig force as a complex force. F ˆF cos ω ˆF si ω ˆF cos ω ϕ ˆF ˆF ˆF ϕ arca ˆF ˆF F ˆFe jω jϕ jω ˆFe F Re jω ˆFe 5 6 Displaceme, Velociy ad Acceleraio v ˆve j ωϕ v a d d v jω ˆve j ωϕ v ω ˆve j ωϕ vπ / v x d jω ˆve j ωϕ v ω ˆve j ωϕ v π / Complex Cojugae F FF * j ωϕ j ωϕ ˆFe ˆFe ˆF ˆF ˆF Re F F F* ˆF cos ω jsi ω ˆF cos ω jsi ω ˆF cos ω 7 8

4 Derivaio echaical Power F ˆF cos ω ϕ F Re ˆFe j ωϕ F v ˆvcos ω ϕ v Re ˆve j ωϕ v W F v Re F Re v * v W F F v * v Re F * v Re ˆFˆve j ϕ F ϕ v W 4 F v F * v * F * v F v * W Re F W Re ˆFˆve j ωϕ Fϕ v W ˆFˆv cos ω ϕ F ϕ v cos ϕ F ϕ v Derivaio echaical Power ˆFˆv cos ω ϕ F ϕ v cos ϕ F ϕ v W W W d W Re Fv* Re F* v ˆFˆvcos ϕ F ϕ v 9 he rasfer Fucio rasfer Fucios H i a i oupu F ipu easure usig impac esig or shaker. Simulae by usig ui force ad deermiig acceleraio. easure usig impac esig or shaker. Simulae by usig ui force ad deermiig acceleraio. a i F Ui Force Assumpio Superposiio Assumpio Lieariy Doublig he force will double he respose. F F a i I F a i II a i I a i II F a i I a i II H i F H i F a i F a i F 4 4

5 Liear Superposiio Sigle DOF Sysems I Operaio Equaio of oio: x Bx Kx f F cos ω f x K B F F oal respose rasie Respose Seady Sae Respose F F 4 rasie respose depeds o a he sysem parameers, B, ad K ad b he I.C s Seady sae respose depeds o a he sysem parameers ad b he ampliude ad frequecy of he forcig fucio 5 6 rasie ad Seady Sae Respose Complex Expoeial Form f ad x rasie Seady Sae oal Respose Forcig Fucio ime s ζω Ae cos ω θ X cos ω φ x D x B x Kx f ˆF cos ω x ˆXe jω jω f ˆFe ω jωb K ˆXe jω ˆF ˆX K ω ˆFe jω f jωb e jϕ K ω ωb ˆF jω x Re ˆXe ωb ϕ arca K ω ˆF cos ω ϕ K ω ωb x K B 7 8 ω δ B g K F he Amplificaio Facor ˆX ĝ ω ω δω δω ϕ arca ω π,,,,... ω he Amplificaio Facor ˆX ω φ ˆX ω ω ω 4 δ ω ωω Siusoidal Seady Sae Respose f F cos ω Frequecy Respose Fucio H s s jω H jω ϕ X ϕ F H jω ϕ Vibraig Sysem wih rasfer fucio Hs Ipu, Oupu x X cos ω ϕ ime s f x 9 5

6 Frequecy Respose Fucios Y ω X ω H ω Dyamic Flexibiliy or Recepace obiliy or echaical Admiace Accelerace Dyamic Siffess echaical Impedace Acousic Impedace Specific Impedace H ω x ω F ω H ω v ω F ω A ω a ω F ω κ ω F ω x ω Z ω F ω v ω Z ω p ω Q ω Z ω p ω u ω H ω Frequecy Respose Fucios X ω F ω ω jωδ K K ω ω δ ω δ > ω δ < ω δ Criically Damped Overdamped Uderdamped Udamped ξ δ Ofe a dampig raio is defied ω j ωδ ω Dyamic Flexibiliy agiude Dyamic Flexibiliy Phase 4 Nyquis Diagram Derivaio of Loss Facor E ki ω x cos ω ϕ E po K x si ω ϕ dx E dis F d dx B d dx B dx d d Bx ω cos ω ϕd Bx ω Bx ω π ω Bx ωπ η E dis π max E po ωb K ωδ ω 5 6 6

7 aerial Dampig aerial Dampig ε σ Creep es σ Relaxaio es ε ε ε σ σ σ J ε E ε σ he Kelvi-Voig -Parameer odel E R Creep Compliace ε J J σ E R e E Sress Relaxaio σ ε E E E 7 8 aerial Dampig aerial Dampig he -Parameer odel E R E Cosas p R E q E E E q R E Creep Compliace ε σ J p q J e q q q q Sress Relaxaio σ E E ε q q q e p p For Low Frequecy Harmoic Processes he Kelvi-Voig odel is Applicable E ω E jωr E jη he aerial Loss Facor Seel ~. Eergy Dissipaed per Cycle η ω aximum Poeial Eergy W D ωr πu E 9 4 Compoe ad Srucural Dampig Cook e al., Srucural Dampig Compoe loss facor differs because sress ad disorio codiios are posiio-depede. Uless paricular measures are ake o icrease compoe dampig, dampig a he coac surfaces of jois boled, riveed, clamped is he predomia dampig mechaism. Dampig is due o relaive moio of he maig surfaces. his ca happe i he form of macro-slip relaive moio across he eire face of he joi, micro-slip ierfacial slip of small areas, ad gas pumpig losses. η ~. Dampig i srucures is ormally o viscous. I is due o mechaisms such as hyseresis i he maerial ad slip i coecios. I realiy, due o compuaio easiess, dampig i srucures are approximaed by viscous dampig. Pheomeological dampig mehods physical dissipaive mechaisms are modeled. Specral dampig mehods viscous dampig is iroduced by meas of specified fracios of criical dampig

8 odels for Damped Srucures odels for Damped Srucures x Cx Kx f x ξω x ω x C K f x f a jω Xe jω Fe x f Viscous Dampig Represeaio x ξωx ω x F ω X ξω jωx ω X F A Resoace η ξ Used i SEA Codes Loss Facor Represeaio x K jη x f K x jη x f ω X jηω X ω X F Used i FE Codes 4 44 VDI 8 odels for Damped Srucures odels for Damped Srucures [ ] x [ C] x [ K] x { f } [ C] α [ K] β[ ] a Assumig special dampig marices which simplify calculaio wih viscous dampig or wih srucural dampig, he equaios of moio ca be decoupled by modal rasformaio. his yields equaios of moio for simple damped sysems which ca be solved i he familiar maer. If his assumpio is o jusified by he physical dampig behavior of a srucure, he he couplig effecs should be icorporaed i he model. Proporioal Dampig Represeaio x αk β x Kx f ξω αω β αω β x ω x f x x ξωx ω x f Relaio o Viscous Dampig Raio ξ αω β ω Cook e al., Rayleigh or Proporioal Dampig Srucural Vibraio a a Sigle DOF A a sigle degree of freedom ξ ξ Desig Specrum β ξ αω ω Esimaig α ad β ξω ξω α ω ω ωω ξω ξω β ω ω u ϕ cos ω u displaceme of a odal DOF ϕ ampliude ω ω

9 Srucural Vibraio for he Eire Srucure { u} { ϕ}cos ω ode Shapes he vecor of ampliudes is a mode shape φ φ φ φ 4 5 φ 6 φ 7 φ 8 { u} Vecor of odal displacemes { ϕ} vecor of ampliudes for each DOF Use subscrip i o differeiae mode shapes ad aural frequecies { u} { ϕ} i cos ω i 49 5 Appropriae Iiial Codiios Deermiig Naural Frequecies If he I.C.s are a scalar a muliple of a specific mode shape he he srucure will vibrae i he correspodig mode ad aural frequecy ICs a{ ϕ} i Cosider a muli DOF sysem [ ]{ u } [ K] { u} { } Noe ha he sysem is ü Udamped ü No excied by ay exeral forces 5 5 odal Aalysis If he sysem vibraes accordig o a paricular mode shape ad frequecy { u} { ϕ} i cos ω i { ϕ} i mode shape i ω i aural frequecy i odal Aalysis g Firs derivaive Velociy { u } ω i { ϕ} i si ω i g Secod derivaive Acceleraio { u } ω i { φ} cos ω i i

10 odal Aalysis Plug velociy ad acceleraio io he equaio of moio ω K i φ wo possible soluios { φ} i { } de ω K i { } i { } he Eige Problem de ω K i he Eige Problem A x { } λ x { } Eigevalue Eigevecor λ { x} odal Aalysis A Eige Problem ω K i { φ} K { } ω i { } φ K Naural frequecies are eigevalues he mode shapes are eigevecors φ φ { } ω i { φ} ω i { φ} odal Aalysis A Eige Problem K { φ } ω i { φ} { } λ{ x} A x Solvig he eige problem A { x } λ x A λ I x de A λ I { } { } Example Eige Problem de * x x λ i x x - λ * i,, Fid Eigevalues de * λ λ Eige values λ λ,. - 9 λ 4λ 5 λ λ

11 λ * * For Eige Vecor,/. - x x,/. - / 4 5 x x / For Eige Vecor *- /,. / x ca be ay value x x x x / -/. / / / / λ 5 * * For Eige Vecor 5 5 5,/. - x x,/ x /. 4 - x 5 / * For Eige Vecor x x. -/, x ca be ay value x x x x. / Example arix Form 5 kg x x kg K K K 8 N/m K 4 N/m Equaios of moios x K x K x x K x K x K x x K, K x,, * x - K K K * x - * - 5 x, 8 8 x,, * x - 8 * x - *

12 Se Up Eige Problem { } ω i { φ} K φ K de 6 6 λ. -, 8 / ω Solve Eige Problem de * λ 5. rad/s f Hz π ω λ rad/s f Hz π 6 λ 6 8 λ φ φ φ φ, ode ode K K K K 69 7 Weighed Orhogoaliy odal Vecor Scalig ϕ ϕ ϕ ϕ [ ] [ K] ϕ ϕ ϕ ϕ g Uiy odal ass ϕ ϕ [ ] ϕ ϕ 7 7

13 odal Vecor Scalig Example Vibraig achie Isolaio g Uiy odal ass.7645x.6446x ϕ ϕ 5 *, X X.6446X m kg m 5 kg κ 5E6 N/m κ E6 N/m d v kg/s d v kg/s 7 74 Eigevalues ad Eigevecors Forces Wih / Wihou Exciaio ω 4.7 rad/s f 6.48 Hz ψ 5.8 ω 45.6 rad/s f 9. Hz ψ.4 F ˆF e jω F ˆF e jω x p ˆx p e jω x p ˆx p e jω ˆF wihou ˆF exc ˆF wih κ ˆx e jω jωd vˆx e jω Simulaeous Differeial Equaios m ω ˆx p jω d v d v ˆx p κ κ ˆx p jωd v ˆx p κ ˆx p ˆF exc jωd v ˆx p κ ˆx p m ω ˆx p jωd v ˆx p κ ˆx p Simulaeous Differeial Equaios γ ω ˆF wihou κ jωd ω v m κ κ jω d v d v ω m κ jωd v ˆF wih κ jωd v κ jωd v 77 78

14 he Frequecy Specrum Dyamic sigals ca be represeed i he frequecy domai by a frequecy specrum ime domai ampliude vs. ime frequecy domai Fudameal firs harmoic Secod harmoic f Hz ampliude vs. frequecy Fourier Aalysis Used o deermie he frequecy specrum of dyamic sigals specrum aalyzer hardware y A o A cos ω B si ω A o C cos ω ϕ A A y d y cos ω d B y si ω d C A B ϕ a B A π/ω Is he period of he sigal, ω is he fudameal frequecy firs harmoic, ω is he secod harmoic, ec y A Example Example Fourier Series Expasio 4A si ω si ω si 5ω si ω π 5 y A i.e., B -A y cos ω d Acos ω d Acos ω d A si ω ω A ω si π * / 4A π, odd y si ω d, eve C A B B ϕ a B A π Acos ω d,. - Phase bewee A B 4A π 4A π 8 8 Example Fourier Series Expasio Example Fourier Lie Specrum 4A si ω si ω si 5ω si ω π 5 y ime domai frequecy domai y A -A 4A π ω 4A π 4A 5π ω 5ω ec. ampliude vs. ime ampliude vs.frequecy

15 ypical Siuaio ypical Siuaio How much is he vibraio? Peak.5 mm Peak-o-peak.4 mm Average - 8.x -4 mm rms.5 mm Specrum reveals several harmoics uder Hz, probably relaed o egie rpm ad firig frequecy Digiizaio of Aalog Sigals ime sigals are sampled usig a aalog-o-digial coverer a digial logic hardware device o yield a digiized versio of he waveform for furher processig. Aalog volage sigal from rasducer ADC ime s ec. Volage V Volage values sored i compuer memory Samplig Parameers Δ sampleierval f sample rae / Δ samples/sec or Hz N oal umber of samples oal sample period NΔ r sampleidex umber,, N rδ ime of ay give sample y s yrδ y Discree Fourier rasform Coiuous aalog form: A o A cos ω B si ω y A y cos ω d B y si ω Discree digial form: he discree form will yield frequecy values up o N/ oly. N N π πr N A y r cos rδ Δ y rδ cos,, NΔ r NΔ N r N N πr B y rδ si N r N d Frequecies i Discreely Sampled Sigals. Like is aalog couerpar, he discree Fourier series coais frequecy compoes spaced / apar. his is called he frequecy resoluio Δf : Fourier Coefficies Δf / Δf / Hz f

16 Frequecies i Discreely Sampled Sigals he Aliasig Pheomeo. Sigals mus be sampled a a miimum rae f s wice he highes frequecy of ieres he Nyquis frequecy f Nyq. he highes frequecy resolved i a DF is: fs f Nyq If his is o doe, aliasig error occurs. a f s f b f > f s > f c f s f he sigal f will appear as f/ i he specrum miimum sample rae 9 9 he Aliasig Pheomeo No Periodic rasie, radom Sigals f Nyq f Nyq he Fourier Iegral rasform: rue specrum f easured specrum f Fω f π f e jω d Fωe jω dω Fω is coiuous, complex lim f Aliasig resuls i harmoics above he Nyquis frequecy beig folded back oo heir eighbors. Commercial specrum aalyzers se he samplig frequecy a leas.5 imes he highes frequecy of ieres. Sigal is low-pass filered before samplig. f ime Domai Fω ω Frequecy Domai 9 94 he Fourier Iegral he Fourier Iegral Example: f A Example: 5 / -A Fω Ae jω d Ae jω d ja ω e jω / e jω / Fω FωF *ω A ω 6 8cos π ω ω o cos π ω ω o where ω o π / Fω A /ω o 4 Compare wih Fourier series resuls ω/ω o

17 Frequecy Respose Fucios FRF s Frequecy Respose Fucios FRF s A FRF is he raio of wo Fourier rasforms: H ω F ω F ω Wha is he FRF bewee he soud pressures a wo pois i a duc carryig a x wave? A A x x H ω P ω P ω Ae jkx e Ae jkx x jkx Noe: kx x ωδ where Δ is he propagaio ime x x