Solutions Manual 4.1. nonlinear. 4.2 The Fourier Series is: and the fundamental frequency is ω 2π

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1 Soluios Maual. (a) (b) (c) (d) (e) (f) (g) liear oliear liear liear oliear oliear liear. The Fourier Series is: F () 5si( ) ad he fudameal frequecy is ω f H z.3 Sice V rms V ad f 6Hz, he Fourier Series is: F () V rms si( f) 69.7si( ) ad he fudameal frequecy is 6Hz.. A -- F () ω T cos( ) d So T T ω T cos( ) d T T -- cos( ω ) d T -- T A [ si( ω ω T ) [ si( ω ) T -- Bu ω -----, so T A ( si( ) si( ) + si( ) ) Bu sie of ay muliple of is, so A Iroducio o Mecharoics ad Measureme Sysems

2 Soluios Maual.5 Usig MahCAD:, Va( ),.. Vb( ) m,.. 5 Vc( ) si(.. ) si(.. ). si(.. ). m cos (... ) ( ).( ) cos (. m.. ) ( m ).( m ).5.5 Va( ) Vb( ) Vc( ) (a) f H ( 6) 7.7Hz So he badwidh is: Hz o 7.7 Hz Iroducio o Mecharoics ad Measureme Sysems

3 Soluios Maual (b) T s f o -- H z, ω T f rad sec ω (rad/sec) f (Hz) A i A ou /A i A ou (A ou /A i )A i ω / / 3ω 3 /3 /3 3 5ω 5 /5 /5 7ω 7 /7 3/ 3/7 5 9ω 9 /9 / /9 >5 (-)ω (-) /f V ou -- si( ) si( 6) si( ) si( ) si( 8) (c) Usig MahCAD:,... Vou( ). si.. ( ) 3.. si( 6.. ) 5.. si(.. 3 ) 7.. si(.. ) 9.. si( 8.. ). Vou( ).7 (a) ω c rad RC sec Iroducio o Mecharoics ad Measureme Sysems

4 Soluios Maual (b) T s, f Hz, ω F () A i ( ) A ou ( ) si( ω ) where A i A i ( ) ( ) A ou A i ( ) ω ω c ω ( ) (c) Usig MahCAD:.. ω c ω (. )..,... F().. si ω. (. ). ω ω c F ( ).8 ω L rad rad Hz sec sec ω H rad rad Hz sec sec ω L badwidh ω H Iroducio o Mecharoics ad Measureme Sysems 3

5 Soluios Maual.9 (a) (b) rad ω 5 rad sec sec ad (c) A i A ou V o R V i R jωc V o V i jωrc jωrc + To fid he cu-off frequecy, se he ampliude raio magiude o : V o V i ωrc ( ωrc) + Solvig for he frequecy gives ω c RC Usig his expressio gives: V o V i ω ω c ω + ω c Iroducio o Mecharoics ad Measureme Sysems

6 Soluios Maual ad ow we ca plo he frequecy respose i erms of he dimesioless frequecy raio ω : ω r ω c ω r ω r, A r ω r ω r A r ω r.5 6 ω r. φ V o ωrc arg ( ) ( + ωrcj) aa aa( ωrc) V i ω Usig ω c ad ω, RC r ωrc ω c φ aa( ω r ) φ ω r ω r.5 Iroducio o Mecharoics ad Measureme Sysems 5

7 Soluios Maual. Usig MahCAD:..,... ω (. ).. F(). si ω. (. ). ATT ATT.5 ATT.5 ATT 3.5 F low () (. ).. ATT. si ω. ATT ATT 7.5 ATT 8.5 ATT.5 F high () (. ).. ATT. si ω. F ( ) F low ( ) F high ( ).3 Whe he mass is i saic equilibrium, so M i g X ou X ou g -- Mi 6 Iroducio o Mecharoics ad Measureme Sysems

8 Soluios Maual ad he saic sesiiviy is K g --. We geerally assume ha he displayed volage is he gai imes he ipu volage. This assumpio will be i error if he oscilloscope is dc coupled ad some of he frequecies i he sigal exceed he badwidh of he oscilloscope..5 KVL gives: where q is he charge o he capacior. Puig his i sadard form gives: where q is he depede variable, he ime cosa (τ) is RC, ad he sesiiviy (K) is C. Usig he geeral soluio for a s order sysem, Therefore, he sep respose oupu volage (which is he volage across he capacior) is.6 The rae of chage of ieral eergy is equal o he rae of hea rasfer: so V ou () IR + C ---q V i ( RC)q + q CV i q () CA i e Defiig C mc (hermal capaciace) ad R (hermal resisace) ad ha coverig io sadard form gives: where he ime cosa is τ R C mc ad he sesiiviy is K. ha - τ ---q() A C i e ---- d ( E d i ) Q i - τ ( mc) dt ou ( ha) ( T d i T ou ) ( C R ) dt ou T d ou ( )T i Iroducio o Mecharoics ad Measureme Sysems 7

9 Soluios Maual.7 Ploig he daa X ou () shows a seady sae asympoe of approximaely 5 idicaig ha: KA i 5 X Ploig Z () ou () l shows a ear liear relaio idicaig ha he sysem KA i ca be modeled as s order. The slope of he lie is approximaely /3 idicaig a ime cosa of τ 3 sec.8 The damped aural frequecy is always smaller if here is dampig i he sysem..9 (a) (b) (c) mechaical roary (applied orque, orsio sprig, roary damper, ad roary ieria): Jθ + Bθ + θ τ ex elecrical (volage source ad series resisor, iducor, ad capacior): Lq + Rq + ---q V or C s LI + RI Id C V s hydraulic (pump wih ile i reservoir, log pipe wih fricio loss ad fluid ieria, ad a): I dq RQ + V P d C --- where V Qd. Give he differeial equaio: τx ou Applyig he procedure resuls i: + x ou Kx i ( τs+ )X ou ( s) KX i ( s) Gs ( ) X ou ( s) X i ( s) K ( τs+ ) Gjω ( ) K + τωj φ X ou X i G ( j ω) K + ( τω) arg( Gjω ( )) aa( τω) aa( τω) 8 Iroducio o Mecharoics ad Measureme Sysems

10 Soluios Maual. F N ω rad Sec ω , ω ω, m r ς ω b.56 m F X [ ω r ς + ω r X X F m F φ ς a rad ω r ω r Therefore, he seady sae respose is: x () X si( ω + φ).3si( )m. The equaio of moio is Taig he Laplace rasform of boh sides gives he rasfer fucio: Now mx ou + bx ou + x ou x i Gs ( ) X ou ( s) X i ( s) Gjω ( ) ms + bs ( mω ) + bωj X ou X i G ( j ω) ( mω ) + ( bω) So he seady sae respose is φ a bω mω X ou x ou () X i si( ω + φ) X i Iroducio o Mecharoics ad Measureme Sysems 9

11 Soluios Maual Usig MahCAD, m. b X i.5 X ou ( ω ) m. ω X. i Oly he firs ipu resuls i a accepable oupu. φ ( ω) agle m. ω, b. ω ( b. ω ) X ou ( ).5 φ ( ).57 deg X ou ( ).5 φ ( ) 9 deg X ou ( ) 5.5. φ ( ) 79. deg.3 x h () e ςω [ Acos( ω d ) + Bsi( ω d ) x p () C x () x h () + x p () x( ) F gives C x() gives A -C x ( ) Therefore, gives B x () F e ςω ( cos( ω d ) ). The volume i he a is D V h The pressure a he boom of he a is P ρgh γh Solvig he volume expressio for h ad subsiuig io he pressure expressio gives γ P V --- V D C 3 Iroducio o Mecharoics ad Measureme Sysems

12 Soluios Maual so C D γ.5 Sice F ma, a x ad x Q ---, A PA Q ( ρla) --- A P ρl Q A IQ.6 Eleme [flow aalogies: F i [ v i V i [ I i [ v i v m C [ I i I m mv [ m LI [ m b [ v m R [ I m [ v m C [ I m Aalogous free body diagram equaios [KVL: V i V C The resulig aalogous elecrical circui follows: V C V R + V C + V L + L V i C R C.7 Eleme [flow aalogies: F [ v V [ I mv [ LI [ [ v C [ I Iroducio o Mecharoics ad Measureme Sysems 3

13 Soluios Maual b [ v v R [ I I [ v v C [ I I Aalogous free body diagram equaios [KVL: V The resulig aalogous elecrical circui follows: F [ v V [ I V C V R V C V L V R + V C V R C C V + L + V.8 Hydraulic elemes are direc aalogies o elecrical elemes. The capaciors are replaced by as, ad he resisor ad iducor are replaced by a log pipe wih flow resisace ad ierace..9 Eleme [flow aalogies: F i [ v V i [ I b [ v v R [ I I [ v v C [ I I mv [ LI [ [ v v 3 C [ I I 3 b [ v 3 R [ I 3 Aalogous free body diagram equaios [KVL: V R V i V R + V C + V C V L + V C 3 Iroducio o Mecharoics ad Measureme Sysems

14 Soluios Maual V C V R The resulig aalogous elecrical circui follows: I I I 3 V i + R (I -I ) L (I-I 3) C C R.3 z ( x) + bz ( x ) µm gsg( x ) m x z ( x) + bz ( x ) F m z rf I θ where z y rθ, z y rθ, z y rθ Iroducio o Mecharoics ad Measureme Sysems 33