ME242 MECHANICAL ENGINEERING SYSTEMS LECTURE 27: Ideal Machines: Transformers and Gyrators 2.4 IDEAL MACHINES. Machine
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1 E4 ECHANICAL ENGINEERING SYSES LECURE 7: Idal achins: ransformrs and Gyrators.4 E4 - Spring Eugnio Schustr Idal IDEAL ACHINES achin An idal machin is a two port dvic that transmits work from on port to th othr No nrgy is stord, gnratd or dissipatd Entropy is not gnratd Can b run in ithr dirction E4 - Spring Eugnio Schustr 344
2 1 Idal IDEAL ACHINES achin Powr Consrvation q& = q& 1 1 E4 - Spring Eugnio Schustr 345 IDEAL ACHINES Physical or chanical systms modld as idal machins Lvrs Gars Elctric motors Piston pumps Elctric ransformrs or accurat (and mor complx) modls of ths dvics might includ othr lmnts. Exampl: Ral Elctric otor E4 - Spring Eugnio Schustr 346
3 IDEAL ACHINES wo spcial cass: ransformrs wo-port dvics Gyrators G G E4 - Spring Eugnio Schustr IDEAL ACHINES - RANSFORER Dfining Condition: q& or = q& 1 1 ransformr odulus (constant) h modulus of th ransformr,, is dfind as th ratio of th gnralizd vlocity or flow on th bond with th outward powr arrow to th gnralizd vlocity or flow on th bond with th inward powr convntion arrow E4 - Spring Eugnio Schustr 348
4 IDEAL ACHINES - RANSFORER Combining th Idal achin condition: q& = q& 1 1 with th ransformr condition: q& = q& 1 yilds an additional condition: = 1 q& = = q& 1 1 h ratio of th gnralizd forcs of an idal tranformr quals th invrs of th ratio of th rspctiv gnralizd vlocitis E4 - Spring Eugnio Schustr 349 IDEAL ACHINES RANSFORER - EXAPLES Friction (shar forcs) Drivs: Pully Drivs 1 r 1 r x& Rolling Contact Drivs = 1 = 1 r = E4 - Spring Eugnio Schustr 350 as x& = r = r 1 1 r
5 IDEAL ACHINES RANSFORER - EXAPLES Positiv Action (normal forcs) Drivs: oothd Drivs E4 - Spring Eugnio Schustr 351 IDEAL ACHINES RANSFORER - EXAPLES Elctric ransformr: E4 - Spring Eugnio Schustr 35
6 IDEAL ACHINES RANSFORER - EXAPLES Positiv-displacmnt mchanical to fluid transducr: piston-and-cylindr / ram Consrvation of Enrgy ( or Powr Balanc) Consrvation of ass ( or Kinmatic Constraint) Cons. of omntum ( or Forc Equilibrium) F x& = PQ E4 - Spring Eugnio Schustr 353 c Q Fc = Ax& = PA IDEAL ACHINES RANSFORER - EXAPLES Pumps and Actuators: Pump Actuator E4 - Spring Eugnio Schustr 354
7 IDEAL ACHINES RANSFORER - EXAPLES Pump: Consrvation of Enrgy ( or Powr Balanc) Consrvation of ass (or Kinmatic Constraint) Cons. of omntum ( or Forc Equilibrium) = PQ Radian Displacmnt Q = Dφ& = D P E4 - Spring Eugnio Schustr 355 IDEAL ACHINES - GYRAORS Dfining Condition: 1 G = Gq& 1 or G 1 Gyrator odulus (constant) h modulus of th Gyrator, G, is dfind as th ratio of th ffort on on of th bonds ithr on to th flow on th othr bond E4 - Spring Eugnio Schustr 356
8 IDEAL ACHINES - GYRAOR Combining th Idal achin condition: q& = q& 1 1 with th Gyrator condition: 1 yilds an additional condition: = Gq& 1 G = q& = = G q& 1 h ratio of th gnralizd forcs of an idal tranformr quals th invrs of th ratio of th rspctiv gnralizd vlocitis E4 - Spring Eugnio Schustr 357 IDEAL ACHINES GYRAOR - EXAPLES Spinning op (A typ of Gyroscop): op Spinning with angular vlocity ω Has angular momntum H = Iω r ω H E4 - Spring Eugnio Schustr 358
9 IDEAL ACHINES GYRAOR - EXAPLES Apply a prpndicular impulsiv forc F t Causs an impulsiv momnt prpndicular to F, t =r F t h momnt quals th changinangular momntum t = H H = t r E4 - Spring Eugnio Schustr 359 ω H F t IDEAL ACHINES GYRAOR - EXAPLES o gt a H rquirs a ω h ω is prpndicular to both ω and F ω r H ω H F t If F t is in dirction 1 and r is indirction 3 thn ω is indirction E4 - Spring Eugnio Schustr 360
10 IDEAL ACHINES GYRAOR - EXAPLES Gyroscop: E4 - Spring Eugnio Schustr 361 IDEAL ACHINES GYRAOR - EXAPLES Gyroscop: top viw : & = 0 φ φ & is th prcssion rat sid viw : & 1 φ 1 = 0 support L mg ω 1 & G φ & 1 φ G = Iω = mr g ω 1 = Lmg = Gφ& 1 = 0 & 1 Lmg Lg φ = = = G mr ω r ω E4 - Spring Eugnio Schustr 36 g g
11 IDEAL ACHINES GYRAOR - EXAPLES Elctric otor / Gnrator (achomtr) E4 - Spring Eugnio Schustr 363 IDEAL ACHINES GYRAOR - EXAPLES DC motor with N coils of radius r rotating in magntic fild B: F = π rnbi F = Gi G1 = π rnb 1 = rf 1 = 1/ r = Gi = G π r NB = Gφ& G1 i F x& 1 G i E4 - Spring Eugnio Schustr 364 or
System variables. Basic Modeling Concepts. Basic elements single and. Power = effort x flow. Power = F x v. Power = V x i. Power = T x w.
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