Lecture 7 Fluid Systems I. System Analysis Spring

Transcription

1 Lecture 7 Flud Systems I 1

2 Brake systems Font Wheel Brake Pedal Vacuum Booster Master Cylnder Proportonng Valve Vacuum Booster ear Wheel Master Cylnder Proportonng Valve Brake Pedal Fundamental structure of a hydraulc brake

3 Conservaton of Mass, Force and Pressure 3

4 Conservaton of Mass, Force and Pressure m m mo ) Volume A dx A dx dx A 1 1 A ) Force f p A f 1 1 dx ) Pressue p p gh pa A f f ( P P) 1 1 A1 L L A f f f L L A1 4

5 Flud esstance Electrcal systems, resstance s defned as v, v h For flud systems, s defned as D m P, P m L Mass flow rate, pressure dfference 5

6 Flud esstance change n pressure dp changen flowrate d m mr 6

7 Flud Capactance Electrcal systems, dv dt 1 C Flud systems, dp 1 1 dt m m, dp mdt, C dt C C dp C change n stored mass change pressure dm dp 7

8 Flud Capactance C change n stored mass change pressure dm dp For a tank, General flud capactance dh 1 A C A A dp g g dm dt dp C m dt mo dp dh C A m dt dt mo For lud level system, C s defned as C 3 change n lud stored m change n head m use h(pressure head), nstead of p. A 8

9 Flud Systems vs Electrcal Systems Electrcal systems, Flud systems, esstance Capactance Inductance 9

10 Flud Systems vs Electrcal Systems Electrcal systems, Flud systems, Voltage V Pressure P Current I (= dq/dt) Mass flow rate Charge Q Flud mass 10

11 Lud Level Systems Control valve Q Q esstance Capactance H H h Q Q o o change n level dfference m C change n lud stored m changen flow rate m 3 / s changenhead m 3 Steady state Q Q Q : 0 H H consder, Qo, Q Qo C dh dt 11

12 Basc Concepts Lamnar flow : When the vscous forces are domnant (slow flow, low e) they are suffcent enough to keep all the flud partcles n lne, then the flow s lamnar. Turbulent flow : It can be nterpreted that when the nertal forces domnate over the vscous forces (when the flud s flowng faster and e s larger) then the flow s turbulent. eynolds number (e) : The rato of nertal forces to vscous forces. It s used to determne whether a flow wll be lamnar or turbulent. e vd : : D : the dynamc vscousty of the flud densty of the flud dameter e 000 : e 4000 : always lamnar always turbulent 1

13 Lamnar Flow Cylndrcal ppe h P1 P D 18 L P1 P gh, Q h 4 g D : vscosty, L: lengthof ppe D : dameter of ppe L h 18 L Q Kl h s m g D, [ / ] 4 Q : steady state flow rate K l : constant h : steady state head 13

14 Lamnar Flow Lud level dynamcs : dh C Q Q dt o Q ( Q ) o o dh d dh ( H h) dt dt dt H h h Qo Qo, o dh 1 1 h dt C C 14

15 Turbulent Flow Q Q H H h C Q Q 0 0 Q Cd a ( P1P ) : densty, a : area, C : dschargecoeffcent d steady state Q Q Q K H : 0 t 15

16 Turbulent Flow Lud level dynamcs dh C Q Q0 Q K H dt dh 1 1 Q K H f( Q, H) dt C C f 1 f f( Q, H) f( Q, H) ( Q Q) ( Q Q) Q! Q QH, f 1 ( H H) ( H H) H QH,! H K Q K H h hgh order term C C C C H f 16

17 Turbulent Flow dh dh dh dt dt dt dh 1 K dt C C H h h H H,, o Q K H K Q dh 1 1 h dt C C h o for small 17

18 Flud esstance(lamnar flow vs turbulent flow) change n pressure dp changen flowrate d m mr m p For lamnar flow m p l For turbulent flow For lud level system, s defned as change n level dfference m dh changen flow rate m s d 3 / m 18

19 Lnearzaton Q Q H h H K H QQ K H ( ) h y f( x), y f( x), x xx f 1 f y f x x x ( ) x x! x x f y x x x f y y ( xx) K( xx) x x f f y f( x, x ) f( x, x ) ( x x ) ( x x ) x1 x x, x x, x yk ( x x ) K ( x x ) y y K ( x x ) K ( x x )

20 Summary of Lud Level Systems Lamnar flow Turbulent flow dh 1 1 h dt C C H Q KH Q dh 1 1 h dt C C o h 18 L g D 4 o h H H H Q K Ca g d Q K H Cda gh Cda g H 0

21 Lud Level Systems wth Interacton Q Q Steady state : Q QQ Q 1 H 1 H1 h1 C 1 Q 1 H H h C 1 Q 1 Q Q H H H 1 1 Lud level dynamcs : dh dh C Q Q C dt dt h Q Q H h ( H h ) h dh dh C Q Q C dt dt 1 1 H h h Q Q 1

22 Lud Level Systems wth Interacton dh1 1 h1 h 1 dt C C dh 1 h h 1 h dt C C 1 1 Transfer Functon Q ( S) 1 Q( S) C C S C C C S

23 End of Lecture 7 3