Class 03 Systems modelling

Transcript

1 Class 03 Systems mdelling

2 Systems mdelling input utput

3 spring / mass / damper

4 Systems mdelling spring / mass / damper

5 Systems mdelling spring / mass / damper applied frce displacement input utput

6 Systems mdelling Newtn s nd Law Sir Isaac Newtn, lg m k µ u,

7 Systems mdelling spring / mass / damper and therefre, m µ k u, r d m µ d k u,

8 Systems mdelling carr / massa / mla Nw, giving values t m, µ and k: m 1 kg µ 4 N s/m k 3 N/m

9 Systems mdelling spring / mass / damper m 1 kg µ 4 N s/m k 3 N/m d m (0) µ d a, k (0) m b µ k u,

10 Systems mdelling spring / mass / damper m 1 kg µ 4 N s/m k 3 N/m and the mdel becmes: d (0) d 4 a, 3 (0) b 4 3 u,

11 translatinal mechanical mtin

12 Systems mdelling translatinal mechanical mtin prblem similar t the previus ne spring / mass / damper

13 Systems mdelling translatinal mechanical mtin prblem similar t the previus ne spring / mass / damper (equivalent vertical)

14 Systems mdelling translatinal mechanical mtin applied frce displacement input utput

15 Systems mdelling translatinal mechanical mtin Again, using the Newtn s nd Law we btain: m µ k u, u d m µ d k u,

16 Systems mdelling spring / mass / damper r translatinal mechanical mtin Thus, these tw systems are described by the same differential equatin ( nd rder), that is, have the same mdel: d m µ d k m µ k u initial cnditins: (0) a, (0) b

17 Systems mdelling spring / mass / damper r translatinal mechanical mtin d m (0) µ a, d k (0) m b µ k u,

18 Systems mdelling mviment translacinal mecânic Nw, giving the same values t m, µ and k that has been given t the prblem spring / mass / damper, we have: m 1 kg µ 4 N s/m k 3 N/m

19 Systems mdelling spring / mass / damper r translatinal mechanical mtin m 1 kg µ 4 N s/m k 3 N/m d m (0) µ a, d k (0) m b µ k u,

20 Systems mdelling spring / mass / damper r (bth have the same mdel) translatinal mechanical mtin m 1 kg µ 4 N s/m k 3 N/m d (0) 4 d a, 3 (0) b 4 3 u,

21 Systems mdelling translatinal mechanical mtin Observatin: Nte that if µ 0 this system becmes the harmnic scillatr.

22 circuit RLC série

23 Systems mdelling RLC series circuit input vltage utput vltage

24 Systems mdelling RLC series circuit input vltage utput vltage input utput

25 Systems mdelling Kirchhff Law (lp rule): Gustav Kirchhff, thus v LC v RC v v i 0,

26 Systems mdelling RLC series circuit and therefre, LC v RC v v v i, r LC d v RC dv v v i, Then, this system is als described by ne differential equatin (f nd rder).

27 Systems mdelling RLC series circuit That is, the mdel f this system is a differential equatin (f nd rder): LC d v RC dv v RC v LCv v v i initial cnditins: v (0) a, v (0) b

28 b (0) v a, (0) v, v v RCv LCv v dv RC v d LC i RLC series circuit Systems mdelling

29 Systems mdelling circuit RLC série Dand valres para R, L e C: R 1000 Ω L 50 H C 1, F

30 Systems mdelling RLC series circuit d LC v v (0) RC a, dv v v (0) b LCv RCv v R 1000 Ω L 50 H C 1, F v i,

31 Systems mdelling circuit RLC série d LC v v (0) RC a, dv v v (0) b LCv RCv v v R 1000 Ω L 50 H C 1, F i,

32 Systems mdelling circuit RLC série d v v (0) 4 dv a, v 3v (0) b v 4v 3v 3v R 1000 Ω L 50 H C 1, F i,

33 rtatinal mechanical mtin

34 Systems mdelling rtatinal mechanical mtin (t) trque applied t the system input [N m]; ω(t) angular velcity utput [rad/s]; J mment f inertia [kg m ]; µ frictin cefficient [N m /rad/s]

35 Systems mdelling rtatinal mechanical mtin trque applied angular velcity input utput

36 Systems mdelling rtatinal mechanical mtin Using the Newtn s Law fr rtatinal systems mments J ω', we btain J ω µ ω,

37 Systems mdelling rtatinal mechanical mtin Thus, this system is described by a differential equatin (f 1 st rder): J dω µ ω Jω µω initial cnditin: ω( 0) a

38 Systems mdelling rtatinal mechanical mtin that is, the mdel f this system is a differential equatin (f 1 st rder): J dω ω(0) µω a Jω µω,

39 Systems mdelling rtatinal mechanical mtin Nw, giving values t J and µ: J 0,5 kg/m µ N m /rad/s J dω ω(0) µω a Jω µω,

40 Systems mdelling rtatinal mechanical mtin Nw, giving values t J and µ: J 0,5 kg/m µ N m /rad/s J dω ω(0) µω a Jω µω,

41 Systems mdelling rtatinal mechanical mtin dω ω(0) 4 ω a ω 4 ω,

42 a seismgraph

43 Systems mdelling seismgraph i (t) b displacement with respect t inertial space; (t) mass displacement with respect t inertial space; y(t) mass displacement with respect t the b. y(t) [ (t) - i (t)]

44 Systems mdelling seismgraph b displacement mass m displacement input utput

45 Systems mdelling Again, by Newtn s nd Law m and therefre, µ ( ) i k( Sir Isaac Newtn, i ), m( ) µ ( i ) i k( i ) m i, y (t) y (t) y(t)

46 Systems mdellling seismgraph thus, r, m y µ y k y m i, d y dy d m µ k y m i,

47 seismgraph Systems mdelling µ µ b (0) y, a y(0), m y k y m y ky dy y d m i

48 a hydraulic serv-mtr

49 Systems mdelling hydraulic serv-mtr input f the system utput f the system

50 Systems mdelling hydraulic serv-mtr we btain, r, m y (t) µ A K ρ y (t) AK K 1 (t), m d y µ A K ρ dy AK K 1 (t),

51 Systems mdelling hydraulic serv-mtr m y (t) µ A K ρ y (t) AK K 1 (t), A pistn area [m ]; ρ il density [kg/m 3 ]; Q flw rate f il that ges t the pwer cylinder (mass flw rate) [kg/s]; P (P 1 P ) pressure difference in the pwer cylinder (pressure drp) [N/m ]. Q K 1 K P

52 a thermal system

53 Systems mdelling sistema thermal térmic system input utput

54 Systems mdelling thermal system we btain, where, dθ RC θ(t) h i (t) heat input rate [cal/s]; R h (t), θ(t) heat utput temperature [ºC]; R thermal resistance (gain f the system) [ºC s/cal]; C thermal capacitance (heat capacity) [cal/ºc]; T RC time cnstant f the system [s]. i

55 ther eamples

56 Systems mdelling by partial differential equatins: t u k u y u z u k ( u u yy u zz ) system linear cntinuus, time invariant, with memry and causal this system describes the space wave prpagatin

57 Systems mdelling r by difference equatins: [ n ] [ n ] ( ) n [ n ] y 4 discrete system, nnlinear, time variant, withut memry and causal.

58 This system describes the dynamic f the evlutin f AIDS Mre cmple systems are represented nt nly by ne, but by several equatins. Systems mdelling [ ] [ ] [ ] v m k t N k m k k m k m k s µ µ µ ω µ ω µ λ θ θ ) ( ) ( ) ( ) (1 ) ( ) ( ),, ( & & & &

59 Thank yu! Felippe de Suza