Class 03 Systems mdelling
Systems mdelling input utput
spring / mass / damper
Systems mdelling spring / mass / damper
Systems mdelling spring / mass / damper applied frce displacement input utput
Systems mdelling Newtn s nd Law Sir Isaac Newtn, 1643-177 lg m k µ u,
Systems mdelling spring / mass / damper and therefre, m µ k u, r d m µ d k u,
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
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,
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,
translatinal mechanical mtin
Systems mdelling translatinal mechanical mtin prblem similar t the previus ne spring / mass / damper
Systems mdelling translatinal mechanical mtin prblem similar t the previus ne spring / mass / damper (equivalent vertical)
Systems mdelling translatinal mechanical mtin applied frce displacement input utput
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,
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
Systems mdelling spring / mass / damper r translatinal mechanical mtin d m (0) µ a, d k (0) m b µ k u,
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
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,
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,
Systems mdelling translatinal mechanical mtin Observatin: Nte that if µ 0 this system becmes the harmnic scillatr.
circuit RLC série
Systems mdelling RLC series circuit input vltage utput vltage
Systems mdelling RLC series circuit input vltage utput vltage input utput
Systems mdelling Kirchhff Law (lp rule): Gustav Kirchhff, 184-1887 thus v LC v RC v v i 0,
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).
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
b (0) v a, (0) v, v v RCv LCv v dv RC v d LC i RLC series circuit Systems mdelling
Systems mdelling circuit RLC série Dand valres para R, L e C: R 1000 Ω L 50 H C 1,333 10-3 F
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 1,333 10-3 F v i,
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 1,333 10-3 F i,
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 1,333 10-3 F i,
rtatinal mechanical mtin
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]
Systems mdelling rtatinal mechanical mtin trque applied angular velcity input utput
Systems mdelling rtatinal mechanical mtin Using the Newtn s Law fr rtatinal systems mments J ω', we btain J ω µ ω,
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
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,
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,
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,
Systems mdelling rtatinal mechanical mtin dω ω(0) 4 ω a ω 4 ω,
a seismgraph
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)]
Systems mdelling seismgraph b displacement mass m displacement input utput
Systems mdelling Again, by Newtn s nd Law m and therefre, µ ( ) i k( Sir Isaac Newtn, 1643-177 i ), m( ) µ ( i ) i k( i ) m i, y (t) y (t) y(t)
Systems mdellling seismgraph thus, r, m y µ y k y m i, d y dy d m µ k y m i,
seismgraph Systems mdelling µ µ b (0) y, a y(0), m y k y m y ky dy y d m i
a hydraulic serv-mtr
Systems mdelling hydraulic serv-mtr input f the system utput f the system
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),
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
a thermal system
Systems mdelling sistema thermal térmic system input utput
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
ther eamples
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
Systems mdelling r by difference equatins: [ n ] [ n ] ( ) n [ n ] y 4 discrete system, nnlinear, time variant, withut memry and causal.
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 1 1 4 3 3 4 3 3 0 1 4 1 1 3 0 1 4 1 1 4 1 1 1 1 1 3 1 4 1 ) ( ) ( ) ( ) (1 ) ( ) ( ),, ( & & & &
Thank yu! Felippe de Suza felippe@ubi.pt