Sping-endulum Dynmic System echtonics Sping-endulum Dynmic System 1
esuements, Clcultions, nufctue's Specifictions odel mete ID Which metes to Identify? Wht Tests to efom? hysicl System hysicl odel th odel Expeimentl Anlysis Assumptions nd Engineeing Judgement hysicl Lws odel Indequte: odify Eqution Solution: Anlyticl nd Numeicl Solution Actul Dynmic Behvio Compe edicted Dynmic Behvio odify o Augment ke Design Decisions odel Adequte, efomnce Indequte odel Adequte, efomnce Adequte Design Complete Dynmic System Investigtion echtonics Sping-endulum Dynmic System
hysicl odel Simplifying Assumptions pue sping, i.e., negligible ineti nd dmping line sping fictionless pivot neglect ll mteil dmping nd i dmping point mss, i.e., neglect ottionl ineti of mss two degees of feedom, i.e., nd e the genelized coodintes (this ssumes no out-ofplne motion nd no bending of the sping) suppot stuctue is igid echtonics Sping-endulum Dynmic System 3
hysicl odel k l + m = pendulum mss = 1.815 kg m sping = sping mss =.1445 kg l = unstetched sping length =.333 m k = sping constnt = 17.8 N/m g = cceletion due to gvity = 9.81 m/s F t = 5.71 N = pe-tension of sping s = sttic sping stetch, i.e., s = (mg-f t )/k =.7 m d = dynmic sping stetch = totl sping stetch = s + d m echtonics Sping-endulum Dynmic System 4
Sping Clibtion F sping (N) mg = 17.85 N K = 17.8 N/m Sping e-tension F t = 5.71 N.7 m Sping Displcement (metes) echtonics Sping-endulum Dynmic System 5
O de d de d echtonics Sping-endulum Dynmic System ol Coodintes: osition, Velocity, Acceletion = e = e e e ATH v = e d = = e + e = ve + ve dt dv = = c h b e + + e dt g = e + e + mgnitude chnge diection chnge mgnitude chnge diection chnge 6 v v
echtonics Sping-endulum Dynmic System 7 Rigid Body Kinemtics m l + k X Y x y O XY: R efeence fme (gound) xy: R 1 efeence fme (pendulum) x y z X Y Z i j k I J K L N O Q = L N O Q L N O Q L N O Q = L N O Q L N O Q cos sin sin cos cos sin sin cos 1 1 R R O R R R R O R R O R R R R v = + + + + ω ω α ω 1 1 1 1 1 1 c h
Rigid Body Kinemtics R R O R1 ω = = O j I J R R1 α = = R R 1 1 k K = + = + sin + cos k K v j I J = f f = = sin + cos j I J = = sin + cos Afte substitution nd evlution: f f R = i + + + j + + echtonics Sping-endulum Dynmic System 8
themticl odel Fee Body Digm k+f t + F = m = m + F = m = m + + k F + mgcos= m + t mgsin= m + + f f f f + mg + + f m m + + k + Ft mgcos= + + + gsin= f Nonline Equtions of otion echtonics Sping-endulum Dynmic System 9
q q themticl odel: Lgnge s Equtions 1 = = Genelized Coodintes Q d dt F HG I T KJ T V + = Q q q q i i i Lgnge s Equtions = F Q = t Genelized Foces i 1 f T= m + + Kinetic Enegy 1 V= k mg + fcos otentil Enegy f m m + + k + Ft mgcos = + + + gsin= Nonline Equtions of otion f echtonics Sping-endulum Dynmic System 1
Lineized Equtions of otion k k + = ω = = 9.76 d/sec=1.55 Hz d d m m g g + = ω = = 4.93 d/sec=.78 Hz 1 + + s s These equtions do not pedict the motion of the system except fo pticul sets of initil conditions!! echtonics Sping-endulum Dynmic System 11
edicted Dynmic Behvio sin(u) Sum oduct 1/s Integte thet cc 1/s Integte thet vel sin 9.81 gvity (m/s^) cos(u) cos oduct oduct Sping endulum Dynmic System Gin oduct u^ sque thet thet position oduct Sum 1/s Integte cc 1/s Integte vel position t Clock time 5.71/1.815 Ft=5.71 N m=1.815 kg 95.1 k/m k=17.8 N/m m=1.815 kg echtonics Sping-endulum Dynmic System.333 sping length unstetched (metes) Sum tlb Simulink Block Digm u^(-1) invese 1
Simultion Results.3 Simultion Results with Initil Conditions: thet = -.74 d, =.46 m. dil nd ngul position (d o m).1 -.1 -. -.3 -.4 1 3 4 5 6 time (sec) echtonics Sping-endulum Dynmic System 13
Simultion Results.5 Simultion Results with Initil Conditions: thet =.1 d, =.115 m..15 dil nd ngul position (d o m).1.5 -.5 -.1 -.15 -. -.5 1 3 4 5 6 time (sec) echtonics Sping-endulum Dynmic System 14
Senso Clibtion 5 otentiomete Clibtion Cuve (Thet = 75.388 V - 196.51) 15 1 Thet (degees) 5-5 -1-15 - -5.3.4.5.6.7.8.9 volts echtonics Sping-endulum Dynmic System 15
Senso Clibtion Ultsonic Senso Clibtion Cuve (X= -8.547 V + 6.41) 16 14 1 1 x (mm) 8 6 4 1 3 4 5 6 7 8 volts echtonics Sping-endulum Dynmic System 16
Actul esued Dynmic Behvio.3 Expeimentl Results with Initil Conditions: thet = -.74 d, =.46 m. dil nd ngul position (d o m).1 -.1 -. -.3 -.4 1 3 4 5 6 time (sec) echtonics Sping-endulum Dynmic System 17
Actul esued Dynmic Behvio. Expeimentl Results with Initil Conditions: thet =.1 d, =.115 m.15 dil nd ngul position (d o m).1.5 -.5 -.1 -.15 -. 1 3 4 5 6 time (sec) echtonics Sping-endulum Dynmic System 18