Spring-Pendulum Dynamic System

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Allan Marsh
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1 Sping-endulum Dynmic System echtonics Sping-endulum Dynmic System 1

2 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

3 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

4 hysicl odel k l + m = pendulum mss = 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

5 Sping Clibtion F sping (N) mg = 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

6 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

7 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 c h

8 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

9 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

10 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

11 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 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

12 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

13 Simultion Results.3 Simultion Results with Initil Conditions: thet = -.74 d, =.46 m. dil nd ngul position (d o m) time (sec) echtonics Sping-endulum Dynmic System 13

14 Simultion Results.5 Simultion Results with Initil Conditions: thet =.1 d, =.115 m..15 dil nd ngul position (d o m) time (sec) echtonics Sping-endulum Dynmic System 14

15 Senso Clibtion 5 otentiomete Clibtion Cuve (Thet = V ) 15 1 Thet (degees) volts echtonics Sping-endulum Dynmic System 15

16 Senso Clibtion Ultsonic Senso Clibtion Cuve (X= V ) x (mm) volts echtonics Sping-endulum Dynmic System 16

17 Actul esued Dynmic Behvio.3 Expeimentl Results with Initil Conditions: thet = -.74 d, =.46 m. dil nd ngul position (d o m) time (sec) echtonics Sping-endulum Dynmic System 17

18 Actul esued Dynmic Behvio. Expeimentl Results with Initil Conditions: thet =.1 d, =.115 m.15 dil nd ngul position (d o m) time (sec) echtonics Sping-endulum Dynmic System 18

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque