Two-ass, Three-Spring Dynamic System Investigation Case Study
easurements, Calculations, anufacturer's Specifications odel Parameter Identification Which Parameters to Identify? What Tests to Perform? Physical System Physical odel athematical odel Experimental Analysis Assumptions and Engineering Judgement Physical Laws odel Inadequate: odify Equation Solution: Analytical and Numerical Solution Actual Dynamic Behavior Compare Predicted Dynamic Behavior odify or Augment ake Design Decisions odel Adequate, Performance Inadequate odel Adequate, Performance Adequate Design Complete Dynamic System Investigation
Objective The objective was to design, build, and demonstrate a dynamic system that: Would demonstrate dynamic system behavior below, at, and above a system resonance Would demonstrate the physical significance of transfer function poles and zeros Would show clearly the relationship between the time domain and frequency domain Would show the difference between collocated and non-collocated control systems 3
Physical System Picture 4
Physical System Schematic otor with Encoder Rack and Pinion Infrared Position Sensor Springs Infrared Position Sensor Connecting Bar Linear Bearings Guideway Two-ass Three-Spring Dynamic System 5
Physical System Components Translating masses () Tension springs (3) Linear bearings (3) and guide-way Rack and pinion gear system Infrared position sensors () Permanent magnet brushed DC motor with optical encoder PW servoamp and power supply 6
Springs Physical odel Assumptions three tension springs are identical linear springs neglect mass of springs neglect structural damping in springs constant spring rate springs are always in tension 7
asses Physical odel Assumptions two masses are rigid bodies one degree of freedom (translation) for each mass neglect air damping on masses Coulomb friction and viscous damping in linear bearings supporting masses and rack Coulomb friction: independent of position and velocity Viscous damping: coefficient is constant 8
Structure Physical odel Assumptions rigid and fixed independent of system motions straight track level track (gravity acts perpendicular to track) 9
Physical odel Assumptions Rack and Pinion Gear System neglect elasticity of gear teeth neglect backlash of gear teeth neglect friction of gear teeth rigid connection to motor shaft and to translating mass 0
Physical odel Assumptions otor / Amplifier PW servoamp operates in current mode neglect amplifier dynamics Coulomb friction and viscous damping in motor Coulomb friction: independent of angular position and angular velocity Viscous damping: coefficient is constant
Sensors Physical odel Assumptions Optical encoder: 000 counts per revolution with quadrature decoding Analog infrared position sensor time constant:.5 ms range: 0.9 m output: 0 to 0 V
Diagram of Physical odel J motor B motor T friction J pinion X X θ r p T m K K K B F f B F f Two-ass Three-Spring Dynamic System Physical odel 3
odel Parameter Identification Translating asses m =.675 kg (includes mass of rack) m =.75 kg Pinion Gear r p = 0.07 m J m p = 0.009 kg p Infrared Position Sensors Calibration Curves m r p p 4
odel Parameter Identification Position (mm), "+" when moves away from the motor 00 80 60 40 0 0-0 -40-60 ( ) Sensor II X(mm)= -0.86*V 3 +.667*V -.74*V+5.0637 Sensor I (near the motor) Sensor II -80 0 3 4 5 6 7 8 9 0 Sensor Volts 5
otor odel Parameter Identification J m = 0.0078 oz-in-s = 5.5080E-5 kg-m B m = 0.0 oz-in/krpm =.3487E-5 N-m-s T f m = 0.9 in-lb = 0.05 N-m K t =.8 oz-in/a = 8.336E- N-m/A Springs K = 494 N/m PW Servo-Amplifier K a = A/V 6
odel Parameter Identification Friction in System ass Viscous Damping B =.8 N-s/m x + Bx + Kx = 0 δ ln n = F H G x x n n+ I KJ δ ζ = a f π + δ B= ζω m Amplitude(m) Approximate Exponential Decay 0.065 0.06 0.055 0.05 0.045 0.04 0 0.5.5.5 3 3.5 4 4.5 5 Time(sec) 7
odel Parameter Identification Friction in System ass Viscous Damping B =.8 N-m/s x + Bx + Kx = 0 δ ζ ln n = F H G = x x δ a f π B= ζω n+ m n I KJ + δ Amplitude(m) Approximate Exponential Decay 0.045 0.04 0.035 0.03 0.05 0.0 0.05 0.0 0.005 0 0.5.5.5 3 3.5 4 4.5 5 Time(sec) 8
odel Parameter Identification Friction in System ass + Rack/Pinion + otor Coulomb Damping F f-eff =.4 N 0.04 0.03 0.0 Approximate Linear Decay x + Kx = F F f K = 4 f Amplitude(m) 0.0 0-0.0-0.0-0.03-0.04 0 0. 0.4 0.6 0.8..4.6.8 t 9
athematical odel: FBD s otor Rotor + Pinion Tm = Kti= KtKaVin JT = Jmotor + Jpinion B m θ r p J T θ o +θ T f bg sgn θ F c o = 0 bg J θ + B θ + T sgn θ = K K V Fr T m f t a in c p 0
athematical odel: FBD s ass x F static F c x F static Kx Kx x Bx F sgn x af f a f F x = 0 af x + Bx + Kx = Kx + F F sgn x c f
athematical odel: FBD s ass x F static x F static a f x Kx Kx Bx F sgn x a f f F x = 0 af x + Bx + Kx = Kx F sgn x f
athematical odel: Equations of otion bg [] J θ + B θ + T sgn θ = K K V Fr T m f t a in c p x Bx Kx Kx F F sgn x + + = + af [] c f x Bx Kx Kx F sgn x + + = af [3] f Kinematic Relations: x x x = = = r r r p p p θ θ θ 3
athematical odel: Equations of otion L N Solve for F c in equation [] using the kinematic relations: + F J r c J x B x Tf sgn θ = T m + r r r p p p bg KK t r Substitute this expression into equation []: O L P Q + N + x B B r P x + Kx T m p p O Q a f p a V Tf sgn x = Kx Ff sgn x + r p af in KK t r p a V in 4
athematical odel: Equations of otion Summary af x + B x + Kx = Kx F sgn x + eff eff f eff af x + Bx + Kx = Kx F sgn x f J eff = + r B B B eff = + r F T p m p T = F + r f eff f f p KK t r p 5 a V in
athematical odel: Linear odel KK t eff x+ B effx + Kx = Kx + rp x + B x + Kx = Kx a V in Laplace Transform: L N s + B s+ K K eff eff K s + B s+ K O QP L N X s af X s afo L Q P = N KK t rp 0 a O QP V in 6
athematical odel: Linear odel - Transfer Functions X s af = KK t a Vin K rp 0 s + Bs+ K s + B s+ K K eff eff K s + B s+ K X af s = s B s K KK t a eff + eff + rp K 0 s + B s+ K K eff eff V K s + B s+ K in 7
athematical odel: Linear odel - Transfer Functions X s V s in af af = F HG X V KK t r in p af s af s a I c KJ s + Bs + K = F HG Ds () KK t r p a I KJ Ds () K h a f a f eff eff eff a f a f 4 3 Ds ()= s + B + B s + + K+ B B s + B + B K s+ 3K eff eff eff 8
9 athematical odel: Linear odel - State Space Equations q Aq Bu y Cq Du = + = + State Variables: State Space Equations: q x q x q x q x 3 4 = = = = q q q q K B K K K B q q q q KK r V y y q q q q eff eff eff eff t a p eff in 3 4 3 4 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L N O Q P P P P = L N O Q P P P P P L N O Q P P P P + L N O Q P P P P P L N O QP = L N O Q P L N O Q P P P P + 0 L N O Q PV in
athematical odel: Transfer Functions and State Space Equations A L N X() s = V () s in 3. 503s + 4. 5887s+ 58. 7 4 3 s +. 3449s + 65. 7s + 44. s+ 84460 X () s 59. 3 = 4 3 V () s s +. 3449s + 65. 7s + 44. s+ 84460 in 0 0 0 489. 45 0. 9333 44. 7 0 = 0 0 0 387. 45 0 774. 90 48. 0 C 0 0 0 = L N O Q P 0 0 0 D = 0 0 O B = QP L N O Q P L N 0 3.503 0 O QP 30
athematical odel: Transfer Functions and State Space Equations Poles: 0. 536 ± 7. i ω = 7. rad / s ζ = 0.033 0.637 ± 3.i ω= 3. rad / s ζ= 0.004 Zeros: af af ± ω = af s af s None X s V s X V in in 0. 706 7. 8i 7. 8 rad / s ζ = 0.054 3
Frequency Response Plots: Analytical X V in -0-30 -40 Bode Diagrams Phase (deg); agnitude (db) -50-60 -70 0-50 -00-50 0 0 0 0 Frequency (rad/sec) 3
Frequency Response Plots: Analytical X V in -0-40 Bode Diagrams Phase (deg); agnitude (db) -60-80 0-00 -00-300 0 0 0 0 Frequency (rad/sec) 33
Frequency Response Plots: Experimental X V in Encoder Network Analyzer agnitude (db) -0-30 -40-50 -60-70 0 0 0 0 0 Frequency (rad/s) Phase (degrees) -50-00 -50-00 0 0 0 0 Frequency (rad/s) 34
Frequency Response Plots: Experimental X V in agnitude (db) -0-30 -40-50 Bode Plot for ass (Optical Encoder) Encoder Sine Sweep -60 0-0 0 0 0 Frequency (Hz) Phase (degrees) -50-00 -50-00 0-0 0 0 Frequency (Hz) 35
Frequency Response Plots: Experimental X V in agnitude (db) -0-30 -40-50 -60 Bode Plot for ass (IR Sensor) IR Sensor Sine Sweep -70 0-0 0 0 Frequency (Hz) 0 Phase (degrees) -50-00 -50-00 -50 0-0 0 0 Frequency (Hz) 36
Frequency Response Plots: Experimental X V in IR Sensor Network Analyzer agnitude (db) -5-30 -35-40 -45-50 -55 0-0 0 0 0 0 Frequency (rad/s) Phase (degrees) -00-00 -300-400 0-0 0 0 Frequency (rad/s) 37
Frequency Response Plots: Experimental X V in agnitude (db) -0-30 -40-50 -60-70 Bode Plot for ass (IR Sensor) IR Sensor Sine Sweep -80 0-0 0 0 Frequency (Hz) 0 Phase (degrees) -00-00 -300-400 -500 0-0 0 0 Frequency (Hz) 38
Collocated and Non-Collocated Control Systems Collocated Control System All energy storage elements that exist in the system exist outside of the control loop. For purely mechanical systems, separation between sensor and actuator is at most a rigid link. Non-Collocated Control System At least one storage element exists inside the control loop For purely mechanical systems, separating link between sensor and actuator is flexible. 39
Poles and Zeros of Transfer Functions: Physical Interpretation Complex Poles of a collocated control system and those of a non-collocated control system are identical. Complex Poles represent the resonant frequencies associated with the energy storage characteristics of the entire system. Complex Poles, which are the natural frequencies of the system, are independent of the locations of sensors and actuators. 40
Poles and Zeros of Transfer Functions: Physical Interpretation At a frequency of a complex pole, even if the system input is zero, there can be a nonzero output. Complex Poles represent the frequencies at which energy can freely transfer back and forth between the various internal energy storage elements of the system such that even in the absence of any external input, there can be nonzero output. Complex Poles correspond to the frequencies where the system behaves as an energy reservoir. 4
Poles and Zeros of Transfer Functions: Physical Interpretation Complex Zeros of the two control systems are quite different and they represent the resonant frequencies associated with the energy storage characteristics of a sub-portion of the system defined by artificial constraints imposed by the sensors and actuators. Complex Zeros correspond to the frequencies where the system behaves as an energy sink. 4
Poles and Zeros of Transfer Functions: Physical Interpretation Complex Zeros represent frequencies at which energy being applied by the input is completely trapped in the energy storage elements of a sub-portion of the original system such that no output can ever be detected at the point of measurement. Complex Zeros are the resonant frequencies of a subsystem constrained by the sensors and actuators. 43