Michigan State University College of Engineering East Lansing, MI

Transcription

1 Michigan State niversity College of Engineering East Lansing, MI MEMO NMBER: PPBS0001 DATE: 2Sept97 TO: FROM: Distribution E. LaBudde SBJECT: PPBS Preliminary Math Model ABSTRACT: This memo contains a first cut at a PPBS model. The model is expressed in the Graphical Description Language TM format. The Graphical files are generated by Visio V 4.1. Most of the model coefficients are gain terms derived from standard engineering models for DC motors, fluid flow and kinematics of the ball. DISTRIBTION: D. Rover, T. Bickart, PPBS modeling team REVIEWED AND APPROVED BY: RELATED DOCMENTS: PPBS.VSD ANALYSIS MODELS: None KEY WORDS: PPBS, Fan, Fluid, Ball Edward V. LaBudde TM LaBudde Systems, Inc. All rights resevered PAGE 1 of 9

2 Objective The purpose of this memo is to identify a first cut model of the PPBS for math modeling of the system. Mission Objectives The objective of the math model for the PPBS is to provide an executable model for CPS479 students that can be run in the LabView environment for purposes of proving their design will work before the actual hardware is designed. The basic requirements for the model will be: 1. Accuracy The velocity of the ball at the center of the tube shall be less than +/ 10% of full scale from the true response from any user input. 2. Time Response The time response shall be within +/ 20% of the true response over the center 50% of the full operating range of any user input. 3. Noise The random time variation in ball position shall be accurate to +/ 50% at the center of the tube. The ser shall be able to disable the noise model by external command. 4. Interfaces The model shall have the exact characteristics of the hardware interfaces. Cumulative response time less than 10 milliseconds can be neglected. 5. Environment The environment shall assumed to be 70 F, and Standard Pressure for MS Altitude. Only relevant environments need be modeled to meet the accuracy requirements. 6. Executable Model The run time model shall require no more than 10 milliseconds of execution time when run in the LabView environment. All model parameters shall be data inputs to the model so that variations between units can be accommodated. 7. Validation Environment A test environment shall be constructed with the ability to simultaneously command the run time model and the real hardware for validation comparisons. The test should allow pulse changes in fan voltage and valve angle and return initial values after a specified time period. The final requirements will be confirmed after a modeling team review. Background The need for a math model of the PPBS arises from the Top Down design method being used in this term in CPS479. It requires that students must prove their design approach works before they build the hardware and software. The best way to achieve this is to model the function and behavior of the overall system. One way to achieve this is to do a simulation of the system. The primary thrust of the model (no pun intended) is to simulate the response of the ball position from user inputs. Glossary of Terms sed Symbol Definition nits Value Ab Area of ball m 2 Ap Area of pipe m 2 PAGE 2 of 9

3 α Acceleration of fan motor rad/sec 2 Cd Ball drag coefficient Db Diameter of Ball m Dp Diameter of Pipe m F Force on the ball due to air flow N Fe Force input to ball mass N Fg Force of gravity on ball N Fn Force due to noise N Fs Force from ball at end stops N Fv Force of ball velocity on ball N g Acceleration of gravity m/sec 3 Im Fan motor current A Jm Fan moment of inertia kgm 2 Kqτ Fluid flow fan torque model Nmsec/m 3 Ksi Ball spring constant inner limit N/m Kso Ball spring constant outer limit N/m Kt Fan motor torque constant Nm/A Kuf Fluid velocity ball force model Nsec/m Kuu Mean Fluid velocity ball velocity model 1 Kv Fan motor back EMF constant Vsec/rad Kw Ballpipe wall correction constant Kωp Fan Speed Fluid pressure Model PAsec/rad Kxr Ball displacement fluid resistance model PAsec/m 2 M Mass of ball kg µ Viscosity of air at ambient temperature kg/msec Po Ambient Atmospheric Pressure (at 228 meters) PA ρo Ambient Atmospheric Density (at 228 meters) kg/m ρ Air Density at the ball kg/m3 P Pressure at the fan output PA Q Volume flow rate in the tube m3/sec Θ Angle of pipe from earth normal Radian Re Reynolds number Rm Fan motor electrical resistance Ohm Rt Total system resistance to flow PAsec/m Rs Resistance to flow of the piping system PAsec/m Rv Resistance to flow at the valve, PA PAsec/m Rx Resistance to flow at the ball, PA PAsec/m Τ Ambient air temperature K 294 Τd motor initialization time delay Sec Τs motor step time delay to 90% of angle Sec τe Torque input to fan inertia Nm τf Friction torque from motor Nm PAGE 3 of 9

4 τm Torque from fan motor Nm τq Torque from air flow Nm Mean velocity of the fluid at the ball m/sec i Instantaneous Mean velocity of the fluid at the ball m/sec Vcc sensor supply voltage V Ve Voltage into fan resistance V Vin ser voltage input to fan V Vf Back EMF from motor V ω Angular speed of fan radian/sec ωf Corner frequency build up of flow in the pipes radian/sec on ball relative to a reference surface m Velocity of ball relative m/s Acceleration of ball m/s This glossary will be put into an excel spreadsheet and updated as required. Level 1 PPBS System SER Ping Pong Ball System Level 1 (Black Box) SER Fan Voltage Command PPB System (double click me) PPB Fan Current Ambient Air Earth g,θ Air Output Figure 1 PPBS Level 1 Block Diagram PAGE 4 of 9

5 MODE OFF MODE 1 MODE 2 MODE 3 MODE 4 MODE 5 MODE 6 MODE 7 MODE? MODE N MODE? MODE MATRI V CMD t>td POSITION 1 FAN OFF 1 ON 2 ON 3 ON 4 ON 5 ON 6 ON 7 ON? ON N ON??? STATE DIAGRAM PPBS Level 1 MODES Black Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) Off Mode 1 Mode 2 Mode 3 Mode 4 Power Off Note: The number of modes is deterimed by the steps needed to repeat valve control effect Mode N Mode? Mode 6 Mode 5 Figure 2 PPBS Level 1 Mode Matrix and State Diagram TIMELINE LEVEL 1 (BLACK BO) 90% Angle INITIAL MODE NET MODE TD WAIT Ts POWER ON OR ADVANCE RETARD Figure 3 PPBS Level 1 Timeline PAGE 5 of 9

6 SER PPBS Level 2 Gray Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) SER Fan Voltage Fan Current Fan Model Ball Model PPB ω τ F Command FluidFan Interface FluidBall Interface g,θ P Rx Earth Q Ambient Air Fluid Model Air Output Figure 4 PPBS Level 2 Block Diagram SER FAN Level 3 White Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) FLIDBALL Im τq Torque Vin + Ve 1/Rm Im Kt τm τe 1/Jm α 1/S + ω Speed τf FRICTION Vf Kv I 2 Rm Figure 5 Fan Level 3 Block Diagram PAGE 6 of 9

7 FAN FLIDFAN Level 3 Gray Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) FLID ω Kωp (Speed Pressure Model) P τq Kqτ (Flow Torque Model) Q Figure 6 Fluid Fan Interface Level 3 Block Diagram Ambient Air FLIDFAN FLID Level 3 GrayWhite Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) BALL Mean Velocity FLIDFAN P 1/Rt i 1/(S/ωf+1) Ap Q Volume Flow SER α Rt Q Command 1 of? Rv Rs Shape Model Geometry Air Output FLIDBALL Rx Figure 7 Fluid level 3 Block Diagram PAGE 7 of 9

8 BALL Level 3 GrayWhite Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) SER g Θ MgCosΘ M Sb1 Inner Mid Ksi FLIDBALL F FLID Off Sb2 Fn On Outer Fg Fs Kso Fe 1/M '' 1/S ' 1/S Vcc 1 0f 16 PPB Noise Fv Cd 1/2*ρ Ab/Kw (') 2 Figure 8 Ball Level 3 Block Diagram BALL FLIDBALL Level 3 Gray Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) FLID Kxr (Displacement Resistance Model) Rx Kuu (Velocity Velocity Model)?? F Cd 1/2*ρ Ab/Kw () 2 Kuf (Velocity Force Model) Figure 9 Fluid Ball Interface Level 3 Block Diagram PAGE 8 of 9

9 BALL Level 4 MODES White Box (DOBLE CLICK ME TO RETRN TO HIGHER LEVEL) MODE Sb1 Sb2 <i <o In Crash Out Crash Inner Outer In Crash Ball Mid Out Crash Ball Mid Noise On Mid On =>i =>o Noise Off Off N_CMD On Noise On Noise off N_CMD Off Figure 10 Ball Level 4 Mode Matrix And State Diagram The Figures above are linked to a Visio file PPBS.VSD. Pages can be accesses by double clicking the red objects in the drawing. The model will be developed using the MKS system of units. Raw test data can use any convent units. Future memos will document verification test plans and model validation and run time model construction. PAGE 9 of 9