MOHAMMAD AKTERUZZAMAN

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MODELING and SIMULATION OF HYDRAULIC POWER
STEERING SYSTEM WITH MATLAB

• MOHAMMAD AKTERUZZAMAN
• Advisor DR. SHUVRA DAS

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MODELING and SIMULATION OF HYDRAULIC POWER
STEERING SYSTEM

• MODELING OF the MECHANICAL and HYDRAULIC
  COMPONENTS of a POWER STEERING SYSTEM.
• SIMULATION OF THE MODEL BY MATLAB.
• Model REPRESENTS THE DYNAMIC RESPONSES OF THE
  power Steering System AND is CAPABLE OF
  ESTIMATING the effect of parameters on system
  response.
• Model is used to study the effect of various
  system parameters on system response.

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PRIOR WORK

• Discussion of Reference Model
• Ali Keyhani He presents the identification of
  the dynamic model for a power steering system
  constructed using a rotary valve based on
  Mathematical (ODE).
• Jose J. Granda Analyze a multi energy non
  linear system using a bond graph model.
• Joel E. Birching He describes a method of
  applying the orifice equation to a steering valve
  along with the procedure for experimentally
  determining the flow Co-efficient for this
  equation.

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Prior Work

• AMESim (Object oriented software) This case
  study gives us a good understanding of how AMESim
  can be used to construct parameterize and analyze
  complex hydro-mechanical dynamic model like power
  steering system.
• N.Riva, E.Suraci (ADAMS based work) A methology
  has developed to simulate the vehicle dynamics
  through Adams Car and Matlab co-simulation.

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Prior Work

• We took Ali Keyhanis dynamic power steering
  model consisting of ordinary differential
  equations for Mechanical and Hydraulic system.
• Some of the design Parameters are difficult to
  obtain. Ali Keyhani used experimental data
  least square approach to determine these
  parameters.

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ACTUAL POWER STEERING SYSTEM
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SCHEMATIC DIAGRAM OF HYDRO-POWERSTEERING SYSTEM
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ALI KEYHANI MODEL Mechanical subsystem
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ALI KEYHANI MODEL Mechanical subsystem

• The equations for the steering column, pinion and
  rack can be written
• Equation 1
• Equation 2

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ALI KEYHANI MODEL Mechanical subsystem

• Where TdTorque generated by the driver,
• Theta1rotational displacement for the steering
  column,
• K2tire stiffness
• B2Viscous damping coefficient
• B1friction constant of the upper-steering column
• Xdisplacement of the rack
• m mass of pinion
• Ap Piston area
• K1torsion bar stiffness
• J1Inertia constant of the upper steering column

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ALI KEYHANI MODEL Mechanical subsystem

• The following assumptions were made
• -the pressure forces on the spool are neglected.
• -the stiffness of the steering column is
  infinite.
• -the inertia of the lower steering column (valve
  spool and pinion) is lumped into the rack mass.

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ALI KEYHANI MODEL hydraulic subsystem
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ALI KEYHANI MODEL hydraulic subsystem

• By applying the orifice equations to the rotary
  valve metering orifices and mass conservation
  equations to the entire hydraulic subsystem the
  following equation are obtained
• Equation 1
• Equation 2
• Equation 3

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ALI KEYHANI MODEL hydraulic subsystem

• Where Ps and Po supply and return pressure of
  the pump.
• Pl and Pr cylinder pressure on the left and
  right side.
• Q supply flow rate of the pump
• A1 and A2 are the metering orifice area
• Rho density of the fluid
• Betabulk modulus of fluid
• Llength of the cylinder
• Cd discharge co-efficient

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ALI KEYHANI MODEL hydraulic subsystem

• The following assumption were made
• -there is no pressure drop on the fluid
  transmission lines between the pump and the valve
  and the cylinder.
• -the wave dynamics on the fluid transmission
  lines are neglected
• -the bulk modulus of the fluid is considered
  constant
• -the inertance of the fluid is neglected
• -there is no leakage at the piston-cylinder
  interface
• -the return pressure dynamics are negligible

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ALI KEYHANIS PARAMETER TABLE from experimental
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ALI KEYHANIS PARAMETER TABLE from experimental
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Information lacking in ALI KEYHANIs Work

• -Missing relationship for variation of A(theta),
  Torque and Flow rate Q.
• -His established parameters do not say from which
  type of vehicle they were obtained.

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How We got the value of A(theta)
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Value of Q

• Q1.5 GPM (gallon per minute) for reasonable
  minimum with the quicker steering ratios for
  pavement cars.
• Q2.5 GPM for dirt .
• (reference power steering Tech,
  www.woodwardsteering.com)
• ? Q.0002 m3/s
• (reference H.Chai. Electromechanical Motion
  Devices, Upper addle River, NJPrentice Hall
  PTR,1998)

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Value of Torque

• Td0-8 N-m is not enough to excite the lower
  steering column modes.
• (reference Ali Keyhani)
• Td0-2 N-m is required at the handwheel during
  normal driving ranges.
• Td15 N-m in extreme cases.
• (reference H.Chai. Electromechanical Motion
  Devices, Upper addle River, NJPrentice Hall
  PTR,1998)

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Model

• Using the equations and input data a MATLAB based
  program was written
• Model parameters were adjusted to obtain the
  results reported by Ali Keyhani

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ResultsComparison from Ali-keyani model
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ResultsComparison from Ali-keyani model
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ResultsComparism from ali-keyani model
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Resultsfrom different study (Tom Wong)
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Results from different study

• Include the other results that I had suggested.
  (driver torque Vs. assist torques, also in
  previous pages copy fig from reference)

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Response graph Effect on theta(radians) Torque
Td2, 9, 15 N-m
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Response graph Effect on cylinder pressure
Torque Td2, 9, 15 N-m
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Response graph Effect on rack Assist pressure
vs. rotation angle (theta)Torque Td2, 9, 15 N-m
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Response graph Effect on pump pressureTorque
Td2, 9, 15 N-m
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Response graph Effect on assist pressure w.r.t
rotation on degreePump flow rate,
Q0.00014,0.00016,0.00024 m3/S.
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Response graph Effect on Pump pressurePump
flow rate, Q0.00014,0.00016,0.00024 m3/S.
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Response graph Effect on displacement (X)Pump
flow rate, Q0.00014,0.00016,0.00024 m3/S.
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Response graph Effect on cylinder pressurePump
flow rate, Q0.00014,0.00016,0.00024 m3/S.
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Response graph Effect on thetaPump flow rate,
Q0.00014,0.00016,0.00024 m3/S.
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Response graph Effect on Assist pressure on
rotation angle (degree)J1.0000322, .0000598
N-m-s2/rad
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Response graph Effect on pump pressure
J1.0000322, .0000598 N-m-s2/rad
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Response graph Effect on rack displacement (X)
in meterJ1.0000322, .0000598 N-m-s2/rad
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Response graph Effect on right cylinder
pressure( N/m2)J1.0000322, .0000598 N-m-s2/rad
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Response graph Effect on Rotation(
radians)J1.0000322, .0000598 N-m-s2/rad
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Response graph Effect assist pressure( N/m2) on
rotation angle ( degree)When m4.76, 8.84 Kg
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Response graph Effect pump pressure( N/m2)
When m4.76, 8.84 Kg
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Response graph Effect displacement(X) When
m4.76, 8.84 Kg
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Response graph Effect on cylinder pressure
When m4.76, 8.84 Kg
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Response graph Effect on assist pressure with
rotation when K127.651,31.33N-m/rad
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Response graph Effect on pump pressure When
K127.651,31.33N-m/rad
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Response graph Effect on rack displacement
When K127.651,31.33N-m/rad
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Response graph Effect cylinder pressure When
K127.651,31.33N-m/rad
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Response graph Effect on theta When
K127.651,31.33N-m/rad
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Summery and conclusion

• A model has been developed for the Hydraulic
  Power steering system
• Several realistic assumptions were used in model
  development.
• The model uses driver torque and pump flow rate
  as inputs.
• The rotation of the torsion bar, the displacement
  of the rack, and the pressures in the cylinder
  are outputs from model.

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Summery and conclusion

• The model was used to develop response curves
  similar to published work
• The model was used to simulate the effect of
  driver torque on the system response. As the
  torque increases.....
• The model was also used to simulate the effect of
  pump flow rate on the system response. As the
  flow rate increases....

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Our Model description

• We got five equation from Ali-keyhani model of
  hydraulic power power steering system
• Equation 1
• Which is second order equation.
• For Matlab programing we can break the second
  order differential equation by two first order
  equation which is below
• Let, y(1)theta, y(2)theta , so we can write
• Y(1)y(2)
• y(2)1/J1(Td-B1 y(2)K1(y(1)-y(3)/r))
• Where y(3) X ( comes from equation 3)

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Our Model description

• Similarly for equation 2 also second order
  equation can write by two first order
  differential equation,
• If y(3)X, y(4)X
• So, y(3)y(4)
• Y(4)1/m(K1/r(y(1)-y(3)/r)P.Ap-B2.y(4)K2.y(3))

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Our Model description

• For hydraulic equation 3,4 and 5,
• If we let y(5)Ps
• y(6)Pr
• y(7)Pl
• Then
• y(5)Beta/Vs(Q-A1CdSqrt 2(y(5)-y(6))/d-A2CdS
  qrt 2(y(5)-y(7))/d)

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Our Model description

• y(6)Beta/(Ap(L/2-y(3)))(A1 Cd Sqrt
  2(y(5)-y(6))/d-A2CdSqrt 2(y(5)-Po)/d)Apy(4))
  
• y(7)Beta/(Ap(L/2y(3)))(A2CdSqrt
  2(y(5)-y(7))/d-A2CdSqrt 2(y(7)-Po)/d)Apy(4))
  

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Our Model description

• So, our equation is seven and variable is seven,
  see below
• Variables are
• y(1)theta, y(2)theta,y(3)X,y(4)X , y(5)Ps
  ,y(6)Pr,y(7)Pl
• Equations are
• Y(1)y(2)
• y(2)1/J1(Td-B1 y(2)K1(y(1)-y(3)/r))

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Model description

• 3. y(3)y(4)
• 4. Y(4)1/m(K1/r(y(1)-y(3)/r)PAp-B2y(4)K2y(3
  ))
• 5. y(5)Beta/Vs(Q-A1 Cd Sqrt 2(y(5)-y(6))/d-A2C
  dSqrt 2(y(5)-y(7))/d)
• 6. y(6)Beta/(Ap(L/2-y(3)))(A1CdSqrt
  2(y(5)-y(6))/d-A2 Cd Sqrt 2(y(5)-Po)/d)Apy(4))
  
• 7. y(7)Beta/(Ap(L/2y(3)))(A2 Cd Sqrt
  2(y(5)-y(7))/d-A2 Cd Sqrt 2(y(7)-Po)/d)Apy(4))
  

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Showing program and Simulation in Matlab

• function Fodefile(t,y)
• thetay(1),theta'y(2),Xy(3),X'y(4),Psy(5),Pr
  y(6),Ply(7)
• Ap12.60.0001
• Td15
• B1.10831200
• B22679
• J1.000046
• m7.5
• K11.843420
• K216072117
• r.008
• Q0.0002
• Q.00027
• d825
• Vs8.190.000001
• beta5515100000
• L.15
• Po0.0
• Cd0.6

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Showing program and Simulation in Matab

• if y(1) lt -63.14159/180
• r13 0.0
• r23 200.000001
• end
• if y(1) gt 63.14159/180
• r13 200.000001
• r23 0.0
• end
• if -63.14159/180 lt y(1) lt 0.0
• r13 0.0001909y(1)200.000001
• r23 200.000001
• end
• if 63.14159/180 gt y(1) gt 0.0
• r13 200.000001
• r23 -0.0001909y(1)200.000001
• end
• r13
• r23

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Showing program and Simulation in Matab

• clear
• y00000000
• tspanlinspace(0,2,5000)
• optionsodeset('reltol',1e-6,'abstol',1e-8,'output
  fcn','odeplot')
• t yode23tb('odefile13',tspan,y0,options)
• figure(1)
• plot(t,y(,1),'r')
• axis(0 2 -0.2 1.2)
• xlabel('time')
• ylabel('theta in radians')
• title('theta Vs time')
• hold on
• figure(2)
• plot(t,y(,6),'b')
• axis(0 2 -10 8000000)
• xlabel('time')
• ylabel('right cylinder pressure in N/m2')
• title(' right cylinder pressure Vs time')
• hold on

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Showing program and Simulation in Matab

• figure(4)
• plot(t,y(,5),'b')
• axis(0 2 -10 8000000)
• xlabel('time')
• ylabel('pump pressure in N/m2')
• title(' pump pressure Vs time')
• hold on
• figure(5)
• plot(((180/3.14159)y(,1)),y(,6),'g')
• axis(0 2 0 8000000)
• xlabel('valve rotation')
• ylabel('assit pressure')
• title('assit pressure Vs rotation')
• hold off

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References

• Dr.Christan Ebner, Steer-by-wire BMW technik,
  May-00, seite-1
• Paul Yih, Toward Steer by wire Dynamic Design
  lab, November 30, 2001
• Tom Wong, Hydraulic power system design and
  optimization simulation SAE technical paper
  series, 2001-01-0479

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References

1. Ali Keyhani Identification of Power steering
   system Dynamic Models ,Mchatronics Journal,
   February 1998
2. Granda J.J. Computer Aided Simulation of a
   Hydraulic Power Steering System with Mechanical
   Feedback
3. Joel E. Birching Two Dimensional Modeling of a
   Rotary Power steering valve International
   Congress and Exposition, Detroit, March 1-4, 1999
4. AMESim Power steering system studied Technical
   Bulletin n 107

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References

• Sanket Amberkar, Mark Kushion, Diagnostic
  Development for an Wlectric Power steering
  system, SAE 2000 World Congress, Detroit ,
  Michigan,March 6-9,2000.
• Paper No. 993079, An ASAE Meeting
  Presentation,Adaptivecontrol of Electric
  Steering system fro wheel-type Agricultural
  Tractors by D.Wu, Q.Zhang.

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references
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references
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Response graph Effect on rack displacement (X)
mTorque Td2, 9, 15 N-m