Basic Concept of Axiomatic Design

					                    One-FR Design




                          Prof. W. Hwang
                 Dept. of Mechanical Engineering
                             Postech


POSTECH ME PCM                                     Chapter 04 FR Design   1
 1. One-FR Design?


    Only one functional requirement is to be satisfied by a
     proposed design without further decomposition.

    When we consider one-FR design, the only relevant axiom is
     Axiom 2 (Information Axiom) because Axiom 1 is automatically
     satisfied.




POSTECH ME PCM                                         Chapter 04 FR Design   2
 2. Design conditions


    The design must be robust so that we can allow the largest
     possible tolerances for the DPs and/or PVs when the FRs and
     their design range are satisfied.


    In addition to Axiom 1 and Axiom 2, the design must also satisfy
     constraints such as cost and geometric bounds imposed by
     external factors.




POSTECH ME PCM                                           Chapter 04 FR Design   3
 3. The key issue of one-FR



    In one-FR design, Independence Axiom is always satisfied.
    Therefore, we only need to concentrate on the minimizing of
    the information content.


                          The key issue is


  “ How do you choose an appropriate DP and appropriate
    PV, so that we can have a robust design?”




POSTECH ME PCM                                        Chapter 04 FR Design   4
 4. One-FR decomposition


 Can a one-FR design become a multi-FR design?


 Sometimes, a one-FR design task becomes a multi-FR design when
 a one-FR design is decomposed to be able to implement the design
 task.




POSTECH ME PCM                                       Chapter 04 FR Design   5
 5.1 Information Content


 Four ways of making the system range lie within the design range:


    1. Reduce the stiffness of the system
    2. Make the system totally immune to random variation of DPs or
       PVs
    3. If the design is redundant, make the extra DP or PV fixed
    4. Make the design range larger.




POSTECH ME PCM                                          Chapter 04 FR Design   6
 5.2 Information Content


 When there are many FRs, the information content can be used as
 the decision-making tool.


 Information content may be reduced when the design is either
 uncoupled or decoupled, i.e., when the design satisfies the
 Independence Axiom.




POSTECH ME PCM                                          Chapter 04 FR Design   7
 5.3 Information Content


 The best values of DPs can be obtained by finding where the value
 of the information reaches its minimum when the following two
 conditions are satisfied:

                             n
                                  I
                              DP  0
                             j 1    j

                             n
                                  2I
                              DP 2  0
                             j 1     j




POSTECH ME PCM                                         Chapter 04 FR Design   8
 6. Design Parameters


    How do we select the best DP in a one-FR design?
     The DP with the minimum information content is the best one.
     That is, the one with largest common range provides the best
     design.

                   FR1  $ f ( DP1 , DP1 , DP1 ,, DP1 )
                                      a      b       c         n


                   Select DP1 over DP1 if ( Acr ) a  ( Acr )b
                                 a               b



     $ signifies the fact that any one of the DP1s in parentheses can satisfy FR1
     Acr is common range




POSTECH ME PCM                                                         Chapter 04 FR Design   9
 7. Optimum in Axiomatic




    In Axiomatic Design, “optimum design” refers to the design that
       satisfied the FR and constraints with zero information content.




POSTECH ME PCM                                                Chapter 04 FR Design 10
  8.1 One-FR design with no constraints(1)


 When there are random variations and physical limitations in
 implementing the design intent, the system probability density
 function(pdf) of the FR can vary over a wide range for a given
 value of the DP.

 Consider a design with one FR and one DP, for which the design
 equation may be written as

                         FR1=A11DP1 (*)

 or in differential form as
                                     FR1
                              dFR1       dDP
                                     DP
                                             1
                                        1


POSTECH ME PCM                                          Chapter 04 FR Design 11
  8.1 One-FR design with no constraints(2)


 The relationship between A11 and (FR1/ DP1) is obtained by
 differentiating equation(*) with respect to DP1 as



                      FR1        A
                            A11  11 DP1
                      DP1        DP1


 Then, the equation (*) may be written as


                               A11            FR1
              dFR1  ( A11          DP )dDP       dDP
                               DP             DP
                                       1    1          1
                                   1              1




POSTECH ME PCM                                         Chapter 04 FR Design 12
  8.1 One-FR design with no constraints(3)


 In the case of a linear design, A11 is constant, i.e.,


                                   FR1
                           A11          constant
                                   DP1



 In the case of non-linear design, A11 will vary as a function of DP1.


                            A11  f ( DP )
                                        1




POSTECH ME PCM                                            Chapter 04 FR Design 13
  8.1 One-FR design with no constraints(4)


 For a one-FR design, it is easy to remove the bias by changing
 the value of DP1.

                                            To reduce the variance, we must
                                            reduce the random variation of
                                            DP1 and the magnitudes of the
                                            higher-order derivatives of FR1
                                            with respect to DP1.



    Fig.4.1 Design range and system range




POSTECH ME PCM                                                 Chapter 04 FR Design 14
  8.1 One-FR design with no constraints(5)


 The variance     2 is the square of the standard deviation.
 The estimated variance    s 2 may be expressed as
                              1 N
                     s 
                       2
                                  
                          N  1 i 1
                                     ( FR1i  FR1avg ) 2

          where FR1i  the i th value of N measurements of FR1
                 FR1avg  the average value of N measurements

 When there are many independent contributions to the variance
 of a system, the total variance of the system is given by

                               stot   si
                                  2          2




POSTECH ME PCM                                             Chapter 04 FR Design 15
 Example 1.1 Measurement of Air Velocity


 The instrument must be able to measure the air velocity within 1% of
 the absolute air velocity.
 Specify the random variations you can allow in the key design
 parameter

                                    The design equation:

                                           FR1 = A11 DP1
                                                or
                                               FR1
                                        dFR1       dDP
                                               DP
                                                       1
                                                  1
                                     FR1= Measure the velocity of air (V)
                                                                             ~
                                     DP1= The relative stagnation pressure ( Ps )
      Fig. 4.2 A Pitot Tube

POSTECH ME PCM                                              Chapter 04 FR Design 16
 Example 1.2 Measurement of Air Velocity


 The relative stagnation pressure
                ~
    ( Ps  P)  Ps    where P is the ambient pressure and
                              Ps is the absolute stagnation pressure

                     From Bernoulli’s Equation:

                       V 2 p ps
                           
                        2  
                              2( ps  p)       2 ~s
                                                 p
                       V                            (*)
                                               




POSTECH ME PCM                                              Chapter 04 FR Design 17
 Example 1.3 Measurement of Air Velocity


 From Bernoulli’s equation, we can find the value of A11 that
                                                         ~
 relates the FR(V) to the relative stagnation pressure ( Ps ) .

                                2 ~
                  FR1  V       ~  ps  f ( ps ) ~s  A11DP1
                                                  p
                                ps
                                                    2
                         where A11  f ( ps ) 
                                                    ~s
                                                     p

 This is a nonlinear design.




POSTECH ME PCM                                                  Chapter 04 FR Design 18
 Example 1.4 Measurement of Air Velocity


 By differentiating equation (*), we can get the differential

                                   V       1
                                       
                                   ~s
                                    p    2  ~s
                                              p

 By using random variation of equation (*), we can get the solution.


             2 ~            2     ~             V   1 ~s
                                                          p
     V         p s   
                           ~
                                    ps
                                                      ~  0.01
             ~s
              p              ps                 V    2 ps
              1                                   ~s
                                                   p
               ~ p
            2  ps
                   ~s                             ~  0.02
                                                   p
                                                   s




POSTECH ME PCM                                              Chapter 04 FR Design 19
 Example 1.5 Measurement of Air Velocity


    The allowable error in pressure measurement depends on the
     absolute magnitude of the stagnation pressure of the air.

    The higher the pressure, the larger is the allowable random
     variation of the pressure measurement.

    If there is a bias, it may be due to the fact that the Pitot tube was
     not located parallel to the direction of the flow, which can be
     corrected to eliminate the bias.

    To reduce the variance, the source of the variance must be
     determined.



POSTECH ME PCM                                              Chapter 04 FR Design 20
 8.1.1 Lower stiffness(1)


 In axiomatic design, robust design is defined as a design
 that always satisfies the functional requirements, i.e.,

  FR  sr (FR)      and the bias b=0, even when there is large
 random variation in the design parameters DPj.


 The specified tolerance DP is determined by the magnitude of A11
 and the magnitude of the design range of FR, that is,

                            DP  FR / A11



POSTECH ME PCM                                              Chapter 04 FR Design 21
 8.1.1 Lower stiffness(2)


 The idea is to make DP as large as possible so that the effect of
 the random variation of DP on FR is always much smaller than the
 specified design range FR.


 Making the stiffness, A11, small can minimize the variation of the FR
 caused by random variation of the DP.


 The stiffness of the system should be reduced to enhance the
 design robustness even when there is random variation.




POSTECH ME PCM                                          Chapter 04 FR Design 22
 8.1.1 Lower stiffness(3)


  Can we make the stiffness infinitesimally small?
   No. The stiffness cannot be reduced indefinitely, since the signal (i.e., FR1) must be
       much larger than the noise (i.e., FR1) to make the signal-to-noise(S/N)ratio
       larger than the minimum S/N ratio.


                                                          2
                                              Signal 
                                 10 log 10         
                                              Noise 

 The actual S/N ratio is greater than the minimum S/N ratio.
                                          2                               2
                                FR1                      FR1 
               sys  10 log10         min  10 log10       
                                FR1                     FR1 

POSTECH ME PCM                                                          Chapter 04 FR Design 23
 8.1.2 Stiffness and Response Rate


 In some, cases, we may need rapid response. However, a robust
 design with low stiffness may be too slow to respond in time.


            dFR1       dDP            dFR1  dFR1 
                  A11     1
                                                
             dt         dt             dt    dt  c
                    dFR1 
             where        is the critical response rate
                    dt  c
 To have a rapid response rate, either dDP1/dt or A11 must be large.




POSTECH ME PCM                                          Chapter 04 FR Design 24
 Example 2.1 Measuring the height of a house


 Two Ladder: L1=24 feet and L2=30 feet
 Which ladder does the error make minimum when you measure
 the height of a house?


                                             H  L sin 
                                             H  H  sin(    ) L
                                                     (sin  cos   cos  sin  ) L
                                             For small  ,
                                                   H  H  sin  L  L cos  
                                                        H  L cos  
   Fig.4.3 Measuring the height of a house




POSTECH ME PCM                                                        Chapter 04 FR Design 25
 Example 2.2 Measuring the height of a house


 FR= Measure H
 DP= The angle     
                   H  L cos 
 The Stiffness :   L(cos  )

    The error term is governed by the stiffness.

    The shorter ladder is used, this error term is smaller than when the
     longer ladder is used.

    However, we can not choose the shorter ladder than the length of
     a house.


POSTECH ME PCM                                            Chapter 04 FR Design 26
 8.1.3 Immune to variation


    When FR1 must remain constant and insensitive to the random
     variation of DP1, the desired design solution is the one that will
     make FR1 “immune” to the variation of DP1.

    This can be done by letting A11 and all higher order derivatives of
     FR1 be equal to zero at the set value of FR1 and DP1 even when
     DP1 fluctuates about, or drifts from, the set value.




POSTECH ME PCM                                              Chapter 04 FR Design 27
 8.2 One-FR design with constraints


    When there are constraints(Cs), the design must satisfy both
     FRs and Cs.

    At the first stage, it is better to ignore the Cs.

    Once appropriate DPs are chosen, we can go back and check
     whether the Cs are violated.




POSTECH ME PCM                                            Chapter 04 FR Design 28
 Example 3.1 Electric Circuit Breaker Box


 Make a new circuit breaker that can transmit twice the power
 of the original design in the same available space.
 C1 = The space available
 C2 = The temperature rise


 FR = Increase the power to double
 DP = The contact area of the circuit breaker plate




                        Fig.4.4a Original Electric Contact

POSTECH ME PCM                                               Chapter 04 FR Design 29
 Example 3.2 Electric Circuit Breaker Box




                                                  Fig.4.4b Comb-like structure

                                      Solution



 Fig.4.4a Original Electric Contact




                                                 Fig.4.4c Hemispheric surface


POSTECH ME PCM                                                    Chapter 04 FR Design 30
 8.3.1 Nonlinear One-FR design with constraints


    Regardless of whether the design is linear or non-linear, after the
     FR is satisfied by choosing a right DP, the designer must check
     the design to determine whether it violates any Cs.

    For some nonlinear designs, the problem can be posed as an
     optimization problem of finding a maximum or minimum of an
     objective function, subject to a set of Cs.




POSTECH ME PCM                                             Chapter 04 FR Design 31
 8.3.2 Nonlinear One-FR design with constraints


 The design equation and Cs may be expressed as


         Maximize
                      FR  f ( DP a )
         Subject to
                      {Ci ( DP b )}  0
                      {Ci ( DP b )}  0
         where {} indicates a vector consisting of many constraints


 We can find the solution by mathematical methods or Numerical
 analysis.

POSTECH ME PCM                                               Chapter 04 FR Design 32
 Example 4.1 Van Seat Assembly


 An automobile company has designed for van which the seat can be
 removed from the vehicle. However, they found that 5% of the seats
 could not be installed without forcing the pins.
 How would you solve this problem?

                                             To install the seat, the front leg engages the front
                                             pin first while the seat is partially folded, and then
                                             the seat is lowered to engage the rear pin with
                                             the rear latch.

                                             When the rear latch hits the pin, the latch open.
                                             When the rear pin is fully engaged in the latch,
                                             the latch closes.



   Fig.4.5 Schematic drawing of a van seat
POSTECH ME PCM                                                             Chapter 04 FR Design 33
 Example 4.2 Van Seat Assembly


 FR = The distance between the front leg and the rear latch,340mm




                   Fig.4.6 Linkage Arrangement of the seat




POSTECH ME PCM                                               Chapter 04 FR Design 34
 Example 4.3 Van Seat Assembly


 Table 1. Length of Linkages and Sensitivity Analysis

              Links    Nominal length(mm)   Sensitivity(mm/mm)

                 L12        370.00                3.29

                 L14         41.43                3.74

                 L23        134.00                6.32

                 L24        334.86                1.48

                 L27         35.75                6.55

                 L37        162.00                5.94

                 L45         51.55                11.72

                 L46         33.50                10.17

                 L56         83.00                12.06

                 L67        334.70                3.71




POSTECH ME PCM                                                   Chapter 04 FR Design 35
 Example 4.4 Van Seat Assembly


 The pin must hit the sweet spot without transmitting any reaction
 force to the pin of the hinge.




                   Fig. 4.7 Impact must be made at the sweet
                           spot(3) so that the jaw will simply rotate

POSTECH ME PCM                                                          Chapter 04 FR Design 36
  Example 4.5 Van Seat Assembly


  Traditional SPC (Statistical Process Control) Approach to
  Reliability and Quality
                                               (a)   Analysis the linkage to determine the
                                                     sensitivity of the error to eliminate the major
                                                     source of error. The most sensitive linkages are
                                                     L45, L46 and L56 (See Table 1)

                                               (b)   Access uncertainly through prototyping
                                                     and measurement: The mean value of the
                                                     distance from the front to rear leg span is
                                                     determined to be 339.5mm. And a standard
                                                     deviation is 3.37mm. The reliability R and the
Fig.4.8 Traditional Implementation Method of         value are given by
        Trying to Assess Uncertainty                       346
                                                                  1
                                                           
                                                                              ( FR FR ) 2 / 2 F
                                                      R                  e                          dFR = 95%
                                                           334   2  F

POSTECH ME PCM                                                                              Chapter 04 FR Design 37
   Example 4.6 Van Seat Assembly


                                                         (c) Develop fixtures and gages to make
                                                             sure that the critical dimensions are
                                                             controlled carefully.

                                                         (d) Hire inspectors to monitor and
                                                             control the key characteristics using
                                                             SPC

                                                         The new data obtained by the use of
                                                         SPC gives us 100% reliability. However,
Fig.4.9 The Improvement Implementation Made in           this is a very expensive way of mass-
        Reliability by Following Traditional SPC Steps   producing the product.
                                                         Moreover, since the assembly process
                                                         may introduce new errors, this method
                                                         can’t guarantee 100% reliability.


POSTECH ME PCM                                                                 Chapter 04 FR Design 38
 Example 4.7 Van Seat Assembly


 New Manufacturing Paradigm – Robust Design
 Instead of using the upper and lower bounds for the functional requirement FR, our
 task is to select the sweet spot and make sure that all the linkages are identical.


                                **
       The sweet spot FR({DPi }) as
                                                  FR
                                                       DPi  DPi ** 
                                            n
         FR ({DPi })  FR ({DPi })  
                                 **

                                           i 1   DPi
      {DPi } denotes a vector consisting of DP1 , DP2 , etc
       The mean value
                                                       FR
                                                            DP i  DPi ** 
                                                  n
           FR ({DPi })  FR ({DPi })  
                                      **

                                                i 1   DPi
       where DP i is the mean value of DPi



POSTECH ME PCM                                                                 Chapter 04 FR Design 39
 Example 4.8 Van Seat Assembly


 The variance
                                               2
                                   n
                                        FR  2
                          2
                                             DPi
                                  i 1  DP 
                           FR
                                            i



 The variance can be minimized if the derivative of FR with respect to
 DPi and (DPi – DPi**) are made small
 The FR is a function of 10 DPs, i.e., 10 linkages


       FR = f (DP1, DP2, …, DP10)



POSTECH ME PCM                                          Chapter 04 FR Design 40
 Example 4.9 Van Seat Assembly


 The FR variance

                           f          10
                                            f
                    FR       DPi           DPj
                          DPi        i 1 DPj
                                           i j

 If we assemble all the linkages of the seat and fix them except one
 DPi (which is equivalent to determining the second term of the right-
 hand side of the above equation constant), and finally adjust DPi
 so that
                       f           10
                                         f
                           DPi           DPj
                      DPi         i 1 DPj
                                    i j




POSTECH ME PCM                                          Chapter 04 FR Design 41
 Example 4.10 Van Seat Assembly


 The engineers of the automobile company did indeed minimized the
 variance by assembling all the linkages except one. Then, the seat
 was folded in a vertical position and put in a fixture that fixed the
 distance F between the front leg and the rear latch. Then, the last
 linkage (DPi) was welded in place to satisfy the FR (the distance F
 between the front leg and the rear latch).




                     Fig.4.10 FR Distribution by a new design
POSTECH ME PCM                                                  Chapter 04 FR Design 42
 9. Elimination of Bias and Reduction of variance


    Decrease the stiffness of the system

    Minimize random variation of DP and PV

    Make the system immune to the variation of DP and PV by
     lowering the sensitivity of the FR with respect to the DP or the
     sensitivity of the DP with respect to the PV

    If the design has more DPs than FRs, fix the values of the extra
     DPs

    Make the design range larger




POSTECH ME PCM                                             Chapter 04 FR Design 43
 10. Robust Design


  It is defined as the design that satisfies the functional
  requirements even though the design parameters and the process
  variables have large tolerances for ease of manufacture and
  assembly.




POSTECH ME PCM                                            Chapter 04 FR Design 44
 10.1 Determination of tolerances(1)


 Consider a one-FR design with a specified design range PV1.
 Then, the design equation may be written as

                      FR1  FR1  A11[ DP  DP ]
                                          1     1

                      DP  DP  B11[ PV1  PV1 ]
                        1     1

                  or FR1  A11DP  ( A11 )( B11 )PV1
                                  1



 DP1 and PV1 are the maximum allowable tolerances for DP1 and PV1.
 The last design equation states that the smaller the coefficients A11 and
 B11, the larger are the maximum allowable tolerances DP1 and PV1.




POSTECH ME PCM                                            Chapter 04 FR Design 45
 10.1 Determination of tolerances(2)



                                        Therefore, to get a robust
                                        design, we must use small
                                        coefficients or design a low
                                        “stiffness” system.


     Fig.4.11 DP vs. FR for stiffness




POSTECH ME PCM                                             Chapter 04 FR Design 46
 10.2 Effect of “Noise”


 Unexpected random variations introduced during manufacture and
 use of a product are called “noise”.

 Noise may be due to random variations introduced by machining
 processes, the temperature fluctuations the product is subjected
 to in use, and other environmental factors, all which contribute to
 the random variation of DP.




POSTECH ME PCM                                          Chapter 04 FR Design 47
 Example 5.1 Joining of Aluminum Tube to Steel Shaft


 The part must maintain a tight fit in the temperature range of –30oC to +70oC.
 The required interference fit between the cylinder and the tube is 500psi to 1,000psi.
 The machining accuracy of the mass-production machines selected to make these
  parts is ±0.001inch.


 The radius of the cylinder is 0.5 inch and the wall thickness of the cylinder is 0.5inch.
 Determine the cause of the failure. Suggest a robust design of the part so that the
 functional requirement can always be satisfied.

               Steel                                               Aluminum




                                 Fig.4.12 Aluminum Tube

POSTECH ME PCM                                                           Chapter 04 FR Design 48
 Example 5.2 Joining of Aluminum Tube to Steel Shaft


 The properties of these materials are:
                 Co.Th. Exp.   Yield Strength   E(x106psi)    G(x106psi)
    Aluminum     25x10-6 /C     47,000 psi        10.4            3.9
      Steel       15x10-6/C     51,000 psi        30.0           11.6



 FR = Exist the compressive stress between the steel shaft and the
      aluminum tube between 500psi and 1,000psi.
 DP = The interference fit r (i.e., the difference between the nominal
      value of the shaft and the nominal value of the cylinder)




POSTECH ME PCM                                               Chapter 04 FR Design 49
  Example 5.3 Joining of Aluminum Tube to Steel Shaft


  Noise : The random variation of the interference fit (r) due to the
             manufacturing variability of the shaft diameter and the inner
             diameter of the tube and also during service by the temperature
             fluctuation
Interfacial Pressure                               (r )   due to machining error : 0.002 inch
                                                            due to temperature :
1,000psi
                                                            [ al   st ] r0 (Tr  T )  0.00025
(srr)r=ro
                                                                         Tr is assumed to be 20o C
  500psi

                                                        The total maximum random variation
                                     Interference Fit
                                                        (r) is ±0.00225inch
            Dr -d(Dr)   Dr   Dr +d(Dr)

Fig.4.13 Interfacial Compressive stress vs.
         interference Fit

POSTECH ME PCM                                                                  Chapter 04 FR Design 50
 Example 5.4 Joining of Aluminum Tube to Steel Shaft


 The design equation:


         FR1=A11 DP1
         ( rr ) r r0  f (r0 , t )[r   r ]      (a)

 The function f can be obtained from stress analysis.
 Eq.(a) can be written at the two bounds

            1,000 psi  f ro , t (r  0.00225)
                                                        (b)
             500 psi  f ro , t (r  0.00225)




POSTECH ME PCM                                           Chapter 04 FR Design 51
 Example 5.5 Joining of Aluminum Tube to Steel Shaft


 Solving Eq. (b)
              f (r0 , t )  1.11  105 lb / in 3           (c)
              r  0.00675 in
 The aluminum tube may yield.
 The Tresca yield criterion for the aluminum tube is

             [    rr ]r r0   y ,al  g (r0 , t )   (d)

 Yielding will occur at r = ro, since sqq is the max. tensile stress at r = ro
 and srr is the min. compressive stress at r = ro.
 From Eqs. (c) and (d), we can solve for ro and t.


POSTECH ME PCM                                                   Chapter 04 FR Design 52
 Example 5.6 Joining of Aluminum Tube to Steel Shaft


 The functions f and g are obtained by analyzing the stress distribution
 in thick wall tube.
                   1                                      where
  f (r0 , t )            1.1110 lb / in
                                         5         3
                                                          b  r0  t
                  1 1
                                                             E Al   r 2  b 2        
                  A B                                     A        2
                                                              r  b r 2           Al 
                                                              0             0         
                                                                  Est
                   2b 2                                 B
    g (r0 , t )   2     2
                             ( rr ) r r0  47,000 psi      r0 (1   st )
                   b  r0                                       E
                                                          G
                                                               2(1   )


                       We obtain ro=3 inches and t=0.1 inch



POSTECH ME PCM                                                             Chapter 04 FR Design 53
 10.3 Robustness and the rate of response
                                in nonlinear design

                                                     At point a, the design responds quickly but
                                                     very sensitive to any variation in DP.


                                                     Design point b may provide a combination
                                                     of reasonable robustness and good response-
                                                     time characteristics.


                                                     At point c, the design responds slowly but
 Fig.4.14 Variation of FR as a Function of DR in a   immune to DP.
          nonlinear Design

                                                     Depending on the design task, we can choose
                                                     different design points by considering the
                                                     signal-to-noise ratio, the rate of response,
                                                     and the sensitivity.

POSTECH ME PCM                                                                 Chapter 04 FR Design 54
 11. Design Process(1)


    Understand customers

    Formulate the FRs and Cs

    Confirm that the selected FRs are right ones

    Map the FR into the DP

    Write the design equation

    Examine whether the constraints is violated

    Make sketches and drawings of the DPs

    Write why the DPs are selected

    Decompose and go back the FR domain


POSTECH ME PCM                                      Chapter 04 FR Design 55
 11. Design Process(2)


    Consider appropriate manufacturing issues in terms of PVs
    At any time during the design process, the designer can change
     his/her mind and go back and re-do the entire design, including
     modification of FRs, DPs, and PVs.
    Go to the implementation stage, including detailing the
     manufacturing method, schedule, cost, and human resources
     required.
    The design range and the system range should be estimated to
     determine which DP is a more suitable choice.
    After the design is complete, one should go back to the original
     customer needs.
    Benchmarking is a good practice if the product is to be sold
     competitively with existing products in a given market.

POSTECH ME PCM                                           Chapter 04 FR Design 56
 12. Summary


    The issues related to a one-FR design were presented. Because
     the one-FR design always satisfies the Independence Axiom,
     the critical task is to map from the functional domain to the
     physical domain properly.


    Once the DP is selected, the design task involves two things:
     Satisfying FR within bounds established by constraints
     Reducing information content to zero to satisfy 2nd Axiom


    Robust design is defined. Robustness and rate of response can
     be two opposing requirements in some designs.


POSTECH ME PCM                                             Chapter 04 FR Design 57
 More Discussion on Joining of Aluminum Tube
                                  to Steel Shaft (1)

 The part must maintain a tight fit in the temperature range of
 –30oC to +70oC.
 The design equation:
       ( rr ) r r0  f (r0 , t )[r   r ]   (a)
         due to machining error : 0.002 inch

          [ al   st ] r0 (Tr  T )  0.00025
 The function f can be obtained from stress analysis.
 Eq.(a) can be written at the two bounds
       1,000 psi  f ro , t (r  0.00225)
                                                   (b)
        500 psi  f ro , t (r  0.00225)
 Strictly speaking, the above equation can not be true.

POSTECH ME PCM                                            Chapter 04 FR Design 58
 More Discussion on Joining of Aluminum Tube
                                  to Steel Shaft (2)

 The eq. (b) should be rewritten
     1,000 psi  f ro , t (r  0.002  0.0005ro )              (i)
      500 psi  f ro , t (r  0.002  0.0005ro )                (ii)
 The functions f and g are                                      where
                                                                b  r0  t
                      1
     f (r0 , t )               (iii)                               E Al   r 2  b 2        
                     1 1                                        A        2
                                                                    r  b r 2           Al 
                                                                   0             0         
                     A B                                                Est
                                                                B
                 2b 2                                            r0 (1   st )
  g (r0 , t )   2    2  ( rr ) r  r0  47,000 psi   (iv)           E
                b  r0                                        G
                                                                     2(1   )

 We get ro = xxx inches and t = yyy inches.
 In this case f(ro, t) = zzzzand r = mmm

POSTECH ME PCM                                                            Chapter 04 FR Design 59

				
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