# 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

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
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

   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
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

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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