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 A11DP ( 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 ) 1.1110 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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