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Multi-FR Design II Prof. W. Hwang Dept. of Mechanical Engineering Postech POSTECH ME PCM Chapter 06 Multi-FR Design 2 1 Example 1.1 Design and assembly of the injection- molded vacuum cleaner wheel The wheel must rotate freely and should withstand a pulling force of 500N. It must be easily assembled during the manufacturing operation with an axial force of less than 50N. Clean Vac Corp. has found that many parts of the Cleaner was broken when it was assembled by a hammer. Clean Vac is concerned about long-term durability. Fig. 6.1 Cross-sectional View of the Wheel/Shank For this purpose, you are asked to provide the following: a. Define the FRs b. Develop DPs c. For your chosen DPs, determine the design matrix d. Modeling the relationship between FRs and DPs e. Optimize the design based on the design matrix and the model POSTECH ME PCM Chapter 06 Multi-FR Design 2 2 Example 1.2 Design and assembly of the injection- molded vacuum cleaner wheel Determination of FRs and Cs FR1 = Make the wheel rotate easily by maintaining low friction between the wheel and the vacuum cleaner body and by making the torque exerted by the wheel/floor contact larger than the friction of the contact of plastic components FR2 = Retain the wheel in the vacuum cleaner body under 500N of pulling force FR3 = Provide a means of easy assembly with an axial force of less than 50N FR4 = Carry the weight of the vacuum cleaner and accidental load applied when people step on the vacuum cleaner(200lbs) POSTECH ME PCM Chapter 06 Multi-FR Design 2 3 Example 1.3 Design and assembly of the injection- molded vacuum cleaner wheel C1 = No fracture C2 = No fatigue failure C3 = No plastic yielding of the wheel C4 = Torque due to the traction at the wheel and the floor > torque at the shaft surface due to the friction between plastic components C5 = Manufacturing considerations, e.g., injection-molded part should have approximately a constant thickness to prevent secondary flow caused by non-uniform cooling C6 = Minimize the manufacturing cost POSTECH ME PCM Chapter 06 Multi-FR Design 2 4 Example 1.4 Design and assembly of the injection- molded vacuum cleaner wheel Selection of DPs DP1 ( D1 D2 ) / 2 (i.e., the clearance between the diameters of the wheel DP2 t ( D2 D3 ) / 2 and the vacuum cleaner body) DP3 DP4 D2t (i.e., the area of the tubular stem without the axial cut) Fig. 6.2 End View of the Shank for the Clean Fig. 6.3 Free-body Diagram of one of the curved Vac Design Beams. POSTECH ME PCM Chapter 06 Multi-FR Design 2 5 Example 1.5 Design and assembly of the injection- molded vacuum cleaner wheel Design equation DP 1( ) DP 2(t ) DP 3() DP 4( D 2 t) FR1 A11 0 0 0 DP1 FR 0 A22 A23 A24 DP2 FR 1 X 0 0 0 2 FR 2 0 X X X FR 3 0 X X 0 FR3 0 A32 A33 0 DP3 FR 4 0 X 0 X FR4 0 A42 0 A44 DP4 To make the design a decoupled design we must make the off-diagonal elements A23 and A24 zero. A23 can be made to be zero if we make the total circular length of all the curved beams remain constant by adding more sections of the curved beams, but this may increase the manufacturing cost. Another way of decoupling the design is to choose either the height h of the interlock key or the length L of the beam as DP3. A24 can be made zero by choosing the area of the unslitted section A as DP4. POSTECH ME PCM Chapter 06 Multi-FR Design 2 6 Example 1.6 Design and assembly of the injection- molded vacuum cleaner wheel We will set the length L=2D2. We will then select the height h of the interlock key as DP3. FR1 A11 0 0 0 DP1 FR 0 A 0 0 DP t 2 2 22 FR3 0 A32 A33 0 DP3 h FR4 0 0 0 A44 DP4 A POSTECH ME PCM Chapter 06 Multi-FR Design 2 7 Example 1.7 Design and assembly of the injection- molded vacuum cleaner wheel Modeling the relationship between the FRs and DPs a. Evaluating A11 We will choose the clearance to be 0.010 inch(0.25cm) in each side. The friction force F is FR1 F W Then A11 is given by W A11 constant If the friction coefficient is the same, C4 is satisfied as long as the diameter of the wheel is larger than the shaft diameter POSTECH ME PCM Chapter 06 Multi-FR Design 2 8 Example 1.8 Design and assembly of the injection- molded vacuum cleaner wheel b. Evaluating A22 We assume that there are three circular sections and that is the included angle. 3 FR2 Fpull 3( )( D22 D32 ) ( )( D2 t )t 8 2 If we want to avoid fatigue, then a good rule of thumb is that should not exceed y / 2 . FR2 3 A22 ( )( D2 2t ) y DP2 4 POSTECH ME PCM Chapter 06 Multi-FR Design 2 9 Example 1.9 Design and assembly of the injection- molded vacuum cleaner wheel c. Evaluating A33 FR3 P tan DP3 h The deflection at the end of the cantilever is given by PL3 h 3EI 3EI A33 ( 3 ) tan L POSTECH ME PCM Chapter 06 Multi-FR Design 2 10 Example 1.10 Design and assembly of the injection- molded vacuum cleaner wheel d. Evaluating A44 y FR4 W A A 2 DP4 A y A44 ( ) 2 e. Evaluating A32 3 tan 3EIh A32 FR3 L 3Eh tan I DP2 t L3 t POSTECH ME PCM Chapter 06 Multi-FR Design 2 11 Example 1.11 Design and assembly of the injection- molded vacuum cleaner wheel We can numerically determine t, h, diameters and the length of the circular cantilever beam after setting the value of , which was set to be 100 degrees after trying several possibilities. The material properties for nylon are: Coefficient of friction = 0.4 E=362,590 psi y = 7,250 psi The solution of the design equations for the dimensions are approximated as: D2 0.375 inch t 0.030 inch D3 0.315 inch D1 0.395 inch h 0.0625 inch D4 0.437 inch POSTECH ME PCM Chapter 06 Multi-FR Design 2 12 1. The relationship between complexity and information content The design that requires more information content is more complex. Next example proves the following statements. (1) Complexity is related to the probability of achieving the functional requirement. (2) Even the same design can have very different information content and complexity, depending on the stiffness of the system. (3) Information content , Complexity , Probability of Success Information content is a measure of design complexity. (4) A design that violates the Independence Axiom is more complex and requires more information content than a design that satisfy the Independence Axiom. POSTECH ME PCM Chapter 06 Multi-FR Design 2 13 Example 2.1 Knob Design FR1 = Grasp the end of the shaft tightly with axial force of 30N FR2 = Turn the shaft by applying 15 N-m of torque DP1 = Interference fit between the shaft and the inside diameter of the knob DP2 = The flat surface The design equation may be written as FR1 X X DP1 FR2 x X DP2 The lower-case x is used to signify the fact that the effect of DP1 on FR2 is much less than the other effects indicated by upper-case X. Eventually, when the grip force is less than the required force to keep the knob on the shaft the knob will slide off the shaft. How do we solve this problem? POSTECH ME PCM Chapter 06 Multi-FR Design 2 14 Example 2.2 Knob Design Some will suggest that the solution to this coupled design problem is to make the outer diameter of the knob shaft thicker, which will make the slot open up less and thus minimize the reduction of the gripping force. However, this solution has its cost; not only does it require more materials but also higher information content, which ultimately means a higher manufacturing cost. Depending on the stiffness, the same bell-shaped distribution along the DP axis translates into very different system distributions in the functional domain . When the stiffness is lower, the system pdf fit in the design range, but when the stiffness increases, the system pdf is outside the design range. When the thickness of the cylinder wall increases, Fig. 6.4 Dependence of System pdf on “Stiffness.” the stiffness increase. POSTECH ME PCM Chapter 06 Multi-FR Design 2 15 Example 2.3 Knob Design New Design The slot terminates where the flat part of the knob begins. Since the flat surface is completely away from the slot, the turning action does not force the slot to open and therefore, the axia l grip is not affected. This is a completely uncoupled design. Fig. 6.5 A New Uncoupled Design. How do we actually determine the wall thickness and the desired interference? The thickness must be determined by considering two limiting factors: Manufacturability and failure of the knob under stress. Can it be manufactured by injection molding? Does the maximum stress at the bottom corner of the slit, which is the stress Concentration point, cause either fracture or plastic deformation? POSTECH ME PCM Chapter 06 Multi-FR Design 2 16 Example 2.4 Knob Design Fig. 6.6 Modified Shaft. Fig. 6.7 A Cantilever Beam Loaded at the End The maximum deflection is given by FL3 bh3 ymax where I 3EI 12 The maximum stress is given by The stiffness K max / ymax is given by FLh 6 FL 3Eh max 2 K 2 2I bh 2L To minimize K for robustness, h should be made as small as possible. The limit is reached when max reached y . 1 yb 2 Then the smallest h is obtained as 6 FL hmin POSTECH ME PCM Chapter 06 Multi-FR Design 2 17 The foregoing example illustrates the following aspects of the Information Axiom and the Independence Axiom: Complexity is related to the probability of achieving the functional requirement. The coupled design made it much more difficult to make the knob. Even the same design can have a very different information content and complexity, depending on the stiffness of the system. The greater the information content, the more complex is the task of achieving the FR since the probability of success decreases. Therefore, information content is a measure of design complexity. A design that violates the Independence Axiom, i.e., a coupled design, is more complex and requires more information content than a design that satisfies the Independence Axiom. POSTECH ME PCM Chapter 06 Multi-FR Design 2 18 2.1 Determination of Information Content Uncoupled design The probability that all m FRs are satisfied by uncoupled designs can be computed by the product of the probabilities for each FR. m I log 2 Pi (m : the number of FR) i 1 Theorem 13 (Information Content of the Total System) If each DP is probabilistically independent of other DPs, the information content of the total system is the sum of the information of all individual events associated with the set of FRs that must be satisfied. POSTECH ME PCM Chapter 06 Multi-FR Design 2 19 2.2 Determination of Information Content Decoupled design The probability that all m FRs are satisfied by decoupled designs can be computed by the product of the probabilities for each FR, provided that appropriate conditional probabilities are used where necessary. m I log 2 Pi| j (m : the number of FR) i 1 Theorem 12 (Sum of Information) The sum of information for a set of events is also information, provided that proper conditional probabilities are used when the events are not statistically independent. POSTECH ME PCM Chapter 06 Multi-FR Design 2 20 2.3.1 Determination of Information Content For example, consider the following design matrix FR1 A11 0 DP1 FR2 A21 A22 DP2 The information content of FR1 can be determined by computing the area of the system pdf in the common range just as for an uncoupled design, since DP2 does not affect FR1. However, to compute the information content associated with FR2, we have to include the change in the information content due to the off-diagonal element. POSTECH ME PCM Chapter 06 Multi-FR Design 2 21 2.3.2 Determination of Information Content Shift of Mean Value The solid curve is the system pdf of FR2 when the off-diagonal element A21 is equal to zero. Fig. 6.7 Shift of the FR2 System pdf Due to change in DP1 However, the system pdf of FR2 for a decoupled design may be shifted to the right or left by the off-diagonal element A21, which changes the mean of the system pdf for FR2 when DP1 changes because FR2 is affected by DP1. POSTECH ME PCM Chapter 06 Multi-FR Design 2 22 2.3.3 Determination of Information Content Change of Variance In some case, the variance of the system pdf can change as well as the mean. Fig. 6.9 Change in Variance of System pdf of FR2 Due to change in DP1. When the system pdf is symmetrical with its mean in the middle of the design range, the effect of the off-diagonal element is to change the spread of the system pdf, as shown in next example. POSTECH ME PCM Chapter 06 Multi-FR Design 2 23 2.3.4 Determination of Information Content In the results, the information content of the decoupled design can increase or decrease by the off-diagonal element. However, in most cases, the information content of a decoupled design is expected to be larger than that of an uncoupled design, since a decoupled design can not be as robust as an uncoupled design. POSTECH ME PCM Chapter 06 Multi-FR Design 2 24 Example 3.1 Information Content of a Decoupled Design FR1 = Turn the shaft The design ranges for FR1 and FR2 are +/- 5%, i.e., FR2 = Grip the shaft FR1 = 1 +/- 0.05 DP1 = Flat surface FR2 = 1 +/- 0.05 DP2 = Interference fit FR1 1 0 DP 1 FR2 0.2 1 DP2 Supplier A : +/- 5% for the DP1 tolerance Supplier B: +/- 10% for the DP1 tolerance Both suppliers : the same 6% for DP2 tolerance Fig. 6.10 System pdf and Common Range Fig. 6.11 System pdf and Common Range of FR1 for Supplier A. of FR1 for Supplier B. POSTECH ME PCM Chapter 06 Multi-FR Design 2 25 Example 3.2 Information Content of a Decoupled Design From the measurements, it was determined that of the parts that were within the DP1 tolerance, only 90% were also within the DP2 tolerance. Determine the information content of this design. The DPs are determined as: DP1 FR1 / A11 1.0 DP2 ( FR2 A21 DP1 ) / A22 (1 0.2(1)) / 1 0.8 DP1 FR1 / A11 0.05 DP2 (FR2 A21DP1 ) / 1 0.04 If the manufacturing process cannot hold the DP1 tolerance to within DP1,two things will happen for those parts that are outside the DP1 tolerance. (1) FR1 will not be satisfied. (2) The established DP2 tolerance will be too large to satisfy FR2 POSTECH ME PCM Chapter 06 Multi-FR Design 2 26 Example 3.3 Information Content of a Decoupled Design In order to calculate the probability that FR2 is satisfied, we must first determine the probability that FR1 is satisfied. Supplier A I log 2 (1) log 2 (0.833) 0.263 Supplier B I log 2 [Pr(satisfy FR1 )] log 2 [Pr(satisfy FR2 | satisfy FR1 )] Fig. 6.12 Specified tolerance for DP2 I log 2 (0.5) log 2 (0.833) 1.264 and actual pdf of DP2. The probability that FR2 is satisfied must be computed conditional upon FR1 being satisfied. POSTECH ME PCM Chapter 06 Multi-FR Design 2 27 3.1 Accommodating “Noise” in the design process During manufacturing and use of a product, random variation from various sources affects the performances of a machine or system. The variation so introduced is given the generic name “Noise”. There are five generic noise sources: Manufacturing variation Customer usage Environmental variation Degradation/Wear-out System-to-system iteration POSTECH ME PCM Chapter 06 Multi-FR Design 2 28 3.2 Accommodating “Noise” in the design process Recall example of one-FR design(Joining of aluminum Tube to Steel shaft). Noise was introduced by the random variation of the machining processes and by the temperature fluctuation in service. Muti-FR designs must also accommodate noise by adjusting the “stiffness”. POSTECH ME PCM Chapter 06 Multi-FR Design 2 29 3.3 Accommodating “Noise” in the design process When the design is a decoupled design with a triangular design matrix [A], the variation of FRi is caused by the random variations of many DPs, which may be expressed as i δ FRi Aijδ DPj j 1 To satisfy FRi, the elements Aij that correspond to large values of the DPs must be made smaller. δ FRi M iδ DPi Where Mi is defined as a module which is equal to i DPj M i Aij j 1 DPi To Minimize the effect of random noise, Mi must be decreased if the random variation in FR is larger than the specified design range of FR. POSTECH ME PCM Chapter 06 Multi-FR Design 2 30 4.1 Integration of DPs to Minimize the Information Content In general, physical integration reduces the information content by removing the uncertainly associated with assembling several physical pieces. Providing that the Independence Axiom is not violated, DPs may be integrated in a single physical part under the following circumstances; 1) DPs do not undergo relative motion 2) DPs can be made of the same material 3) Integration does not create a problem such as excess stress and fracture 4) Integration does not violate a cost contraint 5) The integrated parts can be manufactured POSTECH ME PCM Chapter 06 Multi-FR Design 2 31 4.2 Integration of DPs to Minimize the Information Content The integration of the physical part must be consistent with the DP hierarchy in the physical domain, where all leaf-level DPs are related to other leaf-level DPs according to the specified relationship. POSTECH ME PCM Chapter 06 Multi-FR Design 2 32 5.1 Nonlinear Multi-FR Design When the elements of the design matrix are not constants, but instead are the function of DPs, the design is a nonlinear design. There are three kinds of situation in nonlinear design. (1) The design matrix is always either diagonal or triangular regardless of how DP change. (2) The elements of the matrix may vary, depending on the specific values of DPs, so that the design behaves as a coupled, uncoupled, or decoupled design in different parts of the design window. (3) The design is always coupled regardless of the specific values of DPs. POSTECH ME PCM Chapter 06 Multi-FR Design 2 33 5.2 Nonlinear Multi-FR Design The difference between the linear and the nonlinear design of the second kind is that in the case of nonlinear design, we may strive to find a better design window, because of the elements of the design matrix changes as functions of DPs. This can be illustrated graphically. FR1 and FR2 are independent each other by definition. So they are mutual orthogonal. Uncoupled linear case: DP1 affects only on FR1 and DP2 affects only on FR2. So, DP1 parallels to FR1 and DP2 parallels to FR2. Fig. 6.13(a) A Completely Uncoupled Two-FR Design. POSTECH ME PCM Chapter 06 Multi-FR Design 2 34 5.3 Nonlinear Multi-FR Design Decoupled linear case: DP1 affects only on FR1 but DP2 affects on FR1 and FR2. So, DP1 parallels to FR1 but DP2 doesn’t parallel to FR2. Fig. 6.13(b) A Decoupled Two-FR Design. Coupled linear case: DP1 affects on FR1 and FR2 and DP2 affects on FR1 and FR2. So, DP1 doesn’t parallels to FR1 and DP2 doesn’t parallels to FR2. Fig. 6.13(c) A Coupled Design. POSTECH ME PCM Chapter 06 Multi-FR Design 2 35 5.4 Nonlinear Multi-FR Design In nonlinear design, the lines of constant DPs are curved since the elements of the design matrix are function of DPs as shown in Fig. 6.13(d). Fig. 6.13(d) A Case of Nonlinear Dsign. Nonlinear Case Region A – The magnitude of FR1 is small and that of FR2 is large. Nearly uncoupled region. Diagonal elements are zero or very small. Region B – Nearly decoupled region Region C – Coupled region POSTECH ME PCM Chapter 06 Multi-FR Design 2 36 5.5 Nonlinear Multi-FR Design When there are more than two FRs, it is difficult to use a graphical means. As an alternate means of measuring the independence of FRs, We define two scalar metrices – reangularity R and semangularity S. 2 2 n Aki Akj n 1 k 1 A jj R n S n 1/ 2 n i 1, n 1 j 1 j i i, n k 1 Aki2 k 1 Akj2 2 Akj k 1 These measures are useful when there are many FRs and DPs. POSTECH ME PCM Chapter 06 Multi-FR Design 2 37 5.6 Nonlinear Multi-FR Design Reangularity R measures the angular relationship between the DP axes. Semangularity S measures the magnitude of the diagonal elements of a normalized design matrix. R=S=1 The design is an uncoupled design R=S The design approaches a decoupled design. [When there are only two FRs and two DPs, R=S represents a decoupled design.] In all other cases, the design is a coupled design. In fig. 3.3 (d) , Region A : 1 R=S Region B : RS Region C : R< 1, S< 1 POSTECH ME PCM Chapter 06 Multi-FR Design 2 38 6.1 Axiomatic Design Basis for Robust Design Why robust? 1. The original design goals can be achieved easily and faithfully. 2. The product must be reliable and durable. Necessary condition for robust design The fulfillment of FRs within the bounds established by constraints under all operating conditions POSTECH ME PCM Chapter 06 Multi-FR Design 2 39 6.2 Axiomatic Design Basis for Robust Design One-FR Design (Review) Consider redundant design with one-FR FR1 f ( DP a , DP b , DP c ,, DP n ) Task: Make this design robust under all conditions, if possible. The desired change of FR, f f f FR DP a DP ... b DP n DP a DP b DP n In an ideal design for one-FR, only one DP is needed. All other DPs are possible sources of rand variations POSTECH ME PCM Chapter 06 Multi-FR Design 2 40 6.3 Axiomatic Design Basis for Robust Design Robust Design 1. Make the coefficient (f/DP) associated with extra DPs to be zero. – Immune to random variation [one of the basic concept of robust design practiced in industry today] e.g. Windshield Wiper – Robust Mounting Design 2. Fix the values of all DPs except one DP chosen e.g. Van Seat Assembly POSTECH ME PCM Chapter 06 Multi-FR Design 2 41 6.4 Axiomatic Design Basis for Robust Design The desired change of FR may be expressed as f i n f FR DP c DP i DP c i a DP i i c f DP c [Extra terms] DP c [Module]DP c [Extra terms] [stiffness ]DP c [Extra terms] Where DPc is the DP chosen to satisfy FR. POSTECH ME PCM Chapter 06 Multi-FR Design 2 42 6.5 Axiomatic Design Basis for Robust Design How to select DP? The magnitude of the term (f/DP)DP of the chosen DP should be larger than the sum of the constant terms so that the accumulated errors can be compensated. If the magnitudes of two or more terms are approximately the same, the one with smaller f/DP should be chosen to minimize the sensitivity of FR to the variation of DP. POSTECH ME PCM Chapter 06 Multi-FR Design 2 43 6.6 Axiomatic Design Basis for Robust Design Multi-FR Design The robust design concept discussed with respect to one-FR design does apply, if the design satisfies the Independence Axiom. Consider redundant design with multi-FR {FR} = [Square DM]{DP} + [Extra Matrix]{DP}extra {DP} = the vector of DPs chosen to satisfy the vector {FR} {DP}extra = the vector of the redundant DPs [Square DM] = must be either diagonal or triangular matrix to satisfy the Independence Axiom [Extra Matrix] = can be any matrix, including a full matrix POSTECH ME PCM Chapter 06 Multi-FR Design 2 44 6.7 Axiomatic Design Basis for Robust Design Consider a special case of three FR design, DP4 DP 5 FR1 X 0 0 DP1 X X X . . . X DP6 FR2 0 X 0 DP2 X X X . . . X . FR 0 DP X X 3 . . . X . 3 0 X X . DP 7 The above design can be treated as uncoupled design, if the values of DP4 through DPn are fixed. POSTECH ME PCM Chapter 06 Multi-FR Design 2 45 6.8 Axiomatic Design Basis for Robust Design The desired change of each FRi may be expressed as FRi n FRi FRi DPi DPj DPi j 4 DPj FRi DP [Extra terms] DPi The above equation is similar to one-FR design case with similar implica- -tions for compensation by fixing the values of the extra DPs. POSTECH ME PCM Chapter 06 Multi-FR Design 2 46 6.9 Axiomatic Design Basis for Robust Design If [Square DM] is a triangular matrix, DP4 DP 5 FR1 X 0 0 DP1 X X X . . . X DP6 FR2 X X 0 DP2 X X X . . . X . FR X DP X X 3 . . . X . 3 X X X . DP 7 The above design can be treated as decoupled design, if the values of DP4 through DPn are fixed. POSTECH ME PCM Chapter 06 Multi-FR Design 2 47 6.10 Axiomatic Design Basis for Robust Design The desired change of each FRi may be expressed as FRi 3 FRi n FRi FRi DPi DPj DPk DPi j 1 DPj k 4 DPk j i FRi 3 FRi DP DPj [Extra terms] DPi j 1 DPj j i In compensating for this design, DPj must be set first According to the sequence defined by the triangular matrix. POSTECH ME PCM Chapter 06 Multi-FR Design 2 48 6.11 Axiomatic Design Basis for Robust Design How to select the primary DPs? The selection of DP1, DP2, and DP3 in a multi-FR design Must satisfying the same set of conditions as those discussed For one-FR design; robustness and sensitivity. POSTECH ME PCM Chapter 06 Multi-FR Design 2 49 Example 4.1 Robust design of a micro-gyroscope It is made of silicon by means of photo-lithograph and etching. It measures the motion by resonant vibration responses of MEMS in response to external motion. The mechanism 1. Measurement of bending resonance by means of electric potential 2. Measurement of angular velocity 3. Measurement of Coriolis acceleration 4. Sensing of torsional resonance of the sensing plate 5. Sensing by capacitance It measures motions in one translational direction(the x-direction) and the rotational motion about the x-axis. POSTECH ME PCM Chapter 06 Multi-FR Design 2 50 Electrostatic Comb Drive POSTECH ME PCM Chapter 06 Multi-FR Design 2 51 Example 4.2 Robust design of a micro-gyroscope It measures motions in one translational direction(the x-direction) and the rotational motion about the x-axis. When three of these gyroscopes are mounted along the three orthogonal directions, they can measure motion in six directions. The driving force generates the translational motion, deforming the four bending springs, which in turn induces the rotational motion of the central plate(gimbal) that is attached to the translational plate by the sensing spring. The actual measurement of the relative motion is done by means of the capacitance change between series of capacitor plates between the stationary part and the moving part. POSTECH ME PCM Chapter 06 Multi-FR Design 2 52 Example 4.3 Robust design of a micro-gyroscope The gyroscope is designed to have two specific natural frequencies, f1 and f 2 . f1is the driving force mode and f2 is the sensing mode. These two frequencies must be exactly the same to have the best response and provide the most accurate measurement. Because the tolerance of the manufacturing processes is only 10%, there is a frequency mismatch between the two modes. The current design has large random variation in dimensional tolerances due to the manufacturing accuracy, so tuning is extremely difficult. How can you manufacture more easily and reliably, increasing the yield of good gyroscopes? POSTECH ME PCM Chapter 06 Multi-FR Design 2 53 Example 4.4 Robust design of a micro-gyroscope Fig. 6.14 Design of the original resonant Fig. 6.15 Finite element model of the gyroscope vibratory gyroscope POSTECH ME PCM Chapter 06 Multi-FR Design 2 54 Example 4.5 Robust design of a micro-gyroscope FR1= Set the frequency of the driving mode- the translational motion of the moving plate- at f1 FR2=Set the frequency of the sensing mode- the torsional motion of the central plate - of the central plate at f2 FR3=Let the distribution of the frequency difference f2-f1 be insensitive to geometric tolerances of the gyroscope(i.e., set the mean of (f2-f1) to be within the lower bound aL and the upper bound aU) DP1=Stiffness of the four bending springs DP2=Stiffness of the two torsional springs In the original design, DP3 was absent, and thus Fig. 6.16 The driving mode(upper) and the design was a coupled design. the sensing mode(lower) POSTECH ME PCM Chapter 06 Multi-FR Design 2 55 Example 4.6 Robust design of a micro-gyroscope If we select another DP3, the design equation may be written as FR1 X 0 ? DP 1 FR2 0 X ? DP2 FR ? X DP3 3 ? POSTECH ME PCM Chapter 06 Multi-FR Design 2 56 Example 4.7 Robust design of a micro-gyroscope Finding DP3 To satisfy FR3, the variance of (f2-f1) must be made very small, the best being zero. This can be done if we can make the design immune to the variations introduced by manufacturing operations so that the variance of FR is equal to zero. 2 0 f 2 f1 The natural frequencies of both f1 and f2 are affected by the distribution of the moments of inertia of the gimbal plate. the gimbal plate has three dimensions (a,b,c), all of which affect the FR1, FR2 and FR3. Fig. 6.17 Gimbal plate with three dimensions POSTECH ME PCM Chapter 06 Multi-FR Design 2 57 Example 4.8 Robust design of a micro-gyroscope f1 f1 f1 f1 DP1 0 FR1 DP1 a b c DP2 f 2 f 2 f 2 f 2 DP a FR2 0 3 (a) FR DP2 a b c 3 ( f 2 f1 ) ( f 2 f1 ) ( f 2 f1 ) ( f 2 f1 ) ( f 2 f1 ) DP3b DP1 DP2 a b c DP3c If the design equation elements are under following conditions, f1 f f da 1 db 1 dc 0 a b c The design is a decoupled design. f 2 f 2 f 2 da db dc 0 a b c POSTECH ME PCM Chapter 06 Multi-FR Design 2 58 Example 4.9 Robust design of a micro-gyroscope If the design equation elements are under following conditions, ( f 2 f1) ( f 2 f1) ( f 2 f1 ) ( f 2 f1) ( f 2 f1) dDP 1 dDP2 da db dc 0 DP 1 DP2 a b c Can be reduced ( f 2 f1) ( f 2 f1) ( f 2 f1) da db dc 0 a b c FR1 and FR2 can be satisfied by varying DP1 and DP2 to reach target frequencies and . POSTECH ME PCM Chapter 06 Multi-FR Design 2 59 Example 4.10 Robust design of a micro-gyroscope Set up an orthogonal array “experiment” and FR3 was evaluated for 27 combinations of (a,b,c) at three level of DPs as shown in Table ex4.1, using the finite element method. By various trials, they recommended values as given in table ex5.2. Table. Ex 4.1 Level DP3a:a DP3b:b DP3c:c 1 Lower bound Lower bound Lower bound 2 Current Current Current 3 Upper bound Upper bound Upper bound Table. Ex 4.2 Design f1 f2 (f2-f1)/f1 (f2-f1) (f2-f1) Improvement Original bO gO 0.287 334.33 84.84 Recom. bR gR 0.119 127.15 51.81 38.9% POSTECH ME PCM Chapter 06 Multi-FR Design 2 60 Example 4.11 Robust design of a micro-gyroscope They determined the probability of success of their proposed design. The results show a significant increase. Table. Ex 4.3 Design Probability of success Information content Original 5.7% 4.14 Rrecom. 86.0% 2.18 POSTECH ME PCM Chapter 06 Multi-FR Design 2 61 Example 4.12 Robust design of a micro-gyroscope Fig. 5 The fabricated de-coupled vertical gyroscope (a) The perspective view (b) the wafer level vaccum packaged gyroscope (c) The closed view of comb electrode (d) pad and interconnection feedthrough POSTECH ME PCM Chapter 06 Multi-FR Design 2 62 7.1 Design of Dispatching Rules and Schedules Dispatching and scheduling are important tasks in many situations such as production of mechanical parts in job shops, scheduling of robot tasks in automated manufacturing system, and scheduling of airline flights. So far, the mathematical tools of operations researcher simulations of the actual situation have not always been successful. Because the design that violates the Independence Axiom can not be improved through optimization, these techniques do not always yield sufficiently improved results. POSTECH ME PCM Chapter 06 Multi-FR Design 2 63 7.2 Design of Dispatching Rules and Schedules They must be designed right ( to satisfy the Independence Axiom ) before the parameters can be adjusted to obtain the correct FRs, and the Information Axiom should be applied to minimize the information content. All dispatching and scheduling algorithms must satisfy the Independence Axiom and the Information Axiom to be able to come up with a rational strategy. POSTECH ME PCM Chapter 06 Multi-FR Design 2 64 7.3-1 Design of Dispatching Rules and Schedules When an identical set of parts is processed through a variety of different machines but the same set of processes, rational scheduling and dispatching algorithms can be developed based on the Independence Axiom so that the scheduling and transport of the part will be uncoupled from the manufacturing processes. In this case, we can come up with a “Push type” process that can maximize the productivity. POSTECH ME PCM Chapter 06 Multi-FR Design 2 65 7.3-2 Design of Dispatching Rules and Schedules When a random set of parts is process through a variety of different processes, a “push” system can no longer maximize the throughput rate. In this case, independence of FRs can be satisfied by designing a cellular manufacturing system – a “Pull” system. This “pull” system will control the production rate based on the demand rate, an approach which satisfies the Independence Axiom. POSTECH ME PCM Chapter 06 Multi-FR Design 2 66 7.4 Design of Dispatching Rules and Schedules Decoupler The system must be designed correctly by decoupling the tasks from each other using “ decoupler”. The role of the decoupler is to eliminate coupling when more than one part requires the attention of the same robot(or person) at the same time, or when a machine is not ready to accept the next part which has just been complete by a preceding machine. POSTECH ME PCM Chapter 06 Multi-FR Design 2 67 8.1 Dispatching Rules and the Independence Axiom There are several special cases of dispatching situations; (a) Frequency of dispatches for identical parts Consider a manufacturing system where a certain part must be processed by N machines in a sequential arrangement. τ i the processing time at each machine τ t the transport time between the machines τ m the longest processing time ( Machine m) τ d τ m τ t (1) Then, the part should be dispatched for processing at an interval τ d given by, POSTECH ME PCM Chapter 06 Multi-FR Design 2 68 8.2 Dispatching Rules and the Independence Axiom If the dispatching rate must be increased to a higher rate than that given equation(1), the number n of the slowest machines must be increased to τ n int m τ (2) d where int(x) is an integer rounded to the next whole number for any x. (b) Dispatch rate when random parts are processed Dispatch rate can not exceed the “dispatching” rate given by equation (1). When the demand rate is greater than that given by equation (1), more machines must be added according to equation (2). POSTECH ME PCM Chapter 06 Multi-FR Design 2 69 9. Scheduling Scheduling depends on whether an identical set of parts or different random parts are being processed by the system. POSTECH ME PCM Chapter 06 Multi-FR Design 2 70 Summary The implications of the Independence Axiom and the Information Axiom are presented with relevant theories that govern multi-FR designs. The robust-design concept is given for the multi-FR case as well as discussing the relationship between the complexity and the information content of a design. POSTECH ME PCM Chapter 06 Multi-FR Design 2 71