Decomposition Analysis of an Automotive Powertrain Design ...

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Regarding valve timing, a number of design limitations exist for the valvetrain to function successfully. First the events are bounded so they occur in the appropriate region of the thermodynamic cycle. Second, the difference, g 25 : (i vo - e vc ) / 50 - 1.0 ≤ 0 (88) is bounded to preclude excessive residual fraction at idle conditions. Third, limits exist on the valvetrain acceleration. For both intake and exhaust cams, the acceleration is defined as the second derivative of the lift profile as a function of cam angle and expressed in units of cm/deg 2 . For decomposition analysis purposes they are expressed as g 26 : -.008 ≤ acc iv (i vo , i vc, i lift ) ≤ 0.011 (89) g 27 : -.008 ≤ acc ev (e vo, e vc, e lift ) ≤ 0.011 (90) The specific bounds depend on the valvetrain type, -0.008 and 0.011 correspond to the pushrod valvetrain used in the example. Appendix A summarizes the complete model as presented. Defining equations are treated as zero-valued equalities and each function is given a unique name and number. These identifiers are used in annotating the FDT and corresponding adjacency matrices of the optimization model. The optimal design problem statement is then as follows. maximize MPG m-h Metro-highway fuel economy subject to g 1 : NOx/0.4 -1 ≤ 0 CAA driven constraint on NOx g 2 : τ 0-60 /τ ο -1 ≤ 0 0-60 mph time; τ ο = 11.0 sec g 3 : τ 5-20 /τ 1 -1 ≤ 0 5-20 mph time; τ 1 = 1.81 sec. g 4 : S base /S 0-4 - 1 ≤ 0 Launch distance; S base =88.4 ft. g 5 : 30.0 /α s - 1 ≤ 0 Starting gradability ( > than 30%) g 6 : 65 / V c - 1 ≤ 0 Cruising velocity ( > 65 mph @ 6% grade) g 7 : T e (N e )/T emax (N e ) -1 ≤ 0 Lugging limits g 8 : c r - 13.2 + 0.045 b ≤ 0 Knock limited bore g 9 : θ 0−90 / 70 -1 ≤ 0 Combustion stability constraint g 10 : 2. - ξ fd ≤ 0 Final drive ratio constraint g 11 : ξ fd - 5 ≤ 0 Final drive ratio constraint g 12 : 1.6 - ξ 1 / ξ 2 ≤ 0 Step down ratio constraint
g 13 : ξ 1 / ξ 2 - 2.0 ≤ 0 Step down ratio constraint g 14 : 1.2 - ξ 2 / ξ 3 ≤ 0 Step down ratio constraint g 15 : ξ 2 / ξ 3 -1.6 ≤ 0 Step down ratio constraint g 16 : n gears - 0.5 - ξ 1 / ξ ngears ≤ 0 Gear span constraint g 17 : ξ 1 / ξ ngears - n gears - 0.5 ≤ 0 Gear span constraint g 18 : 0.8 - b/s ≤ 0 Bore-stroke ratio constraint g 19 : b/s - 1.2 ≤ 0 Bore-stroke ratio constraint g 20 : 400 - π b 2 s/ (4 n c ) x 10 -3 ≤ 0 Displacement/cylinder constraint g 21 : π b 2 s/ (4 n c ) x 10 -3 - 600 ≤ 0 Displacement/cylinder constraint g 22 : d i + d e - 0.88 b ≤ 0 Valve size constraint g 23 : 0.85 - d e /d i ≤ 0 Valve size constraint g 24 : d e /d i - 0.87 ≤ 0 Valve size constraint (91) g 25 : (i vo - e vc ) / 50 - 1.0 ≤ 0 Valve event constraint g 26 : acc iv (i vo , i vc, i lift ) - K acci ≤ 0 Valve acceleration constraint g 27 : acc ev (e vo, e vc, e lift ) - K acce ≤ 0 Vavle acceleration constraint h 1 : d bm /b - K dm = 0 Main bearing diameter constraint h 2 : l bm / b - K lm = 0 Main bearing length constraint h 3 : d br / b - K dr = 0 Rod bearing diameter constraint h 4 : l br / b - K lr = 0 Rod bearing diameter constraint h 5+i : i = 0, ..., 56 Defining model equations 4 PARTITIONING The FDT of the optimization model and its undirected graph (adjacency matrix) are shown in Figure 9. Partition metrics include: (i) preference for a feasible coordination strategy which implies a search for linking variables, (ii) at least two disjoint partitions having at least 20 functions each, (iii) the smallest number of linking variables (but not to exceed ten). The heuristic search scheme for acceptable partitions is: 1) Rank variables according to the number of times each appears in the FDT, initialize counters i and j to 1. 2) Construct a vector of linking variables, y i , starting with the first highest ranked variable. 3) Delete y i from FDT, perform depth-first-search on the graph of the modified FDT, and test partitions against acceptability criteria. 4) If acceptable, store y i as a candidate vector to be enumerated.
Page 1 and 2: UNIVERSITY OF MICHIGAN DEPARTMENT O
Page 3 and 4: epresentation facilitates decomposi
Page 5 and 6: of changes in these parameters on t
Page 7 and 8: The U.S. metro-highway fuel economy
Page 9 and 10: Powertrain Relationships The princi
Page 11 and 12: N ds = ξ fd N d (23) dN ds /dt =
Page 13 and 14: to be stalled (Rs = 0). Equation (3
Page 15 and 16: the surface-to-volume ratio at a vo
Page 17 and 18: intake and exhaust valve, and acros
Page 19 and 20: driveability. These values are stor
Page 21 and 22: τ dV τ 0-60 = τ :⌡ ⌠ dt dt =
Page 23: There are also several geometric co
Page 27 and 28: x 2 h 2 (y, x 2 ) = 0 h 2 ( x 2 )=
Page 29 and 30: Master Problem Subproblem min f(y,
Page 31 and 32: 6 COORDINATED OPTIMIZATION SOLUTION
Page 33 and 34: educes fuel flow by 1.3%; the const
Page 35 and 36: Assanis, D.N., and Polishak, M. 198
Page 37 and 38: Patton, K.J., Nitschke, R.G., and H
Page 39 and 40: DECOMPOSITION ANALYSIS MODEL -BASED
Page 41 and 42: Master Problem Subproblem (i = 1, .
Page 43 and 44: 1 2 Impeller Stator Vortex flow dir
Page 45 and 46: Partitioned Graph Resultant FDT 111
Page 47 and 48: Table I Summary of Coordination Str
Page 49 and 50: Index Variable Partition Descriptio
Page 51 and 52: Index Variable Partition Descriptio
Page 53 and 54: 23 Fnd Fnd - ( Wd M g - ∆Wd) (fro
Page 55 and 56: 66 67 68 69 70 71 72 73 74 75 76 77
Page 57 and 58: Appendix B: Parameters in Optimizat
Page 59 and 60: Impeller Driving Table B.4: Base to
Page 61 and 62: Table B.8: Accessory losses. Amps @

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