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TECHNICAL PAPERS

Decomposition-Based Assembly Synthesis for Structural Stiffness

[+] Author and Article Information
Naesung Lyu, Kazuhiro Saitou

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI

J. Mech. Des 125(3), 452-463 (Sep 04, 2003) (12 pages) doi:10.1115/1.1582879 History: Received May 01, 2002; Revised November 01, 2002; Online September 04, 2003
Copyright © 2003 by ASME
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References

Figures

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Outline of the decomposition procedure. (a) structure to be decomposed, (b) basic members and potential joint locations, (c) structural topology graph G, (d) optimal decomposition of G, and (e) resulting decomposition of the original structure.
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Example decompositions of a graph and the corresponding values of vector x. (a) the original graph with x=(1,1,1), and (b) two component decomposition with x=(0,0,1).
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Chromosome representation of design variables x=(xi) and y=(yi), where the elements of these vectors are simply laid out to form a linear chromosome of length (1+n)*|E|. Note n is the number of design variables that determine the design of one joint.
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A side frame of a passenger car used in Case Study 1 (adopted from 17 with authors’ permission)
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An under body frame of a passenger car used in Case Study 2 (adopted from 17 with authors’ permission)
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Flowchart of optimal decomposition software. (a) overall flow, and (b) fitness calculation.
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Definition of basic members and potential joint locations of side frame structure
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Structural topology graph of the side frame, with nodes 0∼7 represent basic members, and edges e0∼e11 represent potential joints between two basic members
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Loading condition of basic bending requirement. Loading F=10,000 [N], which is the weight of a typical passenger vehicle 17.
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Definition of DISPLACEMENTS (G,y) USED IN CASE STUDY 1. OVERALL DISPLACEMENT OF SIDE FRAME IS max{d1,d2}, WHERE d1 AND d2 are the displacements of upper and lower right corners of the door, respectively, measured with respect to un-deformed door geometry attached to deformed hinge OP1.
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Baseline result. (a) one piece structure (k=1) and (b) its deformation with DISPLACEMENTS=1.411 [mm].
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Baseline result. (a) fully decomposed structure (k=9) and (b) its deformation with DISPLACEMENTS=8.251 [mm].
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4-component decomposition (k=4) with constant joint rates in Table 2 (Case 1-1). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.075 [mm]. Ci IN (A) INDICATES i-th component.
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5-component decomposition (k=5) with constant joint rates in Table 2 (Case 1-1). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.109 [mm]. Ci IN (A) INDICATES i-th component.
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4-component decomposition (k=4) WITH VARIABLE JOINT RATES (CASE 1-2). (A) OPTIMAL DECOMPOSITION AND (B) ITS DEFORMATION WITH DISPLACEMENTS =0.062 [mm]. The number at each joint in (a) indicates the optimal joint rate in [104Nm/rad].Ci in (a) indicates i-th component.
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5-component decomposition (k=5) with variable joint rates (Case 1-2). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.065 [mm]. The number at each joint in (a) indicates the optimal joint rates in [104Nm/rad].Ci in (a) indicates i-th component.
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Evolution history of two typical cases. The solid line indicates the history of 4-component decomposition (k=4) with variable joint rates, and the dotted line indicates the 5-component decomposition (k=5) with variable joint rates. The fitness value converges approximately after 200 generations, which is used as the termination criterion (Case 1-2).
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Definition of basic members and potential joint locations of under body frame structure
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Structural topology graph of the under body side frame, with nodes 0∼12 represent basic members, and edge e0∼e17 represent potential joints between two basic members
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Loading condition of torsion load case. Applied torque value T is 3750 [Nm] assuming the static wheel reaction in the pure torsion analysis case (Eq. (12)) 17.
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Definition of DISPLACEMENTS (G,y) USED IN CASE STUDY 2. OVERALL DISPLACEMENT OF UNDER BODY FRAME IS ϕ=Δh/w, WHERE Δh AND w are the vertical distance between P1 and Q1 and width of under body frame, respectively.
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Baseline result. (a) one piece (k=1) STRUCTURE AND (B) ITS DEFORMATION WITH DISPLACEMENTS =0.3488 [rad].
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Baseline result. (a) fully decomposed structure (k=13) and (b) its deformation with DISPLACEMENTS =0.9825 [rad].
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6-component decomposition (k=6) with constant joint rate in Table 6 (Case 2-1). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.3724 [rad]. Ci IN (A) INDICATES i-th component.
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7-component decomposition (k=7) with constant joint rate in Table 6 (Case 2-1). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.3825 [rad]. Ci IN (A) INDICATES i-th component.
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6-component decomposition (k=6) WITH VARIABLE JOINT RATES (CASE 2-2). (A) OPTIMAL DECOMPOSITION AND (B) ITS DEFORMATION WITH DISPLACEMENTS =0.3696 [rad]. The value at each joint in (a) indicates the optimal joint rate (kx,ky,kz) IN [104Nm/rad].Ci in (a) indicates i-th component.
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7-component decomposition (k=7) with variable joint rates (Case 2-2). (a) optimal decomposition and (b) its deformation with DISPLACEMENTS =0.3724 [rad]. The value at each joint in (a) indicates the optimal joint rate (kx,ky,kz) IN [104Nm/rad].Ci in (a) indicates i-th component.
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Evolution history of two typical cases. The solid line indicates the history of 6-component decomposition (k=6) with variable joint rates, and the dotted line indicates the 7-component decomposition (k=7) with variable joint rates. The fitness value converges approximately after 200 generations, which is used as the termination criterion (Case 2-2).

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