In this paper, four convex models (interval, parallelepiped, ellipsoidal, and super-ellipsoidal analyses) are introduced to describe the available data on uncertainty parameters for the engine’s crankshaft. The paper first evaluates the smallest area, such as box, ellipse, parallelogram, and super ellipse, which enclose the available data. Then, the Tchebycheff inequality is employed to inflate the uncertain domain to address the problem of forecasting data, i.e., variables taking values beyond the recorded uncertain data, as a limited amount of samples are used to construct the convex models. The minimum areas before and after the inflation are evaluated. Subsequently, the maximum stresses before and after the inflation of uncertain domain based on the areas of the convex models are obtained. The domain that predicts the minimum of the maximum stresses is declared as the best bounding figure, along with the attendant uncertainty model.

References

References
1.
Moore
,
R. E.
,
1979
,
Methods and Applications of Interval Analysis
,
Prentice Hall
,
London
.
2.
Moore
,
R. E.
,
Kearfott
,
R. B.
, and
Cloud
,
M. J.
,
2009
,
Introduction to Interval Analysis
,
SIAM Press
,
Philadelphia
.
3.
Ben-Haim
,
Y.
, and
Elishakoff
,
I.
,
1990
,
Convex Models of Uncertainty in Applied Mechanics
,
Elsevier
,
Amsterdam
.
4.
Chernousko
,
F. L.
,
1993
,
State Estimation of Dynamic Systems
,
CRC Press
,
Boca Raton
.
5.
Jiang
,
C.
,
Zhang
,
Q. F.
,
Han
,
X.
,
Liu
,
J.
, and
Hu
,
D. A.
2015
, “
Multidimensional Parallelepiped Model: A New Type of Non-Probabilistic Convex Model for Structural Uncertainty Analysis
,”
Int. J. Numer. Methods
,
103
(
1
), pp.
31
59
.10.1002/nme.4877
6.
Jiang
,
C.
,
Zhang
,
Q. F.
,
Han
,
X.
, and
Qian
,
Y. H.
,
2014
, “
A Non-Probabilistic Structural Reliability Analysis Method Based on a Multidimensional Parallelepiped Convex Model
,”
Acta Mech.
,
225
(
2
), pp. 
383
395
.10.1007/s00707-013-0975-2
7.
Elishakoff
,
I.
, and
Bekel
,
Y.
,
2013
, “
Application of Lamé’s Super Ellipsoids to Model Initial Imperfections
,”
ASME J. Appl. Mech.
,
80
(
6
), p. 
061006-9
.10.1115/1.4023679
8.
Elishakoff
,
I.
, and
Ohsaki
,
M.
,
2010
,
Optimization and Anti-optimization of Structures under Uncertainty
,
Imperial College Press
,
London
.
9.
Elishakoff
,
I.
, and
Elettro
,
F.
,
2014
, “
Interval, Ellipsoidal, and Super-Ellipsoidal Calculi for Experimental and Theoretical Treatment of Uncertainty: Which One Ought to be Preferred?
Int. J. Solids Struct.
,
51
(
7
), pp. 
1576
1586
.10.1016/j.ijsolstr.2014.01.010
10.
Jiang
,
C.
,
Ni
,
B. Y.
,
Han
,
X.
, and
Tao
,
Y. R.
,
2014
, “
Non-Probabilistic Convex Model Process: A New Method of Time-Variant Uncertainty Analysis and its Application to Structural Dynamic Reliability Problems
,”
Comput. Methods Appl. Mech. Eng.
,
268
(
1
), pp. 
656
676
.10.1016/j.cma.2013.10.016
11.
Guan
,
F. J.
,
Han
,
X.
, and
Jiang
,
C.
,
2008
, “
Uncertain Optimization of Engine Crankshaft Using Interval Methods
,”
Eng. Mech.
,
25
(
9
), pp. 
198
202
(in Chinese).
12.
Hang
,
X. P.
,
Jiang
,
C.
, and
Han
,
X.
,
2013
, “
A Time-Variant Reliability Analysis Method for Non-Linear Limit-State Functions
,”
Chin. J. Theor. Appl. Mech.
,
46
(
2
), pp. 
264
272
(in Chinese).
13.
Muhanna
,
R. L.
, and
Mullen
,
R. L.
,
2001
, “
Uncertainty in Mechanics Problems––Interval Based Approach
,”
J. Eng. Mech. ASCE
,
127
(
6
), pp. 
557
566
.10.1061/(ASCE)0733-9399(2001)127:6(557)
14.
Rao
,
S. S.
, and
Berke
,
L.
,
1997
, “
Analysis of Uncertain Structural Systems Using Interval Analysis
,”
AIAA J.
,
35
(
4
), pp. 
727
735
.10.2514/2.164
15.
Elishakoff
,
I.
, and
Miglis
,
Y.
,
2012
, “
Overestimate-Free Computation Version of Interval Analysis
,”
Int. J. Comput. Methods Eng. Sci. Mech.
,
13
(
5
), pp. 
319
328
.10.1080/15502287.2012.683134
16.
Zhu
,
L. P.
,
Elishakoff
,
I.
, and
Starnes
,
J. H.
,
1996
, “
Derivation of Multi-Dimensional Ellipsoidal Convex Model for Experimental Data
,”
Math. Comput. Model.
,
24
(
2
), pp. 
103
114
.10.1016/0895-7177(96)00094-5
17.
Silverman
,
B. W.
, and
Titterington
,
D. M.
,
1980
, “
Minimum Covering Ellipses
,”
SIAM J. Sci. Stat. Comput.
,
1
(
4
), pp. 
401
409
.10.1137/0901028
18.
Rublev
,
B. V.
, and
Petunin
,
Y. I.
,
1998
, “
Minimum-Area Ellipse Containing a Finite Set of Points, II
,”
Ukrainian Math. J.
,
50
(
8
), pp. 
1253
1261
.10.1007/BF02513081
19.
Feynman
,
R. P.
,
1986
,
Personal Observations on the Reliability of the Shuttle, Report of the Presidential Commission on the Space Shuttle Challenger Accident
(Appendix F, Vol. 
1
),
U.S. Government Printing Office
,
Washington, DC
.
20.
Wikipedia
,
2013
,
Tchebycheff Inequality
, Available: http://en.wikipedia.org/wiki/Tchebycheff_inequality (Accessed: Dec. 23, 2013).
21.
Wikipedia
,
2013
,
Superellipse
, Available: http://en.wikipedia.org/wiki/Superellipse (Accessed: Dec. 22, 2013).
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