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Here I(z) = th3 12 I(y) = ht3 12 J= ht3 3 (4)
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Using Eq. (4), we may write Eq. (3) as ( M)2 or, from Eq. (2), ( M)2 2EG 6M (6L)2 h3
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2EG t (6L)2 h
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1 (th)4 1 (t/h)2
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1 1 [(6M/(h3 )]2
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6M h
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Since we seek to minimize th and maximize h, it may be seen from Eqs. (5) and (6) that the inequality sign may be replaced by the equality sign in those two equations. In Eq. (6), h is the only unspecified quantity. Further, since the square of t/h may be expected to be small compared to unity, we can obtain substantially simpler approximations of reasonable accuracy. As a first step, we have 1 t =1+ 1 (t/h)2 h
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If we retain only the first two terms in the right side of Eq. (7), we have M = (EG)1/2 6M 1+ 3 6L h
6M h3
6M h
If we also neglect the square of t/h in comparison with unity, we obtain h= (6M)2 (EG)1/2 L 3
as a reasonable first approximation. Thus if we take the factor of safety as 1.5, we have, for a steel member with E = 30 Mpsi, G = 12 Mpsi, and = 30 kpsi, h= (36)[(30 106 )(12 106 )]1/2 M 2 (1.5)(30 000) L
= 8.62
M2 L
(10)
This is a reasonable approximation to the optimal height of the beam cross section. It may also be used as a starting point for an iterative solution to the exact expression, Eq. (6). For the purpose of iteration, we rewrite Eq. (6) as h= (6M)2 L 3 EG 1 [(6M)/(h3 )]2
1/2 1/5
(11)
The value of h obtained from Eq. (9) is substituted into the right side of Eq. (11).The resultant value of h thus obtained is then resubstituted into the right side of Eq. (11); the iterative process is continued until the computed value of h coincides with the value substituted into the right side to the desired degree of accuracy. Having determined h, we can determine t from Eq. (2).
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INSTABILITIES IN BEAMS AND COLUMNS 30.18
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REFERENCES
30.1 H. Herman, On the Analysis of Uniform Prismatic Columns, Transactions of the ASME, Journal of Mechanical Design, vol. 103, 1981, pp. 274 276. 30.2 C. R. Mischke, Mathematical Model Building, 2d rev. ed., Iowa State University Press, Ames, 1980. 30.3 American Institute of Steel Construction, LRFD Manual of Steel Construction, 3d ed., AISC, Chicago, 2003.
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Source: STANDARD HANDBOOK OF MACHINE DESIGN
VIBRATION AND CONTROL OF VIBRATION
T. S. Sankar, Ph.D., Eng.
Professor and Chairman Department of Mechanical Engineering Concordia University Montreal, Quebec, Canada
R. B. Bhat, Ph.D.
Associate Professor Department of Mechanical Engineering Concordia University Montreal, Quebec, Canada
31.1 INTRODUCTION / 31.1 31.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS / 31.1 31.3 SYSTEMS WITH SEVERAL DEGREES OF FREEDOM / 31.19 31.4 VIBRATION ISOLATION / 31.28 REFERENCES / 31.30
31.1 INTRODUCTION
Vibration analysis and control of vibrations are important and integral aspects of every machine design procedure. Establishing an appropriate mathematical model, its analysis, interpretation of the solutions, and incorporation of these results in the design, testing, evaluation, maintenance, and troubleshooting require a sound understanding of the principles of vibration. All the essential materials dealing with various aspects of machine vibrations are presented here in a form suitable for most design applications. Readers are encouraged to consult the references for more details.
31.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS
31.2.1 Free Vibration A single-degree-of-freedom system is shown in Fig. 31.1. It consists of a mass m constrained by a spring of stiffness k, and a damper with viscous damping coefficient c. The stiffness coefficient k is defined as the spring force per unit deflection. The coef-
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