Applied Calculus of Variations for Engineers, Second Edition by Louis Komzsik

By Louis Komzsik

The objective of the calculus of diversifications is to discover optimum suggestions to engineering difficulties whose optimal could be a certain amount, form, or functionality. Applied Calculus of diversifications for Engineers addresses this significant mathematical region acceptable to many engineering disciplines. Its exact, application-oriented method units it except the theoretical treatises of so much texts, because it is geared toward improving the engineer’s realizing of the topic.

This Second Edition text:

  • Contains new chapters discussing analytic recommendations of variational difficulties and Lagrange-Hamilton equations of movement in depth
  • Provides new sections detailing the boundary essential and finite aspect equipment and their calculation techniques
  • Includes enlightening new examples, akin to the compression of a beam, the optimum pass portion of beam less than bending strength, the answer of Laplace’s equation, and Poisson’s equation with a variety of methods

Applied Calculus of diversifications for Engineers, moment variation extends the gathering of suggestions helping the engineer within the program of the suggestions of the calculus of variations.

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Additional info for Applied Calculus of Variations for Engineers, Second Edition

Example text

Y ∂x ∂x ∂y ∂x∂y ∂x ∂y This is the differential equation of minimal surfaces, originally obtained by Lagrange himself. The equation is mainly of verification value as this is one of the most relevant examples for the need of a numerical solution. Most of the problems of finding minimal surfaces are solved by Ritz type methods, the subject of Chapter 6. 1, whose equation is z = x3 − 2xy 2 . It is easy to verify that this satisfies the equation. The figure also shows the level curves of the surface projected to the x − y plane.

The area under any curve going from the start point to the endpoint in the upper half-plane is x1 I(y) = ydx. x0 The constraint of the given length L is presented by the equation x1 1 + y 2 dx = L. J(y) = x0 The Lagrange multiplier method brings the function h(x, y, y ) = y(x) + λ 1 + y 2. The constrained variational problem is x1 I(y) = h(x, y, y )dx x0 whose Euler-Lagrange equation becomes 1−λ d dx y 1+y2 = 0. Integration yields λy 1+y2 © 2009 by Taylor & Francis Group, LLC = x − c1 . 30 Applied calculus of variations for engineers First we separate the variables dy = ± λ2 x − c1 − (x − c1 )2 dx, and integrate again to produce y(x) = ± λ2 − (x − c1 )2 + c2 .

Another integration results in the solution of the so-called catenary curve y=− λ c1 ρ(x − c2 ) − cosh( ), ρ ρ c1 with c2 being another constant of integration. The constants of integration may be determined by the boundary conditions albeit the calculation, due to the presence of the hyperbolic function, is rather tedious. Let us consider the specific case of the suspension points being at the same height and symmetric with respect to the origin. This is a typical engineering scenario for the span of suspension cables.

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