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Analytical solution for the problem of maximum exit velocity under Coulomb friction in gravity flow discharge chutes

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Abstract

In this paper, an analytical solution for the problem of finding profiles of gravity flow discharge chutes required to achieve maximum exit velocity under Coulomb friction is obtained by application of variational calculus. The model of a particle which moves down a rough curve in a uniform gravitational field is used to obtain a solution of the problem for various boundary conditions. The projection sign of the normal reaction force of the rough curve onto the normal to the curve and the restriction requiring that the tangential acceleration be non-negative are introduced as the additional constraints in the form of inequalities. These inequalities are transformed into equalities by introducing new state variables. Although this is fundamentally a constrained variational problem, by further introducing a new functional with an expanded set of unknown functions, it is transformed into an unconstrained problem where broken extremals appear. The obtained equations of the chute profiles contain a certain number of unknown constants which are determined from a corresponding system of nonlinear algebraic equations. The obtained results are compared with the known results from the literature.

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Abbreviations

m :

Mass of the particle

\({\overrightarrow{v}}\) :

Velocity of the particle

\({\overrightarrow{a}}\) :

Acceleration of the particle

\({\overrightarrow{t}}\) :

Unit vector of the tangent

\({\overrightarrow{u}}\) :

Unit vector

v :

Projection of \({\overrightarrow{v}}\) onto \({\overrightarrow{t}}\)

μ :

Coefficient of friction

g :

Acceleration of gravity

\({\overrightarrow{N}}\) :

Normal component of the constraint reaction force

\({\overrightarrow{F}_{\mu}}\) :

Coulomb friction force

\({\varphi}\) :

Slope angle of the tangent

\({\dot{w}_{i}\,(i=1,2)}\) :

Slack variables

λ i  (i = 1, . . . , 5):

Lagrange multipliers

J :

Functional

F :

Integrand of the functional

Δ:

Noncontemporaneous variation

||:

Absolute value

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  1. Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36000, Kraljevo, Serbia

    Slaviša Šalinić

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  1. Slaviša Šalinić
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Correspondence to Slaviša Šalinić.

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Šalinić, S. Analytical solution for the problem of maximum exit velocity under Coulomb friction in gravity flow discharge chutes. Arch Appl Mech 80, 1149–1161 (2010). https://doi.org/10.1007/s00419-010-0432-9

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  • DOI: https://doi.org/10.1007/s00419-010-0432-9

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