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Abstract

Deformations of a rotating beam (of a circular cross-section and the lenght L = 1) may be described (after an appropriate choice of a time scale) by the differential equation
$$ {{z}_{{tt}}} + {{z}_{{xxxx}}} + a(t,x){{z}_{{xxx}}} + b(t,x){{z}_{{xx}}} + c(t,x){{z}_{x}} + d(t,x)z + + e(t,x){{z}_{t}} = f(t,x,z,{{z}_{t}},{{z}_{x}},{{z}_{{xx}}},{{z}_{{xxx}}}) $$
(1)
where z is a complex function of variables t and x defined for t ≥ 0 and x ∈ <0, 1> (see [2], [3]). The coefficients a, b, c, d, e as well as the nonlinear function f represent forces and moments acting on the beam. The function z is supposed to satisfy boundary conditions
$$ z(t,0) = {{v}_{1}}{{z}_{{xx}}}(t,0) + {{v}_{2}}{{z}_{x}}(t,0) = 0, $$
(20)
$$ z(t,1) = {{v}_{3}}{{z}_{{xx}}}(t,1) + {{v}_{4}}{{z}_{x}}(t,1) = 0 $$
(21)
for t ≥ 0. We assume that | v1| + |v2| > 0, |v3| + |v4| > 0. The function f is assumed to be continuous and bounded together with its second derivatives with respect to x, z, z t , z x , z xx , z xxx on the set A = = <0,+ ∞) × <0,1> } <- R, R>5 (where R is a positive real number).

Keywords

Real Axis Elastic System Zero Solution Fundamental System Satisfy Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Neustupa J., (1983) The linearized uniform asymptotic stability of evolution differential equations. Czech. Math. J. 34, 257–284Google Scholar
  2. [2]
    Vejvoda O. et al., (1981) Partial differential solutions. Sijthoof & Noordhoff International Publishers B.V., Alphen aan den Rijn, NetherlandsGoogle Scholar
  3. [3]
    Volmir A.S., (1963) Stability of elastic systems. Gos. Izdat. Fyz.- Mat. Lit., Moscow, USSR (Russian)Google Scholar

Copyright information

© Springer Basel AG 1991

Authors and Affiliations

  • Jiří Neustupa
    • 1
  1. 1.Faculty of Mechanical EngineeringCzech Technical UniversityPraha 6CSFR

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