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  • 1
    Electronic Resource
    Electronic Resource
    Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations (1991)
    Springer
    Nonlinear dynamics 2 (1991), S. 1-34 
    Details
    ISSN: 1573-269X
    Keywords: Autoparametric resonance ; composite beams ; flapwise excitations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045053
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  • 2
    Electronic Resource
    Electronic Resource
    Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations (1991)
    Springer
    Nonlinear dynamics 2 (1991), S. 137-156 
    Details
    ISSN: 1573-269X
    Keywords: Autoparametric resonance ; composite beams ; chordwise excitations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053833
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  • 3
    Electronic Resource
    Electronic Resource
    A nonlinear composite shell theory (1992)
    Springer
    Nonlinear dynamics 3 (1992), S. 431-463 
    Details
    ISSN: 1573-269X
    Keywords: Anisotropic shell ; geometric nonlinearity ; third-order shear deformation theory ; extension-bending-shear vibrations ; general nonlinear equations of motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045647
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  • 4
    Electronic Resource
    Electronic Resource
    A nonlinear composite beam theory (1992)
    Springer
    Nonlinear dynamics 3 (1992), S. 273-303 
    Details
    ISSN: 1573-269X
    Keywords: Composite beams ; geometric nonlinearity ; third-order shear deformation theory ; extensional-flexural-flexuraltorsional-shearing-shearing vibrations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045486
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