Large Amplitude Free Flexural Vibration of Structures
01 Apr 1987-Journal of Reinforced Plastics and Composites (SAGE Publications)-Vol. 6, Iss: 2, pp 153-161
TL;DR: In this article, a brief history of the analysis of large amplitude vibration is given, and nonlinearities exhibited by beams, plates, rings, and shells undergoing large amplitude vibrations are discussed.
Abstract: First, a brief history of the development of the analyses of large amplitude vibration is given. Second, nonlinearities exhibited by beams, plates, rings, and shells undergoing large amplitude vibrations are discussed. It is shown that, in case of large amplitude vibra tions, the modal equations for multilayered beams and plates are qualitatively the same as the modal equation for shells.
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TL;DR: A review of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations is presented in this paper, where a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials.
Abstract: A summary of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations in presented. First, a review of the recent studies on the free-vibration analysis of symmetrically laminated plates is given. These studies have been conducted for various geometric shapes and edge conditions. Both analytical (closed-form, Galerkin, Rayleigh-Ritz) and numerical methods have been used. Because of the importance of unsymmetrically laminated structural components in many applications, a detailed review of the various developments in the analysis of unsymmetrical ly laminated beams and plates also is given. A survey of the nonlinear vibrations of the perfect and geometrically laminated plates is presented next. It is seen that due to the bending-stretching coupling, the nonlinear behavior of the unsymmetrically laminated perfect and imperfect plates, depending upon the boundary conditions, may be hardening or softening type. Similar behavior also is observed for imperfect isotropic and laminated plates. Lastly, the developments in studying the wave propagation in laminated materials are reviewed. It is seen that a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials. Some recent studies on the linear and nonlinear transient response of laminated materials also are described.
288 citations
TL;DR: A model to explain complex nonminimum phase (CNMP) zeros seen in the noncollocated frequency response of a large-displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFMs) as buildingblocks demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros.
Abstract: This paper presents a model to explain complex nonminimum phase (CNMP) zeros seen in the noncollocated frequency response of a large-displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFMs) as buildingblocks. Geometric nonlinearities associated with large displacement along with the kinematic under-constraint in the DPFM lead to a coupling between the X and Y direction displacements. Via a lumped-parameter model that captures the most relevant geometric nonlinearity, it is shown that specific combinations of the operating point (i.e., flexure displacement) and mass asymmetry (due to manufacturing tolerances) give rise to CNMP zeros. This model demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros. Furthermore, a study of how the eigenvalues and eigenvectors of the flexure mechanism vary with the operating point and mass asymmetry indicates the presence of curve veering when the system transitions from minimum phase to CNMP. Based on this, the hypothesis of an inherent correlation between CNMP zeros and curve veering is proposed. [DOI: 10.1115/1.4036032]
9 citations
Cites background from "Large Amplitude Free Flexural Vibra..."
...The impact of these nonlinearities on the dynamics of flexible beams and structures has been studied extensively, as reported in the review papers by Modi [24] and Pandalai [25]....
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12 Oct 2016
TL;DR: In this paper, a model to explain complex non-minimum phase (CNMP) zeros seen in the non-collocated frequency response of a large displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFM) as building-blocks, is presented.
Abstract: This paper presents a model to explain complex nonminimum phase (CNMP) zeros seen in the non-collocated frequency response of a large displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFM) as building-blocks. Geometric nonlinearities associated with large displacement along with the kinematic under-constraint in the DPFM, lead to a coupling between the X and Y direction displacements. Via a lumpedparameter model that captures the most relevant geometric nonlinearity, it is shown that specific combinations of the operating point (i.e. flexure displacement) and mass asymmetry (due to manufacturing tolerances) give rise to CNMP zeros. This model demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros. Furthermore, a study of how the eigenvalues and eigenvectors of the flexure mechanism vary with the operating point and mass asymmetry indicates the presence of curve veering when the system transitions from minimum phase to CNMP. Based on this, the hypothesis of an inherent correlation between CNMP zeros and curve veering is proposed.
3 citations
Cites background from "Large Amplitude Free Flexural Vibra..."
...The impact of these non-linearities on the dynamics of flexible beams and structures has been studied extensively, as reported in the review papers by Modi [24] and Pandalai [25]....
[...]
Dissertation•
01 Jan 2017
2 citations
Cites background from "Large Amplitude Free Flexural Vibra..."
...While the resulting non-linear equations of dynamics are solved in time-domain via perturbation, homotopy, or computational methods, this prior work [32] does not pursue the frequency domain investigation relevant to the present work....
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...The impact of these non-linearities on the dynamics of flexible beams and structures has been studied extensively, as reported in the review papers by Modi [31] and Pandalai [32]....
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References
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TL;DR: In this article, the authors derived general conclusions regarding the non-linear vibration of structural components like curved beams, rings and thin shells from the study of two specific examples, the circular ring and shallow spherical shell, and showed that with careful judgment in the use of mode shapes of one or more terms, the resulting modal equations help one to appreciate much better the physics of the problem.
18 citations
"Large Amplitude Free Flexural Vibra..." refers background in this paper
...The f vs T behaviour as shown in the figure represents nonlinearity of the hardening or softening type depending upon whether the time for half-period TB is greater or less than the half-period T* corresponding to the linear theory [4]....
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...A typical plot of (bla)2 vs (cla) as given in [4], which gives the transition curve for an assumed initial amplitude of fA = 1....
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...) [2], [4] Softening type of nonlinearity or hardening type as in the case of cylindrical shells where the ratio of the axial to circumferential half-waves of the vibration mode is large....
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TL;DR: In this article, the large amplitude (non-linear) free flexural vibration of thin, elastic, orthotropic skew plates clamped along all four edges was analyzed using the Galerkin's method.
15 citations
Additional excerpts
...FLAT PLATES: [1], [2], [3] (Rectangular, circular, elliptical, triangular, skew, sandwich type, etc....
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01 Jan 1973
TL;DR: In this paper, the modal equation is derived for the large amplitude flexural vibration of flat plates, and it is shown that the nonlinearity associated with this type is of the hardening type, that is, the frequency increases with amplitude.
Abstract: Use is made of the two coupled partial differential equations that govern the large amplitude flexural vibration of flat plates. On the basis of an assumed vibration mode taken to be separable in the space coordinates and time, the modal equation is obtained. In the case of a one-term solution, it is shown that the modal equation is the same as the dynamic equation of a mass-spring combination where the restoring force of the nonlinear spring is a cubic and an odd function of the displacement. Since the modal equation is a nonlinear, ordinary differential equation with positive coefficients, it is shown that the nonlinearity associated with the large amplitude flexural vibration of beams and plates is of the hardening type, that is, the frequency increases with amplitude.
7 citations