Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems

We apply two analytical approaches to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane mode. First, we apply the method of multiple scales directly to the governing two integral-partial-differential equations and associated boundary conditions. Reconstitution of the solvability conditions at second and third orders leads to a system of four coupled non-linear complex-valued equations describing the modulation of the amplitudes and phases of the interacting modes. The homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problem are needed to make the reconstituted modulation equations derivable from a Lagrangian. However, this procedure leads to an indeterminacy, indicating a likely inconsistency with this specific application of the method of multiple scales. Then, we apply the method to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problems are required to make the reconstituted modulation equations derivable from a Lagrangian. Frequency–response curves obtained using the two generated asymptotic models, for a specific choice of the arbitrary constant appearing in both models, show different qualitative as well as quantitative predictions for some classes of motions. The effects of an inconsistent reconstitution in the direct approach are also investigated.

Multiple resonances in suspended cables: direct versus reduced-order models

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integro-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach.

Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second...

Multimode Interactions in Suspended Cables

The nonlinear, single-mode responses of a fixed-fixed, buckled beam are investigated under the case of a uniform, transverse, harmonic excitation. In order to avoid axial slipping and to obtain meaningful data, a clamping apparatus was designed to maximize the clamping force applied to the beam. To fully characterize the single-mode responses, data were obtained at various levels of buckling up to 3.3 times the thickness of the beam. The data demonstrate that at a low level of buckling, supercritical period doubling occurs during an amplitude sweep in which the first mode is directly excited. However, as the buckling level increases, the period-doubling bifurcation becomes subcritical during such amplitude sweeps. In addition, a period-five motion, broadband responses, and responses with an unexplained sideband structure were observed.

https://link.springer.com/content/pdf/10.1023/A:1008231012968.pdf

Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam

The linear modes of vibration of buckled beams are investigated analytically and experimentally. An exact solution is obtained for the linear modes and associated frequencies of initially buckled beams with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions. The analytical solution is validated experimentally for the fixed-fixed case. The analytically obtained natural frequencies are in excellent agreement with those obtained experimentally. These exact linear modes provide a basis from which to study the nonlinear vibrations of buckled beams.

An analytical and experimental investigation of the natural frequencies and mode shapes of buckled beams

Experimental investigation of single-mode responses in a fixed-fixed buckled beam

http://asmedigitalcollection.asme.org/IDETC-CIE/proceedings-pdf/doi/10.1115/DETC1995-0274/6655341/345_1_detc1995-0274.pdf

Wayne Kreider

Papers

Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam

Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam

An analytical and experimental investigation of the natural frequencies and mode shapes of buckled beams

Experimental investigation of single-mode responses in a fixed-fixed buckled beam

Two-to-One Internal Resonances in Buckled Beams