Explanation of the self-adaptive dynamics of a harmonically forced beam with a sliding mass

The self-adaptive behavior of a clamped–clamped beam with an attached slider has been experimentally demonstrated by several research groups. In a wide range of excitation frequencies, the system shows its signature move: The slider first slowly moves away from the beam’s center, at a certain point the vibrations jump to a high level, then the slider slowly moves back toward the center and stops at some point, while the system further increases its high vibration level. In our previous work, we explained the unexpected movement of the slider away from the beam’s vibration antinode at the center by the unilateral and frictional contact interactions permitted via a small clearance between slider and beam. However, this model did not predict the signature move correctly. In simulations, the vibration level did not increase significantly and the slider did not turn around. In the present work, we explain, for the first time, the complete signature move. We show that the timescales of vibration and slider movement along the beam are well separated, such that the adaptive system closely follows the periodic vibration response obtained for axially fixed slider. We demonstrate that the beam’s geometric stiffening nonlinearity, which we neglected in our previous work, is of utmost importance for the vibration levels encountered in the experiments. This stiffening nonlinearity leads to coexisting periodic vibration responses and to a turning point bifurcation with respect to the slider position. We associate the experimentally observed jump phenomenon to this turning point and explain why the slider moves back toward the center and stops at some point.

Coupling of Geometric and Material Stiffening Mechanisms in Origami Design

Origami, the ancient art of paper folding, has recently garnered attention from the scientific community for its capacity for unique 2D – 3D shape change and programmable mechanical properties. Application areas of such properties include packaging, self-assembly, shock absorption and deployable structures. Recent studies have highlighted the role of the folded geometry to regulate the mechanical response of the origami structures, such as the increased compression stiffness of origami tubes or the tunable in-plane stiffness through select inversion of bi-stable fold vertices. In addition to geometric re-enforcement, the mechanical response of an origami structure can also be programmed through spatial patterning of the individual fold line stiffnesses. However, the coupling between the geometric and material stiffening design spaces for origami structures is poorly understood and design rules are needed to guide the use of material stiffening to enhance or mitigate a geometric stiffening effect. In this computational study, a modal analysis of a corrugated fold with varying degrees of pre-fold and different sets of fold stiffness distributions is evaluated to highlight the interaction between geometric and material stiffness mechanisms.

Study on the Stress-Stiffening Effect and Modal Synthesis Methods for the Dynamics of a Spatial Curved Beam

In this paper, based on the nonlinear strain–deformation relationship, the dynamics equation of a spatial curved beam undergoing large displacement and small deformation is deduced using the finite-element method of floating frame of reference (FEMFFR) and Hamiltonian variation principle. The stress-stiffening effect, which is also called geometric stiffening effect, is accounted for in the dynamics equation, which makes it possible for the dynamics simulation of the spatial curved beam with high rotational speed. A numerical example is carried out by using the deduced dynamics equation to analyze the stress-stiffening effect of the curved beam and then verified by abaqus software. Then, the modal synthesis methods, which result in much fewer numbers of coordinates, are employed to improve the computational efficiency.

Period Doubling Motions of a Nonlinear Rotating Beam at 1:1 Resonance

A rotating beam subjected to a torsional excitation is studied in this paper. Both quadratic and cubic geometric stiffening nonlinearities are retained in the equation of motion, and the reduced model is obtained via the Galerkin method. Saddle-node bifurcations and Hopf bifurcations of the period-1 motions of the model were obtained via the higher order harmonic balance method. The period-2 and period-4 solutions, which are emanated from the period-1 and period-2 motions, respectively, are obtained by the combined implementation of the harmonic balance method, Floquet theory, and Discrete Fourier transform (DFT). The analytical periodic solutions and their stabilities are verified through numerical simulation.

MODEL REDUCTION WITH GEOMETRIC STIFFENING NONLINEARITIES FOR DYNAMIC SIMULATIONS OF MULTIBODY SYSTEMS

This work investigates the implementation of nonlinear model reduction to flexible multibody dynamics. Linear elastic theory will lead to instability issues with rotating beam-like structures, due to the neglecting of the membrane-bending coupling on the beam cross-section. During the past decade, considerable efforts have been focused on the derivation of geometric nonlinear formulation based on nodal coordinates. In order to reduce the computation cost in flexible multibody dynamics, which is extremely important for complex large system simulations, modal reduction is usually implemented to a rotating flexible structure with geometric nonlinearities retained in the model. In this work, a standard model reduction process based on matrix operation is developed, and the essential geometric stiffening nonlinearities are retained in the reduced model. The time responses of a tip point on a rotating Euler–Bernoulli blade are calculated based on two nonlinear reduced models. The first reduced model is derived by the standard matrix operation from a full finite element model and the second reduced model is obtained via the Galerkin method. The matrix operation model reduction process is validated through the comparison of the simulation results obtained from these two different reduced models. An interesting phenomenon is observed in this work: In the nonlinear model, if only quadratic geometric stiffing term is retained, the two reduced models converge to the full finite element model with only one bending mode and two axial modes. While if both quadratic and cubic geometric stiffing terms are retained in the nonlinear equation, the modal-based reduced model will not converge to the finite element model unless all eigenmodes are retained, that is the reduced model has no degree of freedom reduction at all.

Large deflection effects in flexible energy harvesters

The effect of large deflection on the mechanical and electrical behaviors of flexible piezoelectric energy harvesters has not been well studied. A generalized nonlinear coupled finite element circuit simulation approach is presented in this article to study the performance of energy harvesters subjected to large deflections. The method presented is validated experimentally using three test examples consisting of (a) a static case, (b) a free vibration case, and (c) a forced vibration case. Under static conditions (when the transverse tip deflection exceeds a quarter of the cantilever length), large deflections cause geometric stiffening of the structure that reduces the tip deflection of the generator when compared to linear (i.e. small-deflection) behavior. For a cantilever generator under dynamic conditions, geometric stiffening, inertial softening, and nonlinear damping effects become significant. Large deflections both shift the resonant frequency and increase damping and can thus cause a significant reduction in output voltage when compared with small-deflection linear theory. In the finite element generator model studied in this article, these nonlinear dynamic effects become significant when the transverse tip deflection exceeds 35% of the beam length.

On Stability of a Nonlinear, Periodically Time Varying, Rotating Blade

This paper discusses the stability of a periodically time-varying, spinning blade with cubic geometric nonlinearity. The modal reduction method is adopted to simplify the nonlinear partial differential equations to the ordinary differential equations, and the geometric stiffening is approximated by the axial inertia membrane force. The method of multiple time scale is employed to study the steady state motions, the corresponding stability and bifurcation for such a periodically time-varying rotating blade. The backbone curves for steady-state motions are achieved, and the parameter map for stability and bifurcation is developed. Illustration of the steady-state motions is presented for an understanding of rotational motions of the rotating blade.