Narrow Results

This short paper takes a close look at a relatively simple harmonically-excited mechanical oscillator. Throughout the range of forcing frequencies the basins of attraction are investigated by applying strong perturbations to steady-state behavior. In this way, a more general solution space is mapped out. Numerical simulation of the equation of motion agrees very closely with data generated from a laboratory experiment.

Based on the Euler beam theory and a Galerkin formulation using natural modes, the nonlinear vibration behavior and stability of electrostatic driving fluid-conveying micro (straight or curved) beams are studied in the paper. The focus of this study is on the critical coupling of fluidic, mechanical and electrostatic effects in the nonlinear system. Under these effects, micro devices may exhibit (dynamic) snap-through or/and pull-in instabilities. Our study reveals, for the first time, the effects of the velocity of the inner fluid on the bifurcation diagrams for complex nonlinear systems. It is also found that fluid can be utilized to efficiently tune the frequency of straight beams over a wide range. For curved beams, the tuning can be achieved easily by adjusting the voltage. These findings are beneficial in many applications of the electrostatic microelectronic mechanical systems (MEMS). In addition, phase plane analyses are performed in this study, and more complicated phase portraits for different initial conditions are obtained as well. It is found that the homoclinic connections on the phase plane are directly related to the dynamic snap-through or dynamic pull-in instabilities; and the periodic orbits are directly related to the periodic motions of micro beams. These findings can provide reasonable explanations for the experimentally-observed phenomena for micro sensors, and are beneficial to the optimization design of MEMS.

The curse of dimensionality looms over many studies in science and engineering. Low-order systems provide conceptual clarity but often fail to reveal the extent of possible complexity, whereas high-order systems present a host of daunting challenges to the analyst, not least the classification and visualization of typical behavior. In this paper, we detail the behavior of systems that fall somewhere between a classification of low- and high-order.

We present both theoretical and experimental investigations into the nonlinear behavior of a couple of mechanical systems with three mechanical/structural degrees-of-freedom (DOF), with a special focus on bifurcation and multiple equilibria. Useful insight is provided by observation of transient trajectories as they meander about and between equilibria, especially revealing the influence of unstable equilibria, not normally accessible in an experimental context. For instance, the influences of index-1 saddles are mainly detected in three aspects: determining the systems capability to snap-through by generating accessible snap-though tubes, attracting nearby trajectories temporarily oscillating around it, and separating adjacent trajectories. Iso-potentials are 3D-printed to present the energy landscape. For these systems, the 3D configuration space allows considerable complexity, but is also somewhat amenable to geometric interpretation. By varying a mass/stiffness ratio as a control parameter, bifurcation structures and morphing potential energy landscapes exhibiting up to 11 equilibria are obtained. Finally, analytical and experimental studies reveal that parametric excitations can stabilize some unstable equilibria under the right amplitudes and frequencies.

This paper presents large deflection, post-buckling analysis of plane and spatial elastic frames from a dynamic point of view. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A constant arc length method combined with the modified Newton–Raphson iteration method and the extrapolation technique to improve the convergence behaviour are employed to trace the non-linear equilibrium path up to the limit point. The change in the sign of the incremental work done is used to determine the occurrence of a limit point. As the limit state being examined is passed, the previous converged solution is adopted to start the non-linear dynamic analysis based on the average acceleration of the Newmark algorithm with a slow rate of load increment and in order to trace the post-buckling load-deflection path. As a result, the snap through problem is overcome without decreasing the external load. Numerical examples are presented to demonstrate the performance of the method.

This paper presents a simplified model for predicting the nonlinear random response of flat and buckled plates. Based on a single mode representation of vibration response, r.m.s. values of the strain response to broadband excitation are evaluated for different static buckled configurations using the equivalent linearization technique. The dynamic effects on the overall strain response due to instability motion of snap-through are included. Parametric studies are performed in which the influences of the clamped and simply-supported boundaries, aspect ratio of the plate, thickness and length of the plate are considered. Using a simple single-model formula, the results of dynamic buckling motion were compared with finite-element models as well as experimental results. The comparisons between analytical results and experimental results help to assess the accuracy of the theory and the conditions under which deviations from the theory due to effects of imperfection and higher modes are significant. It is found that the theoretical model is useful for design and checking of computer results for curved plates in a slightly deflected form (initial deflection less than twice the thickness of plate) and the prediction accuracy on nonlinear analysis is higher than existing design formula based on linear response.

In this technical note, the post-buckling behavior of a simply supported elastic column with various rotational end conditions of the supports is investigated. The compressive force is applied at the tip of the column. The characteristic equation for solving the critical loads is obtained from the boundary value problem of linear systems. In the post-buckling state, a set of nonlinear differential equations with boundary conditions is established and numerically solved by the shooting method. The interesting features associated with this problem such as the limit load point, snap-through phenomenon and the secondary bifurcation point will be highlighted herein.

This paper presents the snap-through and bifurcation elastic stability analysis of nano-arch type structures with the Winkler foundation under transverse loadings by the strain gradient and stress gradient (nonlocal) theories. The equations of equilibrium are derived by using the variational method and virtual displacement theorem of minimum total potential energy. In the elastic stability analysis, von Karman's nonlinear strain component is included, with the deformation represented by a series solution. It is concluded that in general, the strain gradient theory pushes the system away from instability as compared to the classical theory. However, the nonlocal theory does the reverse and causes the system to experience instability earlier than that of the classical theory. Moreover, theories with different small-size considerations change the mechanism of instability in different ways. For example, in similar conditions, the strain gradient theory causes the system to reach a snap-through point, while the nonlocal theory causes the system to stop at a bifurcation critical point.

In this paper, we investigate the in-plane stability and post-buckling response of deep parabolic arches with high slenderness ratios subjected to a concentrated load on the apex, using the finite element implementation of a geometrically exact rod model and the cylindrical version of the arc-length continuation method enabled with pivot-monitored branch-switching. The rod model used here includes geometrically exact kinematics of the cross-section, exact kinetics, and a linear elastic constitutive law; and advantageously employs quaternion parameters to treat the cross-sectional rotations and to compute the exponential map in the configurational update of rotations. The evolution of the Frenet frame along the centroidal curve is used to determine the initial curvature of the rod. Three sets of boundary conditions, i.e. fixed–fixed (FF), fixed–pinned (FP) and pinned–pinned (PP), are considered, and arches with a wide range of rise-to-span ratios are analyzed for each set. Complete post-buckling response has been derived for all cases. The results reveal that although all the PP arches and all the FF arches (with the exception of FF arches with rise-to-span ratio less than 0.3) considered in this study buckle via bifurcation, the nature of stability of the different solution branches in the post-buckling regime differs from case to case. All FP slender parabolic arches exhibit limit-point buckling, again with several markedly different post-buckling behaviors. Also, some arches in the FF and PP case are shown to exhibit a clear bistable behavior in the post-buckled state.

This paper investigates the stability behavior of a mechanically coupled bi-stable mechanism made of two parallel and initially curved microbeams with focus on the influence of the coupled beams’ initial curvatures on the system. First, the nonlinear and coupled force–displacement equation is derived. Then, a parametric study of the coupled beams system is studied with the system categorized into different structural compartment types according to the initial curvature of the coupled beams. It is concluded that the snap-through of such a coupled bi-stable system is governed by the beams’ initial curvatures difference. The simulation results showed that these two parameters (the beams’ initial curvatures) essentially govern the structural behavior of the coupled system in satisfying the necessary structural stability condition. It is found that the smallest (critical) value of the minimum force amplitude occurs only when both initial beams’ mid-point elevations are equal to each other. Furthermore, it is shown that any probability to increase or decrease the curvature of any beam will alter the nonlinear behavior of the coupled beam system from a simple regular snap-through to a constrained-snap-through, and even to the disappearance of the snap-through. Finally, a finite element method is conducted to investigate the stability of the coupled mechanism, of which the results show a good agreement with the analytical results of this paper.

This paper presents a three-dimensional finite deformation theory for the geometric nonlinear analysis of both the curved and twisted beams using the meshfree method based on the Timoshenko beam hypothesis. The theory presented is simple, but it is capable of solving the stability, postbuckling, snap-through, and large deformation problems effectively. Clear physical meanings will be revealed in derivation of the three-dimensional finite deformation theory. A meshfree method based on the differential reproducing kernel (DRK) approximation collocation method combined with the Newton–Raphson method is employed to solve the strong forms of the geometrically nonlinear problems. Numerical examples are given to illustrate the validity of the method presented.

Latticed shells and domes usually consist of hundreds, sometimes thousands, beam elements connected by rigid or semi-rigid joints. These connecting elements result, generally, very sophisticated, made with different materials and constituted by disparate connection systems. Recently, the stiffness connections were studied, numerically and experimentally, as one of the most important factors influencing significantly the structural response of space structures and domes. Very often, in the design process, the joints are assumed to be hinged or clamped. This assumption may result significantly far from the actual condition of in-service structure and components, leading to not understanding or not being able to prevent sudden catastrophic collapses (buckling, snap-through). Thus, the inclusion of joint stiffness reduction in the numerical model is necessary, more and more also due to the types of external loads, such as overloads that occur during the life of the structure or, especially, seismic solicitations. In this paper, the stability of an existent timber dome has been studied increasing the yieldingness of the connecting nodes according to an original approach. In addition, sensitivity of this kind of structure to the amplitude and the geometrical imperfections shape have been also considered. Numerical analyses have been conducted with local displacement controls, to take into account the geometric nonlinearity effects. Results evidenced that the dome is affected by instability interaction for particular slenderness and stiffness reduction of the connections.

This paper considers the load–deflection behavior of a pyramid-like, shallow lattice structure. It consists of four beams that join at a central apex and when subject to a lateral load, it exhibits a propensity to snap-through: a classical buckling phenomenon. Whether this structural inversion occurs, and the routes by which it happens, depends sensitively on geometry. Given the often sudden nature of the instability, the behavior is also examined within a dynamics context. The outcome of numerical simulations are favorably compared with experimental data extracted from the testing of three-dimensional (3D)-printed specimens. The key contributions of this paper are that despite the continuous nature of the physical system, its behavior (transient and equilibria) can be adequately described using a discrete model, and the paper also illustrates the utility of 3D-printing in an accessible research context.

Multistable laminates have been actively studied in recent years due to its potential applications in morphing and energy harvesting devices. Variable stiffness (VS) bistable laminates provide opportunities for further improvements in design space in comparison with constant stiffness bistable laminates. The snap-through process involving shape transition between the stable configurations is highly nonlinear in nature and exhibits rich dynamics. Exploiting the dynamic characteristics during the snap-through transition is of considerable interest in designing the morphing structural components. In this paper, we present a semi-analytical model based on Rayleigh–Ritz approach in conjunction with Hamilton’s principle to predict the natural frequencies of bistable VS laminates. The obtained results are compared with the results from the full geometrically nonlinear finite element (FE) model. The proposed FE model is further extended to study the dynamics of VS laminates subjected to external forces with different amplitudes. Subsequently, a parametric study is performed to explore the effect of different curvilinear fiber alignments on natural frequencies, mode shapes, free vibration characteristics and forced vibration characteristics (single-well and cross-well vibrations).

This paper presents a dynamic analysis of trusses with an initial length imperfection of the elements, considering geometrical nonlinearity. In the nonlinear analysis of trusses, the hybrid finite-element formulation considers the initial length imperfection of the elements as a dependent boundary constraint in the master equation of stiffness. Moreover, it was incorporated into the establishment of a modified system of equations. To overcome the mathematical complexity of dealing with initial length imperfections, this study proposes a novel approach for solving nonlinear dynamic problems based on a hybrid finite-element formulation. In this study, the unknowns of the dynamic equilibrium equations were displacements and forces, which were obtained using virtual work. The hybrid matrix of elements of the truss is established based on the hybrid variation formulation with length imperfections of elements, considering large displacements. The authors applied Newmark integration and Newton–Raphson iteration methods to solve the dynamic equations with geometrical nonlinearity. An incremental iterative algorithm and calculation programming routine were developed to illustrate the dynamic responses of trusses with initial-length imperfections. The results verified the accuracy and effectiveness of the proposed approach. The uniqueness of the proposed method is that the length imperfection of the truss element is included in the stiffness matrix and is considered a parameter that affects the dynamic response of the system. This helps to solve the problem of the dynamic response of trusses with length imperfections becoming simpler. The numerical results show that the effect of length imperfection on the dynamic response of the trusses is significant, particularly on the dynamic limit load. In addition, to completely evaluate the behavior of the trusses, this study also developed formulas and analyses to consider the inelastic and local buckling of the truss structures, named ‘Inelastic post-buckling analysis (IPB).’

Capacity to autonomously respond to external stimuli with some switchable structural shapes/properties/functions is highly desirable in many occasions where either system service environments or functional requirements are dynamically changing over time. In this paper, we conceptually propose a new type of smart sandwich structure with the ability to dynamically switch in-plane coefficients of thermal expansion (CTEs) from initially positive to negative or even zero value through internal microstructural transformation triggered solely by a certain temperature stimulus. To this end, a thermally driven snap-through action is purposely added into the design of the microstructure of periodic face-sheets by introducing an active spherical shell component constituted by two materials with different positive CTEs. The lattice core is connected to the upper and lower face-sheets for preventing the overall transverse deformation of face-sheet during temperature variation. Numerical simulations are subsequently carried out to demonstrate the completely reversible snap-through behavior, and the designed function of dynamically switchable in-plane CTE is also validated. Numerical results also reveal that the increasing thickness ratio of high CTE layer to low ones causes a decreased tendency for effective in-plane CTE before snapping, but the influence on after-snapping CTE is negligible. Similarly, the larger shell span brings an obvious increase in effective after-snapping in-plane CTE but without influence on before-snapping CTE. These significant results are beneficial to be summarized as practical design skills for simultaneously designing customized snap-through temperature and effective before or after-snapping in-plane CTE, all of which enable the proposed smart sandwich structure to be flexible to satisfy various requirements in more potential applications.