International Journal of Structural Stability and Dynamics

In dynamic atomic force microscopy (AFM), as the tip of the base-excited micro-cantilever approaches the sample surface, this cantilever tip typically experiences long-range attractive and short-range repulsive forces. Due to the strongly nonlinear nature of these tip-sample interaction forces, various nonlinear phenomena including grazing contact-related phenomena have been observed. With the aim of obtaining a better understanding of such phenomena, a macro-scale experimental system has been constructed with attractive and repulsive tip interactions. In this arrangement, the macro-scale cantilever structure is harmonically excited at its base, a combination of magnets is used to generate the attractive forces, and impacts with a compliant surface are used to produce the repulsive forces. The separation between the tip and the compliant surface is used as a control parameter and the associated qualitative changes in the system dynamics are studied on Poincaré sections constructed by using the excitation frequency as the clock frequency. Off-resonance excitations are considered and a period- doubling window is observed for near-grazing impacts when the excitation frequency is in between the first and the second natural frequencies of the system. To explore the experimental observations further, a reduced-order model of the macro-scale system is developed with Derjaguin–Muller–Toporov (DMT) contact mechanics. The dynamics of the reduced-order system is found to agree well with the corresponding experimental observations. The nonlinear phenomenon observed during off-resonance excitations can form a basis for engineering grazing (zero-speed) impacts or low-speed impacts between a tapping mode AFM cantilever tip and the considered sample. In addition, the macro-scale experimental arrangement can serve as a vehicle to understand the dynamics of elastic structures subjected to different combinations of attractive and repulsive tip forces.

Bifurcation morphing created by means of nonlinear feedback excitation is a vibration-based method for sensing. In this work, several new studies are presented to connect previously demonstrated theory to increasingly more practical applications. In particular, in the process of designing nonlinear feedback auxiliary signals, time delays in the circuitry are unavoidable. Advantages as well as possible side effects and disadvantages of time delays are discussed. Furthermore, additional time delay is considered as a new design parameter. Increasing the time delay has advantages such as enhanced robustness and sensitivity, which are demonstrated computationally. Moreover, calibration using multiple sensor locations is discussed. A clamped-free cantilever beam structure is modeled and used for computational validation. The beam model resembles cantilever-based resonant sensor systems. Thus, potential applications of the new algorithm are discussed and the results stress the importance of nonlinear features for enhancing sensing performance in mechanical sensors without structural modifications.

This paper deals with the nonlinear response of electrostatically actuated cantilever beam microresonators near-half natural frequency. A first-order fringe correction of the electrostatic force, viscous damping, and Casimir effect are included in the model. Both forces, electrostatic and Casimir, are nonlinear. The dynamics of the resonator is investigated using the method of multiple scales (MMS) in a direct approach of the problem. The reduced order model (ROM) method, based on Galerkin procedure, is used as well. Steady-state motions are found. Numerical simulations are conducted for uniform microresonators. The influences of damping, actuation, and fringe effect on the resonator response are found.

Nonlinear vibrations of rectangular laminated composite plates with different boundary conditions are studied by using different nonlinear plate theories. In particular, numerical results for (i) the classical Von Kárman theory, (ii) the first-order shear deformation theory (SDT), and (iii) the third-order SDT are compared. The nonlinear response to harmonic excitation in the frequency neighborhood of the fundamental mode is investigated. Numerical investigation is carried out by using pseudo-arclength continuation method and bifurcation analysis. The boundary conditions of the plates are: simply supported with movable edges, simply supported with immovable edges, and clamped (CL) edges. For thick plates (thickness ratio 0.1), the strongest hardening nonlinear behavior is observed for CL plates, while the simply supported movable plates are the ones with the weakest nonlinearity among the three different boundary conditions studied here. Differences among the three nonlinear plate theories are large for thick laminated plates. For all the other cases, the first-order SDT, with shear correction factor , and the higher-order SDT give almost coincident results.

Sandwich panels have been developed to either produce lighter structures capable of carrying prescribed loads or increase the load-carrying capacity subject to limitations on weight. In these panels, facings carry bending and in-plane loads while the core functions similarly to the web of a beam, mostly resisting transverse shear. Improvements in the load-carrying capacity of sandwich panels can be achieved through modifications in their geometry, boundary conditions, and material distribution. One of the methods recently considered by the authors is based on using facings with a step-wise variable thickness that increases at the critical region of the structure.¹ It was illustrated that the strength of a sandwich panel can be considerably enhanced using such stepped facings, without a detrimental increase of the weight. The present paper expands the study of the feasibility of the stepped-facing sandwich panel concept concentrating on three structural problems, i.e. a possible improvement in stability, changes in the natural frequencies, and forced dynamic response to the explosive blast. It is illustrated that the stepped-facing design can improve stability of the panel and its response to blast loading. However, fundamental frequencies of stepped-facing panels decrease compared to those in their conventional equal-weight counterparts. Such decrease is detrimental in the majority of engineering applications representing a limitation of stepped-facing panels. Nevertheless, the usefulness of the stepped-facing design is proven in the problems of bending, stability, and blast loading. Numerous examples presented in the paper validate our suggestion that the combination of a relatively simple manufacturing process and an improved structural response of sandwich panels with stepped facings may present the designer with an attractive alternative to conventional sandwich structures.

In this paper, a finite element-based approach for nonlinear vibration analysis of shell structures is presented. The approach makes use of a perturbation method that gives an approximation for the amplitude-frequency relation of the structure. The method is formulated using a functional notation and is subsequently converted to a finite element notation. After the determination of the linear natural frequency and corresponding vibration mode, the perturbation approach yields the initial curvature of the amplitude–frequency relation with a modest additional computational cost. The implementation of the perturbation approach in a general purpose finite element code using a laminated curved shell element is described. The effectiveness of the approach is illustrated by application to single-mode and coupled-mode nonlinear vibration analyses of cylindrical shells. Results for isotropic and composite cylindrical shells are presented and compared with results obtained via alternative approaches.

Modern fiber placement machines allow laminates with spatially varying stiffness properties to be manufactured. In earlier research, the authors optimized variable stiffness plates for maximum buckling load, demonstrating significant improvements in load-carrying capacity. In aerospace applications, panel structures are often permitted to enter the postbuckling regime during service. It is, therefore, not only important to understand their postbuckling behavior, but also to develop fast analysis methods that can subsequently be used in a design optimization framework. The aim of the present research is to study the postbuckling behavior of the optimized plates using a perturbation method that has been developed earlier within a general-purpose finite element environment. The perturbation approach is used to compute postbuckling coefficients, which are used to make a quick estimate of the postbuckling stiffness of the panel and to establish a reduced-order model. In the present work, the postbuckling analysis of variable stiffness plates is carried out using the reduced-order model, and the potential of the approach for incorporation within the optimization process is demonstrated.

In classic cable theory, vibrations are usually analyzed by writing the equations of motion in the neighborhood of the initial equilibrium configuration. Furthermore, a fundamental difference exists between out-of-plane motions, which basically corresponds to the linear behavior of a taut string and in-plane motion, where self-weight determines a sagged initial profile. This work makes use of a continuous approach to establish the initial shape of the cable when it is subjected to wind or fluid flow arbitrarily directed and employed a novel nonlinear finite element technique in order to investigate the dynamics present around the initial equilibrium shape of the cable. Stochastic solutions in the frequency domain are derived for a wind-exposed cable after linearization of the problem. By applying the proper orthogonal decomposition (POD) technique with the aim of reducing computational effort, an approach to simulate modal wind forces is proposed and applied to the nonlinear equations of motion.

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas.

To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.