Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams

Analytic approximations are presented for the response of buckling-mode electrothermal actuators with very slender beams with a width-to-length ratio of W/L≤0.001 of the type found in nanoelectromechanical systems (NEMS). The results are found as closed-form solutions to the Euler beam bending theory rather than by an iterative numerical solution or a time-consuming finite element analysis. Expressions for transverse deflections and stiffness are presented for actuators with the common raised cosine and chevron pre-buckled shapes. The approximations are valid when the effects of bending dominate over those of axial compression. A few higher-order approximations are also presented for less slender beams with 0.001≤W/L≤0.01.

Theoretical Analysis on Nonlinear Buckling, Post-buckling of Slender Beams and Bi-stable Mechanisms

Compliant Mechanisms (CMs) are used to transfer motion, force and energy, taking advantages of the elastic deforma- tion of the involved compliant members. A branch of spe- cial type of elastic phenomenon called (post) buckling has been widely considered in CMs: avoiding buckling for better payload-bearing capacity and utilizing post-buckling to pro- duce multi-stable states. This paper digs into the essence of beam's bucking and post-bucking behaviors where we start from the famous Euler–Bernoulli beam theory and then ex- tend the mentioned linear theory into geometrically nonlin- ear one to handle multi-mode buckling problems via intro- ducing the concept of bifurcation theory. Five representative beam buckling cases are studied in this paper, followed by detailed theoretical investigations of their post-buckling be- haviors where the multi-state property has been proved. We finally propose a novel type of bi-stable mechanisms termed as Pre-buckled Bi-stable Mechanisms (PBMs) that integrate the features of both rigid and compliant mechanisms. The theoretical insights of PBMs are presented in detail for future studies. To the best of our knowledge, this paper is the first ever study on the theoretical derivation of the kinematic models of PBMs, which could be an important contribution to this field.

Behavior of under and over-reinforced concrete slender beams at failure

The focus of this paper is to examine the behavior of under and over-reinforced concrete slender beams at failure. The total number of the beams were five, with the provision of the following percentage of tension reinforcements: 1.01% for beam 1 (B1), 1.51% for beam 2 (B2), 2.01% for beam 3 (B3), 2.62% for beam 4 (B4) and 3.01% for beam 5 (B5). The beams were loaded with point loads at the center, with shear span/depth ratio of 3.8. The actual ultimate load of the experimental beam B1 was 141% of the estimated ultimate, while for beams B2, B3, B4 and B5, the actual ultimate loads were between 68% and 87% of the estimated ultimate loads for the beams respectively. The reinforced concrete beams B1, B2 and B3 had the capacity to sustain large deformation under constant loads before their ultimate failure, hence will give warning about the impending failure. For beams B4 and B5, although failed at higher loads had limited rotation capacity, hence will not give warnings about the impending failure. Therefore, 2.01% tension reinforcement is recommended as the maximum to be provided, so that the beam section can behave as a ductile section.

Automatic locking of a parametrically resonating, base-excited, nonlinear beam

Described is a closed-loop control scheme capable of stabilizing a parametrically excited nonlinear structure in several vibration modes. By setting the relative phase between the spatially filtered response and the excitation, the open-loop unstable solution branches are stabilized under a 2:1 parametric excitation of a chosen mode of vibration. For a given phase, the closed-loop automatically locks on a limit cycle, through an Autoresonance scheme, at any desired point on the solution branches. Axially driven slender beams and nanowires develop large transverse vibration under suitable amplitudes and frequency base-excitation that are sensitive to small potential coupled field. To utilize such a structure as a sensor, stable and robust operation are made possible by the control scheme. In addition, an optimal operating point with large sensitivity to the sensed potential field can be set using phase as a tunable parameter. Detailed analysis of the dynamical behavior, experimental verifications, and demonstrations sheds light on some features of the system dynamics.

Automatic locking of a parametrically resonating, base excited, nonlinear beam

Presented is a closed-loop, phase control scheme of a parametrically excited nonlinear structure, capable of stabilizing open-loop unstable solutions while automatically locking onto a desired point on any solution branch. Axially driven slender beams develop large transverse vibration under suitable amplitude and precise frequency base-excitation. The latter can induce parametric excitation along with a nonlinear response. The phase-lag of the 2:1 response over the excitation serves as a tunable parameter affecting the operating point of steady vibrations of a limit cycle. The operating point is tuned to exhibit great sensitivity to small interaction forces thus paving the way towards an ultrasensitive sensor. The paper analyzes the behavior of the mentioned configuration using asymptotic analysis, numerical simulations and an experimental system. Detailed analysis of the dynamical behavior, experimental verifications and demonstrations shed light on some features of the system dynamics.

Analysis of fracture processes in reinforced concrete beams without stirrups

The analysis of fracture processes which led to shear failure in reinforced concrete beams without transverse reinforcement was performed on the basis of test results from the author’s own experimental investigation and numerical simulations. The variable parameters during the experiment were a beam’s length and a shear span. It was observed that the character of failure in the beams depended on the beam’s length and the span-to-depth ratio. In slender beams characterized by the shear span-to-depth ratio 3.4 and 4.1, the formation of the critical diagonal crack caused a brittle, sudden failure and the shear capacity was low. In short beams, when the shear span-to-depth ratio was 1.8 and 2.3, the failure process had a more stable character with a slow developing of inclined cracks and the significantly higher load capacity was reached. The activation of various shear transfer mechanisms was examined with regard to the slenderness of the member and the transition between a beam action which took place in slender beams to an arch action which predominated in short beams was described.

Detecting internal resonances during model reduction

Model order reduction of geometrically nonlinear dynamic structures is often achieved via a static condensation procedure, whereby high-frequency modes are assumed to be quasi-statically coupled to a small set of lower frequency modes, which form the reduction basis. This approach is mathematically justifiable for structures characterized by slow/fast dynamics, such as thin plates and slender beams, and has been shown to provide highly accurate results. Nevertheless, selecting the reduction basis without a priori knowledge of the full-order dynamics is a challenging task; retaining redundant modes will lead to computationally suboptimal reduced-order models (ROMs), while omitting dynamically significant modes will lead to inaccurate results, and important features such as internal resonances may not be captured. In this study, we demonstrate how the error associated with static condensation can be efficiently approximated during model reduction. This approximate error can then be used as the basis of a method for predicting when dynamic modal interactions will occur, which will guide the reduction basis selection process. Equivalently, this may serve as a tool for verifying the accuracy of ROMs without the need for full-order simulations. The proposed method is demonstrated using a simple oscillator and a finite element model of a clamped–clamped beam.

Dynamics of Stress-Driven Two-Phase Elastic Beams

The dynamic behaviour of micro- and nano-beams is investigated by the nonlocal continuum mechanics, a computationally convenient approach with respect to atomistic strategies. Specifically, size effects are modelled by expressing elastic curvatures in terms of the integral mixture of stress-driven local and nonlocal phases, which leads to well-posed structural problems. Relevant nonlocal equations of the motion of slender beams are formulated and integrated by an analytical approach. The presented strategy is applied to simple case-problems of nanotechnological interest. Validation of the proposed nonlocal methodology is provided by comparing natural frequencies with the ones obtained by the classical strain gradient model of elasticity. The obtained outcomes can be useful for the design and optimisation of micro- and nano-electro-mechanical systems (M/NEMS).