Nonlinear Force-displacement Characteristics Research Articles

An adjustable device to adaptively realise diverse nonlinear force-displacement characteristics

The benefit of introducing nonlinear stiffness elements to improve the dynamic performance of vibration energy harvesters, vibration isolators and vibration absorbers have attracted great attention in the recent literature. A simple adjustable device to tailor nonlinear stiffness for diverse engineering applications is thus proposed in this paper. The device adopts two pairs of linear oblique springs, whose geometric arrangement can be modified to achieve specific nonlinear force-displacement characteristics. Depending on its design configuration, it is able to be adapted and behave as linear, hardening, softening, quasi-zero stiffness for large deflections, quasi-zero stiffness for small deflections, bi-stable and tri-stable. The design strategy and the main parameters involved are discussed.

Nonlinear static modeling of a tip-tilt-piston micropositioning stage comprising leaf-spring flexure hinges

Nonlinear static modeling is crucial for the design of stages with large travel ranges. However, few studies have investigated complex spatial compliant mechanisms. The present study proposes an optimization algorithm based on substructure constraint conditions to formulate the nonlinearity of the force–displacement characteristic of a tip-tilt-piston stage comprising leaf-spring flexure hinges. First, the nonlinear force–displacement characteristics of the compound basic parallelogram mechanism are derived using an optimization algorithm based on two constraint conditions (I and II). Second, the nonlinear static modeling of the tip-tilt-piston stage is conducted based on the modeling results of the compound basic parallelogram mechanism. The stage is divided into three parts, and force analyses are conducted for all three parts. The vertical displacement of the compound basic parallelogram mechanism in part 1 and the rotational angle of the rotational plate in part 2 are calculated. Subsequently, the force–displacement characteristics of the tip-tilt-piston stage are obtained based on a third constraint condition (III). A comparison of the finite element analysis results and the theoretical calculation indicates less than 4% errors. In the experimental tests conducted on the proposed stage and four compound basic parallelogram mechanisms, the displacements were evidently larger than those calculated using the finite element analysis. Therefore, a weight coefficient of the axial force w is introduced in the theoretical calculation to solve the problem of the large deviations between the experimental results and the finite element analysis results. When w is set to 1, the theoretical results are in good agreement with the finite element analysis results; when w is set to 0.05 for the tip-tilt-piston stage and 0.15 for compound basic parallelogram mechanisms, the theoretical results are consistent with the experimental results (less than 8.5% errors).

Considering Operating Point Variation in 3-D Analytical Charge Model Used for Design of Low-Stiffness Permanent Magnet Vibration Isolator

Vibration isolators are a key technology in ultraprecision machining and ultraprecision measurements. The increasing performance requirements of magnetic vibration isolation systems (MVISs) continue to demand improvements in the accuracy on which these devices are modeled. However, a serious constraint exists in designing MVISs, as high stiffnesses will be introduced by nonlinear force–displacement characteristics. The traditional analytic magnetic charge model assumes that the relative magnetic permeability $\mu _{r} = 1$ , which causes the accuracy of the magnetic analytic model of the MVIS to decrease. To solve this defect, in this article, low-stiffness MVISs are designed by using the three-dimensional analytic magnetic charge model while considering the variations of the operating point. The optimized parameters of the MVIS are then obtained by combining the genetic algorithm. The feasibility of the proposed analytical charge method is verified from the results of experimental measurements and finite-element methods. The results of this article provide a powerful tool for the design of a high-performance MVIS and provide important support for the ultimate goal of ultraprecision machining and measurements.

Chord line force versus displacement for thin shallow arc pre-curved bimetallic strip

A pre-curved bimetallic strip that is applied with a force in an axial orientation, i.e. along its chord line, exhibits nonlinear force–displacement characteristics. For thin bimetallic strips, whereby the radius of curvature is large compared to the thickness of the strip, the non-linearity tends to be tangent related. The new theoretical formula introduced here was correlated to the results of a set of force–displacement tests, and a good overall fit of the theory to the test data was achieved. The formula put forward in this work enables the evaluation of large chord line displacements but is limited to the permissible stress limits of the material. This work can also be directly applied to thin shallow arc beams of a single material. The application of this work is in the field of bimetallic force–displacement actuators.

Mechanical-Magnetic Coupling Analysis of a Novel Large Stroke Penta-Stable Mechanism Possessing Multistability Transforming Capability

Considering the nonlinear mechanical-magnetic coupling effects, an accurate mathematical model was established for analyzing large stroke penta-stable mechanism possessing multistability transforming capability, with which the mechanism can be switched from pentastability to quadristability. The multistability with any number of stable states can be achieved by integrating spatially arranged magnets and large deformation beams as the fundamental energy storage elements to maintain stable states. By theoretically analyzing the influence of the large mechanical deformation on the magnetic field distribution and system energy, the nonlinear force–displacement characteristics of the multistable mechanism were obtained numerically, which were in good agreement with those obtained by experiments and finite element simulation. Then, an energy-based design criterion for magnetic-mechanical multistable mechanisms was proposed according to the stability theory and energy variation principle. In addition, the multistable transformability was theoretically analyzed, which can transform the proposed mechanism from penta-stability to quadristability by only changing the magnetization direction of moving magnets without varying the structure parameters.

Dynamic isolation systems using tunable nonlinear stiffness beams

Vibration isolation devices are required to reduce the forcing into the supporting structure or to protect sensitive equipment from base excitation. A suspension system with a low natural frequency is required to improve isolation, but with linear supports the minimum stiffness is bounded by the static stiffness required to support the equipment. However, nonlinear high-static-low-dynamic-stiffness (HSLDS) mounts may be designed, for example by combining elastic springs in particular geometries, to give the required nonlinear force-displacement characteristics. Current approaches to realise the required nonlinear characteristics are often inconvenient. Furthermore, the weight of the supported equipment, the environment, or the structural stiffness may change. This paper investigates the design of HSLDS isolation mounts using beams of tunable geometric nonlinear stiffness. In order to obtain the nonlinear response required, we first study the case of generic beams subject to static loads that are able to tune their nonlinear force-displacement characteristics to ensure that the isolators have very low dynamic stiffness. Tuning is achieved by actuators at the ends of the beams that prescribe the axial displacement and rotation. Secondly, we study a composite beam with an initial thermal pre-stress, resulting in internal stresses that give the required nonlinear response.

Simplified expressions for modelling rigid rocking structures on two-spring foundations

This paper presents a new technique for modelling the dynamic response of uplifting rigid structures subjected to base excitation. The proposed technique exploits the use of a two spring foundation, and subsequently an equivalent single-degree-of-freedom procedure is established to model the dynamics of the system. A set of simplified closed-form expressions have been developed to estimate the system’s restoring force-displacement characteristics. The simplified expressions only require details of the system geometry and are shown to predict the nonlinear force-displacement characteristics of a rocking structure as closely as those determined from a complicated pushover analysis. This paper presents two additional numerical examples to demonstrate the use of the proposed technique to simulate the displacement time-histories of a prototype structure under free-vibration-decay or when subjected to earthquake excitations.

Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms

This paper presents parallel kinematic XY flexure mechanism designs based on systematic constraint patterns that allow large ranges of motion without causing over-constraint or significant error motions. Key performance characteristics of XY mechanisms such as mobility, cross-axis coupling, parasitic errors, actuator isolation, drive stiffness, lost motion, and geometric sensitivity, are discussed. The standard double parallelogram flexure module is used as a constraint building-block and its nonlinear force-displacement characteristics are employed in analytically predicting the performance characteristics of two proposed XY flexure mechanism designs. Fundamental performance tradeoffs, including those resulting from the nonlinear load-stiffening and elastokinematic effects, in flexure mechanisms are highlighted. Comparisons between closed-form linear and nonlinear analyses are presented to emphasize the inadequacy of the former. It is shown that geometric symmetry in the constraint arrangement relaxes some of the design tradeoffs, resulting in improved performance. The nonlinear analytical predictions are validated by means of computational finite element analysis and experimental measurements.

An experimental and finite element study of the non-linear force-displacement characteristics of tubular YT-joints under combined axial loading

This paper describes an experimental and finite element investigation of the non-linear elastic-plastic force-displacement response of a simply supported tubular YT-joint under a range of ratios of the T- to Y-brace applied loads, in order to develop an understanding of the dependence of the joint response on loading path. The experiments consist of the static testing of simply supported tin-lead alloy models of tubular YT-joints under combined, proportionally applied T- and Y-brace-end axial forces. A number of different finite element approaches are presented and discussed via comparison with the test results. An improved version of the energy-based elastic-plastic prediction procedure of the present authors is presented; this is shown to represent a framework for the determination of the non-linear local joint T- and Y-brace force-displacement responses to arbitrary radial loading paths in the load space. The work is considered relevant to the simplification of tubular framework analysis methods for the calculation of accurate collapse loads, incorporating non-linear joint behaviour.

Load Distribution in Flexibly Supported Three-Row Roller Slew Bearings

In general, slew bearings are less firmly supported by their mounting structures than small bearings. Hence, the load distribution around the bearing may be vastly different than that predicted by classic bearing formulas. The finite element method has been employed to determine load distribution in a three-row roller slew bearing mounted between two flexible, ring shaped supporting structures composed of beam elements. A special bar element modeling the system of two opposite rollers in contact with its raceways has been elaborated. The influence of the nonlinear force displacement characteristics of each of the rollers, as well as the influence of gaps between the rollers and its raceways have been taken into account. The nonlinear contact problem has been solved utilizing the Newton-Raphson method with continually updated stiffness matrix due to both element nonlinearity and contact status. The effect of a number of design parameters on the load distribution has been investigated.

Analysis of a Nonlinear Dynamic Vibration Absorber

Abstract This paper presents a mathematical analysis of the action of a dynamic vibration absorber. The system analyzed consists of a main mass attached to a rigid foundation by a linear spring coupled to the absorber mass by a spring of nonlinear characteristics. The forced oscillations of the system produced by a harmonic disturbing force acting on the main mass are studied analytically. It is assumed that the coupling absorber spring has nonlinear force-displacement characteristics of the hyperbolic sine type. Expressions for the amplitudes of the vibrations of the two masses as functions of the frequency of the disturbing force are obtained.