Force-deflection Relationship Research Articles

An analytical elastic-perfectly plastic contact model

A new formulation for elastic-perfectly plastic contact in the normal direction between two round surfaces that is solely based on material properties and contact geometries is developed. The problem is formulated as three separate domains: the elastic regime, mixed elastic–plastic behavior, and unconstrained (fully plastic) flow. Solutions for the force–displacement relationship in the elastic regime follow from Hertz’s classical solution. In the fully plastic regime, two well supported assumptions are made: that there is a uniform pressure distribution and there is a linear force–deflection relationship. The force–displacement relationship in the intermediate, mixed elastic–plastic regime is approximated by enforcing continuity between the elastic and fully plastic regimes. Transitions between the three regimes are determined based on empirical quantities: the von Mises yield criterion is used to determine the initiation of mixed elastic–plastic deformation, and Brinell’s hardness for the onset of unconstrained flow. Unloading from each of these three regimes is modeled as an elastic process with different radii of curvature based on the regime in which the maximum force occurred. Simulation results explore the relationship between the impact velocity and coefficient of restitution. Further comparisons are made between the model, experimental results found in the literature, and other existing elastic–plastic models. The new model is well supported by the experimental measurements of compliance curves for elastic–plastic materials and of coefficients of restitution from impact studies, and in elastic-perfectly plastic regimes is demonstrated to be more accurate than existing models found in the literature.

A Compliant Parallel XY Micromotion Stage With Complete Kinematic Decoupling

This paper presents a novel compliant parallel XY micromotion stage driven by piezoelectric actuators (PZT). With the purpose to obtain complete kinematic decoupling and good stiffness performance, the stage is designed using a symmetric 4-PP structure in which double four-bar flexure is chosen as the prismatic joint. Matrix method is employed to establish the compliance model of the mechanism. Based on the model, dynamic analysis is investigated after static analysis is carried out. The dimensions of the mechanism are optimized using the particle swarm optimization (PSO) algorithm in order to maximize the natural frequencies. Finite-element analysis (FEA) result indicates that the mechanism has an almost linear force-deflection relationship, high first natural frequency (720.52 Hz), and ideal decoupling property. To cope with the nonlinearities such as hysteresis that exists in the PZT, the control system is constructed by a proportional-integral-derivative (PID) feedback controller with a feedforward compensator based on Preisach model. The fabricated prototype has a 19.2 μm × 8.8 μm rectangular workspace with coupling less than 5%. The result of the closed-loop test shows that the XY stage can achieve positioning, tracking and contouring tasks with small errors.

Engineering design framework for a shape memory alloy coil spring actuator using a static two-state model

A shape memory alloy (SMA) coil spring actuator is fabricated by annealing an SMA wire wound on a rod. Four design parameters are required for the winding: the wire diameter, the rod diameter, the pitch angle and the number of active coils. These parameters determine the force and stroke produced by the actuator. In this paper, we present an engineering design framework to select these parameters on the basis of the desired force and stoke. The behavior of the SMA coil spring actuator is described in detail to provide information about the inner workings of the actuator and to aid in selecting the design parameters. A new static two-state model, which represents a force–deflection relation of the actuator at the fully martensitic state (M100%) and fully austenitic state (A100%), is derived for use in the design. Two nonlinear effects are considered in the model: the nonlinear detwinning effect of the SMA and the nonlinear geometric effect of the coil spring for large deformations. The design process is organized into six steps and is presented with a flowchart and design equations. By following this systematic approach, an SMA coil spring actuator can be designed for various applications. Experimental results verified the static two-state model for the SMA coil spring actuator and a case study showed that an actuator designed using this framework met the design requirements. The proposed design framework was developed to assist application engineers such as robotics researchers in designing SMA coil spring actuators without the need for full thermomechanical models.

Optimal Active Deflection Compensation of a Hot Leveler

A validated, mathematical deflection model of the work rolls and the machine frame of a hot leveler is presented. The model calculates the roll gap along both planar directions of the plate. The work rolls are loaded by distributed leveling forces, which are calculated based on the local roll gap and the material properties of the leveled plate. Furthermore, the rolls are supported by machine frames, which may be modeled by linear force-deflection relations. The model is well suited for analyzing the influence of the various adjustment and control inputs on the roll gap, which is important for the leveling result. Moreover, control laws providing active symmetric deflection compensation are developed. By minimizing a cost function that depends, inter alia, on the curvature of the plate, optimal reference values for the adjustment cylinders and the bending mechanism are generated. Force constraints are integrated into the optimization problem so that overloading of the leveler is systematically avoided. Numerical results demonstrate the feasibility of the proposed approach.

A new model for precision control of the radius in in-plane roll-bending of strip considering rolls and stand deflections

The rolls and stand deflections induced by rolling force lead to a significant decrease in the radius prediction accuracy of the ring product in the in-plane roll-bending of strip. To precisely predict the deflections and control the radius, a new analytical model is developed. Numerical implementation is presented to solve the theoretical rolling force, the deflections as well as the inherent force–deflection relationship considering the interactions of the strip workpiece, the rolls and the stand. A series of profile measurements of the formed ring parts are made to assess the accuracy of the predicted deflections. Different control approaches are used in the experiments to validate the radius control model. The experimental results show that the model is reliable to control the radii with a maximum relative error of 10.4% deviating from the desired ones. By considering the rolls and stand deflections, the prediction accuracy of the radii of most aluminum alloy AA-3003O rings is improved by 65.9–159.9%.

A 3-D Chain Algorithm with Pseudo-Rigid-Body Model Elements

A chain algorithm element is created from pseudo-rigid-body segments and used in a chain calculation that accurately predicts the force deflection relationship of beams with large 3-D deflections. Each chain element is made up of three superimposed pseudo-rigid-body models acting orthogonally in relation to each other. The chain algorithm can accurately and quickly predict large displacements and the force-deflection relationship of lateral torsional buckled beams. This approach is not intended to compete with finite element analysis, but rather is a supplement tool that may prove particularly useful in the early phases of design when many analysis iterations are required. The 3-D chain algorithm is demonstrated and compared to the finite element analysis for the nonlinear large-deflection, post-buckling path of a flexible beam undergoing lateral-torsional buckling.

Steel–Concrete–Steel sandwich slabs with lightweight core — Static performance

This paper investigates the static performance of Steel–Concrete–Steel (SCS) sandwich slabs which consist of an ultra-lightweight concrete core sandwiched between two steel plates. Special J-hook connectors have been developed to develop composite action between the concrete core and the two steel plates. The core is made of lightweight concrete of density less than 1450 kg/m 3. Laboratory tests were carried out on eight SCS sandwich slabs under centrally applied patch load. Test results showed that the mode of failure and crack pattern of SCS sandwich slabs were very similar to those of reinforced concrete slabs especially when the concrete core and steel plates act in a fully composite manner. Flexural and punching are the primary modes of failure. After flexural yielding, membrane action developed in the slab due to the effectiveness of J-hook connectors in maintaining composite action which further increases its load carrying capacity after flexural yielding. Theoretical models are proposed to predict the flexural and punching resistance and a good correlation with test results is obtained. A large deflection analysis considering plate membrane action is also proposed to predict the force deflection relation of SCS sandwich slabs.

Modeling and Measurement of a Bistable Beam in a Microelectromechanical System

Design and fabrication of microelectromechanical systems (MEMS) can be costly, time consuming, and necessitating accurate models for their behavior. Current theoretical models of bistable beams in MEMS devices are limited to numerical or small deformation models and current measurement techniques are unable to fully characterize these devices as they only determine thresholds or have resolutions that are too coarse to adequately explore the force-deflection relationship of bistable mechanisms. Two analytical models are developed: a stepped Euler–Bernoulli beam and a large deformation model. To validate these models, a new technique for measuring in-plane mechanical properties of MEMS devices is introduced that measures normal and lateral forces against a probe tip, while electrostatic actuation and a force-feedback loop maintain the desired tip position. This allows true displacement-controlled measurements along two axes and facilitates automated positioning. Measurements validate the large deformation model and show that Euler–Bernoulli beam theory is inadequate for modeling the mechanism's bistable behavior. A parameter study in edge width using the large deformation model accounts for discrepancies between predicted and measured forces. The model's utility is further demonstrated by an optimization study that redesigns the mechanism to be less sensitive to the edge width variation introduced in the manufacturing process. <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hfill$</tex></formula> [2010-0026]

Glass microneedles for force measurements: a finite-element analysis model

Changes in developed force (0.1-3.0 microN) observed during contraction of single myofibrils in response to rapidly changing calcium concentrations can be measured using glass microneedles. These microneedles are calibrated for stiffness and deflect on response to developed myofibril force. The precision and accuracy of kinetic measurements are highly dependent on the structural and mechanical characteristics of the microneedles, which are generally assumed to have a linear force-deflection relationship. We present a finite-element analysis (FEA) model used to simulate the effects of measurable geometry on stiffness as a function of applied force and validate our model with actual measured needle properties. In addition, we developed a simple heuristic constitutive equation that best describes the stiffness of our range of microneedles used and define limits of geometry parameters within which our predictions hold true. Our model also maps a relation between the geometry parameters and natural frequencies in air, enabling optimum parametric combinations for microneedle fabrication that would reflect more reliable force measurement in fluids and physiological environments. We propose a use for this model to aid in the design of microneedles to improve calibration time, reproducibility, and precision for measuring myofibrillar, cellular, and supramolecular kinetic forces.

A New Approach to Modeling Surface Defects in Bearing Dynamics Simulations

A dynamic model for deep groove and angular contact ball bearings was developed to investigate the influence of race defects on the motions of bearing components (i.e., inner and outer races, cage, and balls). In order to determine the effects of dents on the bearing dynamics, a model was developed to determine the force-deflection relationship between an ellipsoid and a dented semi-infinite domain. The force-deflection relationship for dented surfaces was then incorporated in the bearing dynamic model by replacing the well-known Hertzian force-deflection relationship whenever a ball/dent interaction occurs. In this investigation, all bearing components have six degrees-of-freedom. Newton’s laws are used to determine the motions of all bearing elements, and an explicit fourth-order Runge–Kutta algorithm with a variable or constant step size was used to integrate the equations of motion. A model was used to study the effect of dent size, dent location, and inner race speed on bearing components. The results indicate that surface defects and irregularities like dent have a severe effect on bearing motion and forces. Furthermore, these effects are even more severe for high-speed applications. The results also demonstrate that a single dent can affect the forces and motion throughout the entire bearing and on all bearing components. However, the location of the dent dictates the magnitude of its influence on each bearing component.

Vehicle/occupant movement in moderate speed in-line collisions

This article presents a simple, but yet quantitative model of vehicle and occupant movement in moderate-speed, in-line collisions. The modelling is based upon established and validated representations of force-deflection relations during frontal/rear-end collisions. The developed model is applicable with head-on, rear-end, and fixed structure collisions. The modelling results show that the vehicle and occupant movement during collision is well described by sinusoidal and linear functions. The model shows that in the beginning stages of an accident, the occupant movement relative to the vehicle is proportional to the cube of the accident time. Thus, the occupant movement in the vehicle is initially slow but then rapidly increasing. Thus, from the perspective of safety, the beginning stages of an accident provide the greatest opportunity for implementation/deployment of air bags, pretensioners, and adaptive head restraint systems.

On the accuracy of the Hertz model to describe the normal contact of soft elastic spheres

The finite element method is used to investigate the validity of the Hertz model for normal contact under conditions that violate some of the assumptions on which the Hertz model was derived. The conditions investigated are: frictional contact, non-flat contact surface and large strains. The investigations were performed for the case of two elastic spheres in contact as well as that of a sphere in contact with a flat plate. The existence of friction for normal contact does not appear to lead to any erroneous predictions of the force-deflection relationship by the Hertz model. The Hertz model also does not appear to predict significant errors when the contact surface is not flat. However, large strains were found to cause important prediction errors by the Hertz model. Calibration curves for the corrections on the Hertz model are proposed.

Force versus Axial Deflection of Pipette-Aspirated Closed Membranes

The axial deformation of a pipette-pressurized fluid membrane bag produces minuscule yet well-defined, reproducible forces. The stiffness of this ultrasensitive force transducer is tunable and largely independent of the constitutive membrane behavior. Based on a rigorous variational treatment, we present both numerical as well as approximate analytical solutions for the force-deflection relation of this unique biophysical force probe. Our numerical results predict a measurably nonlinear force-deflection behavior at moderate-to-large deformations, which we confirm experimentally using red blood cells. Furthermore, considering nearly spherical membrane shapes and enforcing proper boundary conditions, we derive an analytical solution valid at small deformations. In this linear regime the pressurized membrane bag behaves like a Hookean spring, with a spring constant that is significantly larger than previously published for the biomembrane force probe.

Nonlinear dynamic analysis of an elastic beam isolator sliding on frictional supports

The dynamic behavior of an elastic beam free to slide on two frictional supports is studied under sinusoidal and random excitations. The beam force–deflection relationship, originally expressed in terms of elliptic functions, is approximated by a polynomial fit of eleventh order. The friction force is modeled in terms of the sliding velocity and the end slope angle. Under sinusoidal excitation, the equation of motion of the system is solved numerically and the solution is utilized to estimate the system transmissibility. It is found that when the excitation frequency is increased beyond resonance, the friction at the sliding supports improves the transmissibility. The dependence of the response on initial conditions establishes the basins of attraction for different values of friction coefficient and excitation parameters. The dependence of the safety integrity factor on excitation amplitude level and friction coefficient reveals that the friction extends the stable region. Under random excitation, the system response statistics are estimated from Monte Carlo simulation results for different values of friction coefficient and excitation power spectral density level. The friction is found to result in a significant reduction of the system response mean square.

Modeling and Design of a Piezoelectric Forceps Actuator for Meso∕Micro Grasping

Meso∕micro grasping of tiny soft objects such as biological tissues, which ranges from hundreds to thousands of micro-millimeters in dimension, plays a significant role in the fields of tele-surgery, minimally invasive surgery (MIS), and biomedical instrumentation. Recently, the authors proposed a novel piezoelectric forceps actuator (PFA), which is capable of grasping delicate soft objects. One of the advantages of the PFA over conventional MIS forceps lies in that it can be remotely controlled to achieve precision deflection and grasping force. Furthermore, it does not have any moving parts such as gears and hinges, and hence avoids problems in operation like friction, backlash, lubrication, leakage, and sterilization. In this paper, a mathematical model of the PFA is derived, based on which genetic algorithm (GA) is applied to optimize the grasping force-deflection relationship of the actuator. The model developed is experimentally verified on a prototype of the PFA.

Evaluation of damage to offshore platform structures due to collision of large barge

An offshore jacket platform in the South China Sea was impacted by a large derrick and lay barge during installation. This paper presents a non-linear dynamical analysis procedure for firstly determining the impact action based on the forensic evidence from the damaged components, and then evaluating the overall damage effects on the platform structure. The impact action of the barge is simulated with a triangle impulse load with different collision contact times. The curves relating the indentation deformations of the damaged member with different collision contact times are simulated using an estimated velocity of the impacting ship. On the basis of these curves and the actual detected dent damages, the contact time and the maximum impact load on the platform are determined. Taking into account the force–deflection relationship of the local indentation of the damaged cross-diagonal brace, the transmission of the impact load to the platform structure is simulated by a non-linear spring. The added mass coefficient with hydrodynamic effects and the pile–soil-structure interaction are considered in the computational model of the non-linear dynamic response of the platform structure. Subsequently, the dynamic response of the offshore jacket structure is computed and the critical stress and deformation of the tubular joints are obtained as indicators of the damage effects. The results are useful for choosing a feasible and reasonable repairing and strengthening scheme for the damaged platform. The procedure presented in this paper is generally applicable for the evaluation of typical offshore platform structures in the case of impact or collision.

Effect of laminate thickness and of matrix resin on the impact of low fibre-volume, woven roving E-glass composites

Instrumented falling weight tests led to the definition of three behaviour `regimes'; `undamaged', `internal delamination' and `severe' (at low, medium and high incident energies, respectively). The reduction in stiffness due to the internal delamination gave a bi-linear force–deflection relationship. A shear-dominated theory and an energy balance approach gave the relationship between incident energy and maximum force. By considering deviations from this model the complex impact behaviour seen was related to the damage modes observed. At low incident energies bending was significant for all except for the thickest polyester laminates. Delaminated behaviour was shear-dominated for all except the thinnest laminates. At higher incident energies back-face fibre damage controlled behaviour before membrane effects became significant for thinner laminates, and indentation damage of thicker laminates became influential. Subtle differences between the damage modes of the epoxy and polyester laminates were seen; the epoxy laminates suffered more back-face fibre damage, but less internal delamination.

Crushing of an Elastic-Plastic Ring Between Rigid Plates With and Without Unloading

Load-deflection curves are computed for an elastic-plastic ring that is slowly crushed between frictionless, rigid plates (platens). The ring is assumed to be inextensional with plane sections remaining plane and to obey a bi-linear stress-strain law with isotropic hardening. These assumptions lead to a local nonlinear moment-curvature relation identical to that developed by Liu et al. When inserted into the exact equation for moment equilibrium, this constitutive relation yields a second-order, nonlinear ordinary differential equation for the angle α between the deformed centerline of the ring and the horizontal. The numerical solution of this equation, which uses a combined penalty-continuation method, along with an auxiliary equation relating the vertical deflection to α, leads to overall load-deflection curves that depend on two dimensionless parameters, λ and μ. The first is the ratio of the plastic modulus to the elastic modulus; the second measures the ratio of plastic to elastic effects. As μ→0, the overall load-deflection curve of Frish-Fay for the elastica is recovered; as μ→∞, that of DeRuntz and Hodge for a rigid-perfectly plastic ring is recovered. Three scenarios are considered: I0, in which an initially straight, stress-free beam is bent elastically into a ring and then crushed; II0, in which an initially stress-free ring is crushed; and III0, in which an initially straight beam is bent first elastically and then elastically-plastically into a ring and then crushed. Results for scenario II0 are shown to agree well with experiments of Reddy and Reid if λ=0.01 and μ=10 and 20 and with experiments of Avalle and Goglio if λ=0.02 and μ=11. In scenarios I0 and II0, the effects of unloading prove to be small, reinforcing a similar conclusion of Liu et al., who considered the large-deflection of an elastic-plastic cantilever under a tip load. If no unloading is assumed, a more analytical treatment is possible, as shown in the second part of the present paper. The model predicts that the ring always remains in full contact with the platens, in agreement with recent experiments by Avalle and Goglio on annealed aluminum tubes. Pull-away from the platens also observed in experiments is ascribed to end effects which cannot be modeled by a one-dimensional beam theory. However, it is argued that, even if there is pull-away, the effect on the overall force-deflection relation must be small because in both cases the forces exerted by the platens are concentrated at the ends of the contact region. Moving pictures of successive stages of deformation of the ring showing the formation of plastic loading and unloading zones in all three scenarios may be found on the web site www.people.virginia.edu/∼jgs/ring.html.

Improved Vibration Isolating Seat Suspension Designs Based on Position-Dependent Nonlinear Stiffness and Damping Characteristics

The design of seat suspensions having linear stiffness and damping characteristics involves a tradeoff between three performance measures. These measures are: (1) suspension range of motion, (2) improved average vibration isolation (weighted average across a wide exposure spectrum), and (3) improved isolation at the frequency of peak transmissibility. To overcome the limitations associated with this tradeoff, nonlinear mechanical properties are used here in the design of a seat suspension. From the infinite number of possible nonlinear mechanical characteristics, several possibilities that showed promise in previous studies were selected. The selected nonlinear force-deflection relationship (stiffness) of the seat is described by a combination of cubic and linear terms. The selected damping behavior of the seat is described by a combination of a linear term and a position-dependent term. A lumped parameter model (linear-human/nonlinear-seat) of the human/seat-suspension coupled system and a robust direct search routine are used to obtain pseudo-optimal values of the seat design parameters (mass, stiffness, and damping) via simulation in the time domain. Results indicate that the optimal nonlinear seat suspension is significantly better than the optimal linear seat suspension in overall vibration isolation characteristics.

Effective Local Flexural Stiffness of Ball Grid Array Assemblies

Parts on the circuit board serve as reinforcements and produce local stiffness, which influences the deflection of the assembly under vibration. The curvature of the circuit board, combined with material properties, produces stresses that lead to the high-cycle fatigue failure of interconnecting soldering joints. Use of a conventional finite element method (FEM)—referred to as “h-method”—for circuit board analysis is cost prohibitive, as numerous parts, each containing many soldering joints, would need to be analyzed for a typical board. Instead, a direct-stress analysis method—referred to as the multi-domain method (MDM)—can be used to calculate effective local stiffness of ball grid array assemblies. The fast and accurate MDM is based on nested multi-field displacement superposition and is similar in concept to p-type FEM. It is similar to conventional FEM only in its use of the Rayleigh-Ritz methodology. The computational advantages of MDM over conventional FEM for computing thermal stresses caused by thermal coefficient mismatch have been documented previously. In present work, the use of MDM as a direct-stress analysis method to extract the effective local stiffness of ball grid-array assemblies for determining high-cycle fatigue life has been extended. This method simulates a three-point bend test for flexural stiffness calculation. It demonstrates that the force-deflection relationship at the center of the system can be accurately achieved with proper constraints at the ends. The flexural stiffness is then calculated on the basis of beam theory. This calculation produces numerical results for various part-board connections, both with and without underfill. The accuracy of the formulation is examined for layered assembly. The results for long-layered beam theory agree with those based on layered beam theory.