Parallel Spring Research Articles

Characterization and Modelling of Various Sized Mountain Bike Tires and the Effects of Tire Tread Knobs and Inflation Pressure

Mountain bikes continue to be the largest segment of U.S. bicycle sales, totaling some USD 577.5 million in 2017 alone. One of the distinguishing features of the mountain bike is relatively wide tires with thick, knobby treads. Although some work has been done on characterizing street and commuter bicycle tires, little or no data have been published on off-road bicycle tires. This work presents laboratory measurements of inflated tire profiles, tire contact patch footprints, and force and moment data, as well as static lateral and radial stiffness for various modern mountain bike tire sizes including plus size and fat bike tires. Pacejka’s Motorcycle Magic Formula tire model was applied and used to compare results. A basic model of tire lateral stiffness incorporating individual tread knobs as springs in parallel with the overall tread and the inflated carcass as springs in series was derived. Finally, the influence of inflation pressure was also examined. Results demonstrated appreciable differences in tire performance between 29 × 2.3”, 27.5 × 2.8”, 29 × 3”, and 26 × 4” knobby tires. The proposed simple model to combine tread knob and carcass stiffness offered a good approximation, whereas inflation pressure had a strong effect on mountain bike tire behavior.

Transverse vortex-induced vibration of a circular cylinder on a viscoelastic support at low Reynolds number

The effect of a viscoelastic-type structural support on vortex-induced vibration (VIV) of a circular cylinder has been studied computationally for a fixed mass ratio (m∗=2.546) and Reynolds number (Re=150). Unlike the classical case of VIV where the structural support consists of a spring and damper in parallel, this study considers two springs and one damper, where the two springs are in parallel and the damper is in series with one of the springs. This spring/damper arrangement is similar to the Standard Linear Solid (SLS) model used for modelling viscoelastic behaviour. The viscoelastic support (SLS type) is governed by the following two parameters: (a) the ratio of spring constants (R), and (b) the damping ratio (ζ). The focus of the present study is to examine and understand the varied response of the cylinder to VIV as these parameters are varied. For small ζ and R, the cylinder response shows characteristics similar to the classical case, where the amplitude response is composed of an upper- and the lower-type branch. The presence of upper-type branch at low Re is evident through the peak lift force, frequency and phase response of the cylinder. As the damping ratio is increased, the vibration amplitude decreases and hence the upper-type branch disappears. There exists a critical value of ζ=1 beyond which the amplitude again increases asymptotically. The non-monotonic variation of amplitude response with ζ is presented in the form of the ”Griffin plot”. The amplitude, force, frequency and phase-difference response of cylinder were found to be mirror symmetric in log(ζ) about ζ=1. In addition, the effect of R at the critical value of damping, ζ=1, was studied. This show that the amplitude decreases with an increase of R, with suppression of the response branches for high R values. The results suggest that a careful tuning of the damping may be effectively employed both to enhance power output for energy extraction applications or to suppress flow-induced vibration.

DEVELOPMENT AND EXPERIMENTAL INVESTIGATION OF SELF-CENTERING DAMPERS WITH PARALLEL HIGH STRENGTH STEEL RING SPRING GROUPS

Based on the self-centering damper with high-strength steel ring spring, a self-centering damper with parallel high-strength steel ring spring is proposed. The seismic performance of the parallel ring spring self-centering damper under multiple seismic conditions was verified by a hysteretic test. The test results show that the parallel high-strength steel ring spring self-centering damper has good self-centering ability, and will not experience performance degradation after multi-time earthquakes. It does not need to be replaced after the earthquake. Compared with the self-centering damper with a single group of ring springs, the damper bearing capacity can be effectively improved under the same size, and the damper internal space can be fully utilized. The different frictional conditions of the annular spring contact surface have a certain influence on the performance of the self-centering damper. The theoretical prediction model proposed is in a good agreement with the experimental results.

Truck suspension incorporating inerters to minimise road damage

Road damage caused by heavy vehicles is a serious problem experienced worldwide. This paper investigates the potential for reduction in road damage by incorporating the inerter element into truck suspension systems. Initially, quarter-car, pitch-plane and roll-plane models with two low-complexity inerter-based linear suspension layouts are investigated in the frequency domain. Reductions of the J95 road damage index for each model are identified against conventional parallel spring–damper truck suspension layouts. It is also shown that the proposed suspensions are capable of enhancing the roll stability while keeping the road damage at a given level. Subsequently, the nonlinear relationship between force and displacement as manifested by leaf springs is incorporated into the pitch-plane and roll-plane time-domain models. These confirm the potential advantage of inerter-based suspension layouts for road damage reduction.

Failure analysis of anchor bolt of rail fastening system for direct fixation track

The failures of the rail fastening systems have been reported in the curved sections of the subways in South Korea. These failures occur due to the lack of fastening strengths associated with two disc springs arranged in series, causing deformation of the base plate and large shear stresses on the anchor bolts. To demonstrate the failure mechanism, a series of on-site tests and numerical simulations have been conducted. The fastening strength can be enhanced by arranging three disc springs in parallel. When three disc springs were arranged in parallel, a significant reduction in vertical displacements of the rail supporting points, lateral displacements of the base plate and anchor bolts were observed compared to when two disc springs were arranged in series. The numerical simulations revealed that when two disc springs were arranged in series, the base plate can move, resulting in a high shear stress concentration on the anchor bolts. The long-term performance of the proposed setting has also been confirmed.

Ultrasound imaging links soleus muscle neuromechanics and energetics during human walking with elastic ankle exoskeletons

Unpowered exoskeletons with springs in parallel to human plantar flexor muscle-tendons can reduce the metabolic cost of walking. We used ultrasound imaging to look ‘under the skin’ and measure how exoskeleton stiffness alters soleus muscle contractile dynamics and shapes the user’s metabolic rate during walking. Eleven participants (4F, 7M; age: 27.7 ± 3.3 years) walked on a treadmill at 1.25 m s−1 and 0% grade with elastic ankle exoskeletons (rotational stiffness: 0–250 Nm rad−1) in one training and two testing days. Metabolic savings were maximized (4.2%) at a stiffness of 50 Nm rad−1. As exoskeleton stiffness increased, the soleus muscle operated at longer lengths and improved economy (force/activation) during early stance, but this benefit was offset by faster shortening velocity and poorer economy in late stance. Changes in soleus activation rate correlated with changes in users’ metabolic rate (p = 0.038, R2 = 0.44), highlighting a crucial link between muscle neuromechanics and exoskeleton performance; perhaps informing future ‘muscle-in-the loop’ exoskeleton controllers designed to steer contractile dynamics toward more economical force production.

Analysis of block random rocking on nonlinear flexible foundation

In this paper the rocking response of a rigid block randomly excited at its foundation is examined. A nonlinear flexible foundation model is considered accounting for the possibility of uplifting in the case of strong excitation. Specifically, based on an appropriate nonlinear impact force model, the foundation is treated as a bed of continuously distributed springs in parallel with nonlinear dampers. The statistics of the rocking response is examined by an analytical procedure which involves a combination of static condensation and stochastic linearization methods. In this manner, repeated numerical integration of the highly nonlinear differential equations of motion is circumvented, and a computationally efficient semi-analytical solution is obtained. Comparisons with pertinent Monte Carlo simulations demonstrate the efficiency and reliability of the proposed approach. Finally, the survival probability related to toppling of the blocks subject to filtered white noise excitation is investigated, for different kinds of block geometries, foundation materials, and filter parameters.

Analysis, Design, and Preliminary Evaluation of a Parallel Elastic Actuator for Power-Efficient Walking Assistance

This paper introduces the analysis, design and preliminary evaluation of a self-integrated parallel elastic actuator (PEA) with an electric motor and a flat spiral spring in parallel to drive the hip joint of lower limb exoskeletons for power-efficient walking assistance. Firstly, we quantitatively analyze the reason why the parallel elasticity (PE) placed at the sagittal hip joint can reduce the motor power requirement during walking assistance, which contributes to the theory of PEA development and application. The design of the PEA is then introduced in detail. The novelty of the design is that the actuator is reduced in size by integrating the spring into the motor, and the requirement of spring stiffness is significantly reduced by placing the spring directly parallel to the motor shaft. Furthermore, both the simulation based on dynamic modeling and benchtop experiment are conducted preliminarily to evaluate the performance of the PEA with nine spring stiffnesses in a range from 0 – 5.29 mN $\cdot \text{m}$ /rad regarding two indexes including the average and maximum positive electrical power of the motor. Their results show that the two indexes become smaller when the PE is attached and decrease as the spring stiffness increases. When the PE with a stiffness of 5.29 mN $\cdot \text{m}$ /rad is attached, the actuator obtains the largest reduction rate of 11.99 and 16.84 % in the average (root mean square) and maximum positive electrical power of the motor in the simulation and 10.3 and 26.25 % in the experiment, respectively. Those results provide evidence for the applicability of the newly designed PEA in driving a lower limb exoskeleton with high power efficiency during walking assistance for paraplegic patients with complete loss in walking ability.

Research on semi-active vibration isolation system based on electromagnetic spring

This paper proposes a semi-active variable stiffness vibration isolation system based on electromagnetic spring for the low-frequency vibration isolation of mass-varying objects. It is achieved by four straight leaf springs in parallel to an electromagnetic spring system composed of a single electromagnet and a permanent magnet. The equivalent magnetic circuit method is used to compute electromagnetic force of the electromagnetic spring system, and mathematical model of the semi-active vibration isolation system is established according to Maxwell's equations. The nonlinear mathematical model is linearized at the equilibrium point by using the Taylor series expansion theorem to establish linear state-space representation of the system, and then using the traditional PID control method, a double closed-loop feedback control system of the inner current loop and outer location loop is designed. By controlling the current in the coil, the equivalent stiffness and electromagnetic force of the system are variable to achieve semi-active control. Furthermore, the control block diagram of the semi-active vibration isolation system is built based on Simulink software, then make a simulation analysis to the vibration isolation performance of the system and compare the effects of vibration isolation with inner current loop control and without inner current loop control, respectively. Finally, the experiments prove the correctness of the theory. It concludes that this semi-active vibration isolation system is a vibration isolation system with broad application prospects, which has fast current response, high vibration isolation efficiency, and an excellent vibration isolation effect for the low-frequency disturbance of mass-varying objects.

Non-resonant response of the novel airborne photoelectric quasi-zero stiffness platform with friction damping

To suppress the low frequency vibrations of airborne photoelectric system and improve measurement accuracy, a novel passive airborne photoelectric quasi-zero stiffness platform (APQZSP), which is composed of upper/bottom planes, anti-shaking structure and six quasi-zero stiffness (QZS) legs, is designed. The QZS leg is constructed by connecting the folded beam spring with magnetic negative stiffness spring (MNSS) in parallel. According to current model, the magnetic force and negative stiffness of MNSS are derived. As the friction damping is introduced with anti-shaking structure, the isolation performance of the platform under friction damping is investigated based on harmonic balance method. Then the effect of damping and excitation on the isolation performance is analyzed. The results indicate that with the QZS technology, the resonant frequency of the platform is reduced and the low frequency vibrations can be effectively isolated with APQZSP. Moreover, the friction damping can maintain the displacement transmissibility at unity as long as the excitation frequency is lower than the break-loose frequency, and then the resonance can be avoided.

Analysis of extended end plate connection equipped with SMA bolts using component method

Shape Memory Alloys (SMAs) are new materials used in various fields of science and engineering, one of which is civil engineering. Owing to their distinguished capabilities such as super elasticity, energy dissipation, and tolerating cyclic deformations, these materials have been of interest to engineers. On the other hand, the connections of a steel structure are of paramount importance because of their vulnerabilities during an earthquake. Therefore, it is indispensable to find approaches to augment the efficiency and safety of the connection. This research investigates the behavior of steel connections with extended end plates equipped hybridly with 8 rows of high strength bolts as well as Nitinol superelastic SMA bolts. The connections are studied using component method in dual form. In this method, the components affecting the connections behavior, such as beam flange, beam web, column web, extended end plate, and bolts are considered as parallel and series springs according to the Euro-Code3. Then, the nonlinear force- displacement response of the connection is presented in the form of moment-rotation curve. The results obtained from this survey demonstrate that the connection has ductility, in addition to its high strength, due to high ductility of SMA bolts.

Picometer mechanical displacement measurement using heterodyne interferometer with phase-locked loop

In this paper, we show picometer-order mechnical displacmment measurements using a heterodyne interferometer with a phase-locked loop (PLL). A heterodyne light source for the interferometer is implemented with a frequency stabilized HeNe laser and two acousto-optic modulators. A real time phase measurement is performed by the PLL, whose software is programmed in a field-programmable gate array (FPGA). A stiff parallel spring stage combined with a high-voltage piezoelectric actuator is used to generate picometer-order mechanical motion. With the above implementations, mechanical displacement of 10 picometer or less can be measured.

Bespoke extensional elasticity through helical lattice systems.

Nonlinear structural behaviour offers a richness of response that cannot be replicated within a traditional linear design paradigm. However, designing robust and reliable nonlinearity remains a challenge, in part, due to the difficulty in describing the behaviour of nonlinear systems in an intuitive manner. Here, we present an approach that overcomes this difficulty by constructing an effectively one-dimensional system that can be tuned to produce bespoke nonlinear responses in a systematic and understandable manner. Specifically, given a continuous energy function E and a tolerance ϵ > 0, we construct a system whose energy is approximately E up to an additive constant, with L∞-error no more that ϵ. The system is composed of helical lattices that act as one-dimensional nonlinear springs in parallel. We demonstrate that the energy of the system can approximate any polynomial and, thus, by Weierstrass approximation theorem, any continuous function. We implement an algorithm to tune the geometry, stiffness and pre-strain of each lattice to obtain the desired system behaviour systematically. Examples are provided to show the richness of the design space and highlight how the system can exhibit increasingly complex behaviours including tailored deformation-dependent stiffness, snap-through buckling and multi-stability.

Modelling of Roller-raceway Contacts in the Slewing Bearing Taking into Account Asymmetrical Load Transfer Through a Roller

During the operation, a slewing bearing is always subjected to a set of combined loads. It is the source of deformation of ball-raceway contacts, rings, and even supporting structures. In practice, deformation of rings and supporting structures is often neglected for simplification, that is, they are supposed to be ideally stiff. To take elasticity of rings and supporting (fixed) structures into consideration, the finite-element method (FEM) is applied. In slewing bearings, a great number of contact pairs are present on the contact surfaces between the rolling ele-ments and raceways of the bearing. In order to improve the computational efficiency of load distribution of large roller slewing bearing, a computa-tional model using one-dimensional finite elements (nonlinear elements) is presented in this paper. In this model, each roller is simulated by a group of nonlinear elements truss, which has the same load-deformation perfor-mance with solid roller-raceway contacts. The results show that a group of parallel springs can be used to replace the solid roller and simulate the line contact performance between the roller and raceway. Obtained results are presented as graphs.

Fractional derivative models for viscoelastic materials at finite deformations

Fractional derivative models, which are expressed by combining standard dashpots, fractional dashpots and elastic springs in series or parallel, are often utilized to account for the behaviors for viscoelastic materials. Even with the models extended to finite deformation, the precise definition of objective fractional derivative remains challenging. The proposed fractional derivative model is expressed by the combination of an elastic spring in series with two parallel fractional dashpots. We extend the fractional derivative model to finite deformation through a new approach without defining an objective fractional derivative and assuming the decomposition of the deformation rate into the elastic and inelastic parts. This proposed model can be reduced to the Maxwell model for finite deformation. Such reduction results in a model that stands in between the two existing Maxwell models in which the objective rate of the Cauchy stress is taken as the material corotational rate and the relative corotational rate respectively. The proposed model is applied to the simple shear deformation.

Scaled and Dynamic Optimizations of Nudged Elastic Bands.

We present a modified nudged elastic band routine that can reduce the number of force calls by more than 50% for bands with nonuniform convergence. The method, which we call "dyNEB", dynamically and selectively optimizes images on the basis of the perpendicular PES-derived forces and parallel spring forces acting on that region of the band. The convergence criteria are scaled to focus on the region of interest, i.e., the saddle point, while maintaining continuity of the band and avoiding truncation. We show that this method works well for solid state reaction barriers-nonelectrochemical in general and electrochemical in particular-and that the number of force calls can be significantly reduced without loss of resolution at the saddle point.

A Stochastic Damage Model for Bond Stress-Slip Relationship of Rebar-Concrete Interface under Monotonic Loading.

The stochastic bond stress-slip behavior is an essential topic for the rebar-concrete interface. However, few theoretical models incorporating stochastic behavior in current literature can be traced. In this paper, a stochastic damage model based on micro-mechanical approach for bond stress-slip relationship of the interface under monotonic loading was proposed. In order to describe the mechanical behaviors of the rebar-concrete interface, a microscopic damage model was proposed. By introducing a micro-element consists of parallel spring element, friction element and a switch element, the model is formulated. In order to reflect the randomness of the bond stress-slip behavior contributed by the micro-fracture in the interface, a series of paralleled micro-elements are adopted with the failure threshold of individual spring element is set as a random variable. The expression of both mean and variance for the bond stress-slip relationship was derived based on statistical damage mechanics. Furthermore, by utilizing a search heuristic global optimization algorithm (i.e., a genetic algorithm), parameters of the proposed model are able to be identified from experimental results, which a lognormal distribution has adopted. The prediction was verified against experimental results, and it reveals that the proposed model is capable of capturing the random nature of the micro-structure and characterizing the stochastic behavior.

Design of a Soft Composite Finger with Adjustable Joint Stiffness

This article presents the design of a soft composite finger with tunable joint stiffness. The composite finger, made of two different types of silicone, uses hybrid actuation by combining tendon and pneumatic actuation schemes. Tendons control the finger shape in a prescribed direction to demonstrate discrete bending behavior due to different material moduli, similar to that of a human's finger. The pneumatic actuation changes the stiffness of joints using air chambers. The feasibility of adjustable stiffness joints is proven using both the parallel spring model and experiments that demonstrate the stiffening effect when pressurized. A set of experiments were also conducted on fingers with four different chamber shapes to observe the effect of chamber shape on stiffening and the discrete bending capability of the finger. The stiffness control can tune the structural softness of the finger, which leads to firm grasp during higher acceleration object manipulation.

Design and Characterization of an Active Compression Garment for the Upper Extremity

This paper presents the design, development, and testing of an active textile-based wearable compression device that is capable of delivering a controlled pulsatile compression. The device uses low spring index nickel titanium (NiTi) coil actuators to produce an applied dynamic pressure of up to 5.5 kPa. The selected NiTi coil actuators produce pressure when thermally stimulated with Joule heating via an applied current (0.3 A), and generate recoverable strains up to 75% in extension. An optical fiber strain sensor was developed to monitor the textile strain and enable the indirect estimation of the applied pressure. A new approach using a passive NiBR spring (in parallel with the NiTi actuators) was also developed to assist the NiTi coils in recovering the detwinned martensite form after cooling. The pressure distribution around a rigid cylindrical shape was also evaluated, showing higher applied pressures (5.5 kPa) where the NiTi coil actuators were located. The strain sensor exhibits high accuracy compared to a reference commercial sensor (as indicated by the high correlation indexes of up to 0.97 between compression cycle measurements with both solutions).

A Viscoelastic-Plastic Constitutive Model of Shape Memory Polymer

ABSTRACTIt is of practical significance to develop a constitutive model which is able to predict the thermomechanical behaviors of the shape memory effect occurring in a shape memory polymer (SMP) accurately. The mechanism of shape memory effect of SMP is explained based on the assumption that SMP is composed by two phases, reversible phase and stationary phase. Especially the different flow elements are respectively added to the reversible phase and stationary phase in order to express the plastic behavior of SMP. There are two springs in series, one dashpot and one flow element in the reversible phase. There are two springs in parallel, one dashpot and one flow element in the stationary phase. A constitutive equation is developed to express the thermo-mechanical behaviors of shape memory effect in the SMP based on viscous-elastic mechanics and plastic theory. An internal variable, frozen ratio, is defined to follow the shape memory process in SMP, and the material properties are described as the functions of frozen ratio based on phase transition theorem. The developed constitutive model, which includes above constitutive equation and material parameter functions, is used to numerically simulate the thermo-mechanical behaviors of SMP under various load cycles. Results show that the developed constitutive model can not only predict the shape memory process of SMP accurately, but also express the rate-dependent behaviors of SMP effectively.