Torsional Spring Research Articles

An analytical closed-form solution for the in-plane elastic modulus of plain-woven textile composites

Engineering application of textile composites needs simple relations for the estimation of their mechanical properties. Many homogenization and simplified models have been already proposed for the analytical characterization of the elastic moduli of this class of materials. Along this route, the current study introduces an analytical solution for the longitudinal elastic modulus of plain-woven textile composites. A hybrid beam-spring model of the representative unit cell (RUC) is postulated consisting of a horizontal beam connected to an inclined beam, two supporting springs beneath the free end of the horizontal beam and the common end of the horizontal and inclined beams, and a torsion spring at the free end of the inclined beam. The mechanical and geometrical properties of the beams and springs are calculated from the weave structure and the symmetry conditions. A reduction technique from the matrix analysis of structures is borrowed to consider the effect of internal degrees of freedom on the in-plane stiffness coefficient and a closed-form formula is extracted for the elastic modulus. The proposed analytical formula is finally verified against available experimental data and a wide range of FE predictions for the GFRP and CFRP plain-woven composites.

A self-propelling clapping body

We report an experimental study of the motion of a clapping body consisting of two flat plates pivoted at the leading edge by a torsion spring. Clapping motion and forward propulsion of the body are initiated by the sudden release of the plates, initially held apart at an angle $2\theta _o$ . Results are presented for the clapping and forward motions, and for the wake flow field for 24 cases, where depth-to-length ratio ( $d^* = 1.5,1\text { and }0.5$ ), spring stiffness per unit depth ( $Kt$ ), body mass ( $m_b$ ) and initial separation angle ( $2\theta _o = 45^{\circ }\text { and }60^{\circ }$ ) are varied. The body initially accelerates rapidly forward, then slowly retards to nearly zero velocity. Whereas the acceleration phase involves a complex interaction between plate and fluid motions, the retardation phase is simply fluid dynamic drag slowing the body. The wake consists of either a single axis-switching elliptical vortex loop (for $d^* = 1\text { and }1.5$ ) or multiple vortex loops (for $d^* = 0.5$ ). The body motion is nearly independent of $d^*$ and most affected by variation in $\theta _o$ and $Kt$ . Using conservation of linear momentum and conversion of spring strain energy into kinetic energy in the fluid and body, we obtain a relation for the translation velocity of the body in terms of the various parameters. Approximately 80 % of the initial stored energy is transferred to the fluid, only 20 % to the body. The experimentally obtained cost of transport lies between 2 and $8\ \mathrm {J}\ \mathrm {kg}^{-1}\ \mathrm {m}^{-1}$ .

On the Aeroelasticity of a Cantilever Wing Equipped with the Spanwise Morphing Trailing Edge Concept

This paper studies the aeroelastic behavior of a rectangular, cantilever wing equipped with the spanwise morphing trailing edge (SMTE) concept. The SMTE consists of multiple trailing edge flaps that allow controlling the spanwise camber distribution of a wing. The flaps are attached at the wing’s trailing edge using torsional springs. The Rayleigh–Ritz method is used to develop the equations of motion of the wing-flap system. The use of shape functions allows for representing the wing as an equivalent 2D airfoil with generalized coordinates that are defined at the wingtip. Strip theory, based on Theodorsen’s unsteady aerodynamic model, is used to compute the aerodynamic loads acting on the wing. A representative Padé approximation for Theodorsen’s function is utilized to model the aerodynamic behaviors in a state-space form allowing time-domain simulation and analysis. The model is validated using a rectangular cantilever wing and the data are available in the literature. A comprehensive parametric comparison study is conducted to assess the impact of flap stiffness on the aeroelastic boundary. In addition, the potential of the SMTE to provide load alleviation and flutter suppression is assessed for a wide range of flight conditions, using a discrete (1-cosine) gust. Finally, the implementation and validation of a controller for a wing with SMTE for gust load alleviation are studied and controller parameters are tuned for a specific gust model.

Integrated Optimum Design of a Torsion Spring-Compensated Automotive Engine Hood Linkage Mechanism

A torsion spring-assisted automotive hood linkage with joint friction is statically balanced for its entire range of motion. The four-bar linkage dimensions are to be synthesized. Coulomb friction at the joints can assist in the balancing. The magnitude of friction at the joints are unknown, and so are the torsion spring characteristics. All the aforementioned unknowns are determined in an integrated procedure such that the linkage dimensions, joint friction as well as the torsion spring are all designed optimally together in one go. The objective is to require the lowest force to close and to open the engine hood, with the entire design procedure to be performed in just one-step. Only three specifications are known: The mass characteristics (weight and center of gravity location) of the hood, the two acceptable regions of the hinge locations either on the engine hood or on the car body, as well as, the closed and opened positions of the engine hood. Thirty different design configurations (scenarios) are investigated and the results are discussed. The optimal results for a torsion spring-assisted hood linkage when compared to a similar tension spring-assisted linkage, show much better load compensation characteristics: The magnitudes and fluctuations of the external lifting force are smaller. Moreover, problem specification for a torsion spring system is also simpler.

Modelling and experimental identification of spring-damping properties of the off-road vehicle rubber tracks, rubber belts, and rubber-bushed tracks subjected to flexural vibrations

In the article, a model for predicting the energy losses caused by the flexural vibrations of rubber tracks, rubber belts, and rubber-bushed metal link-tracks for off-road vehicles is proposed, and a test stand and an experimental procedure are developed to identify the mechanical parameters of this model. The track or belt is represented by a chain of discrete rigid links connected by revolute joints, and a discrete spring-element is placed in parallel with multiple Maxwell-elements in each joint to capture the flexural rigidity and damping of the real track or belt. The mechanical parameters of the joint are found by testing real tracks or belts under cyclic bending. The models consisting of three, four, or five Maxwell-elements per joint are the most successful in predicting the response of a sample rubber track to cyclic bending. The spring-damping properties of tracks and belts identified with the method discussed herein can be applied in simulation studies on the interaction of tracked vehicles and soil. Furthermore, vehicle elements such as rubber bushings for suspension systems, rubber torsion springs, and oil-filled and rubber torsion dampers can be tested with this method to find their spring-damping properties required by vehicle dynamics simulations.

Maximizing Wave Energy Converter Power Extraction by Utilizing a Variable Negative Stiffness Magnetic Spring

Complex conjugate impedance matching is a key concept for wave energy converter design. Matching the impedance of the power take-off (PTO) system to the complex conjugate of the wave energy converter's (WEC) impedance ensures efficient transfer of energy from the WEC body motion to electrical power. In low frequency waves, impedance matching often requires a negative PTO stiffness. In this paper, an adjustable stiffness magnetic torsion spring will be presented and modeled to understand its potential to improve WEC performance. The spring has the ability to provide a negative stiffness, allowing the PTO impedance to more closely match the complex conjugate of the WEC impedance at low frequencies. The spring also supports an adjustable stiffness value, meaning it can be tuned according to the incoming wave conditions. The spring's tunability may put less stress on the rest of the PTO system in wave conditions outside its normal operation zone without sacrificing electrical power output. The adjustable magnetic spring's effects are modeled and explored in this paper by examining the resultant average annual electrical power and capacity factor. The study suggests that the tunable magnetic spring has the potential to significantly improve capacity factor while maintaining a high average electrical power.

Dynamic characteristics of cracked simply supported bidirectional functionally graded Rayleigh beam

This paper investigates the dynamic behavior of cracked Rayleigh beams constructed from bidirectional functionally graded (BDFG) materials under simple boundary conditions. A torsional massless spring is employed to model the beam's open crack type. The vibration equations are obtained using Hamilton's principle. The graded beam material properties are varied throughout the thickness based on the power-law distribution and in the longitudinal direction using the exponential material distribution. To solve the dynamic equations, Galerkin's approach is employed. The paper evaluates the impacts of the axial index, gradient property index, beam modulus ratio, and crack parameters on the natural frequencies of the FG beam. The results indicate that the dimensionless natural frequencies of intact graded beams decrease with increased gradient index k. In contrast, they increase with a rise in the modulus ratio. Additionally, the results demonstrate that an increase in the crack depth ratio decreases dimensionless natural frequencies.

Lagrange's method applied to a Slinky

The Lagrange method is applied to two dynamic models of a Slinky, one based on point masses and linear springs and a second where the Slinky is represented as a sequence of half hoops connected by torsion springs. For the first time, the use of Lagrange's method applied to a Slinky has produced a multi-body dynamic model that can potentially with minor modification reproduce all the interesting behaviour Slinkies are well known for; descending stairs, pseudo levitation, transmission of longitudinal and transverse waves. In this paper, the models are derived and a limited exploration of the two dynamic models’ behaviour is considered. For unforced oscillation, the point mass model and torsion spring model produce a similar amplitude and frequency. When considering forced oscillations, the point mass model has a very different spectrum of natural frequencies than the torsion spring model. The torsion spring model is considered for different forcing conditions.

Controlled synchronization of coupled pendulums by Koopman Model Predictive Control

We propose a method to solve a class of control problems arising from a system of coupled pendulums. The system considered in this work is a one-dimensional array of pendulums pivoting around a single axis with adjacent pendulums coupled through torsion springs. Only a single torque motor attached to one of the two boundary pendulums actuates the system. This setup of coupled pendulums is a mechanical realization of the Frenkel–Kontorova (FK) model – a spatially discrete version of the sine-Gordon equation describing (nonlinear) waves. The main challenges of controlling this system are high order (the number of pendulums can be high), nonlinear and oscillatory dynamics, and only one actuator. The proposed class of problems can be characterized as controlled synchronization – designing a closed-loop controller that synchronizes the motion of the pendulums. Controlled synchronization is a special case of reference tracking, where all pendulums reach a common point or a trajectory. One can formulate many practically motivated problems within this class; here, we identify three problems: a vibration control of a flexible structure, swing-up control of all pendulums in the array, and low-friction sliding of an atomic-scale structure. We show that the presented problems can be dealt with by the Koopman Model Predictive Control (KMPC). The KMPC allows for controlling nonlinear systems by combining the classical linear model predictive control (MPC) with the Koopman operator approach for nonlinear dynamical systems. The main idea of the method is to construct a linear predictor of a nonlinear system in a higher-dimensional, lifted space and use the predictor within the linear MPC. The optimization problem formulation within the KMPC can then be tailored to a specific problem in the class. Simulations and experiments on a hardware platform realizing the FK model show the effectiveness of the method.

Miniature side-view dual axes confocal endomicroscope for repetitive in vivo imaging.

A side-view dual axes confocal endomicroscope is demonstrated that can be inserted repetitively in hollow organs of genetically engineered mice for in vivo real-time imaging in horizontal and vertical planes. Near infrared (NIR) excitation at λex = 785 nm was used. A monolithic 3-axis parametric resonance scan mirror was fabricated using micro-electro-mechanical systems (MEMS) technology to perform post-objective scanning in the distal end of a 4.19 mm diameter instrument. Torsional and serpentine springs were designed to "switch" the mode of imaging between vertical and horizontal planes by tuning the actuation frequency. This system demonstrated real-time in-vivo images in horizontal and vertical planes with 310 µm depth and 1.75 and 7.5 µm lateral and axial resolution. Individual cells and discrete mucosal structures could be identified.

Modelling Cell Orientation Under Stretch: The Effect of Substrate Elasticity

When cells are seeded on a cyclically deformed substrate like silicon, they tend to reorient their major axis in two ways: either perpendicular to the main stretching direction, or forming an oblique angle with it. However, when the substrate is very soft such as a collagen gel, the oblique orientation is no longer observed, and the cells align either along the stretching direction, or perpendicularly to it. To explain this switch, we propose a simplified model of the cell, consisting of two elastic elements representing the stress fiber/focal adhesion complexes in the main and transverse directions. These elements are connected by a torsional spring that mimics the effect of crosslinking molecules among the stress fibers, which resist shear forces. Our model, consistent with experimental observations, predicts that there is a switch in the asymptotic behaviour of the orientation of the cell determined by the stiffness of the substratum, related to a change from a supercritical bifurcation scenario, whereby the oblique configuration is stable for a sufficiently large stiffness, to a subcritical bifurcation scenario at a lower stiffness. Furthermore, we investigate the effect of cell elongation and find that the region of the parameter space leading to an oblique orientation decreases as the cell becomes more elongated. This implies that elongated cells, such as fibroblasts and smooth muscle cells, are more likely to maintain an oblique orientation with respect to the main stretching direction. Conversely, rounder cells, such as those of epithelial or endothelial origin, are more likely to switch to a perpendicular or parallel orientation on soft substrates.

Interacting effects of AFO stiffness, neutral angle and footplate stiffness on gait in case of plantarflexor weakness: A predictive simulation study

To maximize effects of dorsal leaf ankle foot orthoses (AFOs) on gait in people with bilateral plantarflexor weakness, the AFO properties should be matched to the individual. However, how AFO properties interact regarding their effect on gait function is unknown. We studied the interaction of AFO bending stiffness with neutral angle and footplate stiffness on the effect of bending stiffness on walking energy cost, gait kinematics and kinetics in people with plantarflexor weakness by employing predictive simulations.Our simulation framework consisted of a planar 11 degrees of freedom model, containing 11 muscles activated by a reflex-based neuromuscular controller. The controller was optimized by a comprehensive cost function, predominantly minimizing walking energy cost. The AFO bending and footplate stiffness were modelled as torsional springs around the ankle and metatarsal joint. The neutral angle of the AFO was defined as the angle in the sagittal plane at which the moment of the ankle torsional spring was zero. Simulations without AFO and with AFO for 9 bending stiffnesses (0–14 Nm/degree), 3 neutral angles (0–3-6 degrees dorsiflexion) and 3 footplate stiffnesses (0–0.5–2.0 Nm/degree) were performed.When changing neutral angle towards dorsiflexion, a higher AFO bending stiffness minimized energy cost of walking and normalized joint kinematics and kinetics. Footplate stiffness mainly affected MTP joint kinematics and kinetics, while no systematic and only marginal effects on energy cost were found.In conclusion, the interaction of the AFO bending stiffness and neutral angle in bilateral plantarflexor weakness, suggests that these should both be considered together when matching AFO properties to the individual patient.

Robotic check of a subassembly, and its simulation

This paper discusses a quality inspection process of a subassembly of a battery cover, which is performed with an industrial robot equipped with an intelligent end-effector. The subassembly has a plastic pin and a torsion spring part. During intended use of the unit, the necessary force to actuate the pin is determined with measurements and simulations. In the measurement a self-devised intelligent end-effector is equipped with a microcontroller and a beam type load cell in order to measure the spring force. The main aim is to make an automatic decision without operator intervention at the quality check process, which is related to the quality of the subassembly. In the simulation deformation of the spring part is modeled with geometrically nonlinear 2D beam finite elements in the course of assembly and intended use. Co-rotational approach is used to consider the large displacements and rotations with small strains. Penalty method is applied to treat the contact between the spring and pin parts.

Study on a quasi-zero-stiffness isolator for variable mass load

Generally, the performance of the quasi-zero-stiffness (QZS) isolator is very sensitive to the deviation of mass load. Due to the ultra-low stiffness, even a slight deviation could lead to a large offset from the designed equilibrium. The performance degradation for low frequency vibration would be caused by the increase of the stiffness around the actual equilibrium. In this paper, a QZS isolator with the deviation of mass load is studied. The isolator comprises a nonlinear positive stiffness configuration with a pair of torsion springs, oblique bars and linear bearings, and a negative stiffness configuration with a pair of oblique bars connected to horizontal springs. The dynamic equation of the isolator with the deviation of mass load is derived. The amplitude-frequency property and force transmissibility are investigated. It is found that the deviation of mass load could significantly change the dynamic properties of the QZS isolator. To improve the isolation performance, the parameters of the positive stiffness configuration are considered to be adjustable. The results show that the adjustability can help remain the initial equilibrium, and reduce the response amplitude and force transmissibility in low frequency. The negative effect of the deviation of mass load on the QZS isolator can be decreased to a great extent.

The X-shaped structure with nonlinear positive stiffness compensation for low-frequency vibration isolation

X-shaped structure with quasi-zero stiffness (QZS) properties are suitable for low-frequency vibration isolation because of its simple structure. The effective working range of classic X-shaped structure (CXS) is narrow for the existence of negative stiffness region, and using positive stiffness compensation to overcome this deficiency is a common way. This paper proposes a novel nonlinear positive stiffness compensation method (NPSC) to improve the load capacity and QZS range of CXS. Different from the previous studies that utilize linear or hardening stiffness compensation, the NPSC exhibits the characteristics of softening stiffness first and then hardening stiffness and is provided by torsion springs simply. The static properties of X-shaped structure with NPSC are investigated. The result indicates the NPSC possesses dual significant advantages. Firstly, the X-shaped structure with NPSC can achieve larger load capacity and wider QZS range than that with linear or hardening stiffness compensation in optimal parameter region. Secondly, the X-shaped structure with NPSC can conveniently overlap the equilibrium position with QZS point to obtain ultra-low frequency vibration isolation effect compared to that with linear stiffness compensation. The motion equation of the proposed system is derived, and the dynamic response is solved through harmonic balance method (HBM). The influence of different parameters on displacement transmissibility is investigated and the result is verified by numerical simulation. The transmissibility comparison between the proposed model and the CXS with linear compensation shows that the proposed model possesses better vibration isolation performance. Besides, the static compression test and dynamic sweep frequency experiment are implemented, and the experiment results verify the validity of proposed model. The proposed NPSC is an important inspiration for the stiffness compensation research of X-shaped structures.

Automatic T-Branch Travel of an Articulated Wheeled In-Pipe Inspection Robot Using Joint Angle Response to Environmental Changes

The present paper proposes an automatic T-branch travel method of an articulated wheeled in-pipe inspection robot, AIRo-5.2, using its joint angle response to environmental changes. The robot joints are composed of two passive elastic joints with torsional springs and a single active joint using a polyurethane-based series elastic actuator (SEA). By switching from torque control to angle control of this SEA, the AIRo-5.2 can pass through vertical T-branches. However, the timing needed to switch them has not yet been defined, and thus it was necessary to train the operators. Therefore, a method that uses the middle joint angle change was proposed for automatic T-branch travel. To elucidate the phenomenon of this angle change and to determine the effects of a robot's mechanical parameters, a partial dynamic model was constructed when only the head link contact was released at a T-branch. Finally, the travel performance of our developed in-pipe robot was tested on 10 types of T-branches with different gravity directions. From the experiments, although the probability of completing the course was not 100% in all cases, the effectiveness of our proposed principle and algorithm for automatic T-branch travel was confirmed.

Dynamic Modelling of the Spring Attached Two-Link Planar Manipulator

Robot dynamics is required for simulation, control, analysis of robot motion planners and controllers. There are a number of dynamic modelling investigations of robots with the different approaches and these investigations mostly focus on the link flexibility, joint friction or actuator dynamics. In addition to that these studies present dynamic model without any attached linear or torsion springs. In this study, the dynamic modelling of a two-link rigid manipulator with attached torsion springs is presented using the Lagrangian approach. The Lagrangian approach is a variational method that relies on the potential and kinetic energy of the mechanism, making it well-suited for the analysis of the two-link planar manipulator considered in this study. Initially, the two-link mechanism equations of motion are derived without any attached torsion springs. Subsequently, two torsion springs are attached to the joints of the mechanism, and the existing equations of motion are modified accordingly. The study also presents the kinetic, potential and total energies of the two link-manipulator, angular positions of the links and their velocities. By considering the dynamic modelling of the torsion spring attached two-link rigid manipulator, this study contributes to understanding and analysis of the two-link manipulators and the dynamic effects of the attached springs.

Design and Analysis of an Extended Simply Supported Beam Piezoelectric Energy Harvester

The harvesting efficiency of a cantilevered piezoelectric energy harvester is limited by its uneven strain distribution. Moreover, a cantilevered harvester requires a large workspace due to the large displacement of its free end. To address these issues, a novel piezoelectric energy harvester based on an extended simply supported beam is proposed. The proposed design features a simply supported piezoelectric main beam with an extended beam attached to its roller end and a tip mass to reduce the resonant frequency. The theoretical model of the proposed piezoelectric energy harvester is developed based on the Euler–Bernoulli beam theory. The model has been experimentally validated through the fabrication of a prototype. The extended beam and tip mass are adjusted to see their influence on the performance of the harvester. The resonant frequency can be maintained by shortening the extended beam and increasing the tip mass simultaneously. A shorter extend beam leads to a more even strain distribution in the piezoelectric layer, resulting in an enhanced output voltage. Moreover, the simulation results show that a torsional spring is installed on the roller joint which greatly influences the voltage output. The strain distribution becomes more even when proper compressive preload is applied on the main beam. Experiments have shown that the proposed design enhances the output power by 86% and reduces tip displacement by 63.2% compared to a traditional cantilevered harvester.

Origami-Inspired Variable Stiffness Actuator for Safe Human–Robot Interaction

Abstract In this paper, an origami-inspired variable stiffness actuator (OVSA) is proposed. Innovatively borrowing from the origami principle, OVSA uses torsion springs and hinges to form a transformable hexagonal structure as the special elastic component. While having a light and compact structure, it can also theoretically realize the change of stiffness from zero to infinity. Archimedean Spiral Cam (ASC) is used as a transmission element to connect the motor and elastic components further increasing the compactness of the structure. In addition, the OVSA requires very little torque to maintain stiffness under deflection conditions, which improves energy efficiency. In this paper, the stiffness equation of OVSA is established and verified by experiments. The results show that its stiffness is high when the deflection angle is close to 0 deg and decreases rapidly with the increase of deflection angle, which makes it have sufficient safety.

Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots

Underactuated robots have fewer actuators than degrees of freedom (DOF). Nonactuated joints can be equipped with torsional springs. Underactuated robots can be controlled in a point-to-point motion if they have a particular mass distribution that makes them differentially flat. The trajectory described by the robot moving from the start point to the end point largely depends on the torsional stiffness of the nonactuated joints and on motion time. Thus, the same point-to-point motion can be obtained by sweeping different parts of the workspace. This property increases the dexterity of the robot. This paper focuses on the trajectories of a 3-DOF robot moving in the horizontal plane with two actuators and a torsional spring. Parametric analyses showing the effect of torsional stiffness and motion time are presented. The existence of combinations of torsional stiffness and motion time that minimize the motion torques or the swept area is discussed. The area swept by the underactuated robot is compared with the one swept by an equivalent actuated robot performing the same task. Reductions in the swept area of up to 36% are obtained. Finally, numerical results are validated by means of experimental tests on a simplified prototype.