Quadratic Spring Research Articles

Design and control of a novel variable stiffness actuator based on antagonistic variable radius principle

Variable stiffness actuators (VSAs) are essential for ensuring safe human–robot interactions in robotic applications. This paper proposes a novel rotary VSA using an antagonistic Hoberman linkage mechanism (AHLM), which offers a large stiffness range and a compact structure. The VSA-AHLM consists of two sets of antagonistic-type quadratic springs based on spiral cams connected to the Hoberman linkage mechanism (HLM) through four cables. By simultaneously adjusting both the radius of the HLM and the spring preload, the stiffness of the VSA-AHLM can be varied within a large range. Furthermore, the position and stiffness of the VSA-AHLM can be controlled independently by two rotary motors. The geometric parameters of the spiral cam are determined to achieve the desired linear stiffness–elongation behavior of a quadratic spring, and detailed models of the actuator’s stiffness, elastic energy, and torque are established. An actuator prototype is fabricated to demonstrate the proposed variable stiffness approach. Experiments show that the developed actuator can achieve significant stiffness changes and exhibits excellent positioning and trajectory tracking performance.

A model for hyperelastic rubber-like materials based on micro-mechanical elements

In this work, we propose a microstructurally motivated hyperelastic model to describe the behavior of rubber-like materials. At the scale of the Representative Volume Element (RVE), we assume that, for each macromolecular chain, the segments of the chains are deformable and that there is a bending energy between two consecutive segments. We propose to model each macromolecular chain using micro-mechanical elements: elastic bars to represent the segments between cross-linking points and elastic spire to illustrate the flexibility of rotations around the cross-linking points. We thus suggest to model the behavior of the macrochain, using a quadratic spring like potential for a linear behavior of the chain segments, associated with a nonlinear elastic sigmoidal behavior at the connection points between Kuhn segments. Numerical simulation, on different RVEs, show that the proposed modeling represent the response of hyperelastic rubber-like materials in uniaxial extension, simple shear, pure shear and biaxial extension. In order to validate the proposed model, the results obtained in the case of four RVEs will be compared with the experimental data of Treloar (1944). These comparisons show that the proposed model is able to reproduce the experimental behavior of rubber-like materials.

Design of a Quadratic, Antagonistic, Cable-Driven, Variable Stiffness Actuator

Abstract Antagonistically actuated variable stiffness actuators (VSAs) take inspiration from biological muscle structures to control both the stiffness and positioning of a joint. This paper presents the design of an elastic mechanism that utilizes a cable running through a set of three pulleys to displace a linear spring, yielding quadratic spring behavior in each actuator. A joint antagonistically actuated by two such mechanisms yields a linear relationship between force and deflection from a selectable equilibrium position. A quasi-static model is used to optimize the mechanism. Testing of the fabricated prototype yielded a good match to the desired elastic behavior.

Conceptual mechanical design of antagonistic variable stiffness joint based on equivalent quadratic torsion spring.

The variable stiffness joint is a kind of flexible actuator with variable stiffness characteristics suitable for physical human-robot interaction applications. In the existing variable stiffness joints, the antagonistic variable stiffness joint has the advantages of simple implementation of variable stiffness mechanism and easy modular design of the nonlinear elastic element. The variable stiffness characteristics of antagonistic variable stiffness joints are realized by the antagonistic actuation of two nonlinear springs. A novel design scheme of the equivalent nonlinear torsion spring with compact structure, large angular displacement range, and desired stiffness characteristics is presented in this article. The design calculation for the equivalent quadratic torsion spring is given as an example, and the actuation characteristics of the antagonistic variable stiffness joint based on the equivalent quadratic torsion spring are illustrated. Based on the design idea of constructing the antagonistic variable stiffness joint with compact structure and high compliance, as well as the different design requirements of the joints at different positions of the multi-degrees of freedom robot arm, nine types of mechanical schemes of antagonistic variable stiffness joint with the open design concept are proposed in this article. Finally, the conceptual joint configuration schemes of the robot arm based on the antagonistic variable stiffness joint show the application scheme of the designed antagonistic variable stiffness joint in the multi-degrees of freedom robot.

Silicon carbide zipper photonic crystal optomechanical cavities.

We demonstrate a silicon carbide (SiC) zipper photonic crystal optomechanical cavity. The device is on a 3C-SiC-on-silicon platform and has a compact footprint of ∼30 × 1 μm. The device shows an optical quality of 2800 at telecom and a mechanical quality of 9700 at 12 MHz with an effective mass of ∼3.76 pg. The optical mode and mechanical mode exhibit strong nonlinear interaction, namely, the quadratic spring effect, with a nonlinear spring constant of 3.3 × 104 MHz2/nm. The SiC zipper cavity is potentially useful in sensing and metrology in harsh environments.

Analytical Expressions for Spring Constants of Capillary Bridges and Snap-in Forces of Hydrophobic Surfaces.

When a force probe with a small liquid drop adhered to its tip makes contact with a substrate of interest, the normal force right after contact is called the snap-in force. This snap-in force is related to the advancing contact angle or the contact radius at the substrate. Measuring snap-in forces has been proposed as an alternative to measure the advancing contact angles of surfaces. The snap-in occurs when the distance between the probe surface and the substrate is hS, which is amenable to geometry, assuming the drop was a spherical cap before snap-in. Equilibrium is reached at a distance hE < hS. At equilibrium, the normal force F = 0, and the capillary bridge is a spherical segment, amenable again to geometry. For a small normal displacement Δh = h – hE, the normal force can be approximated with F ≈ −k1Δh or F ≈ −k1Δh – k2Δh2, where k1 = −∂F/∂h and k2 = −1/2·∂2F/∂h2 are the effective linear and quadratic spring constants of the bridge, respectively. Analytical expressions for k1,2 are found using Kenmotsu’s parameterization. Fixed contact angle and fixed contact radius conditions give different forms of k1,2. The expressions for k1 found here are simpler, yet equivalent to the earlier derivation by Kusumaatmaja and Lipowsky (2010). Approximate snap-in forces are obtained by setting Δh = hS – hE. These approximate analytical snap-in forces agree with the experimental data from Liimatainen et al. (2017) and a numerical method based on solving the shape of the interface. In particular, the approximations are most accurate for super liquid-repellent surfaces. For such surfaces, readers may find this new analytical method more convenient than solving the shape of the interface numerically.

Nonlinear vibro-acoustic behaviors of coupled sandwich cylindrical shell and spring-mass-damper systems

Abstract This paper is concerned with the structural-acoustic analysis of a coupled sandwich cylindrical shell and nonlinear spring-mass-damper system immersed in an infinite acoustic medium. The restoring force of each spring in the spring-mass-damper system is assumed as a general polynomial function of the spring deformation, from which the nonlinear quadratic, cubic and quartic springs can be directly recovered. A modified variational approach is employed to formulate the nonlinear dynamics model of the coupled structural system, and the exterior acoustic fluid is computed by a time-domain Kirchhoff boundary integral formulation. The structural and acoustic models are coupled together through the compatibility conditions on their interface. The proposed model is validated by comparing the solutions obtained from the finite element/boundary element method and those available in the literature. Very good agreement between the predicted results and the reference solutions is achieved. Individual contributions of circumferential wave modes of the sandwich cylindrical shell to the vibration and acoustic responses of the coupled nonlinear structural system are examined. The effects of the amplitude and excitation frequency of the external loads, and different types of nonlinear springs (including the quadratic, cubic and quartic nonlinear springs) on the vibration and sound radiation behaviors of the structural system are investigated.

Nonlinear structural and acoustic responses of three-dimensional elastic cylindrical shells with internal mass-spring systems

Prediction of the vibration and sound radiation of an elastic cylindrical shell with nonlinear internal structures is of significant interest to marine and aerospace industries. This paper investigates the structural and acoustic behaviors of a finite cylindrical shell attached with a nonlinear spring-mass-damper system and embedded in an infinite acoustic medium. The restoring forces of the springs are expressed as general polynomial functions of the spring deformation, from which the nonlinear quadratic, cubic and quartic springs can be directly recovered. The exact three-dimensional theory of elasticity is employed for the structural modeling of the cylindrical shell. The discretized equations of motion of the coupled nonlinear structural system are established using an efficient modified variational method, and the fluid surrounding the shell is computed by a time-domain Kirchhoff boundary integral formulation. The two sets of structural and acoustic equations are coupled together through the compatibility conditions on the fluid-structure interface. The present results are validated by comparing the solutions obtained from the finite element method and those available in the literature. The vibration and sound pressure responses of the coupled structural system with nonlinear quadratic, cubic and quartic springs are investigated. The contributions of the circumferential wave modes of the shell to the nonlinear vibro-acoustic responses and sound directivity patterns of the nonlinear coupled structural system are discussed.

Stiffness control of a nonlinear mechanical folded beam for wideband vibration energy harvesters

Abstract This paper presents a novel approach to design and optimize geometric nonlinear springs for wideband vibration energy harvesting. To this end, we designed a spring with several folds to increase its geometric nonlinearities. A numerical analysis is performed using the Finite Element Method to estimate its quadratic and cubic spring stiffness. A nonlinear effective spring constant is then calculated for different values of the main folding angle. We demonstrate that this angle can increase nonlinearities within the structure resulting in higher bandwidths, and that it is possible to control the behavior of the system to have softening-type or hardening-type response depending on the choice of the folding angle. Based on the Lindstedt-Poincaré perturbation technique, a first order approximation is determined to predict the frequency-response of the system. In order to validate the perturbation analysis, numerical solutions based on long-time integration method and mixed VHDL-AMS/Spice simulations are presented. Finally, this method is applied to a previously published device and shows a good agreement with experiments.

Nonlinear analysis for ship-generated waves interaction with mooring line/riser systems

A time-domain hybrid finite element-boundary element (FE-BE) method is developed to investigate nonlinear interaction between ship waves and slender structures. The ship waves are modeled in the context of the three-dimensional potential theory including ship steady state motion, and the velocity potential are computed using a higher-order boundary element method (HOBEM). A special global coordinate-based FEM rod model, when using catenary solution as an initial static profile, is employed in the accurate analysis of slender structures. The internal flow is incorporated in the numerical model by adding a new term to the effective tension formulation while both the seabed support and friction effect are performed by using a nonlinear quadratic spring and a Coulomb friction model. The numerical model is verified with the published experimental and numerical results for the ship waves generation and dynamic responses of mooring line/riser. Numerical simulations are executed to determine the dynamic behavior of two example slender structures such as multi-component mooring line and deepwater steel catenary riser under ship waves. Then the coupled analysis program is applied for a truss Spar with its mooring lines subjected to ship waves, wind-generated waves and their combined action, respectively. The dependence of the top node tensions of the slender structures on ship path, ship speed, ship distance, internal flow and seabed friction, is also examined.

Nonlinear dispersive waves in repulsive lattices.

The propagation of nonlinear waves in a lattice of repelling particles is studied theoretically and experimentally. A simple experimental setup is proposed, consisting of an array of coupled magnetic dipoles. By driving harmonically the lattice at one boundary, we excite propagating waves and demonstrate different regimes of mode conversion into higher harmonics, strongly influenced by dispersion and discreteness. The phenomenon of acoustic dilatation of the chain is also predicted and discussed. The results are compared with the theoretical predictions of the α-Fermi-Pasta-Ulam equation, describing a chain of masses connected by nonlinear quadratic springs and numerical simulations. The results can be extrapolated to other systems described by this equation.

A novel variable stiffness mechanism with linear spring characteristic for machining operations

SUMMARYVariable stiffness mechanisms are able to mechanically reconfigure themselves in order to adjust their system stiffness. It is generally accepted that only antagonistic designs, featuring quadratic springs, can produce linear spring-like behaviour (i.e., a linear relationship between the displacement and its resultant force). However, these antagonistic designs typically are not as energy efficient as series-based designs. In this work, we propose a novel variable stiffness mechanism that can achieve both linear-spring behaviour whilst maintaining an energy efficient characteristic. This paper will present the working principle, mechanical design and characterization of the joints stiffness properties (verified via experimental procedure). The pros and cons of this novel design with reference to the other Variable Stiffness Actuator (VSA) designs will be discussed based on experimental results and in the context of general machining tasks.

Nonlocal continuum analysis of a nonlinear uniaxial elastic lattice system under non-uniform axial load

The static behavior of the Fermi-Pasta-Ulam (FPU) axial chain under distributed loading is examined. The FPU system examined in the paper is a nonlinear elastic lattice with linear and quadratic spring interaction. A dimensionless parameter controls the possible loss of convexity of the associated quadratic and cubic energy. Exact analytical solutions based on Hurwitz zeta functions are developed in presence of linear static loading. It is shown that this nonlinear lattice possesses scale effects and possible localization properties in the absence of energy convexity. A continuous approach is then developed to capture the main phenomena observed regarding the discrete axial problem. The associated continuum is built from a continualization procedure that is mainly based on the asymptotic expansion of the difference operators involved in the lattice problem. This associated continuum is an enriched gradient-based or nonlocal axial medium. A Taylor-based and a rational differential method are both considered in the continualization procedures to approximate the FPU lattice response. The Padé approximant used in the continualization procedure fits the response of the discrete system efficiently, even in the vicinity of the limit load when the non-convex FPU energy is examined. It is concluded that the FPU lattice system behaves as a nonlocal axial system in dynamic but also static loading.

Mechanical Design and Analysis of the Novel 6-DOF Variable Stiffness Robot Arm Based on Antagonistic Driven Joints

This paper proposes four types of conceptual models of the 6-DOF variable stiffness robot arms based on the antagonistic driven joints (ADJs). For convenience of control, the equivalent quadratic torsion spring (EQTS) is selected as the elastic element of the ADJ. The relationship between the output stiffness and the angular displacement of the EQTS is fairly linear. The elastic actuating torque of the ADJ is related to the initial amount of deformation of the EQTS and the angular deflection of the ADJ. The output stiffness of the ADJ is a linear function of the initial amount of deformation of the EQTS. The convenience control of the torque and stiffness of the ADJ will be beneficial to reduce the complexity of the control strategy, and this feature is beneficial for real-time control. In the mechanical solutions, nine types of conceptual models of the EQTSs are presented, and nine types of conceptual models of the ADJs are demonstrated. The cam parameters and the spring parameters of the EQTSs are given. The cam profiles and the pressure angles of the cam-roller mechanisms are illustrated. The elastic actuating torque and output stiffness of the EQTSs and the ADJs are shown. The structure features and actuation characteristics of the EQTSs and the ADJs are compared and analyzed. Since the actuation requirements of the joints of the robot arm differ significantly, four types of conceptual models of the 6-DOF robot arms are assembled based on the different ADJs.

Design and Analysis of Two Types of Variable Stiffness Actuator Based on Adjustable Moment Arm Mechanism

This paper introduces the design and analysis of two types of novel inherently compliant actuators. The two actuation systems are characterized by the property that the output apparent stiffness can be varied independently from the position. The two proposed actuators can regulate the stiffness through the adjustable moment arm mechanism with minimum energy consumption. In the first VSA, this behavior is realized by implementing a lever arm mechanism with variable pivot point position. Compared with the existing VSA that based on the lever arm principle, the way of the spring compressed is more convenient to be implemented by using the tendon and pulley mechanism. Furthermore, the kinematic structure for spring compression is more beneficial for dealing with the unexpected dynamic collision situation. The internal elastic elements of the VSA use the fitted quadratic spring, which improve the characteristics of actuating torque and output stiffness. The stiffness regulation of the second VSA is realized by implementing the connecting rod and slider mechanism. This is a equivalent model of the lever arm mechanism with a variable length moment arm. The stiffness of the VSA is only related to the effective working radius for linear spring configuration, but not to the applied external torques or the deflection angle. This feature makes the stiffness control of the VSA more convenient. The working principles of the two actuators are elaborated and the characteristics of torque and stiffness of the VSA are presented. The mechanical solutions of the two actuators are described.

Route to thermalization in the α-Fermi–Pasta–Ulam system

We study the original α-Fermi-Pasta-Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ(8), where ϵ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension

The variable stiffness actuator (VSA) is an open research field. This paper introduces the conceptual design and analysis of four types of novel variable stiffness actuators (VSAs). The main novelty of this paper is focused on the convenience control of the torque and stiffness of the actuator. We do not need to design complicated control strategies to achieve the desired actuating torque and output stiffness. This feature is beneficial for real-time control. In order to achieve this advantage, three types of approximate quadratic springs and four types of stiffness regulation mechanisms are presented. For the first VSA, the relationship between the output stiffness and angular deflection is close to linear, and the fitted quadratic spring is easy to implement and compact. For the second VSA, the output stiffness is only related to the working radius, and the relationship between the exerted torque and angular deflection is linear. For the third VSA, two equivalent quadratic torsion springs are used. Compared with the existing quadratic torsion springs, the design method is simple and the structure size is compact. The stiffness and torque is a linear function of the kinematic parameters of the VSA. The novelty of the fourth VSA is the use of the tension spring set. The approximate quadratic tension spring is compact and easy to implement. The relationship between the output stiffness and the angular deflection is fairly linear. The conceptual layouts and working principles are elaborated for the four actuators. The characteristics of torque and stiffness of the VSAs are presented, and the mechanical solutions are illustrated.

Stability of the equilibrium positions of an engine with nonlinear quadratic springs

AbstractOur paper realizes a study of the equilibrium positions for an engine supported by four identical nonlinear springs of quadratic characteristic. The systems with quadratic characteristic are generally avoided because they lead to mathematical complications. Our goal is to realize such a study for an engine supported on quadratic springs. For the model purposed, we established the equations of motion and we discussed the possibilities for the equilibrium positions. Because of the quadratic characteristic of the springs and of the approximations made for the small rotations, the equations obtained for the equilibrium lead us to a paradox, which consists in the existence of an open neighborhood in which there exists an infinity of positions of indifferent equilibrium, or a curve where the equilibrium positions are situated. Moreover, the study of the stability shows that the stability is assured for the position at which the springs are not compressed. Finally, a numerical example is presented and completely solved.

Synthesis of a torsional spring mechanism with mechanically adjustable stiffness using wrapping cams

Most adjustable springs that have appeared in the literature are based on the antagonistic actuation of two quadratic springs. When two quadratic springs are coupled antagonistically, the resulting system attains a linear spring behavior enabling the adjustment of the stiffness mechanically. However, a non-linear spring with perfect quadratic behavior is difficult to realize. In this study, the synthesis of a non-linear spring using mechanisms is investigated where the problem is reduced to a function generation problem. A wrapping cam mechanism is proposed as an effective mechanism to be used in non-linear spring synthesis. An analytical solution is developed to calculate the necessary cam profile. Later, the design of the wrapping cam mechanism specific to quadratic springs is investigated. A torsional spring with mechanically adjustable stiffness utilizing a wrapping cam mechanism has been designed, manufactured and experimentally tested. Strong linear behavior is observed and the experimental stiffness values are found to be in good agreement with the analytical solution.

Nonlinear analysis of flexible and steel catenary risers with internal flow and seabed interaction effects

Flexible risers and steel catenary risers often provide unique riser solutions for today’s deepwater field development. Accurate analysis of these slender structures, in which there are high-speed HP/HT internal flows, is critical to ensure personnel and asset safety. In this study, a special global coordinate-based FEM rod model was adopted to identify and quantify the effects of internal flow and hydrostatic pressure on both flexible and deepwater steel catenary risers, with emphasis on the latter. By incorporating internal flow induced forces into the model, it was found that the internal flow contributes a new term to the effective tension expression. For flexible risers in shallow water, internal flow and hydrostatic pressure made virtually no change to effective tension by merely altering the riser wall tension. In deep water the internal pressure wielded a dominant role in governing the riser effective tension and furthering the static configuration, while the effect of inflow velocity was negligible. With respect to the riser seabed interaction, both the seabed support and friction effect were considered, with the former modeled by a nonlinear quadratic spring, allowing for a consistent derivation of the tangent stiffness matrix. The presented application examples show that the nonlinear quadratic spring is, when using the catenary solution as an initial static profile, an efficient way to model the quasi-Winkler-type elastic seabed foundation in this finite element scheme.