Torsional Spring Research Articles

Modeling Flexible Multi-Body Systems Within the Udwadia–Kalaba Framework, a Lumped Parameter Approach

Abstract It is common practice within multi-body dynamic (MBD) modeling to assume that individual bodies are rigid however this can be an oversimplification especially when slender bodies are present resulting in inaccurate estimations of the system's natural frequencies and overall behavior. To address this shortfall, we extend the Udwadia–Kalaba (U–K) MBD formulation in this paper to model flexible multibody systems for purposes of exploring system natural frequencies. To model the flexibility, a lumped parameter approach is proposed, which in this work idealizes a flexible beam as a series of discrete rigid elements connected by torsional springs. In the U–K formulation, a mechanical system can also be discretized into rigid elements and adapted. This is viewed as a benefit for incorporating a lumped parameter approach within the U–K formulation to model flexible multibody systems. A flexible crank-slider mechanism is introduced and modeled within the Udwadia–Kalaba formulation to capture the dynamics of flexibility through linkage compliance. The model is validated against an alternatively formulated MBD model and system natural frequencies and mode shapes numerically predicted. Results of the study show the effectiveness and potential of extending the application of the Udwadia–Kalaba formulation by using a lumped parameter approach to dynamically model flexible multibody systems.

Effect of Mach Number and Pitching Eigenfrequency on Transonic Buffet Onset

The transonic flow around an elastically suspended supercritical airfoil was experimentally investigated in order to enhance knowledge of buffet boundary dependencies and the corresponding self-excited fluid–structure interaction (buffeting). For that purpose, an experimental setup was designed, manufactured, and integrated in the Trisonic Wind Tunnel Munich. The design consisted of a rigid, two-dimensional, supercritical airfoil (OAT15A) with optional pitching degree of freedom and variable torsional spring stiffness. High-speed background-oriented schlieren measurements were used to observe the shock with its dynamics, while a high-speed stereo camera setup for correlation-based deformation measurements was implemented to track the dynamics of the structural motion. Pre-buffet and buffet flows were analyzed by a continuous increase of the angle of attack. The detailed observation of the shock position with increasing angle of attack exhibited a clear effect of Mach number and pitch eigenfrequency on the buffet phenomenon. The corresponding onset boundary clearly shifted into the “pre-buffet” regime given specific structural settings. Furthermore, the resulting fluid–structure interaction of shock and coupled pitch–heave motion showed characteristics of structural frequency lock-in for the pitch-to-buffet frequency ratios of 1.2 and the mode veering region for ratios of 1 and 0.9, indicating the transition region from fluid mode flutter to structural mode flutter.

Wind dynamics and leaf motion: Approaching the design of high-tech devices for energy harvesting for operation on plant leaves.

High-tech sensors, energy harvesters, and robots are increasingly being developed for operation on plant leaves. This introduces an extra load which the leaf must withstand, often under further dynamic forces like wind. Here, we took the example of mechanical energy harvesters that consist of flat artificial "leaves" fixed on the petioles of N. oleander, converting wind energy into electricity. We developed a combined experimental and computational approach to describe the static and dynamic mechanics of the natural and artificial leaves individually and join them together in the typical energy harvesting configuration. The model, in which the leaves are torsional springs with flexible petioles and rigid lamina deforming under the effect of gravity and wind, enables us to design the artificial device in terms of weight, flexibility, and dimensions based on the mechanical properties of the plant leaf. Moreover, it predicts the dynamic motions of the leaf-artificial leaf combination, causing the mechanical-to-electrical energy conversion at a given wind speed. The computational results were validated in dynamic experiments measuring the electrical output of the plant-hybrid energy harvester. Our approach enables us to design the artificial structure for damage-safe operation on leaves (avoiding overloading caused by the interaction between leaves and/or by the wind) and suggests how to improve the combined leaf oscillations affecting the energy harvesting performance. We furthermore discuss how the mathematical model could be extended in future works. In summary, this is a first approach to improve the adaptation of artificial devices to plants, advance their performance, and to counteract damage by mathematical modelling in the device design phase.

Divergence instability in an air-conveying soft tube: Analysis of static zig-zag shapes

This work is devoted to the characterization and modeling of surprising stable states appearing in air-conveying soft tubes. A soft tube differs from a flexible tube by its very thin walls which allows it to fold with a sharp angle. When the upper end of a soft tube is fixed to an air pump while the bottom end is left free, the system exhibits a divergence instability which develops under the form of “zig-zag” shapes constituted by three roughly straight segments. We characterize each stable shape by the lengths and angles of these three segments. We observe, for example, that the intermediate segment is limited to short lengths because its inclined direction with respect to the vertical increases the gravitational torque on the upper fold. In order to theoretically reproduce the zig-zag shapes, we use a model of three articulated straight rigid tubes conveying airflow. The torque exerted by the pressurized tube on a folded part of the pipe is modeled by a nonlinear torsional spring. This model shows that the system can stabilize into zig-zag states qualitatively similar to experimental observations.

The kinematics and static equilibria of a Slinky

A Slinky is a loose helical coil spring and is a well-known educational toy. In this paper a model for a Slinky is presented. The Slinky is represented as a sequence of rigid half coils connected by torsional springs. A range of Slinky configurations in static equilibrium are calculated. Where possible the torsion spring model is compared with the much simpler point mass model of a Slinky, and a more sophisticated model that includes axial and shear deformation. The torsional spring model is shown for the most part to produce similar qualitative behaviour as the more complex model. The simplicity of the torsional spring model makes it a candidate for a discrete stair walking dynamic model of a Slinky, The point mass model is straightforward to derive and solve and makes it possible for an undergraduate engineering student in the early part of their degree course to understand. The torsional spring model is a useful tool for showing what can be done in a more advanced mechanics course to model a system that most students would have previous experience of.

A nonlinear magnetic and torsional spring coupling piezoelectric energy harvester with internal resonance

Magnetically coupled dual-beam energy harvesters have been presented by several researches to achieve a broader response bandwidth. However, in the low frequency operating environment with limited space, the dimension of the piezoelectric beam is limited which leads to high stiffness of the generated beam. Therefore, the output of dual-beam system is relatively low under low frequency and amplitude vibrations. In this paper, a nonlinear magnetic and torsional spring coupling energy harvester is proposed. The stiffness of the torsional spring can be easily adjusted to make the designed harvester suitable for low frequency vibrations. The introduction of torsion springs provides an extra degree of freedom to the system and broadens the response band of the system from 8.7 Hz to 15 Hz. Furthermore, the presence of the 1:3 internal resonance phenomenon in the harvester considerably achieves frequency enhancement and bandwidth extension, significantly improving the harvesting performance in low frequency environments. Numerical analysis, simulation, and experimental are carried out to verify that the proposed energy harvester is capable of harvesting low frequency wideband vibration energy. The results show that magnetic and torsional spring coupling piezoelectric energy harvester is capable of broadband energy harvesting in the range of 5–20 Hz (i.e. 15 Hz bandwidth).

Simulation of membrane deployment accounting for the nonlinear crease effect based on absolute nodal coordinate formulation

With the advantages of lightweight and high stowing efficiency, the origami membrane has a broad application prospect in the space structure. As an improvement of the existing deployment simulations, the behavior of crease is considered and both static and dynamic simulation are conducted in this paper. The origami membrane is modeled as a flexible multibody system and its deployment is simulated based on the absolute nodal coordinate formulation. The nonlinear effect of the crease to resist deployment is integrated into the multibody system via the virtual work principle. The facet is modeled by the thin shell element and the crease is constructed as a nonlinear torsional spring with specific position and gradient constraints. The behavior of the crease is parameterized based on the experimental data. The modeling approach is verified by the numerical/experimental reference of the Z-folding structure in the existing literature. For the multi-crease origami membrane model considering the crease effect, a form-finding method to obtain the initial configuration of the flexible origami is proposed based on the principle of minimum potential energy. The deployment of the classical Miura-ori unit membrane structure is analyzed. The magnitude of driving force is estimated based on the force analysis of rigid deployment. Based on the simulation result of flexible deployment, the driving force is planned to achieve a stable deployment with a constant increase speed in the deployment ratio. This work is expected to provide a reference for the design of space membrane structure, the estimation of driving force, and optimizing the deployment strategy of an origami membrane.

Efficient Fluid–Structure Interaction Model for Twistable Flapping Rotary Wings

Wing flexibility is critical to flapping rotary wings (FRWs), and in that the deformation is bilaterally coupled with aerodynamic forces and thus determines the performance. Conventional solutions to this fluid–structure interaction (FSI) topic require considerable computational resources. In this paper, an efficient FSI model is proposed to calculate the aerodynamic force and passive twisting of FRWs. The passive pitching is regulated by a torsional spring, and the twisting is simplified as a quadratic distribution. A well-verified quasi-steady model is employed to estimate the aerodynamic forces. Our results show that the performance of rigid FRWs is superior to twistable FRWs within an upper limit of the wing-root stiffness , which is around . At higher values, the twistable FRWs generate comparable lift to rigid FRWs at a higher efficiency. An increase in flapping frequency can remarkably reduce the efficiency of twistable FRWs despite the lift enhancement, while a concomitant reduction of flapping amplitude can moderate the loss of efficiency at higher flapping frequencies. Our model provides an efficient tool for the quick estimation of the aeroelastic performance of twistable FRWs and can thus contribute to the wing stiffness design.

Design of the Jump Mechanism for a Biomimetic Robotic Frog.

Frogs are vertebrate amphibians with both efficient swimming and jumping abilities due to their well-developed hind legs. They can jump over obstacles that are many or even tens of times their size on land. However, most of the current jumping mechanisms of biomimetic robotic frogs use simple four-bar linkage mechanisms, which has an unsatisfactory biomimetic effect on the appearance and movement characteristics of frogs. At the same time, multi-joint jumping robots with biomimetic characteristics are subject to high drive power requirements for jumping action. In this paper, a novel jumping mechanism of a biomimetic robotic frog is proposed. Firstly, the structural design of the forelimb and hindlimb of the frog is given, and the hindlimb of the robotic frog is optimized based on the design of a single-degree-of-freedom six-bar linkage. A simplified model is established to simulate the jumping motion. Secondly, a spring energy storage and trigger mechanism is designed, including incomplete gear, one-way bearing, torsion spring, and so on, to realize the complete jumping function of the robot, that is, elastic energy storage and regulation, elastic energy release, and rapid leg retraction. Thirdly, the experimental prototype of the biomimetic robotic frog is fabricated. Finally, the rationality and feasibility of the jumping mechanism are verified by a jumping experiment. This work provides a technical and theoretical basis for the design and development of a high-performance amphibious biomimetic robotic frog.

Optimal design of functionally graded lattice structures using Hencky bar-grid model and topology optimization

Presented herein is a novel design framework for obtaining the optimal design of functionally graded lattice (FGL) structures that involve using a physical discrete structural model called the Hencky bar-grid model (HBM) and topology optimization (TO). The continuous FGL structure is discretized by HBM comprising rigid bars, frictionless hinges, frictionless pulleys, elastic primary and secondary axial springs, and torsional springs. A penalty function is introduced to each of the HBM spring’s stiffnesses to model non-uniform material properties. The gradient-based TO method is applied to find the stiffest structure via minimizing the compliance or elastic strain energy by adjusting the HBM spring stiffnesses subjected to prescribed design constraints. The optimal design of FGL structures is constructed based on the optimal spring stiffnesses of the HBM. The proposed design framework is simple to implement and for obtaining optimal FGL structures as it involves a relatively small number of design variables such as the spring stiffnesses of each grid cell. As illustration of the HBM-TO method, some optimization problems of FGL structures are considered and their optimal solutions obtained. The solutions are shown to converge after a small number of iterations. A Python code is given in the Appendix for interested readers who wish to reproduce the results.

The dynamics of a rigid inverted flag

An ‘inverted flag’ – a flexible plate clamped at its trailing edge – undergoes large-amplitude flow-induced flapping when immersed in a uniform and steady flow. Here, we report direct numerical simulations of a related single degree-of-freedom mechanical system: a rigid plate attached at its trailing edge to a torsional spring. This system is termed a ‘rigid inverted flag’ and exhibits the dynamical states reported for the (flexible) inverted flag, with additional behaviour. This finding shows that the flapping dynamics of inverted flags is not reliant on their continuous flexibility, i.e. many degrees of freedom. The rigid inverted flag exhibits additional, novel states including a heteroclinic-type orbit that results in small-amplitude flapping, and a number of chaotic large-amplitude flapping regimes. We show that the various routes to chaos are driven by a series of periodic states, including at least two which are subharmonic. The instability and competition between these periodic states lead to chaos via type-I intermittency, mode competition and mode locking. The rigid inverted flag allows these periodic states and their subsequent interaction to be explained simply: they arise from an interaction between a preferred vortex shedding frequency and a single natural frequency of the structure. The dynamics of rigid inverted flags is yet to be studied experimentally, and this numerical study provides impetus for such future work.

Structural stability of a lightsail for laser-driven interstellar flight

The structural stability of a lightsail under the intense laser flux necessary for interstellar flight is studied analytically and numerically. A sinusoidal perturbation is introduced into a two-dimensional thin-film sail to determine if the sail remains stable or if the perturbations grow in amplitude. A perfectly reflective sail material that gives specular reflection of the laser illumination is assumed in determining the resulting loading on the sail, although other reflection models can be incorporated as well. The quasi-static solution of the critical point between shape stability and instability is found by equating the bending moments induced on the sail due to radiation pressure with the restoring moments caused by the strength of the sail material and the tension applied at the edges of the sail. From this quasi-static solution, analytical expressions for the critical value of elastic modulus and boundary tension magnitude are found as a function of sail properties (e.g., thickness) and the amplitude and wave number of the initial sinusoidal perturbation. These same expressions are also derived from a more formal variational energy (virtual work) approach. A numerical model of the complete lightsail dynamics is developed by discretizing the lightsail into rectangular finite elements. By introducing torsional and rectilinear springs between the elements into the numerical model, a hierarchy of models is produced that can incorporate the effects of bending and applied tension. The numerical models permit the transient dynamics of a perturbed lightsail to be compared to the analytic results of the quasi-static analysis, visualized as stability maps that show the rate of perturbation growth as a function of sail thickness, elastic modulus, and applied tension. The analytic theory is able to correctly predict the stability boundary found in the numerical simulations. The stiffness required to make a thin lightsail stable against uncontrolled perturbation growth appears to be unfeasible for known materials, however, a relatively modest tensioning of the sail (e.g., via an inflatable structure or spinning of the sail) is able to maintain the sail shape under all wavelengths and amplitudes of perturbations.

Vibration Behaviour Analysis and Vibration Suppression Studies of the Space Robot

Aiming at the problem that the vibration of space robot will reduce the dynamic accuracy of robot, a method of vibration suppression of space robot through trajectory planning is proposed. First, the joint was simplified to a linear torsion spring, and the flexible rod was modelled using the finite element method, resulting in a flexible motion model of the robot. Then, a rigid-soft coupled dynamic model that combined the flexible motion model with the Lagrange method was established. Using the dynamic model, the factors influencing the vibration behaviour of space robots were analysed. Finally, the space robot was subjected to vibration suppression through trajectory planning. As verified by experimental analysis, different trajectories and loads affect the excitation force and inherent characteristics of the robot. In the design process, it is necessary to consider the relationship between the excitation force and natural frequency. The trajectory planning method has apparent effects on vibration suppression. The vibration amplitude was significantly reduced, which can improve the positioning accuracy and work efficiency of the robot.

Energy-based swing-up control for a two-link underactuated robot with flexible first joint

This paper concerns the swing-up control of a two-link robot moving in a vertical plane, which has a single actuator at the second joint and a linear torsional spring at the first joint. First, we present a necessary and sufficient condition such that the robot is linearly controllable at the upright equilibrium point (UEP, where two links are both upright). Second, we prove without any assumption that the robot is at an equilibrium point provided that its actuated joint angle is constant under a constant torque. Third, for the robot with its torsional stiffness of the spring being not greater than a value determined by the coefficients of its gravitational terms, we propose an energy-based swing-up controller without singular points. We conduct a global motion analysis for the robot under the proposed controller. For the case that the total mechanical energy of the robot converges to its desired value, we present the phase portrait of the closed-loop solution. For the case that the convergence is not achieved, we show that the closed-loop solution approaches an equilibrium point belonging to a set of equilibrium points, and give a sufficient condition to check its instability. From the motion analysis, we present a sufficient condition such that the robot can be swung-up close to the UEP under the proposed swing-up controller. Finally, we verify our theoretical results through a numerical simulation.

Insect Wing Buckling Influences Stress and Stability During Collisions

Abstract Flapping insect wings collide with vegetation and other obstacles during flight. Repeated collisions may irreversibly damage the insect wing, thereby compromising the insect's ability to fly. Further, reaction torques caused by the collision may destabilize the insect and hinder its ability to maneuver. To mitigate the adverse effects of impact, some insect wings are equipped with a flexible joint called a “costal break.” The costal break buckles once it exceeds a critical angle, which is believed to improve flight stability and prevent irreversible wing damage. However, to our knowledge, there are no models to predict the dynamics of the costal break. Through this research, we develop a simple model of an insect wing with a costal break. The wing was modeled as two beams interconnected by a torsional spring, where the stiffness of the torsional spring instantaneously decreases once it has exceeded a critical angle. We conducted a series of static tests to approximate model parameters. Then, we used numerical simulation to estimate the reaction moments, angular impulse, and peak stresses experienced by the wing during a collision. When evaluated over the duration of an external load, we found that buckling could reduce reaction moments and angular impulse up to 82% and 99%, respectively, compared to a homogeneous wing. This suggests the costal break can enhance flight stability. On the other hand, buckling maximally increased peak stresses two times compared to a homogeneous wing, indicating the costal break does not reduce likelihood of damage under the simplified loading considered.

Control of a 2 DOF Elbow Prosthesis prototype

This paper presents a 2 DOF elbow prosthesis design and its control, which includes developing the mechanism, adapting a torsional spring for the flexo-extension movement, and a crown pinion transmission with two torsional springs pronosupination movement. In addition, this prototype has planned an elbow joint control method. The control has been obtained using the kinematic of a pendulum with a torsional spring, where the model comes with a set of dynamic simulations to determine how feasible the proposed mechanism is, and the torque needed for daily activities performance. The results show the physical and kinematic characteristics of the designed system.

On the Aeroelasticity of the Active Span and Passive Pitching Polymorphing Wing: A Parametric Study

This paper presents an aeroelastic analysis of a polymorphing wing capable of active span extension and passive pitch variation. The wing is split into two segments: an inboard segment responsible for span extension/retraction and an outboard segment capable of pitch variation. The two segments are connected through an overlapping spar and a torsional spring. A finite element aeroelastic model is developed where the wing structure is discretized into Euler–Bernoulli beam elements and the aerodynamic loads are calculated using Theodorsen’s unsteady model. A comprehensive parametric analysis is carried out with and without span extension to analyze the effect of varying critical design parameters, such as elastic axis position of outboard section and torsional spring rigidity, and conditions for aeroelastic phenomena of flutter and divergence are studied. A gust load analysis is carried out to quantify the capability of the outboard wing passive twist mechanism to alleviate loads. Finally, a nonlinear analysis is carried out by replacing the linear torsional spring with a nonlinear cubic spring to study the effects of cubic hardening and softening on the aeroelasticity of the polymorphing wing.

Lateral Dynamic Simulation of a Bus under Variable Conditions of Camber and Curvature Radius

The objective of this paper is to describe a model for the simulation of the lateral dynamics of a vehicle, specifically buses, under variable trajectory conditions, such as camber and radius of curvature; in addition, a variable speed is added as a simulation parameter. The objective of this study is the prevention of vehicle rollover and sideslip. An 8 degrees of freedom model was developed, considering a front and a rear section of the bus with its respective suspension system, and both sections have been connected by a torsion spring that emulates the torsional stiffness of the vehicle chassis. A Panhard bar is also added at the rear as an additional element to the suspension and the behavior of the bus when it is added is analyzed. This model also allows the evaluation of the force on each suspension component, which allows for future controllability of the active suspension components. The results show the dynamic behavior of the vehicle, and some indicators are introduced to show the possible sideslip or rollover. As a conclusion, the influence of the road parameters on the dynamic behavior of the bus and the effect of the Panhard bar on the dynamic behavior of the bus can be pointed out.

High-accuracy prediction and compensation of industrial robot stiffness deformation

Industrial robots (IRs) are promising options for machining large complex structural parts due to the higher flexibility, larger operating space, and lower cost compared with multi-axis machine tools. However, the relatively low posture-dependent stiffness and large stiffness deformation of IRs significantly deteriorate the contour accuracy of milling in which the cutting force is large generally. It is very complex to achieve a precise stiffness model and predict stiffness deformation of IRs because of the joint clearance, drift of zero-position, and other nonlinear factors. The conventional stiffness model of IRs only takes each joint as a constant linear torsion spring into consideration and ignores other difficult-to-model factors, which leads to low-accuracy identified results and thereafter induces deformation prediction errors. The data-driven approach can be used to obtain an accurate stiffness and deformation model, but a large amount of experimental data is required and it will cost enormous time and effort. In order to circumvent the experimental data deficiency and difficult-to-model issue, a simulation-driven transfer learning method named Adaptive Domain Adversarial Neural Network with Dual-Regressions (ADANN-2R) is designed for robot deformation prediction. Amounts of coarse deformation data, which are generated by the conventional stiffness model, are regarded as source data. And few real deformation data, which are obtained by deformation experiments, are regarded as target data. The Dual-Regressions are designed after the feature extractor, and the weighting parameters are adjusted adaptively to tackle the different magnitude of the regression loss and domain discrimination loss. The ADANN-2R aligns the simulated source data and real target data to perform adversarial training, and an accurate target deformation predictor is achieved. Experimental results indicate that the proposed ADANN-2R can obtain high-accuracy prediction with few real data compared with the conventional stiffness model. Compared with the path without deformation compensation and the pre-compensated path using the conventional stiffness model, the maximum position error of the pre-compensated path using the proposed ADANN-2R is reduced by 78.12% and 32.45%, respectively.

Design and simulation analysis of a bionic ostrich robot.

To look for the reason why the biped animal in nature can run with such high speed and to design a bionic biped prototype which can behave the high speed running and jumping ability, this paper takes the fastest bipedal animal in nature: ostrich as the research subject. Firstly, the body structure and motion characteristics of ostrich are investigated. Secondly, a simple mechanical structure of bionic ostrich robot is designed based on the above biological investigated results. The robot is under-actuated with one actuator each leg, with a spring on the tarsometatarsus and a torsion spring on the metatarsophalangeal joint at the foot end. And then the mechanical design of leg structure is optimized. Finally, the high-speed running and jumping running gait is planned, and comparative simulations are implemented with different design requirements among pure rigid and rigid-flexible coupling scheme, which are rigid, only with spring, only with torsion spring, and with spring and torsion spring both, in detail. Simulation results show that the rigid-flexible coupling design scheme and whole body motion coordination can achieve better high speed performance. It provides an insight for the design and control of legged robots.