Nonlinear Torsional Spring Research Articles

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.

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.

Aeroelastic analysis and nonlinear characterization of three-degree-of-freedom systems with discontinuous nonlinearities

The effects of freeplay, multi-segmented, and impact nonlinearities in the pitch and control surface degrees of freedom on the nonlinear responses of a three-degree of freedom aeroelastic system are investigated to characterize the nonlinear behavior and possible bifurcations. The system consists of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom, a nonlinear torsional spring in the pitch degree of freedom, and a control surface with a nonlinear spring in the trailing edge. The unsteady representation based on the Duhamel formulation is used to model the aerodynamic loads. Nonlinear characterization is performed to identify the system’s response, under different representations of the nonlinear springs, below the linear flutter speed. The results show that grazing/sliding bifurcations are present and nonlinear coupling of the system’s frequencies lead to complex responses including sudden transitions between harmonic and chaotic responses as the freestream velocity is increased. These responses, which include the sudden amplitude jumps, generation of higher harmonics and broadband oscillatory behavior, could result in catastrophic structural failures of aeroelastic systems even when operating below the linear flutter speed.

Nonlinear dynamics and gust response of a two-dimensional wing

In this paper, aeroelastic equations for a two-dimensional wing encountering a wind gust are presented and solved numerically. The indicial aerodynamic theory based on Wagner’s function is adopted to obtain aerodynamic forces and moments acting on the wing in the time domain. The dynamics of the wing is approximated via two degrees-of-freedom, i.e. pitch and plunge. The structural stiffness is modeled by a linear translational and a nonlinear torsional spring. Two different types of stiffness nonlinearities are examined: (i) flat spot or dead zone to model free-play, and (ii) hysteresis. For given system parameters, bifurcation diagrams are presented, and time-history, power-spectral density, phase-plane and Poincaré diagrams are shown at different flow velocities. We show that the dynamics of the system with the free-play nonlinearity may be very complex with the possibility of the occurrence of chaos through period-doubling bifurcations. In contrast, the dynamics with the hysteresis nonlinearity is found to be rather simple, where a Hopf bifurcation is the only bifurcation observed. The response of the nonlinear system to sharp-edged and 1-cosine gust profiles are also obtained at different flow velocities and compared to the time response of the system with no gust input. It was found that the gust input may cause the nonlinear system to get attracted to a periodic orbit in the subcritical flow regime. In addition, basin of attraction is obtained for various amplitudes of the sharp-edged gust. It is discussed that as the gust becomes stronger, the likelihood of the occurrence of limit-cycle oscillation increases while the stable points become less dispersed inside the stability map and form a finite region confined to large values of initial conditions.

Dynamic modeling and control of large flexiblespacecraft combination

<p indent=0mm>In the mission of spacecraft on-orbit assembly and service, a large flexible spacecraft combination will be formed by the capturing spacecraft and the target spacecraft after the space manipulator completes the capturing misssion. First, taking comprehensive consideration of vibration influence of all the flexible components as the joints, links and solar arrays, the dynamic model of the spacecraft combination is established. The non-linear torsional spring models are adopted for flexible joints description; the assumed mode method and the finite element method are adopted for the flexible links and solar arrays; and the dynamic equation of the flexible spacecraft combination is established by use of the momentum conservation and the Lagrange equation. In addition, a composite controller of the flexible spacecraft combination is designed for trajectories tracking and vibration suppressing. The dynamic model of the flexible spacecraft combination is decomposed by use of the singular perturbation method and the two sub-controllers are designed seperately in two different time scales. Finally, the numerical simulations are carried out to verify the effectiveness of the modeling and control methods. The results illustrate that the dynamic model is accurate and effective for representing the kinematic and dynamic properties of the flexible spacecraft combination, and the composite control method is able to guarantee the trajectory tracking pricision as well as effective vibration suppression of the flexible components. The dynamic modeling and control method are applicable for engineering practice.

Vibration of nonlinear bolted lap-jointed beams using Timoshenko theory

This paper investigates the vibrational behavior of a system which consists of two free–free Timoshenko beams interconnected by a nonlinear joint. To model the bolted lap joint interface, a combination of the linear translational spring, linear and nonlinear torsional springs, and a linear torsional damper is used. The governing equations of motion are derived using the Euler–Lagrange equations. The reduced-order model equations are obtained based on Galerkin method. The set of coupled nonlinear equations are then analytically solved using the harmonic balance approach and numerical simulation. A parametric study is carried out to reveal the influence of different parameters such as linear and nonlinear torsional spring, linear translational spring, and linear torsional damper on the vibration and stability of the bolted lap joint structure. It is shown that the effect of the nonlinear torsional spring on the response of the system is significant. Interestingly, it is observed that in the presence of the nonlinear spring the softening behavior could be changed to hardening behavior. In addition, the effects of the different engineering beam theories on the modeling of the substructures are studied and it is observed that considering the effect of the rotary inertia and shear deformations is significant. In addition, it is observed that neglecting each of them can yield completely wrong interpretations of the system behavior and incorrect results.

An Experimental Study and Model Determination of the Mechanical Stiffness of Paper Folds

Over the past few decades, folding paper has extended beyond the origami deployable applications to reach the engineering field. Nevertheless, mechanical information about paper behavior is still lacking, especially during folding/unfolding. This article proposes an approach to characterize the paper fold behavior in order to extract the material data that will be needed for the simulation of folding and to go a step further the single kinematics of origami mechanisms. The model developed herein from simple experiments for the fold behavior relies on a macroscopic local hinge with a nonlinear torsional spring. Though validated with only straight folds, the model is still applicable in the case of curved folds thanks to the locality principle of the mechanical behavior. The influence of both the folding angle and the fold length is extracted automatically from a set of experimental values exhibiting a deterministic behavior and a variability due to the folding process. The goal is also to propose a methodology that may extend the simple case of the paper crease, or even the case of thin material sheets, and may be adapted to other identification problems.

Flutter analysis of a nonlinear airfoil using stochastic approach

In this paper, the dynamic instability of a nonlinear system has been studied using the stochastic vibration analysis and employing statistical properties of the system response. In this method neither the time domain analysis nor limit cycle oscillations were used. A two degrees-of-freedom airfoil subjected to an aerodynamic quasi-steady flow with a nonlinear torsional spring was considered as the case study. The spring nonlinearity was examined in hardening and softening states. A random force in the form of the white noise with Gaussian function was added to the aerodynamic lift force. The statistical linearization and random vibration analysis were applied to the nonlinear system to obtain the equation of one-dimensional nonlinear map in terms of response variance and flow speed. Furthermore, the nonlinear map was solved to analyze the response variance of the system against the flow speed. The flow velocity of the maximum variance of the system response was regarded as the flutter speed. The bifurcation point and the approaching path to the chaos were determined by investigating the nonlinear map through the iteration process. In addition, a good vision of jump phenomenon in velocity-variance diagram was given through the current stochastic analysis using the chaos intermittency cascade and the tangent bifurcation point. The aforementioned result is new in the nonlinear aeroelastic dynamic instability field.

Load and Response Identification for a Nonlinear Flexible Structure Subject to Harmonic Loads

Experimental monitoring of dynamic response is generally limited to few locations in the system. However, the analysis of structural performance and design of control systems would benefit from a complete knowledge of the system dynamic during service. A numerical approach is developed to numerically reconstruct the load and response of a complete structure from few reference points, based on a modal approach for projecting the response at few points on the domain of the structure. This methodology is particularly advantageous when full-field monitoring of a structure is not a possible solution. An assembly of two beams joined by a nonlinear torsional spring is analyzed in case of different load distributions acting on its span. The approach is shown to be robust and reliable.

Nonlinear analysis and enhancement of wing-based piezoaeroelastic energy harvesters

We investigate the level of harvested power from aeroelastic vibrations for an elastically mounted wing supported by nonlinear springs. The energy is harvested by attaching a piezoelectric transducer to the plunge degree of freedom. The considered wing has a low-aspect ratio and hence three dimensional aerodynamic effects cannot be neglected. To this end, the three dimensional unsteady vortex lattice method for the prediction of the unsteady aerodynamic loads is developed. A strong coupling scheme that is based on Hamming's fourth-order predictor–corrector method and accounts for the interaction between the aerodynamic loads and the motion of the wing is employed. The effects of the electrical load resistance, nonlinear torsional spring and eccentricity between the elastic axis and the gravity axis on the level of the harvested power, pitch and plunge amplitudes are investigated for a range of operating wind speeds. The results show that there is a specific wind speed beyond which the pitch motion does not pick any further energy from the incident flow. As such, the displacement in the plunge direction grows significantly and causes enhanced energy harvesting. The results also show that the nonlinear torsional spring plays an important role in enhancing the level of the harvested power. Furthermore, the harvested power can be increased by an order of magnitude by properly choosing the eccentricity and the load resistance. This analysis is helpful in designing piezoaeroelastic energy harvesters that can operate optimally at specific wind speeds.

Nonlinear dynamics of galloping-based piezoaeroelastic energy harvesters

The normal form is proposed as a tool to analyze the performance and reliability of galloping-based piezoaeroelastic energy harvesters. Two different harvesting systems are considered. The first system consists of a tip mass prismatic structure (isosceles 30° or square cross-section geometry) attached to a multilayered cantilever beam. The only source of nonlinearity in this system is the aerodynamic nonlinearity. The second system consists of an equilateral triangle cross-section bar attached to two cantilever beams. This system is designed to have structural and aerodynamic nonlinearities. The coupled governing equations for the structure’s transverse displacement and the generated voltage are derived and analyzed for both systems. The effects of the electrical load resistance and the type of harvester on the onset speed of galloping are quantified. The results show that the onset speed of galloping is strongly affected by the load resistance for both types of harvesters. The normal form of the dynamic system near the onset of galloping (Hopf bifurcation) is then derived. Based on the nonlinear normal form, it is demonstrated that smaller levels of generated voltage or power are obtained for higher absolute values of the effective nonlinearity. For the first harvesting system, the results show a supercritical Hopf bifurcation for both isosceles 30° or square cross-section geometries. The nonlinear normal form shows that the isosceles triangle section (30°) is more efficient than the square section. For the second harvesting system, the normal form is used to identify the values of the nonlinear torsional spring which changes the harvester’s instability. It is demonstrated that this critical value of the nonlinear torsional spring depends strongly on the load resistance.

Modeling and performance analysis of cambered wing-based piezoaeroelastic energy harvesters

We investigate the effects of aerodynamic loads on the performance of wing-based piezoaeroelastic energy harvesters. The rigid airfoil consists of pitch and plunge degrees of freedom supported by flexural and torsional springs with a piezoelectric coupling attached to the plunge degree of freedom. The effects of aerodynamic loads are investigated by considering a camber in the airfoil. A two-dimensional unsteady vortex-lattice method (UVLM) is used to model the unsteady aerodynamic loads. An iterative scheme based on Hamming’s fourth-order predictor–corrector method is employed to solve the governing equations simultaneously and interactively. The effects of varying the camber, its location, and the nonlinear torsional spring coefficient are determined. The results show that, for small values of the camber location, the flutter speed changes greatly on increasing the camber of the airfoil. On the other hand, for large values of the camber location, the variation of the flutter speed when changing the camber is very negligible. We demonstrate that the symmetric airfoil case is the best configuration to design enhanced wing-based piezoaeroelastic energy harvesters. Furthermore, the results show that an increase in the camber results in a decrease in the level of the harvested power. For cambered airfoils, we demonstrate that an increase in the camber location leads to an increase in the level of the harvested power. The results show that an increase in the airfoil camber delays the appearance of a secondary Hopf bifurcation.

State constrained optimal control of a ball pitching robot

We present a method for offline optimal control of a two-link ball pitching robot with the aim of throwing a ball as far as possible. The pitching robot is connected to a motor via a non-linear torsional spring at the shoulder joint. The elbow joint is passive and loaded with a linear torsional spring. We model the system based on an Euler–Lagrange formulation. Constraints on the motor torque and power as well as the angular velocity of the motor shaft are included in the model. By using an interior point method with gradients supplied by a discrete adjoint method, we numerically solve the resulting constrained control problem of finding the optimal piecewise constant motor torque profile and release position. Numerical experiments illustrate the effectiveness of our strategy as well as the effect of the constraints on the objective. In our experiments, the optimal motor torque gives rise to motions comprising an initial backswing; a transition, where the elbow spring accumulates potential energy; and finally a fast acceleration phase leading up the ball release.

Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping

A model for harvesting energy from galloping oscillations of a bar with an equilateral triangle cross-section attached to two cantilever beams is presented. The energy is harvested by attaching piezoelectric sheets to cantilever beams holding the bar. The derived nonlinear distributed-parameter model is validated with previous experimental results. The quasi-steady approximation is used to model the aerodynamic loads. The power levels that can be generated from these vibrations, and the variations of these levels with the load resistance and wind speed, are determined. Linear analysis is performed to validate the onset of galloping speed with experimental measurements. The effects of the electrical load resistance on the onset of galloping are then investigated. The results show that the electrical load resistance affects the onset speed of galloping. A nonlinear analysis is also performed to determine the effects of the electrical load resistance and the nonlinear torsional spring on the level of the harvested power. The results show that maximum levels of harvested power are accompanied by minimum transverse displacement amplitudes. It is also demonstrated that there is an optimum load resistance that maximizes the level of the harvested power.

Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity

We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum value of the load resistance exists, at which the level of the harvested power is maximized.

Modeling and identification of freeplay nonlinearity

A nonlinear analysis is performed for the purpose of identification of the pitch freeplay nonlinearity and its effect on the type of bifurcation of a two degree-of-freedom aeroelastic system. The databases for the identification are generated from experimental investigations of a pitch-plunge rigid airfoil supported by a nonlinear torsional spring. Experimental data and linear analysis are performed to validate the parameters of the linearized equations. Based on the periodic responses of the experimental data which included the flutter frequency and its third harmonics, the freeplay nonlinearity is approximated by a polynomial expansion up to the third order. This representation allows us to use the normal form of the Hopf bifurcation to characterize the type of instability. Based on numerical integrations, the coefficients of the polynomial expansion representing the freeplay nonlinearity are identified.

Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.