Dynamics and control of an active pendulum system

The work focuses on autoparametric vibrations of system composed of a non-linear oscillator with an attached pendulum. A combination of semi-active damper together with a non-linear spring mounted in the suspension of system is proposed. The spring is made from a shape memory alloy (SMA) while damping is realized through a magnetorheological (MR) damper. The MR damper is used for controlling damping of the system, SMA spring is used to change system׳s stiffness. The system is solved numerically and verified experimentally on a custom made experimental rig. Specifically, non-linear resonances are investigated, and their influence on the system dynamics and absorption effect.     

Note on Chaos in Three Degree of Freedom Dynamical System with Double Pendulum

The nonlinear response of a three degree of freedom vibratory system with double pendulum in the neighbourhood internal and external resonances is investigated. The equations of motion have bean solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibration there may also appear chaotic vibration. To prove the character of this vibration and to realise the analysis of transitions from periodic regular motion to quasi-periodic and chaotic, the following have been constructed: bifurcation diagrams and time histories, phase plane portraits, power spectral densities, Poincaré maps and exponents of Lyapunov. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits.     

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.     

A combined analytical,numerical, and experimental study of shape-memory-alloy helical springs

In this paper, the pseudoelastic response of shape memory alloy (SMA) helical springs under axial force is studied both analytically and numerically. In the analytical solution two different approximations are considered. In the first approximation, both the curvature and pitch effects are assumed to be negligible. This is the case for helical springs with large ratios of mean coil radius to the cross sectional radius (spring index) and small pitch angles. Using this assumption, analysis of the helical spring is reduced to that of the pure torsion of a straight bar with circular cross section. A three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. In the next step, the curvature effect is included and the SMA helical spring is analyzed using the exact solution presented for torsion of curved SMA bars. In this refined solution, the effect of the direct shear force is also considered. In the numerical analyses, the three-dimensional constitutive equations are implemented in a finite element method and using solid elements the loading–unloading of an SMA helical spring is simulated. Analytical and numerical results are compared and it is shown that the solution based on the SMA curved bar torsion gives an accurate stress analysis in the cross section of the helical SMA spring in addition to the global load–deflection response. All the results are compared with experimental data for a Nitinol helical spring. Several case studies are presented using the proposed analytical and numerical solutions and the effect of changing different parameters such as the material properties and temperature on the loading–unloading hysteretic response of SMA helical springs is studied. Finally, some practical recommendations are given for improving the performance of SMA helical springs used as energy dissipating devices, for example for seismic applications.     

Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

Dimensionless modeling and nonlinear analysis of a coupled pitch–plunge–leadlag airfoil-based piezoaeroelastic energy harvesting system

In this paper, an airfoil-based piezoaeroelastic energy harvesting system is proposed with an additional supporting device to harvest the mechanical energy from the leadlag motion. A dimensionless dynamic model is built considering the large-effective-angle-of-attack vibrations causing (1) the nonlinear coupling between the pitch–plunge–leadlag motions, (2) the inertia nonlinearity, and (3) the aerodynamic nonlinearity modeled by the ONERA dynamic stall model. Cubic hardening stiffness is introduced in the pitch degree of freedom for persistent vibrations with acceptable amplitude beyond the flutter boundary. The nonlinear aeroelastic response and the average power output are numerically studied. Limit cycle oscillations are observed and, as the flow velocity exceeds a secondary critical speed, the system experiences complex vibrations. The power output from the leadlag motion is smaller than that from the plunge motion, whereas the gap is narrowed with increasing flow velocity. Case studies are performed toward the effects of several dimensionless system parameters, including the nonlinear torsional stiffness, airfoil mass eccentricity, airfoil radius of gyration, mass of the supporting devices, and load resistances in the external circuits. The optimal parameter values for the power outputs from the plunge and leadlag motions are, respectively, obtained. Beyond the secondary critical speed, it is shown that the variations of the power outputs with those parameters become irregular with fluctuations and multiple local maximums. The bifurcation analysis shows the mutual transitions between the limit cycle oscillations, multi-periodic vibrations, and possible chaos. The influences of these complex vibrations on the power outputs are discussed.     

Computational and numerical analysis of a nonlinear mechanical system with bounded delay

Modern structures are increasingly resistant and complex. In many cases, such systems are modeled by numerical approximations methods, due to its complexities. In the study of vibration levels in the response of a system is important to consider issues like reliability and efficient design, since that such vibrations are undesirable phenomena that may cause damage, failure, and sometimes destruction of machines and structures. In this paper we investigated a modeling strategy of nonlinear system with damping, subject the time delayed. From models widely used in literature and with the help of numerical simulations a nonlinear damped system with two degree-of-freedom is analyzed. The system is constituted of a primary mass attached to the ground by a spring and damping with linear or nonlinear characteristics (primary system), and the secondary mass attached to the primary system by a spring and damping with linear or nonlinear characteristics (secondary system). It is well known that time delayed systems, due to its own nature, has singular behavior in its dynamics and that such singularities propagate over the time. Based on this, the main concerns of the present paper is to analyze the stability of a delayed system with two degree of freedom by means of the techniques development in [1] (Hu andWang, 2002). We also obtain the solution using the integration of equations of motions performing a Fourth Order Runge-Kutta Method. The behavior of a nonlinear main system with nonlinear secondary system will be investigated to many cases of resonances. In this case, various time delayed values are used to confirm its influence on the attenuation of vibrations, but, unfortunately, also the increase of nonlinearity (instable responses) of the system in question is observed.     

An influence of temperature on reconstructed middle ear with shape memory prosthesis

In the paper idea of reconstructed middle ear with a prosthesis made of shape memory alloy is proposed. The new design of shape memory prosthesis is used to enable adjusting its length to individual patient’s needs which is a novel contribution of the paper. In order to make a proper fit of prosthesis, a surgeon has to adjust its size and position by cutting step by step classical prosthesis. It takes time of a operation and enlarges a period of patient’s esthesia. Therefore, a shape memory prosthesis (SMP) is proposed to shorten operation time and improve fitting through precisely selected length. A reconstructed middle ear is modelled as a two degree of freedom system with nonlinear shape memory element. Finding advisable periodic and undesirable a periodic and irregular behaviour in various temperature is the main aim of the paper. Results of the study should give an answer whether SMP can be useful in medical practice and should also explain dynamics of the middle ear with SMP. The properties of the shape memory prosthesis, in the form of helical spring, are described here by a polynomial dependence. Firstly, free vibrations are investigated and equilibrium points, next forced vibrations are studied under different parameters of external excitations and temperature range. Bifurcation analysis and stability of periodic solutions are performed in order to reveal the system behaviour. Finally, interesting dynamical findings of chaotic vibrations pure regular and regular oscillations with fluctuation are presented. However, from practical point of view only harmonic response of the stapes is advisable. That can be achieved at the normal temperature of a human body only for small excitation amplitude.     

On the stability of nonlinear vibrations of coupled pendulums

The motion of two identical pendulums connected by a linear elastic spring is studied. The pendulums move in a fixed vertical plane in a homogeneous gravity field. The nonlinear problem of orbital stability of such a periodic motion of the pendulums is considered under the assumption that they vibrate in the same direction with the same amplitude. (This is one of the two possible types of nonlinear normal vibrations.) An analytic investigation is performed in the cases of small vibration amplitude or small rigidity of the spring. In a special case where the spring rigidity and the vibration amplitude are arbitrary, the study is carried out numerically. Arbitrary linear and nonlinear vibrations in the case of small rigidity (the case of sympathetic pendulums) were studied earlier [1, 2].     

Dynamic Analysis of a Two-Mass System with Essentially Nonlinear Vibration Damping

A nonlinear system with two degrees of freedom is considered. The system consists of an oscillator with relatively large mass, which approximates some continuous elastic system, and an oscillator with relatively small mass, which damps the vibrations of the elastic system. A modal analysis reveals a local stable mode that exists within a rather wide range of system parameters and favors vibration damping. In this mode, the vibration amplitudes of the elastic system and the damper are small and high, respectively__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 102–111, January 2005.     

Experimental and theoretical analysis of the transverse vibrations of a beam having bilinear support

Experimental and theoretical investigations have been conducted to study the transverse vibrations of a beam having nonlinear constraint. One end of the beam is fixed while the other is supported on a bilinear spring and carries a concentrated mass. Free-vibration curves, obtained for different values of the spring constants and the end mass, indicate that free periodic vibrations with frequencies which can lie within any one of an infinite number of ranges may occur. Forced harmonic response may exhibit the multiplicity of jump phenomena within the frequency ranges of free vibrations.     

A Bifurcation Analysis of the Dynamics of a Simplified Shell Model

We study parametrically excited vibrations of a shallow cylindrical panel. The mathematical model is a system of two partial differential equations based on Donnell's shallow shell theory. The original equations are discretised using Galerkin approximations and all calculations are performed through symbolic manipulations. A bifurcation analysis of a system model with two degrees of freedom is accomplished by continuation techniques. The importance of the second degree of freedom as well as some open questions concerning the modelling of shell vibrations are discussed.     

Thermodynamics of Shape Memory Alloy Wire: Modeling, Experiments, and Application

A thermomechanical model for a shape memory alloy (SMA) wire under uniaxial loading is implemented in a finite element framework, and simulation results are compared with mechanical and infrared experimental data. The constitutive model is a one–dimensional strain-gradient continuum model of an SMA wire element, including two internal field variables, possible unstable mechanical behavior, and the relevant thermomechanical couplings resulting from latent heat effects. The model is calibrated to recent and new experiments of typical commercially available polycrystalline NiTi wire. The shape memory effect and pseudoelastic behaviors are demonstrated numerically as a function of applied displacement rate and environmental parameters, and the results compare favorably to experimental data. The model is then used to simulate a simple SMA actuator device, and its performance is assessed for different thermal boundary conditions.     

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.     

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy(SMA) composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA’s strain(Ms-Strain) value at the beginning of martensitic phase transformation.It is shown that SMA’s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.     

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.     

The geometrical stability of non-linear normal modes in two degree of freedom systems

The geometrical stability of the non-linear normal mode vibrations of a class of two degree of freedom dynamical systems is studied by utilizing the definitions and analysis of Synge's “Geometry of Dynamics.”It is shown that instabilities can occur for small amplitudes of vibration only if (a) one of the associated linear normal modes possesses a frequency which is nearly a multiple of the frequency of the other linear normal mode, or (b) the frequency of one linear normal mode is nearly zero.     

Chaotic vibrations of a beam with non-linear boundary conditions

Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.     

A closed-form solution procedure for circular cylindrical shell vibrations

A systematic procedure for obtaining the closed-form eigensolution for thin circular cylindrical shell vibrations is presented, which utilizes the computational power of existing commercial software packages. For cylindrical shells, the longitudinal, radial, and circumferential displacements are all coupled with each other due to Poissons ratio and the curvature of the shell. For beam and plate vibrations, the eigensolution can often be found without knowledge of absolute dimensions or material properties. For cylindrical shell vibrations, however, one must know the relative ratios between shell radius, length, and thickness, as well as Poissons ratio of the material. The mode shapes and natural frequencies can be determined analytically to within numerically determined coefficients for a wide variety of boundary conditions, including elastic and rigid ring stiffeners at the boundaries. Excellent agreement is obtained when the computed natural frequencies are compared with known experimental results.     

FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a two-dimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a fourth order Runge-Kutta scheme. Results for soft and hard springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit-cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analysed using analytical techniques developed for nonlinear dynamical systems. The Hopf bifurcation point is determined analytically and the amplitude of the limit-cycle oscillation in post-Hopf bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit-cycle oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit-cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom.