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Dynamics of a rotor composed of a flexible beam attached to a slewing rigid hub is presented in the paper. Dynamics of the structure is studied for a slender beam model, based on extended Bernoulli–Euler theory, which takes into account a nonlinear curvature, coupled transversal and longitudinal oscillations and non-constant angular velocity of the hub. Moreover, to demonstrate a general case for dynamical boundary conditions, lumped mass fixed at the beam tip is added. The partial differential equations (PDEs) are derived from Hamilton principle of the least action. The analytical solutions of the PDEs are obtained by the multiple time scale method applied directly to PDEs. Forced vibrations around selected resonance zones are studied and the influence of beam rotation, preset angle, hub radius, tip mass is presented. Hardening and softening phenomena, respectively for the first and the second mode, are obtained for various angular velocity values.

The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 : ir, $$ir \in \mathbb {N}$$ i r ∈ N , internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).

The main purpose of this paper is to analyze the impact of hysteresis of a piezoelectric subsystem on the nonlinear beam dynamics with active vibration control. The investigated model considers geometric and inertia nonlinearities of the beam (plant), nonlinear coupling of the controller, and hysteresis of the piezoelectric actuator. The developed numerical model of the electromechanical system allows a comparison of responses given by a simplified model and a model with a complete description of the piezoelectric subsystem, including hysteresis. Obtained results provide information about the effect of hysteresis on the beam and controller dynamics. Quantitative and qualitative changes due to considering the effect of hysteresis are presented in the paper.

A rotating system comprising a hub and a thin-walled laminate cantilever beam with embedded nonlinear piezoelectric layers is analysed in the paper. The reinforcing fibres set-up in composite material conforms to circumferentially uniform stiffness lamination scheme. This configuration exhibits the mutual bending couplings in two orthogonal planes. Nonlinear analytical model of a piezoelectric material embedded onto the beam walls is postulated by considering the higher-order constitutive relations with respect to electric field variable. Moreover, to properly model electromechanical structural behaviour, the full two-way coupling piezoelectric effect is considered. To this aim, the assumption of a spanwise electric field variation is postulated in the mathematical model of the structure. Based on previous authors’ research, the system of mutually coupled nonlinear equations of motion is formulated. In the numerical analysis the forced response of the system under zero and nonzero mean value harmonic torque excitation is considered. In particular, the influence of hub inertia, excitation amplitude and mean rotating speed on system dynamics is investigated. The results are presented in the form of appropriate frequency response plots and bifurcation diagrams.

The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.

A nonlinear thermoelastic model of a circular plate is presented in the paper. The model, based on the Mindlin plate theory, is extended by taking into account nonlinear geometrical terms. Partial differential equations of plate’s dynamics are derived for a fully coupled thermal and mechanical fields. Then the model is reduced to a set of ordinary differential equations taking into account the first three natural modes and assuming a constant thermal field. The influence of elevated temperature on the resonance curves and the mode involvement due to nonlinear and thermal couplings is presented. The analysis shows that the increased temperature may lead to various bifurcation scenarios. The buckling phenomenon and post-buckling nonlinear regular and chaotic oscillations are studied.

Herein, a two degree-of-freedom model of the reconstructed middle ear using shape memory prosthesis is investigated near the primary resonance for various prosthesis types. The polynomial model of shape memory alloy is used to characterise the properties of the prosthesis. To solve the differential equations of middle ear motion, the multiple time scales method is applied. The first- and second-order approximations of the primary resonance curves are compared with each other and verified through numerical simulations. The obtained analytical solutions studied for the stiff and flexible prosthesis reveal various accuracies depending on the prosthesis stiffness. The second-order approximation of the multiple scales method reveals unstable harmonic vibrations near the primary resonance zone, but are not found in the first-order approximation.

The main goal of this work is to develop a theoretical and numerical approach, to study the geometrical nonlinear vibration of bi-material beams under the combined action of mechanical and thermal loads. The geometrically nonlinear version of the Timoshenko beam theory is used to describe the theoretical model of the problem. Starting from the geometrical, constitutive and equilibrium equations of each layer the governing equations of the bi-material beam are derived. The beam is subjected to heat flux and dynamic mechanical loading. The influence of the elevated temperature or the heat propagation along the beam length and thickness on the response of the beam was studied.

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged–simply supported beam with a linear axial spring of arbitrary stiffness. The forced-damped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt–Poincare method. A special numerical technique using explicit finite element method to draw the frequency–response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.

A nonlinear two degrees of freedom system with shape memory element (SMA) is presented in the paper. Thermomechanical properties, shape memory effect and pseudoelasticity of the SMA joint, are modelled by phenomenological approach with the help of a fifth order polynomial approximation. The analytical solution of the mathematical model is obtained by means of the multiple time scales method. Next, a numerical study of a biomechanical system of human middle ear is performed. It has been demonstrated that for a selected rod type prothesis the cubic nonlinearity has the most essential effect on the system dynamics in the first order approximation, which converges with direct numerical simulations.

Numerical and experimental studies of a magnetic levitation harvester are presented in the paper. The idea is based on the motion of permanent cylinder magnet in a coil exploited for energy harvesting. The novel model is based on a new definition of the coupling coefficient (inductive coefficient) which relates mechanical and an electrical components. The performed static and dynamics experimental tests show that this coefficient is a nonlinear function of the magnet position, and highly depends on the magnet coordinate in the coil, in such a way that the maximum energy is obtained in a coil ends. The comparison between classical – fixed value model – and novel nonlinear model of the inductive coefficient is presented for selected cases. The most essential differences are presented.

During middle ear reconstruction, surgeons typically have to adjust the size and position of a medical prosthesis to fit the needs of each patient. In this paper, a prosthesis made of shape memory alloy is proposed to reduce operation time while accounting for the individual variability in middle ear lengths. This shape memory prosthesis (SMP) is activated by heat, so it is important to measure the regular and irregular behaviour prior to recommendation for medical practice. We present a two-degrees-of-freedom model of a reconstructed middle ear with an SMP that has two nonlinear springs, representing the properties of the annular ligament and the prosthesis. The properties of the SMP are described via polynomial dependence. Experiments on a temporal bone are used to determine the parameters of an intact ossicular chain and test a conventional titanium prosthesis. The dynamics of the reconstructed middle ear are investigated under different external excitation parameters. Bifurcation analysis was conducted and the stability of periodic solutions was examined to monitor the behaviour of the stapes. The results of this study provide interesting chaotic vibrations and regular oscillations with fluctuation. We determine that only the harmonic response of the stapes is advisable from a practical point of view, and this can only be achieved for small excitation amplitudes. Therefore, we show the importance of selecting appropriate prosthetic parameters to avoid irregular motions and deterministic chaos.

In this paper a new damage index for detection and location of damage is introduced. It is improved version of the earlier developed Poincare map method based on forced response of the structure. The proposed method is compared with the most popular modal based methods for damage detection: the modal displacements, modal curvatures and the strain energy methods. A finite element model of intact and damaged composite beams is used in order to validate correctness and efficiency of the method. Moreover, the modal based and forced response are tested experimentally by using a scanning laser vibrometer. The advantages of the offered method and its application in SHM are demonstrated by numerical simulations and experimental tests.

Modal interactions, stability analysis and time delay control of a composite column loaded by axial periodic force is presented in the paper. The simply supported column is composed of channel sections made of layers of a laminate with an arrangement which couples bending and twisting of the structure. The laminate coupling, together with a nonlinear geometrical displacement field result in a nonlinear model with parametric excitation. It is shown that for the selected configuration different mode couplings and parametric instability zones may occur. In order to stabilise the system a time delay displacement feedback control is proposed. An effect of the time delay control on the first principal parametric resonance zone is studied analytically and numerically.

The paper addresses the nonlinear dynamics of a hinged simply supported beam with an axial elastic support. The investigation is based on the Multiple Time Scales Method (MTSM) applied directly to the governing partial differential equations of motion, and is focused on the first seven bending resonances of a planar system. The effect of axial spring on hardening/softening phenomena is illustrated. Backbone Curves (BbCs) and Frequency Response Curves (FRCs) for selected cases are shown and compared with their numerical counterpart done with the Finite Element Method (FEM) to cross-checking the reliability of analytical and numerical methods.

A nonlinear model of a structure consisting of a rigid hub and three thin-walled flexible laminated beams is analysed in this paper. It is assumed the considered blades have additional piezoceramic layers embedded into master laminate, so due to elastic specimen deformation an electric field is induced. In the mathematical model of the multi-material blades a full two-way piezoelectric coupling effect involving mutual mechanical and electrical interactions is adopted. Moreover, a nonlinear constitutive relation for piezoceramic material with respect to electrical field is considered to properly model the active structure behaviour when operated in near resonance conditions. The governing equations of the system have been sourced from the previous authors’ research. Next, resonance curve plots of the coupled electromechanical system considering both the blades and the hub dynamics have been prepared. Within the performed studies cases of a fully symmetric and next a slightly mistuned rotor configuration have been discussed.

An effect of thermal loading on vibrations of beams is investigated in the paper. A beam is considered as an extended Timoshenko beam model with nonlinear terms resulted from large deflections. Dynamics of the structure is analysed under thermal and mechanical loadings considering transient dynamics due to a heat pulse imposed to the beam. The numerical method for solving coupled thermo-mechanical problem is presented. On this basis the importance of the heat pulse intensity around the first resonance condition is demonstrated. The effect of the heat on the the complex transient dynamics of the beam and its qualitatively different response is shown as well.

The dynamics of a system consisting of a rotating rigid hub and a flexible composite thin-walled beam is discussed. The nonclassical effects like material anisotropy, rotary inertia and transverse shear are considered in the mathematical model of the structure. Moreover, the hub mass moment of inertia is taken into account. The differential equations of motion featuring beam bending–twist elastic coupling are derived using the Hamilton principle, and the Galerkin method is applied in order to reduce the partial differential governing equations to the ordinary differential equations. Parametric studies are conducted to evaluate beam stiffness coefficients depending on the fiber lamination angle. Next, numerical results are obtained to investigate the impact of hub to beam relative inertia on the natural frequencies of the structure. Cases of forced vibrations of the system are examined where the driving torque is considered as the sum of a constant (mean value) and a periodic component. Simulations show the importance of the hub inertia on the complete system dynamics. A shift of the resonance zones and a vibration absorption are observed.

A nonlinear saturation control method applied for reduction of oscillations of a rotating composite beam is presented in this paper. The flexible blade is treated as a thin-walled structure attached to a rigid hub driven by an external periodic torque. The beam model, derived in [1] , [2] , takes into account a strong coupling between the flexural and torsional deformations due to the assumed specific reinforcing fibers perimeter orientation. The proposed saturation control method is used for the reduction of coupled torsional-flexural beam oscillations around the third natural frequency of the system. It is shown that for assumed structural parameters it is possible to find gains of the controller which may essentially reduce vibrations of the beam near the resonance zone.

The aim of this paper is to validate experimentally a nonlinear model of a kinematically excited hinged-simply supported beam with a spring subjected to one end. An experimental setup configuration enables to test different variants of axial boundary conditions: first a typical simply supported beam (no spring), and next two different spring systems. The prototype is kinematically excited with different amplitudes of excitation and then full frequency response curves are drawn wherein hardening/softening dichotomy is recognized. A set of mechanical properties of the system is identified and then used to reproduce tests with finite element simulations. Consequently numerical vs. experiment results are compared. The analysis demonstrates amplitude dependent damping as well as that the natural frequency depends on environmental conditions, and thus may change over experiments.

Numerical investigations of milling operations dynamics are presented in this paper. A two degree of freedom nonlinear model is used to study workpiece-tool vibrations. The analyzed model takes into account both flexibility of the tool and the workpiece. The dynamics of the milling process is described by the discontinuous ordinary differential equation with time delay, which can cause process instability. First, stability lobes diagrams are created on the basis of the parameters determined in impact test of an end mill and workpiece. Next, the bifurcations diagrams are performed for different values of rotational speeds.Numerical investigations of milling operations dynamics are presented in this paper. A two degree of freedom nonlinear model is used to study workpiece-tool vibrations. The analyzed model takes into account both flexibility of the tool and the workpiece. The dynamics of the milling process is described by the discontinuous ordinary differential equation with time delay, which can cause process instability. First, stability lobes diagrams are created on the basis of the parameters determined in impact test of an end mill and workpiece. Next, the bifurcations diagrams are performed for different values of rotational speeds.

The aim of the paper is to study a synchronisation phenomenon as observed in a rotating structure consisting of three composite beams and a hub. The beams are made of eighteen carbon-epoxy prepreg material layers stacked in a specific sequence. In the performed analysis it is assumed one of the beams is de-tuned due to small misalignment of its reinforcing fibers orientation with regard to the two remaining nominal design blades. The non-classical effects like transverse shear, material anisotropy, non-uniform torsion and cross-section warping are taken into account in the mathematical model of the blades. The partial differential equations of motion of the structure are derived by the Hamilton principle; next the reduction to the ordinary differential ones is done by the Galerkin method. Finally, the equations are solved numerically and the resonance curves for the hub and the individual beams are plotted. In the performed studies two possible variants of the rotor excitation are considered: (a) driving torque expressed by a harmonic function or (b) torque given by a chaotic oscillator formula. The analysis of the synchronisation phenomenon of the hub and the blades motion is based on the study of the resonance curves and time histories in the prepared graphs. The analysis of the structure driven by chaotic oscillator revealed the existence of the strange chaotic attractor for every beam of the rotor; in the particular, nominal beams are fully synchronised, but the de-tuned one is synchronised with a small difference in amplitude.

Dynamics of a system composed of an initially straight, planar, linearly elastic hinged-supported beam and (initially) a coaxial spring attached to a supported section is studied in the paper (Fig. 1). The beam model considers coupled axial-transversal displacements of its axis, furthermore rotatory inertia and shear deformation are included [1,2]. The structure is excited by periodic load uniformly distributed in X (transversal) and Z (longitudinal) directions. Partial differential equations of motion, which include external loads and assumed linear viscous damping, are applied to method of multiple scales [3]. The developed model allowed to study resonance response waves for a wide spectrum of beam slenderness and spring stiffness ( varying from 0 to ∞). In particular cases the coupled internal sub-resonances cured. Numerical computations by finite element method (commercial software Abaqus CAE) are compared with selected experimental tests. In numerical simulations composite material is considered, too.

Nonlinear vibrations of a circular plate subjected to mechanical and thermal loadings are presented in the paper. A model of the plate is based on the extended Mindlin theory, taking into account nonlinear geometrical terms and acting heat uniformly distributed along the plate span. The dynamics of a coupled thermo-mechanical problem is reduced from a set of partial differential equations to ordinary differential equations. Considering oscillations around the first natural frequency just one mode reduction is proposed. The analysis shows that elevated temperature shifts the resonance curve and new post-buckling oscillations arise. Depending on initial conditions for the post-buckling state various scenarios of bifurcations take place and transient irregular oscillations may occur. The proposed one degree of freedom model shows a good agreement with response of the model based on three or five-modes reduction.

In this paper the polynomial model of shape memory alloy is used to characterise properties of a micro-actuator which is applied as a new middle ear prosthesis. A two degrees of freedom model of the reconstructed middle ear is solve by means of multiple time scales method. The system has various behaviours near the primary resonance depending on ambient temperature. The special case when relative temperature θ = 1.0 characterises untypical resonance curve. Increasing temperature to the normal human body one the resonance curves are typical. Then the system has only one periodic solution if the excitation is not too strong.

The aim of the paper is to analyze the synchronization phenomenon of a rotating structure composed of three beams attached to a rigid hub. It is assumed in the calculations that one beam is 10% thicker comparing to the remaining ones. Furthermore, two possible variants of excitation are considered: (a) torque given by harmonic function or (b) torque produced by a chaotic oscillator. Next, the equations have been solved numerically and the resonance curves and time series have been analyzed in terms of synchronized motion of the hub and blades of the rotor. The influence of hub’s mass moment of inertia have been checked as well. For the system with chaotic Duffing oscillator Poincare maps have been obtained.The aim of the paper is to analyze the synchronization phenomenon of a rotating structure composed of three beams attached to a rigid hub. It is assumed in the calculations that one beam is 10% thicker comparing to the remaining ones. Furthermore, two possible variants of excitation are considered: (a) torque given by harmonic function or (b) torque produced by a chaotic oscillator. Next, the equations have been solved numerically and the resonance curves and time series have been analyzed in terms of synchronized motion of the hub and blades of the rotor. The influence of hub’s mass moment of inertia have been checked as well. For the system with chaotic Duffing oscillator Poincare maps have been obtained.

The numerical and experimental validation of multistable behavior of cantilever shells is addressed. The design of the laminated composite shells is driven by a recently proposed semi-analytical shell model, whose predictions are verified and critically examined by means of finite element simulations and stability tests on two manufactured demonstrators. In addition, the influence of the main design parameters on the shells stability scenario is discussed. Despite its simplicity, the reduced model allows to depict a fairly faithful picture of the stability scenario; therefore, it proves to be a useful tool in the early design stages of morphing shell structures.

The paper presents a study of a possible application of structure embedded piezoelectric actuators to enhance the performance of a rotating composite beam exhibiting the coupled flexural-flexural vibrations. The discussed transversal and lateral bending modal coupling results from the directional properties of the beam's laminate and ply stacking distribution. The mathematical model of the beam is based on an assumption of cross-sectional non-deformability and it incorporates a number of non-classical effects. The final 1-D governing equations of an active composite beam include both orthotropic properties of the laminate and transversely isotropic properties of piezoelectric layers. The system's control capabilities resulting from embedded Macro Fiber Composite piezoelectric actuators are represented by the boundary bending moment. To enhance the dynamic properties of the composite specimen under consideration a combination of linear proportional control strategies has been used. Comparison studies have been performed, including the impact on modal coupling magnitude and cross-over frequency shift.

Vibrations of a nonlinear self- and parametrically excited MEMS device driven by external excitation and time delay inputs are analysed in the paper. The model of MEMS resonator includes a nonlinear van der Pol function producing self-excitation, a periodically varied coefficient which represents Mathieu type of parametric excitation and furthermore, periodic force acting on the resonator. Analysis of frequency locking zones is presented with suggestions for a strategy of a closed loop control. Interactions between self- and parametric excitation lead to quasi-periodic oscillations but under specific conditions the motion becomes harmonic. The so called frequency locking, near the resonance zones is observed. This is caused by the second kind Hopf bifurcation (Neimark–Sacker bifurcation). The amplitudes of periodic oscillations are determined analytically by the multiple time scale method (MS) in the second order perturbation. The effect of external force has been observed by the internal loop occurring inside the frequency locking zone. The localisation of the zones and existence of the internal loop can be controlled by a selection of gains and time delay of displacement or velocity feedbacks.

In this work, the large amplitude vibration of a heated Timoshenko composite beam having delamination is studied. The model of delamination considers the contact interaction between sublaminates including normal forces, shear forces, and additional damping due to the interaction of sublaminates. This work is an extension of the previous analysis based on a model of the dynamic behavior of a beam with delamination considering additionally the nonlinearities due to large displacements and temperature changes. Numerical calculations are performed in order to estimate the influence of the delamination, the geometrically nonlinear terms, and elevated temperature on the response of the beam.

In this research we consider a saturation adaptive control strategy to suppress vibrations of a system consisting of a rotating rigid hub and a thin-walled composite blade with an embedded piezoelectric active element. The adopted mathematical model of the beam considers non-classical effects like a circumferentially asymmetric stiffness lamination scheme that result in strong mutual coupling of the bending-twisting deformations as well as a higher order piezoceramic constitutive relation. The discussed structure has been investigated for possible levels of original system simplifications starting from the fully linearised one up-to the control applied to the nonlinear structure performing the full rotation. Obtained results of numerical simulations prove the applied nonlinear saturation control to be the robust and effective method for beam vibration suppression in near-by resonance zones for a non-rotating as well as rotating structures. It is shown that vibration of the beam can be suppressed to similar levels independently of the model simplification degree presuming the condition of proper controller tuning is preserved. However, significant differences in the width of vibration suppression zones are observed for studied subcases. Moreover, the analysis of the system response sensitivity to feedback and control gains is discussed.