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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 dynamics of a system consisting of a rotating rigid hub and a thin-walled composite beam with embedded active element is presented. The beam comprises of a generally orthotropic host made of an arbitrary laminate and an additional layer of a transversely isotropic piezoceramic material. The higher-order constitutive relations for piezoelectric are used to properly model its electromechanical structural behaviour when operated in near resonance conditions or subjected to strong electric fields. In the mathematical formulation of the problem, the full two-way coupling piezoelectric effect is considered by adopting the assumption of a spanwise electric field variation. To enhance the generality of the formulation, the model considers also the hub mass moment of inertia and a non-constant rotating speed case. A general nonlinear system of mutually coupled partial differential equations is derived using the Hamilton principle, and the Galerkin method is applied to reduce these governing equations to the ordinary differential ones. A specific case of CAS lamination scheme that exhibits flapwise bending and twist mode elastic coupling is discussed in detail. Numerical results for system free vibrations are obtained to investigate the natural mode shapes and electrical field spatial distribution depending on the system rotation speed and laminae fibre orientation angle. Next, a forced vibration case is studied by applying to the hub a periodic driving torque with zero mean value. Nonlinear frequency response curves for the combined beam–hub system together with time series plots are prepared for different driving torque scenarios.

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.

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 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.

This paper presents a theoretical vs. experimental modal analysis of a composite thin-walled beam featuring a circumfer-entially asymmetric stiffness (CAS) profile characteristics. The adopted lamination scheme results in the complex elastic deformation modes exhibiting mutual coupling of flapwise bending, transverse shear and torsion. The analytical model used in this study is based on the authors previous research and takes into account most classical and non-classical effects specific for thin-walled composite structures. The theoretical outcomes are compared to experimental ones obtained by two different test methods, namely an impact hammer test and a laser vibrometer test. In this second experiment the macro fibre composite (MFC) patch actuators have been used to excite the system. For comparative purposes two different transducer types providing different excitation load have been examined. The performed analytical and laboratory experiments demonstrate extremely high consistent findings irrespective of the means of excitation. Therefore, the laser-based motion analysis system combined with piezo-actuator excitation may be considered as a feasible and accurate method for static and dynamic experiments on systems exhibiting complex deformation modes. These include also highly flexible structures where the deformations can not be measured by conventional contact methods e.g. due to the influence of an accelerometer mass. The discussed laser-based motion measuring technique can also be used in future modal analysis experiments on rotating beams.

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.

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.

Shape memory alloys are a family of metals with the ability to change specimen shape depending on their temperature. This unique property is useful in many areas of mechanical and biomechanical engineering. A new half-ring middle ear prosthesis design made of a shape memory alloy, that is undergoing initial clinical tests, is investigated in this research paper. The analytical model of the studied structure made of nonlinear constitutive material is solved to identify the temperature-dependent stiffness characteristics of the proposed design on the basis of the Crotti-Engesser theorem. The final integral expression for the element deflection is highly complex, thus the solution has to be computed numerically. The final results show the proposed shape memory C-shape element to behave linearly in the analysed range of loadings and temperatures. This is an important observation that significantly simplifies the analysis of the prototype structure and opens wide perspectives for further possible applications of shape memory alloys.

A model of a nonlinear system composed of a hub with attached two pendula rotating in a horizontal plane is studied in the paper. Each single pendulum, treated as a stiff and massless rod with a lumped mass, is connected to the hub by a flapping hinge. Nonlinear stiffness and viscous damping of the hinge is taken into consideration. The system is excited by an external torque generated by a DC motor which is considered as an ideal system with torque given by a harmonic function. For small oscillations the problem is linearised and then solved analytically. An influence of the structural parameters like mass of the hub and pendula length on natural end excited vibrations is presented. Large oscillations are studied by a continuation technique, directly from the original Ordinary Differential Equations of motion (ODE). The complete synchronisation, phase synchronisation, bifurcations and transition through resonances are analysed considering the influence of the mass of the hub. The existence of chaotic oscillations of the system and paths leading to chaos are demonstrated as well.

An electromechanical coupled theory is used to develop the equations of motion of a rotating thin-walled composite beam with surface bonded/embedded piezoelectric transducers. The higher order constitutive relations for the piezoceramic material are used to take into account the impact of a high electric field. In the mathematical model of the hybrid structure, the non-classical effects like material anisotropy, rotary inertia and transverse shear deformation as well as an arbitrary beam pitch angle are incorporated. Moreover, the model considers the hub mass moment of inertia and a non-constant rotating speed case. This approach results in an additional equation of motion for the hub sub-system and enhances the generality of the formulation. It is shown that final equations of motion of the hub–beam system are mutually coupled and form a nonlinear system of partial differential equations. Comparing to the purely mechanical model, the proposed electromechanical one introduces additional stiffness-type couplings between individual degrees of freedom of the system.

Sagged cable vibrations caused by support motion and possible external loading are investigated via the four-degree-of-freedom model proposed in Benedettini et al. (J Sound Vib 182(5):775–798, 1995). The model has a considerable potential in terms of forcing cases to be possibly addressed, with the physical motion of the supports naturally giving rise to a variety of external and parametric excitation terms. Dynamics of the system is studied close to the multiple internal resonance at cable crossover, which involves two in-plane and two out-of plane vibration modes. Solutions are found by the multiple time scale method. In the numerical investigation, attention is focused on the effects of planar support motion (symmetric and/or antisymmetric) at primary resonance, with the addition of planar symmetric external excitation entailing a nice cancellation phenomenon in the system response. Results are discussed also in the background of theoretical and experimental outcomes available in the literature. Comparison with a computer simulation of original equations of motion shows that analytical results are correct for moderately large oscillations, whereas a different scenario of multimodal responses may occur at higher excitation amplitudes. The nonlinear modal coupling is investigated through bifurcation scenarios and other dynamics tools, showing also transitions to complex response regimes.

This study considers forced vibrations of a rotating structure consisting of a rigid hub and three flexible beams. The blades are nominally made of a multilayered laminate with a specific stacking sequence resulting in full isotropic macroscopic material behaviour. However, in the performed analysis it is assumed that the rotor has been mistuned because of manufacturing tolerances of the composite material. These inaccuracies are represented by deviations of reinforcing fibres orientations from their nominal values. The considered tolerances break the intended macroscopic material isotropy and make the laminate to exhibit the fully orthotropic behaviour. Based on previous authors research, the system of four mutually coupled dimensionless ordinary differential governing equations is adopted. Forced responses of the system under regular and chaotic excitations are investigated.

The paper deals with the effect of dimensional imperfections of truss members on the minimum weight design of a structure. It is assumed that for each element its imperfections cannot exceed given a priori maximum values, called tolerances. The incorporation of the considered imperfections into the design is achieved by diminishing the limit values of state functions by the product of assumed imperfections and appropriate sensitivities. Therefore, the given method allows the introduction of tolerances into the design in a relatively simple way. In the submitted paper both members’ cross-section and length imperfections are discussed. The paper is illustrated with several design examples, considering cases with multiple loading conditions and buckling analysis. The achieved optimum solutions for designs with admissible tolerances show significant differences in structural weight and material distribution compared to ideal structures (i.e. having nominal dimensions). The calculations for designs with buckling analysis also reveal changes in material distribution compared to the designs without buckling constraints.

The paper deals with the effect of manufacturing imperfections in fiber orientations on the optimum design of laminate plates exposed to compression load. It is assumed that each dimensional imperfection cannot exceed one of two accepted values, called tolerances. In general, these are the largest allowed deviations of the design variable from its nominal value. The incorporation of tolerances is achieved by diminishing the limit values of state variables by the product of assumed tolerances and appropriate sensitivities. Therefore, the given method allows the introduction of tolerances into design in a relatively simple way and ensures safe results. The presented approach is a deterministic approximation of an exact stochastic solution of the problem. The paper is illustrated with two examples of the rectangular laminated plate design. The first one is maximization of buckling load for a given plate thickness. The second one is minimum weight (thickness) design for a given load. Numerical results show th...

In this paper, the effectiveness of a saturation control strategy in suppressing vibration of a rotating composite thin walled beam is studied. The mathematical model of the flexible beam takes into account a shear deformation effect, a warping function, a centrifugal force and the Coriolis acceleration. To extend the generality of the proposed formulation an inertia of the hub is also considered. Adaptive capability of the beam is achieved through the implementation of the saturation control algorithm. Within the performed tests, the discussed control strategy is applied for different magnitudes of flexural–torsional vibration modes resulting from different orientations of beam laminate-reinforcing fiber’s. The obtained results prove the applied nonlinear control to be the effective method for beam vibration suppression in near-by resonance zones for all studied cases. Parametric studies considered different rotating speeds of the system. It is shown that the vibration of the beam can be suppressed to similar levels independently of the transportation motion rotating speed. However, significant differences in the width of vibration suppression zones are observed.

In the presented paper the equations of motion of a rotating composite Timoshenko beam are derived by utilising the Hamilton principle. The non-classical effects like material anisotropy, transverse shear and both primary and secondary cross-section warpings are taken into account in the analysis. As an extension of the other papers known to the authors a nonconstant rotating speed and an arbitrary beam’s preset (pitch) angle are considered. It is shown that the resulting general equations of motion are coupled together and form a nonlinear system of PDEs. Two cases of an open and closed box-beam cross-section made of symmetric laminate are analysed in details. It is shown that considering different pitch angles there is a strong effect in coupling of flapwise bending with chordwise bending motions due to a centrifugal force. Moreover, a consequence of terms related to nonconstant rotating speed is presented. Therefore it is shown that both the variable rotating speed and nonzero pitch angle have significant impact on systems dynamics and need to be considered in modelling of rotating beams.

The paper discusses the dynamics of a composite thin-walled box-beam built into a rigid hub and performing combined motion of rotation and in-plane translation. The assumed blade spacial orientation and laminate material configuration results in twist/in-plane bending/in-plane shear motions coupling. Ordinary differential equations of motion are derived by adopting the Hamilton's variational principle and next an assumed mode method. It is shown that final equations are mutually coupled and form a highly nonlinear system. The coefficients of governing equations are found to be time-dependent and periodic due to blade temporary angular position and possible base excitation frequency. Within performed studies the effects of changes in selected structural parameters on the dynamic characteristics of the system are investigated.