Nonlinear Dynamics of a Spinning Shaft with Non-Constant Rotating Speed

Reported research on spinning shafts is mostly restricted to cases of constant rotational speed without examining the dynamics thatoccursduringtheir spin-up or spin-down operation. In suchcases, the motion is described by a nonlinear system of Partial Differential Equations (PDEs) coupled with an Integro-Differential Equation (IDE). The nonlinear system of PDEs with IDE, projected onto the infinite basis of the modes of the underlying linear system, results in a system of nonlinear Ordinary Differential Equations (ODEs). In this articleis appliedthe multiple scales perturbation method for dynamic analysis and the system in first order approximation takes the form of two coupled sets of pairedequations. The first pair describes torsional and rigid body rotation whilst the secondconsists of the equations describing the two lateral bending motions. Although in this system non-conservative forces are not considered in terms of damping or explicit externally applied load (torques/forces), the solution of the Is'order approximation of the first set of equations indicates that there are no periodic motions. The solution of the second set of equations of 1st order approximation coincides with the case of constant rotating speed. It isshown, that the Normal Modes in bending motions are the critical speeds of the shaft. It is shown that the frequencies in the Campbell diagram coincide with the frequencies associated with the 1st order solution of the nonlinear system. Moreover, the analytical solution of the first pair of equations is in good agreement with direct numerical simulations. This work paves the way for the development of the Nonlinear Campbell diagram that can be used to determine the dynamic behaviour of rotating structures during spin-up or spin-down operation

Towards the determination of a Nonlinear Campbell diagram of a spinning shaft with non-constant rotating speed

In the literature it is reported only the studies for the steady states of spinning shafts without any information of the shaft’s dynamics during spin-up, spin-down operation. The derivation of the system’s discrete equations of motion is required for the analysis of the behavior of spinning shafts during this type of operation. The equations describing the motion of a spinning shaft with nonconstant rotating speed forms a system of nonlinear Partial Differential Equations (PDEs) coupled with an IntegroDifferential (ID) equation which describes the rigid body motion of the shaft. Towards the determination of the Nonlinear Campbell diagram of the spinning shaft the equations has been discretized by projecting the dynamics to the infinite base of the linear mode shapes of the underlying linear PDEs and the resulted discrete equations forms a nonlinear system in a Non-Cauchy format. Further on, based on the derived formulation the Campbell diagram has been determined by considering constant rotating speed and the results have been compared with those obtained by Finite Element analysis and they are in very good agreement. This work paves the way for the determination of the Nonlinear Campbell diagram by considering the periodic motions of the derived nonlinear discrete system using either numerical (shooting method with continuation) or analytical techniques. It should be mentioned that the same approach can be used also for spin-up, spin-down dynamic analysis with the determination of Nonlinear Campbell diagram in case of rotating blades with nonconstant rotating speed with application in turbomachinery, wind turbines, pumps and many other mechanical applications with rotating components

Linear modal analysis of L-shaped beam structures: parametric studies

Linear modal analysis of L-shaped beam structures indicates that there are two independent motions, these are in-plane bending and out of plane motions including bending and torsion. Natural frequencies of the structure can be determined by finding the roots of two transcendental equations which correspond to in-plane and out-of-plane motions. Due to the complexity of the equations of motion the natural frequencies cannot be determined explicitly. In this article we nondimensionalise the equations of motion in the space and time domains, and then we solve the transcendental equations for selected values of the L-shaped beam parameters in order to determine their natural frequencies. We use a numerical continuation scheme to perform the parametric solutions of the considered transcendental equations. Using plots of the solutions we can determine the natural frequencies for a specific L-shape beam configuration

Mode shapes variation of a composite beam with piezoelectric patches

In this paper the modal shapes of a light, thin laminate beam with active elements were evaluated. Cases with one or two Macro Fiber Composite (MFC) active elements adhered onto a glass-epoxy cantilever beam were analyzed. The systems under consideration were modeled in ABAQUS finite element software to derive mode shapes numerically. Next, the modes were compared to each other to estimate the influence of PZT patches. First 20 modes of natural vibrations were examined including bending, torsion and axial ones. The comparisons of mode shapes were performed according to Modal Assurance Criterion (MAC) analysis. The examination of changes of mode shapes of the original beam with placement of active elements is the starting point in prior of optimal placements of PZTs with final goal the control of dynamics of helicopter blades

Rational placement of a macro fibre composite actuator in composite rotating beams

In the presented research the dynamics of a thin rotating composite beam with surface bonded MFC actuator are considered. A parametric analysis aimed at finding the most efficient location of the actuator on the beam is presented. Gyroscopic effects resulting in the beam’s initial strain and therefore non-zero voltage in PZT are taken into account. Within the frame of the study maximising the system's response observed in vibration modes for uncoupled and coupled motions is examined. The results are compared to the case of a nonrotating beam and also to the maximum response of the beam with the actuator placed at different positions. To perform the analysis an ABAQUS finite element model of an electromechanical system under consideration is developed. The multi-layer composite beam structure is modelled by shell elements according to a layup-ply technique; the MFC actuator is modelled by 3D coupled field piezoelectric elements. Both modal analysis and frequency response spectra are performed to obtain the structural modal parameters and response amplitude, respectively. The analysis is repeated for three different orientations of the beam's cross-section with respect to the plane of rotation (i.e. arbitrary assumed pitch angles); in all cases the condition constant angular speed is preserved. This work is fundamental for continuing the research for control of dynamics of rotating composite beams with active elements

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially nonlinear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption

Vibro-impact attachments as shock absorbers

The use of vibro-impact (VI) attachments as shock absorbers is studied. By considering different configurations of primary linear oscillators with VI attachments, the capacity of these attachments to passively absorb and dissipate significant portions of shock energy applied to the primary systems is investigated. Parametric studies are performed to determine the dependence of energy dissipation by theVI attachment in terms of its parameters.Moreover, non-linear shock spectra are used to demonstrate that appropriately designed VI attachments can significantly reduce the maximum levels of vibration of primary systems over wide frequency ranges. This is in contrast to the classical linear vibration absorber, whose action is narrowband. In addition, it is shown that VI attachments can significantly reduce or even completely eliminate resonances appearing in the linear shock spectra, thus providing strong, robust, and broadband shock protection to the primary structures to which they are attached

Nonlinear modal analysis of an L-shape beam structure

In this work it is derived the nonlinear equations of motion of L-shaped beam structure considering rotary inertia terms for out-of-plane motion in order to be used for nonlinear modal analysis of the structure. The dynamics has been projected in the infinite mode shapes space and it is derived the equations of motion in generalized coordinates. The nonlinear equations of motion indicates that there is coupling between in-plane and out-of-plane motions which in linear case is not the case

Chaotic dynamics in spinning shafts with non-constant rotating speed described by variant Lyapunov exponents

The dynamics of spinning shafts with non-constant rotating speed is described by a nonlinear system that under certain conditions might exhibit also chaotic behavior. In this article chaotic dynamics of the spinning shaft is examined. Initially, the trajectories in phase space around the equilibrium manifolds are determined. Then by choosing a set of initial conditions, nearby to an equilibrium, corresponding to eigenvalues of the Jacobian with a nonzero real part, identification of chaos is examined. Approximations of the trajectory, with the linearization curves around the equilibria, are defined and they are good in a region very close to the associated equilibrium point. It is shown that the eigenvalues, as Lyapunov exponents indicators, are not parameter dependent but state dependent. The eigenvalues of the linearized system within an orbit are varying from positive to zero, therefore the Lyapunov exponent is not defined through this limit as an explicit number but variant. The existence of eigenvalues with positive real parts in certain parts of the orbit is an indication of chaos since it shows a divergence of nearby orbits. One orbit starting from an initial condition which corresponds to eigenvalues with positive real part is crossing the threshold and pass to points that the eigenvalues with zero real parts, therefore this ‘threshold’ is not discriminating chaotic with regular regions as expected. The variant positive Lyapunov exponents have been examined also with numerical investigations and it is an indication of chaos. The Poincare section indicates irregular motion and the approximated Information Entropy is relatively high, and both are indicating chaos. It should be highlighted that this is a mechanical system with variant real parts of eigenvalues as Lyapunov exponents within one orbit and the threshold is insufficient to distinguish chaotic from regular regions. Further work is needed to determine the chaotic regions of the spinning shaft. Further developments in the mathematics of nonlinear dynamical systems associated with the equilibrium manifolds are needed to examine the significance of variant Lyapunov exponents for this kind of systems. Also, the necessity to reexamine the validity of existing algorithms and the development of new ones for the determination of variant Lyapunov exponents, become evident

Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments

We study Targeted Energy Transfers (TETs) and nonlinear modal interactions attachments occurring in the dynamics of a thin cantilever plate on an elastic foundation with strongly nonlinear lightweight attachments of different configurations in a more complicated system towards industrial applications. We examine two types of shock excitations that excite a subset of plate modes, and systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena occurring between the plate and the attachments. The following attachment configurations are considered: (i) a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) attachment – termed nonlinear energy sink (NES); (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single multi-degree-of-freedom (MDOF) NES with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and locations of the NESs, in order to optimize passive TETs from the plate modes to the attachments, and we showed that the optimal position for the NES attachments are at the antinodes of the linear modes of the plate. The parametric study of the damping coefficient of the SDOF NES showed that TETs decreasing with lower values of the coefficient and moreover we showed that the threshold of maximum energy level of the system with strong TETs occured in discrete models is by far beyond the limits of the engineering design of the continua. We examine in detail the underlying dynamical mechanisms influencing TETs by means of Empirical Mode Decomposition (EMD) in combination with Wavelet Transforms. This integrated approach enables us to systematically study the strong modal interactions occurring between the essentially nonlinear NESs and different plate modes, and to detect the dominant resonance captures between the plate modes and the NESs that cause the observed TETs. Moreover, we perform comparative studies of the performance of different types of NESs and of the linear Tuned-Mass-Dampers (TMDs) attached to the plate instead of the NESs. Finally, the efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed