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The derivation of a linear fractional representation (LFR) model for a flexible, spinning and uniform Euler-Bernoulli beam is accomplished using the Lagrange technique, fully capturing the centrifugal force generated by the spinning motion and accounting for its dependence on the angular velocity. This six degrees of freedom (DOF) model accounts for the behavior of deflection in the moving body frame, encompassing the bending, traction and torsion dynamics. The model is also designed to be compliant with the Two-Input-Two-Output Port (TITOP) approach, which offers the possibility to model complex multibody mechanical systems, while keeping the uncertain nature of the plant and condensing all the possible mechanical configurations in a single LFR. To evaluate the effectiveness of the model, various scenarios are considered and their results are tabulated. These scenarios include uniform beams with fixed root boundary conditions for different values of tip mass, root offset and angular velocity. The results from the analysis of the uniform cantilever beam are compared with solutions found in the literature and obtained from a commercial finite element software. Ultimately, this paper presents a multibody model for a spinning spacecraft mission scenario. A comprehensive analysis of the system dynamics is conducted, providing insights into the behavior of the spacecraft under spinning conditions. • Multibody system modeling tool dedicated to the dynamics of spinning spacecraft. • Parametric modeling of a spinning flexible spacecraft for robust control design. • Modeling and analysis of a flexible spinning Euler-Bernoulli beam. • Centrifugal force analysis relative to a spacecraft's angular velocity. [ABSTRACT FROM AUTHOR]

Due to their large and increasing size and the corrosive nature of salt water and high wind speeds, offshore wind turbines are required to be more robust, more rugged and more reliable than their onshore counterparts. The dynamic characteristics of the blade and its response to applied forces may be influenced dramatically by rotor rotational speed, which may even threaten the stability of the wind turbine. An accurate and computationally efficient structural dynamics model is essential for offshore wind turbines. A comprehensive model that takes the centrifugal stiffening effect into consideration could make rapid and accurate decisions with live data sensed from the structure. Moreover, this can enhance both the performance and reliability of wind turbines. When a rotating blade deflects in its plane of rotation or perpendicular to it, the centrifugal force exerts an inertia force that increases the natural frequencies and changes the mode shapes, leading to changes in the dynamic response of the blade. However, in the previous literature, studies of centrifugal stiffening are rarely found. This study investigates the influence of centrifugal stiffening on the free vibrations and dynamic response of offshore wind turbine blades. The National Renewable Energy Laboratory (NREL) 5 MW blade benchmark was considered to study the effect of angular speed in the flap-wise and edge-wise directions. The results demonstrate that the angular speed directly affects the modal features, which directly impacts the dynamic response. The results also show that the angular velocity effect in the flap-wise direction is more significant than its effect in the edge-wise direction. [ABSTRACT FROM AUTHOR]

To pursue high efficiency for wind turbine and harness more energy from wind, the wind power generation system has been designed larger and larger. With length scale increasing, the natural frequency of blade will be lower and changing dramatically with rotational speed, which will threaten the stability of the blade and turbine. However, in the previous research, the parametric, analytical, and quantitative study of centrifugal stiffening effect is rare to see. As a result, the current research has comprehensively studied the centrifugal stiffening effect on the structural characteristics of large wind turbine blade. Specifically, the equivalent rotating wedge beam model has been proposed for composite wind turbine blade. Based on the proposed wedge beam model and Rayleigh-Ritz method, the natural frequency variation curve has been derived and verified by ANSYS 3D simulation results. The parameters, including blade length, stiffness-mass ratio, and aspect ratio, affecting the natural frequency curve have been uncovered and analysed. It was found that the centrifugal stiffening effect has a great impact on the fundamental frequency of blade, e.g. 10% for NREL 5 MW, while it has less impact on other modal frequencies. The variation curve of fundamental natural frequency can be approximated by parabola. The product of the parabola curvature and fundamental frequency is found to be approximately constant for both NREL 5 MW and WindPACT 1.5 MW turbine blades. This research will provide guidelines for the resonance avoidance design of large wind turbine blade in the future. • The equivalent wedge beam model for large wind turbine blade has been constructed. • The factors affecting centrifugal stiffening were found by Rayleigh-Ritz method. • It was found that the variation curve of fundamental natural frequency is parabola. • It was found that the resonance phenomenon of large wind turbine blade more easily occurs. [ABSTRACT FROM AUTHOR]

The modified decomposition method is applied to analyze the transverse vibrations of tapered rotating beams incorporating both axial centrifugal stiffening and flexible end constraints. Unlike prior analyses relying upon a power series expansion about the left end constraint, we instead expand the solution for the transverse deflection about the interval midpoint, which has significant advantages including an increased rate of convergence and readily adaptation to the Robin-type boundary conditions. The modified decomposition method facilitates systematic solution of the mathematical model including parametric simulations where the boundary conditions can be varied without resolving the original equation. [ABSTRACT FROM AUTHOR]

Cantilever beam assumption is widely used to estimate gear flexibility for analytical calculation of transmission error of gear pair. Similar to the rotating beam, it is supposed that centrifugal stiffening will also affect dynamic transmission error of gear pair. This work presents an efficient computational procedure for centrifugal stiffening analysis of gear pair. As inertia centrifugal force is distributively applied on gear body, the finite element grid model of gear pair is firstly established. Then, in order to reduce large number of degrees of freedom in finite element grid, the generalized component mode synthesis is employed to establish dynamic equation of flexible gear with constant mass matrix and co-rotated constant stiffness matrix. In order to avoid highly refined grid near contact zone, the semi-analytic contact technique which incorporates advantage of penalty method and Lagrange multiplier method is employed to calculate contact force between non-matching sparse tooth face meshes. After transforming system dynamic differential algebraic equation into ordinary differential equation, dynamic contact analysis of a helical gear pair is carried out, and dynamic transmission error at different rotational speeds are compared. Inflation deformation of gear under inertia centrifugal force is shown as explanation for influence of centrifugal stiffening on dynamic transmission error. [ABSTRACT FROM AUTHOR]

Rotating beams are usually encountered in various dynamic analyses of rotation machineries, most of the current studies are mainly focused on the classical boundary conditions. From the viewpoint of engineering practice, elastic boundary supports will be of more significance. Motivated by such limitation, a Fourier series solution for vibration analyses of a rotating hub-beam with elastic boundary restraints and centrifugal stiffening is established by solving the governing differential equation and general boundary conditions, simultaneously. For elastic restraints at both ends, two types of boundary springs against translation and rotation are introduced to simulate the general boundary supports. Fourier series supplemented by boundary smoothed terms is constructed for the rotating beam displacement expression. Harmonic balance principle is employed to formulate the system characteristic matrix, from which all the modal information of rotating beam can be derived by solving a standard eigen-value problem. Numerical examples are then presented to verify the correctness and effectiveness of the developed model. The influence of boundary restraining stiffness and rotation speed on modal characteristics of rotating beam are investigated and addressed. The analytical solution established in this work can be used as the benchmark for other solution development, and provide an efficient tool for the parametric studies and/or optimization of dynamic characteristics of rotating beam structure, especially with complicated boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted airfoil is derived. Strain-displacement relationships include the effect of warping of the cross-section due to twist-bend coupling effect introduced as a result of varying stagger angle and camber radius of the blade. The equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil by considering it as an anisotropic shell in full curvilinear coordinates subjected to a centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The fourth-order partial differential equation characterizing the flexural motion of the airfoil is transformed into a matrix- eigenvalue form using a Rayleigh-Ritz technique. The blade deformations are represented by a set of “admissible” sinusoidal trial functions, which fully satisfy all the clamped-end constraints as well as the free-edge boundary conditions. The numerical results presented in a non-dimensional parametric form are directly applicable in determining the static and running frequencies of typical composite blades used in the fan module of an aeroengine. [ABSTRACT FROM AUTHOR]

The Adomian modified decomposition method (AMDM) is employed for the free transverse vibration analysis of rotating non-uniform Euler-Bernoulli beams using several boundary conditions, rotation speeds, and beam lengths. The equation of motion includes the axial force due to centrifugal stiffening. The AMDM allows the governing differential equation to become a recursive algebraic equation and after additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. The AMDM technique is systematic, so it is straightforward to modify the boundary conditions for specific cases. Comparisons with previously reported results demonstrated that the results obtained by the proposed method were in close agreement. [ABSTRACT FROM AUTHOR]

This study analyzes the elastic vibration of a simultaneously spinning and precessing cantilevered rotor for its stability margin and whirl frequency. The governing equations suggest that the stability is largely governed by two counteracting effects – the centrifugal stiffening and the precession softening. The concentrated mass and inertia of the disc as well as the distributed mass of the shaft contribute to both of these effects. A finite element formulation shows that along with the standard matrices for conventional rotor dynamic analysis, two completely new ones are obtained to account for the effect of precession. Two- and four-degrees-of-freedom models indicate that the rotor is always stable irrespective of its precession speed. But, interestingly, results from the converged finite element model show that the rotor will be unstable beyond a moderately high value of precession speed. The reason for this can be attributed to the shape of deformation of the rotor during its motion. This shape is only approximate in two- and four-degrees-of-freedom models. The Campbell diagrams computed using the four-degrees-of-freedom model and the finite element model are compared and presented. [ABSTRACT FROM AUTHOR]

The centrifugal effect is a natural phenomenon in rotor dynamics, significantly changing the deformation responses in high-speed conditions. However, this phenomenon is generally unexplored in high-speed gears. This paper developed an analytical mesh stiffness model to bridge the kinematics and the centrifugal fields of spur gears. Starting from the elasticity of fillet-foundation, this model refined and extended the bi-dimensional formulas to consider gears in simultaneous rotating and meshing. The proposed load-sharing model additionally considered the extended tooth contact and centrifugal expansions, non-iteratively determining the complicated tooth engagement regions. Analytical formulas are further implemented with a fast numerical method, which removed restrictions of classical fillet-foundation formulas and covered both thick-walled and thin-walled gears. Quasi-static and dynamic solutions are compared with existing models, FEM, and experiments, showing the acceptable accuracy of this model and demonstrating noticeable centrifugal stiffening in high-speed conditions. To this end, sensitivities of centrifugal stiffening to multi-parameters are further investigated and characterized. [Display omitted] • Extended analytical formulas for the mesh deformation in high-speed conditions. • Incorporated centrifugal expansion and extended-tooth-contact in load-sharing. • Non-iteratively determined complex tooth contact regions in centrifugal field. • An improved mesh stiffness model with high accuracy and wide application range. • Characterized the centrifugal stiffening of spur gears in statics and dynamics. [ABSTRACT FROM AUTHOR]

The centrifugal softening effect is an alleged and elusive reduction of the natural frequen-cies of a rotating system with increasing speed which is sometimes found infinite element rotordynamics. This reduction may, in some instances, be large enough to cause some of the natural frequencies to vanish, leading to a sort of elastic instability. Some doubts can, however, be cast on the phenomenon itself and on the mathematical models causing it to appear. The aim of the present work is to shed some light on centrifugal softening and to discuss the assumptions that are at the basis of three-dimensional FEM modeling in rotordynamics. One and two degrees of freedom models, such as the ones introduced by Rankine and Jeffcott, are first studied and then the classical rotating beam, ring, disk, and membrane are addressed. Some numerical models, built using the FEM, are then solved using both dedicated and general purpose codes. In all cases no strong centrifugal softening is found. [ABSTRACT FROM AUTHOR]

Many elements of machines, like shafts, blades, connecting rods, etc., are often modeled using the so-called beam theory and, when these elements rotate with respect to an inertial reference frame, this rotation can deeply affect their dynamic behavior. The position of the beam with respect to the rotation axis and the possibility that the rotation is more complicated than a simple constant-rate rotation about a fixed axis influence this effect, and different models are usually employed. As a result, different phenomena, like gyroscopic effect, centrifugal stiffening or softening, and instability due to rotation are often mentioned in reference to the different cases. The aim of this article is that of building a much simplified beam model, and to subject it to a compound, nonconstant rate rotation. Since the model can be solved in closed form, at least in several cases, a general discussion of the revant phenomena can be done to shed some light on some aspects, like the instability ranges due to rotation and to the damping of the system. [ABSTRACT FROM PUBLISHER]

This paper presents a dynamic response analysis of the blade of horizontal axis wind turbines using finite element method. The blade is treated as a cantilever and modeled with two-node beam element. The blade element-momentum theory is applied to calculate the aerodynamic loads. Dynamic inflow and dynamic stall are taken into account to reflect the transient aerodynamics. The centrifugal stiffening is introduced to consider the restoring effects of centrifugal force. An aerodynamic damping model is presented for calculating the overall damping ratio instantaneously during time-domain simulation. The structural dynamic equation is solved using Newmark method and the overall dynamic response of the blade is obtained based on the modal superposition principle. Applying the proposed method, the power production load case of a 1.0 MW wind turbine operating in turbulent wind field is simulated. The simulation results indicate that the blades of large-scale horizontal axis wind turbines undergo significant vibration and deflection during operation, and the centrifugal stiffening and aerodynamic damping both considerably affect the structural response of the blade. [ABSTRACT FROM AUTHOR]

Abstract: We present the generalized multiple shooting method (GMSM) to analyze the dynamics of elastic mechanisms. The GMSM solves a boundary value problem by treating it as an initial value problem. Its accuracy depends on the order of space marching schemes rather than size of discretization. Dynamic equations with joint boundary conditions are derived by using Hamilton’s principle to be systematically solved by the GMSM. Comparing with existing solutions and experiments, the GMSM is shown to be efficient yet it captures deflection precisely. We expect it to serve as a good alternative to existing methods. [Copyright &y& Elsevier]

Abstract: This paper proposes an approach to investigate the along-wind forced vibration response of a wind turbine tower and rotating blades assembly subjected to rotationally sampled stationary wind loading. The wind turbine assembly consists of three rotating rotor blades connected to the top of a flexible annular tower, constituting a multi-body dynamic entity. The tower and rotating blades are each modelled as discretized multi-degree-of-freedom (MDOF) entities, allowing the free vibration characteristics of each to be obtained using a discrete parameter approach. The free vibration properties of the tower include the effect of a rigid mass at the top, representing the nacelle, and those of the blade include the effects of centrifugal stiffening due to rotation and blade gravity loadings. The blades are excited by drag force time-histories derived from discrete Fourier transform (DFT) representations of rotationally sampled wind turbulence spectra. Blade response time-histories are obtained using the mode acceleration method, which allows for the quantification of base shear forces due to flapping for the three blades to be obtained. This resultant base shear is imparted into the top of the tower. Wind drag loading on the tower is also considered, with a series of spatially correlated nodal force time-histories being derived using DFTs of wind force spectra. The tower/nacelle is then coupled with the rotating blades by combining their equations of motion and solving for the displacement at the top of the tower under compatibility conditions in the frequency domain. An inverse Fourier transform of the frequency domain response yields the response time-history of the coupled system. The response of an equivalent system that does not consider the blade/tower interaction is also investigated, and the results are compared. [Copyright &y& Elsevier]

Systems of rigid and flexible bodies undergoing large rigid body motions but small elastic deformations are investigated. In order to get the correct equations of motion one has to consider geometric nonlinearities even in the elastic coordinates. Different possibilities of independently choosing these coordinates are presented. The flexible bodies are discretized using a Ritz-Galerkin approximation. This discretization leads to ordinary differential equations for the description of the clastic vibrations of the flexible bodies as well as for the description of the rigid body motions. The modelling of deformable bodies is one aspect of our investigations. Another aim of our research is the consideration of clastic bodies in existing multibody programs, in particular in the program AUTOLEV. This is a symbol manipulating program for the formulation of the equations of motion running on PCs. Two examples are considered showing that it is possible to treat clastic bodies in this and in similar programs with the concept of modelling presented above. [ABSTRACT FROM AUTHOR]

• A unified analytical model for centrifugal softening and stiffening harvesters is proposed. • The influence mechanism of centrifugal effects is analytically analyzed. • The analytical model is experimentally validated. • Case studies are conducted to comprehensively compare two types of harvesters. Depending on its mounting orientation, the rotational piezoelectric harvester can be subjected to the centrifugal softening or stiffening effects, which may pose a positive or negative influence on the energy harvesting performance. This paper firstly presents the comprehensive comparison of the centrifugal softening and stiffening harvesters with a unified dimensionless analytical model. The analytical expressions of the optimum short-circuit and open-circuit rotational frequencies, the optimum resistances and the optimum centrifugal coefficients are derived to reveal the influence mechanism of the centrifugal effects. The proposed model is validated experimentally and used for case studies to compare the centrifugal softening and stiffening harvesters. Results illustrate that the centrifugal softening harvester has a superior energy harvesting performance at low rotational frequencies while the centrifugal stiffening harvester is better at high rotational frequencies. Furthermore, the peak power of the centrifugal softening harvester is higher while the frequency bandwidth of the centrifugal stiffening harvester is wider. In addition, it is found that the centrifugal stiffening harvester is more sensitive to the centrifugal coefficient. Based on the centrifugal stiffening effect, the self-tuning harvester can be achieved with a wide frequency bandwidth. Meanwhile, this requires that the centrifugal coefficient is precisely determined, and the fundamental natural frequency of the harvester without the centrifugal force should be low enough when operating at low rotational frequencies. Compared with the centrifugal stiffening effect, the centrifugal softening effect may be more suitable to be used in MEMS/NEMS applications since it can reduce the fundamental resonant frequency of the harvester during the rotation. [ABSTRACT FROM AUTHOR]