Linear Mode Shapes Research Articles

Static analysis of wind blades based on weight loads to the gravitation and the centrifugal forces Using Finite Element

Static analysis of wind turbine blades that convert kinetic energy using wind to electrical power has been analyzed using Ansys model. The geometry was imported in ANSYS workbench 2020R1, where the material was changed to Epoxy E-Glass Wet. The mesh independence that occurred at 800 and 600 element sizes and an element size of 800 was chosen for subsequent static analysis. The static analysis shows that the total deformation and von Mises stress of the wind turbine blade was improved at both the vertical position and the horizontal position by 40.0% and 38.61%, respectively. Besides, the static analysis showed an almost linear mode shape increased from mode lines. Nevertheless, Campbell's diagram showed that resonance vibration occurred at an amplitude of 5.5x106Pa with a corresponding frequency of 5Hz, which gave a rotational speed of 262.4x106 rpm. This shows that the impulse force is better applied in wind turbine blade transient analysis at a lower von Mises stress to attain a quicker damping response steady-state, which favors wind turbine blade analysis. Overall, the wind turbine blade is better designed at a vertical position to allow better deformation stress at a static position. Conversely, the von Mises stress showed 44.64% and 38.61% at horizontal and vertical positions. This also agrees with static total deformation, since stress is better managed at a lower percentage range.

Some Unfamiliar Structural Stability Aspects of Unsymmetric Laminated Composite Plates.

It is widely recognized that certain structures, when subjected to static compression, may exhibit a bifurcation point, leading to the potential occurrence of a secondary equilibrium path. Also, there is a tendency of deflection increment without a bifurcation point to occur for imperfect structures. In this paper, some relatively unknown phenomena are investigated. First, it is demonstrated that in some conditions, the linear buckling mode shape may differ from the result of geometrically nonlinear analysis. Second, a mode jumping phenomenon is described as a transition from a secondary equilibrium path to an obscure one as a tertiary equilibrium path or a second bifurcation point. In this regard, some non-square plates with unsymmetric layer arrangements (in the presence of extension-bending coupling) are subjected to a uniaxial in-plane compression. By considering the geometrically linear and nonlinear problems, the bucking modes and post-buckling behaviors, e.g., the out-of-plane displacement of the plate versus the load, are obtained by ANSYS 2023 R1 software. Through a parametric analysis, the possibility of these phenomena is investigated in detail.

Linear and nonlinear vibrations of nonlinearly elastically constrained functionally graded porous microbeams with rough surface

Linear and nonlinear free vibrations of nonlinearly elastically constrained functionally graded porous (FGP) microbeams with rough surface are analytically studied in this paper via the modified couple stress theory. A surface roughness model of the FGP microbeams is developed. The material properties including porous property are supposed to continuously vary along the thickness direction. The higher-order nonlinear governing equation and elastic boundary conditions are derived through the Hamilton principle. A closed-form solution for linear frequency and mode shape is presented. Subsequently, an analytical solution for nonlinear frequency is achieved by employing the Galerkin procedure. Finally, the effects of surface roughness, size parameter, power-law index, porosity volume fraction, and nonlinearly elastic constraints on the linear and nonlinear vibrations of functionally graded porous microbeams are examined.

Formulation of justifiable wind loading distributions up tall buildings derived from HFFB measurements

For the general case of non-linear and combined sway and twist mode shapes, analysis of HFFB measurements gives rise to serious challenges and various techniques have been developed using significant assumptions. This is because it is not possible to directly obtain the distribution of varying fluctuating wind loads throughout the height of tall buildings. This paper proposes a straightforward HFFB-based wind load/twist distribution approach in the time domain to predict the wind-induced dynamic response of tall buildings. Using this framework, it is possible to overcome the deficiencies of mode shape correction factor approach, and the complexity and impracticality of alternative approaches for tall buildings. Two 1:300 scale rigid models of benchmark tall buildings, IAWE Buildings A and B, have been selected to perform a detailed wind tunnel investigation as subjects to evaluate this proposed HFFB approach study. Building A has complex 3D mode shapes combining sway and twist, whereas the simpler Building B has uncoupled linear mode shapes in sway and twist. The high frequency pressure integration (HFPI) wind tunnel technique was utilised to measure surface pressure information in separate tests to obtain the reference wind loading for validation of the HFFB predictions. Furthermore, a commonly used HFFB mode shape correction factor approach was used to estimate the dynamic wind responses which were compared to those from the proposed approach. From the comparison between the aerodynamic and dynamic wind results from proposed HFFB approach and reference HFPI results, it is found that the proposed HFFB approach can predict the vertical distributions of the wind forces and torque (wind loads) with acceptable accuracy for carrying out the conceptual stage of tall building design.

Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.

A novel fabrication method of the three-dimensional forced vibration system for high-rise buildings

The forced vibration wind tunnel test is widely used to estimate the aeroelastic effect of high-rise buildings due to its ease of fabrication and reliable identification of the aeroelastic parameters. However, most traditional forced vibration systems (FVSs) can only realize one-dimensional (1D) or two-dimensional (2D) vibrations at most, and the structural model can only simulate linear or constant mode shapes. The present study proposes a novel three-dimensional (3D i.e., two translational and one torsional vibrations) FVS that can drive the model to vibrate in 3D while maintaining more realistic mode shapes. Employing the proposed 3D FVS, the amplitude, frequency, and phase angle of the model's 3D vibrations can be adjusted independently. Then, the resultant motions of the model are measured by three laser sensors to examine the performance of the system. Results show that the proposed system performs well, providing stable vibration amplitude, frequency, and phase angle in all the three directions. Finally, making use of the 3D FVS, the significant differences between the results of 3D forced vibration and 1D forced vibration are uncovered, in terms of power spectrum density (PSD) of wind forces on the model and the associated aerodynamic stiffness and damping ratios. The proposed system, therefore, provides an alternative of wind tunnel test methods for determining the aeroelastic properties of high-rise buildings.

Thermal Postbuckling Characteristics Of Angle-ply Laminated Truncated Circular Conical Shells

Here, the nonlinear thermoelastic pre- and post-buckling characteristics of angle-ply laminated composite conical shells subjected to uniform temperature rise are studied using semi-analytical finite element approach. The finite element formulation is based on first-order shear deformation theory and field consistency principle. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton-Raphson iteration procedure coupled with displace- ment control method to trace the prebuckling followed by postbuckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, ply-angles, number of layers and number of circumferential waves on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the deformation shape analysis.

A novel dynamic model on nonlinear vibrations of functionally graded graphene platelet reinforced rotating pretwisted composite blade considering subsonic airflow excitation and blade-casing rubbing

A novel dynamic model of the functionally graded graphene platelet (FGGP) reinforced rotating pretwisted composite blade under the aerodynamic force and blade-casing rubbing is established for the first time. The blade is simplified to a FGGP reinforced rotating pretwisted composite cantilever plate. The dynamic model of the axial excitation produced by considering the aerodynamic force in the tip clearance includes two trigonometric functions with different frequencies. The subsonic airflow is considered as a transverse excitation which is derived by utilizing the vortex lattice method (VLM). The dynamic model expresses the changes of the non-uniform axial and contact forces through the tip clearance and rotating pretwisted plate-casing rubbing when the shaft is eccentric. Based on Rayleigh-Ritz method, the linear frequencies and mode shapes are obtained for the FGGP reinforced rotating pretwisted plate. The influences of the graphene distribution pattern, rotating pretwisted plate-casing rubbing, axial excitation in the tip clearance and geometric parameter on the frequencies are investigated. Using the obtained mode shapes, von-Karman type nonlinear geometric assumptions and Lagrange equation, the differential governing equations of motion are derived for the FGGP reinforced rotating pretwisted plate. The nonlinear vibrations under the 1:1 internal resonance at two critical rotating speeds is studied by using Runge-Kutta method. The amplitude-frequency and force-frequency response curves under the low and high critical rotating speeds are investigated by using numerical calculations. The obtained results demonstrate the influence of the rotating speed, frequency ratios and incoming flow speeds on the nonlinear vibrations of the FGGP reinforced rotating pretwisted plate. The dynamic model of the rubbing-impact for the rotating blade provides a theoretical basis for the blade-casing rubbing analysis. At the same time, this study is also used as a theoretical guidance to reduce the damage of the blade-casing rubbing and blade design.

Modified couple stress flexural–flexural quasi-static pull-in analysis of large deformable cantilever-based micro-gyroscopes

This paper aims to develop a size-dependent large deformable model to study the flexural–flexural quasi-static motion of cantilever-based micro-gyroscopes based on the modified couple stress theory. The micro-cantilever is assumed to undergo non-planar bending–bending–twisting motion. Based on this assumption, the displacement field associated with a point placed on the cross-section of the micro-beam is introduced. Employing the introduced displacement field, the components of the Green–Lagrange strain and curvature tensors are obtained. Neglecting the terms with the time derivatives, the nonlinear, coupled and higher-order equations governing on three-dimensional quasi-static motion of in-extensional cantilever-based micro-gyroscopes and their corresponding boundary conditions are then obtained. Normalizing the strain and kinetic energies as well as the works done by the external forces, the reduced equations governing the quasi-static motion of micro-gyroscopes are obtained through employing the Ritz method with linear mode-shapes of clamped-free beams as the approximating functions. Vanishing the Jacobian of the reduced equations, the governing stability equations of the system are then obtained. Afterward, employing the achieved reduced equations, both the stable and unstable branches of the micro-gyroscope equilibrium paths are extracted. To validate the accuracy of the present model, several verifications have been conducted. The performed verifications demonstrate that the current model can be considered as a promising tool in capturing the influence of small scales on micro-gyroscopes undergoing extremely large deformations.

Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models

Strain-driven (ɛD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for ɛD- and σD-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz–Galerkin methods and general differential quadrature method (GDQM) based Newton–Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz–Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz–Galerkin and GDQM based Newton–Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz–Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.

Stress recovery algorithm for reduced order models of mechanical systems in nonlinear dynamic operative conditions

Nonlinear forced response analyses of mechanical systems in the presence of contact interfaces are usually performed in built-in numerical codes on reduced order models (ROM). Most of the cases these derive from complex finite element (FE) models, resulting from the high accuracy the designers require in modeling and meshing the components in commercial FE software. In the technical literature several numerical methods are proposed for the identification of the nonlinear forced response in terms of a kinematic quantity (i.e. displacement, velocity and acceleration) associated either to the master degrees-of-freedom retained in the ROM, or to the slave ones after having expanded the reduced response through the reduction matrix. In fact, the displacement is the quantity usually adopted to monitor the nonlinear response, and to evaluate the effectiveness of a partially loose friction interface in damping vibrations, with respect to a linear case where no friction interfaces exist and no energy dissipation can take place. However, when a ROM is used the engineering quantities directly involved in the mechanical design, i.e. the strains and stresses, cannot be retrieved without a further data processing. Moreover, in the case of a strong nonlinear behavior of the mechanical joints, the distributions of the nonlinear strains and stresses over the structure is likely different than the one obtained as a superposition of linear mode shapes whose definition require a-priori assumptions on the boundary conditions at the contact interface. This means that the mentioned approximation cannot be used to predict the safety margins of a structure working in real (nonlinear) operative conditions. This paper addresses this topic and presents a novel stress recovery algorithm for the identification of the strains and stresses resulting from a nonlinear forced response analysis on a ROM. The algorithm is applied to a bladed disk with friction contacts at the shroud joint, which make the behavior of the blades nonlinear and non-predictable by means of standard linear analyses in commercial FE software.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Nonlinear dynamic analysis of nonlocal composite laminated toroidal shell segments subjected to mechanical shock

The present study deals with the nonlinear forced vibration of composite shallow nano toroidal shell segments. The improved Donnell shell theory is used with the nonlinear geometric strains, i.e., von Karman, Stein, and McElman’s assumptions. A coupled nonlinear partial differential equations using nonlocal theory, including the influence of nano-scales, is derived. The boundary conditions are simply supported and clamped–clamped, and the nano toroid shell is subjected to impulse radial dynamic excitation. The linear solution is verified by comparing natural frequencies with the available results using Galerkin’s method. The approximated linear mode shapes of the free vibration, which satisfy the boundary conditions, are used as the assumed multi-mode displacement functions required to solve the nonlinear vibration. Using Hamilton’s principle, the nonlinear equations of motion in generalized coordinates are solved using the displacement field by applying Galerkin’s and Runge–Kutta’s methods. Using the presented model, the effects of layer stacking, toroidal shell segments curvature (convex and concave toroidal shells), length scale parameter, and material properties are studied in detail, and some conclusions are surveyed. It is observed that the nonlocal parameter demonstrates a considerable effect on the dynamics behavior characteristics response of the toroidal shell segments. Furthermore, different fiber orientation effects, displacement fields, time response, and geometry parameters, i.e., thickness, and length are studied. Results indicate that natural frequencies significantly depend on the toroid dimensions, boundary conditions, layer stacking, curvature change, and considered wave numbers.

Analysis of Axisymmetric Vibration of Functionally-Graded Circular Nano-Plate Based on the Integral Form of the Strain Gradient Model

In this paper, it is aimed to analyze the linear vibrational behavior of functionally-graded (FG) size-dependent circular nano-plates using the integral form of the non-local strain gradient (NSG) model. The linear axisymmetric vibration of the circular FG nano-plates based on the non-local strain gradient (NSG) model is the focal point of this study. In this regard, the non-local elasticity theory (NET) and strain gradient (SG) models are used in conjunction with Hamilton's principle to obtain the governing equations. Discretization of the obtained governing equations is performed with the help of generalized differential quadrature rule (GDQR) and Galerkin weighted residual method (GWRM). The analysis is focused on the effect of non-local and material parameters, as well as the aspect ratio, heterogeneity index of FG material, different boundary conditions, and frequency number on the overall behavior of nano-plate. On using the Galerkin method, a system of linear differential equations is obtained and solved to determine the natural linear frequencies and mode shapes. The obtained results are then compared with the existing results in the literature. On using the proposed procedure in this paper, the dynamic behavior of nano-plate under different boundary conditions can be well described. In addition, the existing deficiencies in other non-local theories can be eliminated. The results of this investigation can be considered as a turning point in the improvement of theoretical results for achieving a better prediction of vibrational behavior in nanostructures.

Wind-resistant structural optimization of supertall buildings based on high-frequency force balance wind tunnel experiment

The high-frequency force balance (HFFB) wind tunnel test is one of the most widely used techniques to estimate wind effects on supertall buildings. This paper introduces a wind-resistant optimal design method of supertall buildings based on HFFB wind tunnel experiments. Based on the HFFB wind tunnel test and wind-induced vibration analysis, it provides a reliable and efficient method for structural optimization of supertall buildings with different types of structural members and multiple constraints. Taking advantage of the classic linear mode shape assumption of the HFFB technique, this paper also presents an alternative routine that transforms all different types of constraints into a uniform frequency constraint, which significantly reduces the computational intensity while still offers acceptable accuracy of optimization results. Moreover, the sensitivity of the optimization results to the adopted basic wind pressure value is discussed. The total material cost of the optimized structure increases exponentially with the increase of the basic wind pressure, indicating that the structural costs of supertall buildings are quite sensitive to the adopted basic wind pressure for structural design and its value should be precisely determined by the wind climate analysis of the construction site.

Nonlinear geometry effects investigation on free vibrations of beams using shear deformation theory

This article presents an analytical procedure to predict the beams nonlinear natural frequencies using the first-order shear deformation theory. The nonlinear kinematics assumptions include the moderately large deformation of the mid-plane stretching and transverse deflection that are defined by using the von Kármán relations. The strains are small and Hooke’s law is used as the constitutive equation. A coupled nonlinear longitudinal-transverse set of motion equations are derived utilizing Hamilton’s principle. The equations are solved analytically using the powerful multiple scales method of the perturbation technique. At first, the linear natural frequencies and mode shapes are determined. Then the nonlinear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the nonlinear frequencies are functions of the axial and transverse amplitudes of vibrations. The influences of the axial load and aspect ratio on the linear and nonlinear frequencies are studied too. The results of the special cases are compared with the available results in the literature and the finite element analysis. The results show the noticeable effect of the axial amplitude as well as the transverse amplitude of vibrations on the nonlinear frequencies.

Full-field strain estimation of subsystems within time-varying and nonlinear systems using modal expansion

Virtual sensing allows for the estimation of stress and/or strain response in unmeasured locations of a system. Often, these virtual sensing techniques assume a linear and time-invariant system with proportional damping. In this article, one virtual sensing technique - the modal expansion - is proven applicable for stress/strain estimation of subsystems within time-varying and nonlinear systems with general viscous damping where the time-varying and nonlinear effects act externally on the subsystems. This technique uses the mode shapes of the subsystem to expand the response by a subspace projection. It is proven that the mode shapes of the underlying undamped and linear system form a basis for the response of the time-varying and nonlinear systems with general viscous damping. Therefore, a truncation of the mode shapes results in modal truncation errors that depend on the span of the applied mode shapes. Thus using an appropriate set of undamped and linear mode shapes of the subsystem, the modal expansion allows for estimation of the stress/strain response for “linear” subsystems within time-varying and nonlinear systems with general viscous damping. This concept is proven both numerically and experimentally.

Vibration of the Biomass Boiler Tube Excited with Impact of the Cleaning Device

In boilers with biomass fuel, a significant problem is caused due to the slag layer formed from the unburned particles during combustion. In the paper, a tube cleaning method from slag is proposed. The method is based on the impact effect of the end of the tube with the aim to produce vibration for slag elimination. The tube is modeled as a clamped-free nonlinear oscillatory system. The initial impact of the tube causes vibrations. The mathematical model of the system is a nonlinear partial differential equation with zero initial deflection. To obtain the ordinary differential equations, the Galerkin method is applied. By discretizing the equation into a finite degree of freedom system, using the undamped linear mode shapes of the straight beam as basic functions, the reduced order model, consisting of ordinary differential equations in time, is obtained. The ordinary time equations are analytically solved by adopting the Krylov–Bogoliubov procedure. Special cases of nonlinear differential equations are investigated. In the paper, the influence of the nonlinear parameters and initial conditions on the vibration properties of the tube is obtained. We use the procedure developed in the paper and the analytical results for computation of the impact parameters of the cleaning device.

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Representation of wind turbine blade responses in power production load cases by linear mode shapes

AbstractThe wind turbine industry is designing large MW size turbines with very long blades, which exhibit large deflections during their operational life. These large deflections decrease the accuracy of linear models such as linear finite element and modal‐based models, in which the structure is represented by linear mode shapes. The aim of this study is to investigate the competence of the mode shapes to represent the large blade responses in normal operation load cases. For this purpose, blade deflections are projected onto the linear modal space, swept by mode shape vectors. The projection shows the contribution of each mode and the projection error. The blade deflections are calculated by a nonlinear aero‐servo‐elastic solver for power production fatigue load cases with normal turbulence. The mode shapes are calculated at the steady‐state deflected blade position computed at different wind speeds. Three reference turbine blades are used in the study to evaluate the effects of various blade design parameters such as length, stiffness, mass, and prebend. The results show that although the linear mode shapes can represent the flapwise and edgewise deflections accurately, axial and torsional deflections cannot be captured with good accuracy. The geometric nonlinear effects are more apparent in the latter directions. The results indicate that the blade deflections occur beyond the linear assumptions.