• Energy Research

Abstract A fundamental issue in turbomachinery design is the dynamical stress assessment of turbine blades. In order to reduce stress peaks in the turbine blades at engine orders corresponding to blade natural frequencies, friction dampers are employed. Blade response calculation requires the solution of a set of non-linear equations originated by the introduction of friction damping. Such a set of non-linear equations is solved using the iterative numerical Newton–Raphson method. However, calculation of the Jacobian matrix of the system using classical numerical finite difference schemes makes frequency domain solver prohibitively expensive for structures with many contact points. Large computation time results from the evaluation of partial derivatives of the non-linear equations with respect to the displacements. In this work a methodology to compute efficiently the Jacobian matrix of a dynamic system having wedge dampers is presented. It is exact and completely analytical. The proposed methods have been successfully applied to a real intermediate pressure turbine (IPT) blade under cyclic symmetry boundary conditions with underplatform wedge dampers. Its implementation showed to be very effective, and allowed to achieve relevant time savings without loss of precision.

In turbomachinery, the perfect detuning of turbine blades in order to avoid high cycle fatigue damage due to resonant vibration is often unfeasible due to the high modal density of bladed disks. To obtain reliable predictions of resonant stress levels of turbine blades, accurate modeling of friction damping is mandatory. Blade root is one of the most common sources of friction damping in turbine blades; energy is dissipated by friction due to microslip between the blade and the disk contact surfaces held in contact by the centrifugal force acting on the blade. In this paper, a method is presented to compute the friction forces occurring at blade root joints and to evaluate their effect on the blade dynamics. The method is based on a refined version of the state-of-the-art contact model, currently used for the nonlinear dynamic analysis of turbine blades. The refined contact model is implemented in a numerical solver based on the harmonic balance method able to compute the steady-state dynamic response of turbine blades The proposed method allows solving the static and the dynamic balance equations of the blade and of the disk, without any preliminary static analysis to compute the static loads acting at the contact interfaces.

The prediction of the aeromechanical behavior of low-pressure blades represents one of the main challenges in the Steam Turbine Industry. The evaluation of forced response and damping is critical for the reliability of new designs and usually requires expensive validation campaigns such as Wheel Box Tests (WBT). A WBT consists of one or more blade rows assembled on a rotor and spun at the desired rotating speed in a vacuum cell, with synchronous excitation provided by various sources. The WBT provides accurate information about the blade modes frequency, the alternating response level, and allows the evaluation of the mechanical damping. Given the large effort in terms of costs and time associated to the experimental activity, the possibility to rely on the output of a numerical code either during the first steps of a new design or to investigate the effect of minor changes to a current design would be extremely beneficial to the development of future products. In order to compute the non-linear forced response of shrouded blades of steam turbines, custom numerical solvers must be developed, since commercial finite element (FE) solvers do not perform this kind of analysis in the frequency domain. In this paper, the forced response of a blade with shrouds of a low pressure steam turbine is computed and numerical results are compared with the experimental Wheel Box Tests performed at GE Oil & Gas. The calculations require a three-step procedure: in the first step, a non-linear static analysis is performed in ANSYS® in order to compute the actual contact area on the shroud surface and the distribution of static normal loads, then a reduced order model of the blade is generated in ANSYS® taking into account the stiffening effect on the blade of the pre-stress due to the centrifugal force, finally the reduced model is imported in a numerical code and the non-linear forced response of the blade is computed. The numerical code solves the balance equations of the system in the frequency domain, by means of the Harmonic Balance Method, imposing cyclic symmetry boundary conditions of the system. An interpolation procedure is implemented in order to manage the non-perfectly matching meshes of the shroud contact surfaces, while the tangential and normal contact stiffness is computed with a numerical model based on the contact mechanics principles. The numerical and the experimental results around some of the critical resonances of the system are compared in order to assess the reliability and accuracy of the numerical tool for its future implementation in the mechanical design practice of the blades.

In this paper a methodology for forced response calculation of bladed disks with underplatform dampers is described. The FE disk model, supposed to be cyclically symmetric, is reduced by means of Component Mode Synthesis and then DOFs lying at interfaces are further reduced by means of interface modes. Underplatform dampers are modeled as rigid bodies translating both in the radial and in the tangential direction of the engine. Contacts between blade platforms and damper are simulated by means of contact elements characterized by both tangential and normal contact stiffness, allowing partial separation of contact surfaces. Differential equilibrium equations are turned in non-linear algebraic equations by means of the Harmonic Balance Method (HBM). The methodology is implemented in a numerical code for forced response calculation of frictionally damped bladed disks. Numerical calculations are performed to evaluate the effectiveness of both the reduced order model and the underplatform model in simulating the dynamic behavior of bladed disks in presence of underplatform dampers.

In this paper, a pre-existing reduction technique suitable for the analysis of mistuned bladed disk dynamics, the Component Mode Mistuning technique (CMM), originally developed exclusively for the use of blade frequency mistuning pattern, is extended in order to allow for the introduction of a sector frequency mistuning pattern. If either mistuning is not confined to the blades (i.e. blades-to-disk interface mistuning), or the blades can not be removed from the bladed disk (i.e. integral bladed disks), sector mistuning rather than blade mistuning is a more suitable choice to perturb the tuned system. As a consequence, the extension of the original technique is referred as Integral Mode Mistuning (IMM). After a theory review of the original technique, the modifications leading to the IMM are described. Finally, the proposed IMM technique is validated in terms of both modal parameters estimation and forced response calculation, by means of a dummy bladed disk developed at Politecnico di Torino.

Assembled structures are characterized by contact interfaces that introduce a local non-linearity and affect the dynamics of the assembly in terms of resonance frequencies and vibration levels. To assess the forced response levels of the assemblies during the design, nonlinear dynamic analyses are performed and, in order to reduce the computation time, spatial and temporal reductions of the governing equations must be used. A classical way to achieve temporal reduction is to implement the harmonic balance method to turn the time-domain differential governing equations into frequency-domain algebraic equations. Due to the local nature of contact interfaces, which usually involve a subset of degrees of freedom (dofs) of the structure, a common strategy to achieve spatial reduction is to use component mode synthesis (CMS), by retaining the contact dofs as master dofs. In this paper, a recent CMS approach, named dual Craig–Bampton method (Rixen in J Comput Appl Math, 2004. doi: 10.1016/j.cam.2003.12.014 ), is applied to the nonlinear forced response of structures with contact interfaces. The spectral orthogonality of the two subsets of mode shapes used as a projection basis is exploited to write a set of algebraic equations of the contact dofs in the frequency domain, with no need to compute the reduced matrices of the system. Different formulations of the governing equations are proposed for different configurations (i.e., outer contacts, inner contacts and structures with floating components), and two academic numerical test cases are used to demonstrate the method.

Highly stressed forced vibrations experienced by Low Pressure Turbine (LPT) blades during their operation can result in high cycle fatigue (HCF) which can ultimately lead to failure. To avoid this occurrence, the vibration amplitudes must be anticipated and reduced. This is generally done by employing friction damping devices like under-platform dampers, shrouds and snubbers. In case of blades with shrouds at their tips, the adjacent shrouded blades are coupled to each other and three-dimensional periodic shroud contact forces acting on shrouds significantly influence the vibration amplitudes as energy is dissipated due to friction. Hence, for the experimental validation of the numerical contact models that are used for the prediction of nonlinear forced response of shrouded blades, it is equally necessary to measure the contact forces acting at the shrouds. Moreover, present-day requirement of more accurate and detailed models requires thorough and comprehensive experimental validation calling for the measurement of three-dimensional shroud contact forces from purposely designed test rigs. This study presents the design, development and operation of an experimental test rig that allows full characterization of the dynamics of shrouded turbine blade, i.e., simultaneous measurement of three-dimensional shroud contact forces and shrouded blade forced response. The starting point was the design of the test rig components based on the design requirements. This is followed by the description of the major components of the proposed test rig i.e., the three-dimensional contact force measurement system and the torque screw mechanism. In the subsequent section, the details of the experimental setup to measure the forced response and three-dimensional shroud contact forces simultaneously are highlighted as the test campaign is performed for different normal preloads and excitation force levels. Consequently, the effects of the variation in the normal preload and excitation force on the measured response and shroud contact forces are discussed. Finally, it is shown how the results establish the efficacy of the proposed test rig in providing a comprehensive depiction of the dynamic response of the shrouded blade and 3D shroud contact forces that will result in more accurate and reliable experimental validation of numerical tools.

Green’s function technique (GFT) is largely used for on-line calculation of thermal stresses in machines and plants; it allows directly turning parameters such as fluid temperatures, pressures and flow rates in thermal stresses. Recently the use of the GFT is extended by the authors to cases having variable convective coefficients. The novel methodology is made of two steps: first of all boundary temperatures are evaluated by time integration of a reduced thermal model and then thermal stresses are calculated by means of the GFT using as inputs the boundary temperatures previously evaluated. The new approach implies a large number of convolution integrals for thermal stress calculation. In order to reduce computation time it is proposed to convert the convolution integrals which characterize the GFT into time integration of an equivalent system of uncoupled first order differential equations, whose coefficients are estimated fitting Green’s functions with a sum of exponential terms.

Abstract This paper addresses two main subjects. First, a novel test setup is described to experimentally study the nonlinear dynamic behavior of a turbine blade coupled with two midspan dampers (MSDs). To this end, a representative turbine blade and midspan friction dampers are originally designed, and they are assembled to a special test rig which has been previously developed at Politecnico di Torino. Second, the variability of the dynamic response is intensively investigated with a purposely defined loading/unloading strategy. To better understand the inherent kinematics of the blade–damper interaction, contact forces are measured through the novel design of the experimental campaign. It is shown that multiple responses, which are obtained in different tests while keeping all user-controlled inputs nominally same, are due to nonunique contact forces that provide different static force equilibria on the damper. This outcome is further supported by the qualitative illustration of hysteresis cycles. This study contributes to the understanding of the response repeatability linked to the nonuniqueness of friction forces.