Optimal Tuning of Centrifugal Pendulum Vibration Absorbers

2013 ◽  
Author(s):  
Chengzhi Shi ◽  
Robert G. Parker ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Turbine Blades ◽  
Vibration Absorbers ◽  
Wind Turbine Blades ◽  
Optimal Tuning ◽  
Translational Vibration ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Rotational Vibration ◽  
Reaction Forces ◽  
Analytical Proof

This note provides an analytical proof of the optimal tuning of centrifugal pendulum vibration absorbers (CPVAs) to reduce in-plane translational and rotational vibration for a rotor with N cyclically symmetric substructures attached to it. The reaction forces that the substructures (helicopter or wind turbine blades, for example) exert on the rotor are first analyzed. The linearized equations of motion for the vibration are then solved by a gyroscopic system modal analysis procedure. The solutions show that the rotor translational vibration at order j is reduced when one group of CPVAs is tuned to order jN − 1 and the other is tuned to order jN + 1. Derivation of this result is not available in the literature. The current derivation also yields the better known result that tuning CPVAs to order jN reduces rotational rotor vibration at order j.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part I: Formulation and Verification of Single-Load-Path Configurations

2013 ◽  
Vol 58 (3) ◽  
pp. 1-9 ◽  
Author(s):  
Zahra Sotoudeh ◽  
Dewey H. Hodges
Keyword(s):  
Structural Dynamics ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Rotor Blades ◽  
Load Path ◽  
Dynamics Modeling ◽  
Governing Equations ◽  
Wind Turbine Blades ◽  
First Order ◽  
Infinite Degree

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. In Part I, the theory and the formulation are presented, along with verification of single-load-path configurations. Part II is devoted to the verification of dual-load-path configurations.

The Effects of Gravity on the Response of Centrifugal Pendulum Vibration Absorbers

2021 ◽  
pp. 1-61
Author(s):  
Darryl Tchokogoue ◽  
Ming Mu ◽  
Brian F. Feeny ◽  
Bruce K. Geist ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Operating Conditions ◽  
Vibration Absorbers ◽  
Cyclic Symmetry ◽  
Passive Devices ◽  
Engine Order ◽  
Gravity Effects ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Automotive Powertrain ◽  
Linearized Model

Abstract This paper describes the effects of gravity on the response of systems of identical, cyclically arranged, centrifugal pendulum vibration absorbers (CPVAs). CPVAs are passive devices composed of movable masses suspended on a rotor, suspended such that they reduce torsional vibrations at a given engine order. These absorbers are becoming prevalent in automotive powertrain components in order to expand fuel-efficient engine operating conditions. Gravitational effects acting on the absorbers can be important for a horizontal rotor/CPVA system spinning at relatively low rotation speeds, for example, during engine idle conditions. The main goal of this investigation is to predict the response of a CPVA/rotor system in the presence of gravity. A linearized model which includes the effects of gravity and an order n torque acting on the rotor is analyzed by exploiting the cyclic symmetry of the system. The results show that the N absorbers respond in one or more groups, where the absorbers in each group respond with identical waveforms but shifted phases. The number of groups depends on the engine order n and the ratio Nn. It is shown that there are special resonant effects if the engine order is n = 1 or n = 2, the latter of which is particularly important in applications. In addition, it is shown that for N > 1 the rotor response is not affected by gravity, due to the symmetry of the gravity effects. The analytical predictions are verified by direct simulations of the equations of motion.

Vibration Analysis of Vertical-Axis Wind-Turbine Blades

2016 ◽  
Author(s):  
Fatemeh Afzali ◽  
Onur Kapucu ◽  
Brian F. Feeny
Keyword(s):  
Equations Of Motion ◽  
Turbine Blades ◽  
Vertical Axis ◽  
Vertical Axis Wind Turbine ◽  
Wind Turbine Blades ◽  
Aerodynamic Model ◽  
Vibration Model ◽  
Nonlinear Equations Of Motion ◽  
Energy Equations ◽  
Twisting Deformation

In this work the derivation of a vibration model for an H-rotor/Giromill blade is investigated. The blade is treated as a uniform straight elastic Euler-Bernoulli beam under transverse bending and twisting deformation. The derivation of the energy equations for the bending and twisting blade and a simplified aerodynamic model is issued. Lagrange’s equations are applied to assumed modal coordinates to obtain nonlinear equations of motion for bend and twist. A single quasi-steady airfoil theory is applied to obtain the aeroelastic loads. The behavior of the linearized equation for bend only is examined.

Accounting for Roller Dynamics in the Design of Bifilar Torsional Vibration Absorbers

2009 ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw ◽  
Alan H. Haddow ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
Tuning Parameter ◽  
System Response ◽  
Vibration Absorbers ◽  
State Response ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
The Common ◽  
Roller System ◽  
First Time ◽  
Selection Of

Centrifugal pendulum vibration absorbers are used for reducing torsional vibrations in rotating machines. The most common configuration of these devices utilizes a bifilar suspension in which the absorber mass rides on a pair of rollers, whose mass is small compared to that of the absorber. These rollers are typically solid steel cylinders that allow the CPVAs to move along a prescribed path relative to the rotor, determined by the shape of machined cutouts on the rotor and the absorber mass. Previous studies have considered how to account for the roller dynamics in selecting the linear tuning characteristics of the absorber system, but have not quantified the errors induced by the common approximations that either ignores their effects completely, or does not account for the nonlinear aspects of their dynamics. In this paper we systematically investigate these effects. Specifically, we first show that there exists an absorber path for which the absorber/roller system maintains the same frequency of free oscillation over all physically possible amplitudes. This tautochronic path has been well known for the case with zero roller inertia, and herein, for the first time, the corresponding path with rollers is shown to exist and is constructed. In addition, we carry out an analysis of the steady-state response of the rotor/absorber/roller system in order to quantify the effects of various approximations commonly used in regards to the roller dynamics. This analysis is based on the equations of motion, scaled in such a manner so that they are amenable to a perturbation analysis, which includes the effects of rollers in the perturbation terms. It is shown that if one accounts for the linear tuning aspects of the rollers, the system response is essentially insensitive to the selection of the nonlinear tuning parameter, so long as it is close to the tautochronic value. This implies that the approximation commonly used for selecting absorber paths with rollers is adequate.

Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers

2011 ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Torque Converter ◽  
Transient Dynamics ◽  
Circular Path ◽  
Vibration Absorbers ◽  
Approximate Analytical Expression ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Nonlinear Transient ◽  
System Equations ◽  
Minimal Mass

This paper describes an analytical and experimental investigation of the transient dynamics of centrifugal pendulum vibration absorbers, which are used for reducing torsional vibrations in rotating machines. Recently these absorbers have been proposed for use in automotive engines, to aid with fuel saving technologies such as cylinder deactivation and torque converter lockup. In order for them to operate effectively with minimal mass, they must be designed to allow for large amplitude, nonlinear responses. In this paper we consider the transient dynamics of these absorbers, focusing on the response during startup. During these transient events the absorbers experience a beating type motion, resulting in overshoot of the absorber response before reaching steady state conditions. Using a perturbation analysis of the system equations of motion, an approximate analytical expression for nonlinear overshoot is derived, relating the overshoot to the system and excitation parameters. These predictive results are derived for a general class of absorbers, and are verified by simulations of the full equations of motion and by experiments using a fully instrumented spin rig. It is found that the overshoot for absorbers with softening nonlinearity, such as circular path absorbers, can be well over the 100% upper limit for a linear absorber, and can be as high as 173%. For absorbers with tautochronic paths, the overshoot remains quite close to that of the linearized system, even for large amplitudes. These results provide a useful tool for the design of absorbers to meet transient response specifications.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part II: Dual-Load-Path Configurations

2013 ◽  
Vol 58 (3) ◽  
pp. 1-9 ◽  
Author(s):  
Zahra Sotoudeh ◽  
Dewey H. Hodges
Keyword(s):  
Structural Dynamics ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Rotor Blades ◽  
Load Path ◽  
Dynamics Modeling ◽  
Governing Equations ◽  
Wind Turbine Blades ◽  
First Order ◽  
Infinite Degree

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. Part II is devoted to verification of certain dual-load-path configurations suggested by Bell Helicopter Textron.

Aeroelastic Modeling of Large Wind Turbines

1976 ◽  
Vol 21 (4) ◽  
pp. 17-27 ◽  
Author(s):  
Peretz P. Friedmann
Keyword(s):  
Wind Turbine ◽  
Turbine Blade ◽  
Differential Form ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Aeroelastic Stability ◽  
Wind Turbine Blades ◽  
Partial Differential ◽  
General Terms ◽  
Large Wind

A set of coupled flap‐lag‐torsional equations of motion for a single wind turbine blade are derived in a general, nonlinear, partial differential form. These equations are suitable for determining the aeroelastic stability or response of large wind turbine blades. Methods for solving the equations together with some possible simplification of the equations are discussed. Finally, the formulation of the complete rotor‐tower aeroelastic problem is considered in general terms.

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