Nonlinear Interactions in Systems of Multiple Order Centrifugal Pendulum Vibration Absorbers

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Brendan J. Vidmar ◽  
Steven W. Shaw ◽  
Brian F. Feeny ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
System Stability ◽  
System Response ◽  
Nonlinear Interactions ◽  
Vibration Absorbers ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
System Equations ◽  
Improved Performance ◽  
Nonlinear Equations Of Motion ◽  
Selection Of

We consider nonlinear interactions in systems of order-tuned torsional vibration absorbers with sets of absorbers tuned to different orders. In all current applications, absorber systems are designed to reduce torsional vibrations at a single order. However, when two or more excitation orders are present and absorbers are introduced to address different orders, nonlinear interactions become possible under certain resonance conditions. Under these conditions, a common example of which occurs for orders n and 2n, crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid these interactions, and to explore possible improved performance with sets of absorbers tuned to different orders, we develop predictive models that allow one to examine the effects of absorber mass distribution and tuning. These models are based on perturbation methods applied to the system equations of motion, and they yield system response features, including absorber and rotor response amplitudes and stability, as a function of parameters of interest. The model-based analytical results are compared against numerical simulations of the complete nonlinear equations of motion, and are shown to be in good agreement. These results are useful for the selection of absorber parameters to achieve desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield optimal performance. It is also found that utilizing multiple order absorber systems can be beneficial for system stability, even when only a single excitation order is present.

Analysis and Design of Multiple Order Centrifugal Pendulum Vibration Absorbers

2012 ◽  
Author(s):  
Brendan J. Vidmar ◽  
Steven W. Shaw ◽  
Brian F. Feeny ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
System Response ◽  
Torsional Vibrations ◽  
Nonlinear Interactions ◽  
Vibration Absorbers ◽  
Analysis And Design ◽  
Engine Order ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
System Equations ◽  
Nonlinear Equations Of Motion

We consider nonlinear interactions in systems of order-tuned torsional vibration absorbers. These absorbers, which consist of centrifugally driven pendulums fitted to a rotor, are used to reduce engine-order torsional vibrations in rotating machines, including automotive engines, helicopter rotors, and light aircraft engines. In all current applications, absorber systems are designed to reduce torsional vibrations at a single order. However, when two or more excitation orders are present and absorbers are introduced to address different orders, undesirable nonlinear interactions become possible under certain resonance conditions. Under these conditions, a common example of which occurs for orders n and 2n, crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid these interactions, we develop predictive models that allow one to select proper tuning and sizing of the absorbers. These models are based on perturbation methods applied to the system equations of motion, and they yield system response features, including absorber and rotor response amplitudes and stability, as a function of parameters of interest. The model-based analytical results are compared against numerical simulations of the complete nonlinear equations of motion, and are shown to be in good agreement. These results are useful for the selection of absorber parameters for desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield optimal performance.

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.

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

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw ◽  
Alan H. Haddow ◽  
Bruce K. Geist
Keyword(s):  
Finite Amplitude ◽  
Tuning Parameter ◽  
System Response ◽  
Vibration Absorbers ◽  
Engine Order ◽  
Constant Rate ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Roller System ◽  
First Time ◽  
Selection Of

Centrifugal pendulum vibration absorbers are used for reducing engine-order torsional vibrations in rotating machines. The most common configuration of these devices utilizes a bifilar suspension in which the absorber mass is suspended by a pair of cylindrical rollers that allow it to move along a prescribed path that is determined by the shape of machined cutouts on the rotor and the absorber mass. Previous studies have considered how to account for the roller inertia in selecting the linear (small amplitude) tuning characteristics of the absorber system. Here, we describe a systematic study of the nonlinear (finite amplitude) aspects of this system and 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 when the rotor spins at a constant rate. 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 a method for its construction is presented. In addition, we carry out a perturbation analysis of the steady-state dynamic response of the rotor/absorber/roller system in order to quantify the effects of various approximations commonly used with regard to the roller dynamics. The results show that if one accounts for the rollers in the linear absorber tuning, the nonlinear system response is essentially insensitive to the selection of the nonlinear tuning parameter, so long as it is close to the tautochronic value.

Optimal Design of Vibration Absorber Systems Supported by Elastic Base

1991 ◽  
Author(s):  
Hashem Ashrafiuon
Keyword(s):  
Dynamic Models ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Base ◽  
Design Parameters ◽  
Primary System ◽  
Vibration Absorbers ◽  
Equivalent Damping ◽  
System Equations ◽  
Different Levels

Abstract This paper presents the effect of foundation flexibility on the optimum design of vibration absorbers. Flexibility of the base is incorporated into the absorber system equations of motion through an equivalent damping ratio and stiffness value in the direction of motion at the connection point. The optimum values of the uncoupled natural frequency and damping ratio of the absorber are determined over a range of excitation frequencies and the primary system damping ratio. The design parameters are computed and compared for the rigid, static, and dynamic models of the base as well as different levels of base flexibility.

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.

Dynamic Response of Spinning Blades Subjected to Gyroscopic Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 6-15 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Spin Axis ◽  
System Response ◽  
First Order ◽  
System Equations ◽  
Element Technique ◽  
Gyroscopic Motion

This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

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.

Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions

2015 ◽  
Vol 23 (12) ◽  
pp. 1873-1889 ◽  
Author(s):  
S Mehrdad Pourkiaee ◽  
Siamak E Khadem ◽  
Majid Shahgholi
Keyword(s):  
Stability Analysis ◽  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Van Der Waals ◽  
Surface Effects ◽  
Van Der Waals Forces ◽  
System Response ◽  
Nonlinear Equations Of Motion ◽  
Piezoelectric Nanobeam

This paper investigates the nonlinear vibration and stability analysis of a doubly clamped piezoelectric nanobeam, as a nano resonator actuated by a combined alternating current and direct current loadings, including surface effects and intermolecular van der Waals forces. The governing equation of motion is obtained using the extended Hamilton principle. The multiple scales method is used to solve nonlinear equations of motion. The influence of van der Waals forces, piezoelectric voltages and surface effects are investigated on the static equilibria, pull-in voltages and dynamic primary resonances of the nano resonator. It is shown that for accurate and exact investigation of the system response, it is necessary to consider the surface effects. To validate the analytical results, numerical simulation is performed. It is seen that the perturbation results are in accordance with numerical results.

Optimal Design of Vibration Absorber Systems Supported by Elastic Base

1992 ◽  
Vol 114 (2) ◽  
pp. 280-283
Author(s):  
H. Ashrafiuon
Keyword(s):  
Optimal Design ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Base ◽  
Design Parameters ◽  
Primary System ◽  
Vibration Absorbers ◽  
Equivalent Damping ◽  
System Equations ◽  
Flexible Models

This paper presents the effect of foundation flexibility on the optimum design of vibration absorbers. Flexibility of the base is incorporated into the absorber system equations of motion through an equivalent damping ratio and stiffness value in the direction of motion at the connection point. The optimum values of the uncoupled natural frequency and damping ratio of the absorber are determined over a range of excitation frequencies and the primary system damping ratio. Optimal design parameters are computed and compared for the rigid, and flexible models of the base as well as different levels of base flexibility.

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