Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration

1999 ◽  
Vol 121 (3) ◽  
pp. 316-321 ◽  
Author(s):  
Jian Lin ◽  
R. G. Parker
Keyword(s):  
Degrees Of Freedom ◽  
Eigenvalue Problems ◽  
Linear Time ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
Factors Affecting ◽  
Linear Time Invariant ◽  
Gyroscopic Effects ◽  
Time Invariant

This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. It includes key factors affecting planetary gear vibration such as gyroscopic effects and time-varying stiffness. For the linear, time-invariant case, examination of the associated eigenvalue problem reveals the well-defined structure of the vibration modes, where the special structure results from the cyclic symmetry of planetary gears. Vibration modes are classified into rotational, translational and planet modes. The unique characteristics of each type of mode are analytically investigated in detail. For each class of mode, reduced-order eigenvalue problems are derived.

Natural Frequency Spectra and Vibration Modes of Planetary Gears

1998 ◽  
Author(s):  
Jian Lin ◽  
Robert G. Parker
Keyword(s):  
Degrees Of Freedom ◽  
Eigenvalue Problems ◽  
Linear Time ◽  
Planetary Gear ◽  
Transmission Error ◽  
Vibration Modes ◽  
Modal Strain Energy ◽  
Frequency Spectra ◽  
Planetary Gears ◽  
Gyroscopic Effects

Abstract This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. It includes key factors affecting planetary gear vibration such as gyroscopic effects, time-varying stiffness, and static transmission error excitation. For the linear time-invariant case, examination of the associated eigenvalue problem reveals the well-defined structure of the vibration modes, where the special structure results from the cyclic symmetry of planetary gears. Vibration modes are classified into rotational, translational and planet modes. The unique characteristics of each type of mode are analytically investigated in detail. For each class of mode, reduced-order eigenvalue problems are derived. The modal strain energy distributions are also discussed.

Vibration Properties of High-Speed Planetary Gears With Gyroscopic Effects

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Eigenvalue Problems ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
High Speeds ◽  
Wide Range ◽  
Modal Property ◽  
Gyroscopic Effects ◽  
Complex Valued

This study investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speed-dependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

scholarly journals VIBRATION ANALYSIS OF PLANETARY GEAR SYSTEM

2013 ◽  
pp. 30-34
Author(s):  
MAJID MEHRABI ◽  
DRV.P. SINGH
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Linear Time ◽  
Planetary Gear ◽  
Vibration Modes ◽  
System Parameters ◽  
Linear Time Invariant ◽  
Inertia Parameters ◽  
Planetary Gear System ◽  
Time Invariant

In this work a dynamic model of a planetary gear transmission is developed to study the sensitivity of the natural frequencies and vibration modes to system parameters in perturbed situation. Parameters under consideration include component masses ,moment of inertia , mesh and support stiff nesses .The model admits three planar degree of freedom for planets ,sun, ring, and carrier. Vibration modes are classified into translational, rotational and planet modes .Well-defined modal properties of tuned (cyclically symmetric) planetary gears for linear ,time-invariant case are used to calculate eigensensitivities and express them in simple formulae .These formulae provide efficient mean to determine the sensitivity to stiffness ,mass and inertia parameters in perturbed situation.

Zeno-free adaptive event-triggered control design

2018 ◽  
Vol 41 (8) ◽  
pp. 2328-2337 ◽  
Author(s):  
Hassan Adloo ◽  
Mohammad Hossein Shafiei
Keyword(s):  
Degrees Of Freedom ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Performance Criterion ◽  
Linear Time Invariant ◽  
Event Triggering ◽  
Linear Time Invariant Systems ◽  
System States ◽  
Time Invariant ◽  
Event Triggered

This paper presents a new general framework for adaptive event-triggered control strategy to extend average inter-event interval, while maintaining the performance of the system. The proposed event-triggering mechanism is acquired from input to state stability conditions, which is defined in terms of system states as well as an adaptation parameter. Under the Lipschitz assumption, a positive lower bound on sampling durations is also established that is essential to restrain the Zeno behavior. Applying the proposed method to linear time-invariant systems, leads to sufficient conditions to guarantee asymptotic stability in the form of matrix inequalities. Moreover, it is shown that there exist more degrees of freedom to improve the performance criterion from theoretical aspects. Finally, in order to show capability of the proposed method and its better performance compared with some recent works, numerical simulations are presented.

scholarly journals Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems

Algorithms ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 136
Author(s):  
Manuel Duarte-Mermoud ◽  
Javier Gallegos ◽  
Norelys Aguila-Camacho ◽  
Rafael Castro-Linares
Keyword(s):  
Fractional Order ◽  
Performance Optimization ◽  
Degrees Of Freedom ◽  
Linear Time ◽  
Minimal Realization ◽  
Linear Time Invariant ◽  
Mixed Order ◽  
Linear Time Invariant Systems ◽  
Minimal Realizations ◽  
Time Invariant

Adaptive and non-adaptive minimal realization (MR) fractional order observers (FOO) for linear time-invariant systems (LTIS) of a possibly different derivation order (mixed order observers, MOO) are studied in this paper. Conditions on the convergence and robustness are provided using a general framework which allows observing systems defined with any type of fractional order derivative (FOD). A qualitative discussion is presented to show that the derivation orders of the observer structure and for the parameter adjustment are relevant degrees of freedom for performance optimization. A control problem is developed to illustrate the application of the proposed observers.

Multivariable Control of an Over-Head Crane System Based on Entire Eigen-Structure Assignment

2010 ◽  
Author(s):  
Hamed Moradi ◽  
K. Haji Hajikolaei ◽  
Firooz Bakhtiari-Nejad ◽  
Aria Alasty
Keyword(s):  
Degrees Of Freedom ◽  
Linear Time ◽  
Control Strategies ◽  
Control Signal ◽  
Space Representation ◽  
Linear Time Invariant ◽  
State Space Representation ◽  
Stabilization Control ◽  
Structure Assignment ◽  
Time Invariant

Motion and stabilization control strategies are required to improve positioning accuracy, transportation time and swing angle of an overhead crane system. In this paper, a controller is designed to enhance both efficiency and safety and to extend the system application to other engineering fields. An over-head crane is modeled as a linear time invariant (LTI) system with two degrees of freedom. Trolley position and cable angle are the controlled outputs while the force exerted on trolley and torque on the load are the control inputs of the system. After state-space representation of the problem, feedback control is designed for tracking objective. An increase in the overall speed of the system time response corresponds to an increase in the control signal and leads to additional cost. Therefore, developing a code in MATLAB, eigenvalues and eigenvectors of the system are chosen optimally until an appropriate response is achieved; while the gains of control signal remain small.

Modal H∞-Control in the Context of a Rotor Being Subject to Unbalance Excitation and Gyroscopic Effect

2013 ◽  
Author(s):  
Rudolf Sebastian Schittenhelm ◽  
Bernd Riemann ◽  
Stephan Rinderknecht
Keyword(s):  
Degrees Of Freedom ◽  
Controller Design ◽  
Linear Time ◽  
Rotational Frequency ◽  
Gyroscopic Effect ◽  
Actual System ◽  
Linear Time Invariant ◽  
Optimal Controllers ◽  
Problem Description ◽  
Time Invariant

H∞-optimal controllers are designed for a rotor being subject to unbalance excitation and gyroscopic effect. The system possesses two unbalance-induced resonances within its operating range. The presence of gyroscopic effect is challenging for linear time invariant controller design because of the associated dependence of the system dynamics on the rotational frequency of the rotor. Controllers thus have to be robust against deviation of the actual system behavior from the controller design point model. For vibration control purposes, there are two piezoelectric actuators installed in one of the two supports of the rotor. The signals of four inductive sensors measuring the displacements of the two discs of the rotor are used for controller design. In this article, H∞-optimal controllers are designed on the basis of input and output weighting as well as weighting of modal degrees of freedom and modal excitations. It is shown that superior control performance is achieved using modal weighting since a more accurate problem description of rotors excited by unbalance is incorporated in controller design. Results in this article show furthermore that it is possible to design well performing H∞-optimal controllers for a gyroscopic rotor by means of iterative controller design without taking model uncertainty directly into account via weighting of certain FRFs of the system to be controlled.

scholarly journals ANALYTICAL MODELING OF PLANETARY GEAR AND SENSITIVITY OF NATURAL FREQUENCIES

2013 ◽  
pp. 35-40
Author(s):  
MAJID MEHRABI ◽  
DR. V.P. SINGH
Keyword(s):  
Natural Frequency ◽  
Degrees Of Freedom ◽  
Analytical Modeling ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
System Parameters ◽  
Inertia Parameters

This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. Vibration modes are classified into rotational, translational and planet modes. The natural frequency sensitivities to system parameters are investigated for tuned (cyclically symmetric) planetary gears. Parameters under consideration include support and mesh stiffnesses, component masses, and moments of inertia. Using the well-defined vibration mode properties of tuned planetary gears, the eigen sensitivities are calculated and expressed in simple exact formulae. These formulae connect natural frequency sensitivity with the modal strain or kinetic energy and provide efficient means to determine the sensitivity to all stiffness and inertia parameters by inspection of the modal energy distribution.

Nonlinear Analysis of Vibro-Impacts for Unloaded Gear Pairs With Various Excitations and System Parameters

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Jong-yun Yoon ◽  
Iljae Lee
Keyword(s):  
Degrees Of Freedom ◽  
Linear Time ◽  
System Model ◽  
System Parameters ◽  
Linear Time Invariant ◽  
Gear Mesh ◽  
The Stability ◽  
Linear Time Invariant System ◽  
Time Invariant ◽  
Gear Rattle

Torsional systems with clearance-type nonlinearities have inherent vibratory problems such as gear rattle. Such vibro-impacts generally occur on the unloaded gear pairs of a vehicle correlated with the firing excitation of an engine. This study investigates the gear rattle phenomena on unloaded gear pairs with different excitation conditions and various system parameters. First, a linear time-invariant system model with six degrees of freedom is constructed and then a numerical analysis is applied to the gear rattle motion. Smoothening factors for clutch stiffness and hysteresis are employed for the stability of numerical simulations. Second, the dynamic characteristics of vibro-impacts are studied by examining the fast Fourier transform (FFT) components of the gear mesh force in a high frequency range. The effects of various system parameters on the vibro-impacts are examined using a nonlinear system model. Finally, the vibro-impacts, in terms of “single-sided” and “double-sided” impacts, are identified in phase planes.

Vibration Structure of Gyroscopic Planetary Gears

2011 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Eigenvalue Problems ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
High Speeds ◽  
Wide Range ◽  
Mode Method ◽  
Complex Valued

This study investigates the vibration structure of high-speed, gyroscopic planetary gears. The vibration modes of these systems are complex-valued and speed dependent. Three mode types exist, and these are classified as planet, rotational, and translational modes. Each mode type is mathematically proven by the use of a candidate mode method. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

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