On Dynamic Tooth Load and Stability of a Spur-Gear System Using the State-Space Approach

1985 ◽  
Vol 107 (1) ◽  
pp. 54-60 ◽  
Author(s):  
A. S. Kumar ◽  
T. S. Sankar ◽  
M. O. M. Osman
Keyword(s):  
State Space ◽  
Dynamic Load ◽  
Initial Conditions ◽  
Gear Tooth ◽  
Spur Gear ◽  
The State ◽  
Gear System ◽  
Computational Time ◽  
Steady State Condition ◽  
The Stability

In this study, a new approach using the state-space method is presented for the dynamic load analysis of spur gear systems. This approach gives the dynamic load on gear tooth in mesh as well as information on the stability of the gear system. Also a procedure is given for the selection of proper initial conditions that enable the steady-state condition to be reached faster, conditions that result in considerable savings in computational time. The variations in the dynamic load with respect to changes in contact position, operating speed, backlash, damping, and stiffness are also investigated. In addition, the stability of the gear system is studied, using the Floquet theory and the well-known stability conditions of difference systems.

On Statistical Analysis of Gear Dynamic Loads

1986 ◽  
Vol 108 (3) ◽  
pp. 362-368 ◽  
Author(s):  
A. S. Kumar ◽  
M. O. M. Osman ◽  
T. S. Sankar
Keyword(s):  
Steady State ◽  
Statistical Analysis ◽  
Dynamic Load ◽  
Initial Conditions ◽  
Gear System ◽  
Computational Time ◽  
Statistical Linearization ◽  
Mesh Stiffness ◽  
Steady State Condition ◽  
Gaussian White Noise Process

Statistical analysis of the gear dynamic load is carried out using piecewise constant mesh stiffness approximation. The dynamics of the spur gear system is modeled as a nonlinear, nonstationary process, and the gear transmission error which acts as a random input to the gear system is generated by passing a Gaussian white noise process through a time invariant shaping filter. The equivalent discrete time state equation and the mean and covariance propagation equations are then written for the augmented system. Then starting from known initial conditions these propagation equations are used to compute the statistics of the steady state response and hence those of the dynamic load. A procedure is presented for the selection of proper initial conditions so as to reach the steady state condition faster, thereby reducing the computational time required. The variations in the statistics of the dynamic load with respect to changes in contact position, random error magnitude, and operating speed are also investigated with the help of a numerical example. The results show that the approach presented in this study provides truer results than the statistical linearization approach used by Tobe et al. [13]. Moreover, the proposed procedure has the advantage that it can be applied to higher-order systems with complex mesh stiffness and torque fluctuations and to systems with symmetrical or nonsymmetrical nonlinearities.

Load Effects on the Dynamics of Spur Gear Transmissions

2010 ◽  
Author(s):  
Alfonso Fernandez del Rincon ◽  
Fernando Viadero Rueda ◽  
Miguel Iglesias Santamaria ◽  
Pablo Garcia Fernandez ◽  
Ana de-Juan de Luna ◽  
...  
Keyword(s):  
Gear Tooth ◽  
Hybrid Approach ◽  
Angular Position ◽  
Spur Gear ◽  
Computational Effort ◽  
Contact Forces ◽  
Mode Shapes ◽  
Computational Time ◽  
Applied Torque ◽  
Rolling Elements

Gear transmissions in general and spur gears in particular exhibit a different dynamic behavior depending on the level of the transmitted load. This fact justifies the interest in the study of the role of the load in gear dynamics not only in the context of design, vibration and noise control but also for condition monitoring. This task requires the development of advanced models achieving a compromise between accuracy and computation time. In this work, gear and bearing non-linearities associated with the contact among teeth and roller elements have been included, taking into account the flexibility of gears, shafts and bearings. Besides, parametric excitations coming both from gear and bearing supports, as well as clearance, were also considered. Gear contact force calculations are carried out following a hybrid approach which combines both analytical and numerical tools. This lets to achieve accurate results with an acceptable computational effort and thus dynamic analysis becomes feasible. This approach was improved and the calculation speeded up from the point of view of computational time. This was performed by using a pre-calculated value for gear tooth stiffness as a function of load and the angular position when it operates under stationary conditions. On the other hand, bearings were formulated just as deflections of Hertzian type. This means that bending and shearing of races and rolling elements are neglected. However, the variation in the number of loaded rolling elements as a function of the load and the angular position was taken into account. Shaft flexibilities were added to gear and bearing models to define a simple transmission that was used to study the vibratory behavior under different levels of applied torque. In a preliminary study, this model was linearized for several loads, obtaining the corresponding frequencies and mode-shapes in order to assess their variation with this parameter. Finally, dynamic simulations were carried out, showing the modifications undergone by the orbits, meshing contact forces and transmitted bearing forces.

Directed Policy Search for Decision Making Using Relevance Vector Machines

2014 ◽  
Vol 23 (04) ◽  
pp. 1460016
Author(s):  
Ioannis Rexakis ◽  
Michail G. Lagoudakis
Keyword(s):  
Decision Making ◽  
State Space ◽  
Previous Method ◽  
The State ◽  
Computational Time ◽  
Support Vector ◽  
Learning Approaches ◽  
Policy Search ◽  
Relevance Vector Machines ◽  
Vector Machines

Several recent learning approaches in decision making under uncertainty suggest the use of classifiers for representing policies compactly. The space of possible policies, even under such structured representations, is huge and must be searched carefully to avoid computationally expensive policy simulations (rollouts). In our recent work, we proposed a method for directed exploration of policy space using support vector classifiers, whereby rollouts are directed to states around the boundaries between different action choices indicated by the separating hyperplanes in the represented policies. While effective, this method suffers from the growing number of support vectors in the underlying classifiers as the number of training examples increases. In this paper, we propose an alternative method for directed policy search based on relevance vector machines. Relevance vector machines are used both for classification (to represent a policy) and regression (to approximate the corresponding relative action advantage function). Classification is enhanced by anomaly detection for accurate policy representation. Exploiting the internal structure of the regressor, we guide the probing of the state space only to critical areas corresponding to changes of action dominance in the underlying policy. This directed focus on critical parts of the state space iteratively leads to refinement and improvement of the underlying policy and delivers excellent control policies in only a few iterations, while the small number of relevance vectors yields significant computational time savings. We demonstrate the proposed approach and compare it with our previous method on standard reinforcement learning domains (inverted pendulum and mountain car).

Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

scholarly journals ‘Intractable and unsolved’: some thoughts on statistical data assimilation with uncertain static parameters

2013 ◽  
Vol 371 (1991) ◽  
pp. 20120297 ◽  
Author(s):  
Jonathan Rougier
Keyword(s):  
Numerical Methods ◽  
Data Assimilation ◽  
State Space ◽  
Stochastic Models ◽  
Environmental Science ◽  
Statistical Data ◽  
Initial Conditions ◽  
The State ◽  
Likelihood Functions ◽  
Sensitive Dependence

If you seem to be able to do data assimilation with uncertain static parameters then you are probably not working in environmental science. In this field, applications are often characterized by sensitive dependence on initial conditions and attracting sets in the state-space, which, taken together, can be a major challenge to numerical methods, leading to very peaky likelihood functions. Inherently stochastic models and uncertain static parameters increase the challenge.

ANALYSIS OF COLD-STANDBY SYSTEMS — CUTTING AND CLUSTERING THE STATE SPACE

1995 ◽  
Vol 02 (03) ◽  
pp. 327-340 ◽  
Author(s):  
M.N. GOPALAN ◽  
U. DINESH KUMAR
Keyword(s):  
State Space ◽  
Integral Equations ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
The State ◽  
Convolution Integral ◽  
Cold Standby ◽  
Convolution Integral Equations ◽  
Systems Of Integral Equations ◽  
Repair Facility

In this paper two methods, namely cutting and clustering the state space are discussed to find approximate solutions to n-unit cold-standby system with a single repair facility. It has been assumed that the failure rate is constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. Explicit expressions for the availability and the reliability of the system are obtained. An iterative method is used to solve the systems of integral equations obtained and a comparative study has been carried out between exact and approximate solutions.

Finite Element Application of Gear Tooth Analysis

2014 ◽  
Vol 889-890 ◽  
pp. 527-531
Author(s):  
V. Balambica ◽  
T. Jayachandra Prabhu ◽  
R. Venkatesh Babu
Keyword(s):  
Dynamic Load ◽  
Gear Tooth ◽  
Design Procedure ◽  
Spur Gear ◽  
Dynamic Loads ◽  
Historical Period ◽  
Simple Design ◽  
Gear Teeth ◽  
Gear Design ◽  
Stiffness Characteristics

Gears play an important role in every aspect of power and motion of transmission from historical period to modern day period . Due to this , gear design has become a complicated art.A considerable amount of research has been carried out to determine the amount of dynamic gear tooth loads acting.The findings of the dynamic load between the gear teeth results in difficulty for the designer.In this paper, an effort has been made to formulate a simple design procedure for calculating the dynamic load .Earlier the stiffness characteristics and deformation of the gear tooth were studied to predict the dynamic load acting. This was developed with the tooth assumed as a short cantilever.Whereas in reality, an involute profile exists in a spur gear tooth.Based on this reality, work has been done to model the exact profile of the tooth.Later ,the stiffness characteristics were carefully analysed and an improvement was thus made. It was proved that FEA is one such technique that can be used for predicting dynamic loads acting on a gear tooth.

scholarly journals Some New Trajectory Patterns and Periodic Behaviors of Unstable Second-Order Digital Filter with Two's Complement Arithmetic

2003 ◽  
Vol 13 (08) ◽  
pp. 2361-2368 ◽  
Author(s):  
Wing-Kuen Ling ◽  
Kwong-Shum Tam
Keyword(s):  
Periodic Orbit ◽  
Digital Filter ◽  
State Vector ◽  
Initial Conditions ◽  
Second Order ◽  
The State ◽  
Simulation Results ◽  
The Stability ◽  
Straight Lines

This Letter shows some counter-intuitive simulation results that for some filter parameters in the extended boundaries of the stability triangle, the state vector will converge to a periodic orbit after some iterations, no matter what the initial conditions. Also, a new pattern, which looks like a rotated letter "X", is found. The center of the rotated letter is located at the origin and the slopes of the "straight lines" of the rotated letter are equal to the values of the pole locations.

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