Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2007 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and non-rotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2005 ◽  
Vol 128 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Effect of Ring-Planet Mesh Phasing and Contact Ratio on the Parametric Instabilities of a Planetary Gear Ring

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Excitation ◽  
Planetary Gear ◽  
Time Varying ◽  
Ring Gear ◽  
Contact Ratio ◽  
System Parameters ◽  
Gear Ring ◽  
Parametric Instabilities ◽  
Planetary Gear System ◽  
Mesh Phasing

Parametric excitation of a rotating ring subject to moving time-varying stiffnesses has previously been investigated and given as closed-form expressions in the system parameters. These conditions are applied to identify ring gear parametric instabilities in a planetary gear system. Certain mesh phasing and contact ratio conditions suppress parametric instabilities, and these conditions are presented with examples.

Vibration Resonances in a Planetary Gear Ring: Effects of Mesh Phasing and Contact Ratio

2007 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Excitation ◽  
Planetary Gear ◽  
Time Varying ◽  
Ring Gear ◽  
Contact Ratio ◽  
System Parameters ◽  
Gear Ring ◽  
Parametric Instabilities ◽  
Planetary Gear System ◽  
Mesh Phasing

Parametric excitation of a rotating ring subject to moving time-varying stiffnesses have previously been investigated and given as closed form expressions in the system parameters. These conditions are applied to identify ring gear parametric instabilities in a planetary gear system. Certain mesh phasing and contact ratio conditions suppress parametric instabilities, and these conditions are presented with examples.

Parametric Instability in a Gear Train System Due to Stiffness Variation

2013 ◽  
Author(s):  
S. Y. Wang ◽  
S. C. Sinha
Keyword(s):  
Differential Equations ◽  
Parametric Instability ◽  
Gear Train ◽  
Elastic Model ◽  
Time Varying ◽  
Ring Gear ◽  
Mesh Stiffness ◽  
Varying Coefficients ◽  
Wide Range ◽  
Stiffness Variation

The excitation from mesh stiffness variation in a tunnel gear driving system can cause excessive noise and vibrations. Since the stiffness variation may induce parametric instability, the system could be damaged on a permanent-basis. Therefore, the study of parametric instability in such system is of paramount importance. In this work, a rigid-elastic model is developed using the energy method, where the ring gear is treated as a rotating thin ring having radial and tangential deflections, whereas the pinions are assumed to be rigid bodies having translational motion relative to the radial directions of the ring gear as well as rotational motions around their centers. All gear meshes are modeled as interactions caused by time-varying springs, and the supports of the pinions are modeled as linear springs in the radial direction relative to the ring gear. The modeling leads to a set of partial-ordinary linear differential equations with time-varying coefficients. For an N planet system, the discretization process yields 2N+2 ordinary differential equations. Stability boundaries are determined using Floque’t theory for a wide range of parameter values. Specifically, the effects of mesh stiffness on the parametric instability are examined. The results show that the instability behaviors are closely related to the basic parameters when considering the time-varying excitation. This could be a serious consideration in the preliminary design of such systems.

Finite-time control of uncertain time-varying systems in p-normal form

2020 ◽  
Author(s):  
Kanya Rattanamongkhonkun ◽  
Radom Pongvuthithum ◽  
Chulin Likasiri
Keyword(s):  
Normal Form ◽  
Finite Time ◽  
Control Law ◽  
Time Varying ◽  
Time Control ◽  
Special Cases ◽  
Time Varying Systems ◽  
A Chain ◽  
Uncertain Time ◽  
Power Integrator

Abstract This paper addresses a finite-time regulation problem for time-varying nonlinear systems in p-normal form. This class of time-varying systems includes a well-known lower-triangular system and a chain of power integrator systems as special cases. No growth condition on time-varying uncertainties is imposed. The control law can guarantee that all closed-loop trajectories are bounded and well defined. Furthermore, all states converge to zero in finite time.

Bending of Thin Ring-Sector Plates

1951 ◽  
Vol 18 (4) ◽  
pp. 359-363
Author(s):  
L. I. Deverall ◽  
C. J. Thorne
Keyword(s):  
Continuous Function ◽  
Bending Moment ◽  
Circular Ring ◽  
Circular Plates ◽  
Thin Ring ◽  
Special Cases ◽  
Edge Conditions

Abstract General expressions for the deflection of plates whose planform is a sector of a circular ring are given for cases in which the straight edges have arbitrary but given deflection and bending moment. The solutions are given for all combinations of physically important edge conditions on the two circular edges. Sectors of circular plates are included as special cases. Solutions are given for a general load which is a continuous function of r, and a sectionally continuous function of θ, where r and θ are the usual polar co-ordinates with the pole at the center of the ring. Several specific examples are given.

Influence of Axial Excitations and of Boundary Conditions on the Parametric Instability of a Beam

1995 ◽  
Author(s):  
Régis Dufour ◽  
Alain Berlioz ◽  
Thomas Streule
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Parametric Instability ◽  
Experimental Tests ◽  
Time Varying ◽  
Periodic Force ◽  
Modal Reduction ◽  
Time Varying Parameter ◽  
The Stability

Abstract In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet’s theory and investigated in detail. The Rayleigh-Ritz results are compared to those of a finite element modal reduction. It shows that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

scholarly journals Optimizing preservation strategies for deteriorating items with time-varying holding cost and stock-dependent demand

2020 ◽  
Vol 30 (2) ◽  
pp. 237-250
Author(s):  
Aditi Khanna ◽  
P Priyamvada ◽  
Chandra Jaggi
Keyword(s):  
Optimal Solution ◽  
Optimal Investment ◽  
Deteriorating Items ◽  
Time Varying ◽  
Preservation Technology ◽  
Special Cases ◽  
Stock Dependent Demand ◽  
Managerial Implications ◽  
Holding Cost ◽  
Dependent Demand

Organizations are keen on rethinking and optimizing their existing inventory strategies so as to attain profitability. The phenomenon of deterioration is a common phenomenon while managing any inventory system. However, it could become a major challenge for the business if not dealt carefully. An investment in preservation technology is by far the most inuential move towards dealing with deterioration proficiently. Additionally, it is noticed that the demand pattern of many products is reliant on its availability and usability. Thus, considering demand of the product to be ?stock-dependent" is a more practical approach. Further, in case of deteriorating items, it is observed that the longer an item stays in the system the higher is its holding cost. Therefore, the model assumes the holding cost to be time varying. Hence, the proposed framework aims to develop an inventory model for deteriorating items with stock-dependent demand and time-varying holding cost under an investment in preservation technology. The objective is to determine the optimal investment in preservation technology and the optimal cycle length so as to minimize the total cost. Numerical example with various special cases have been discussed which signifies the effect of preservation technology investment in controlling the loss due to deterioration. Finally, the effect of key model features on the optimal solution is studied through sensitivity analysis which provides some important managerial implications.

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