Abstract
This paper presents a study on the dynamics of the rattling problem in gearboxes under nonideal excitation. The subject has being analyzed by a number of authors such as Karagiannis and Pfeiffer (1991), for the ideal excitation case. An interesting model of the same problem by Moon (1992) has been recently used by Souza and Caldas (1999) to detect chaotic behavior. We consider two spur gears with different diameters and gaps between the teeth. Suppose the motion of one gear to be given while the motion of the other is governed by its dynamics. In the ideal case, the driving wheel is supposed to undergo a sinusoidal motion with given constant amplitude and frequency. In this paper, we consider the motion to be a function of the system response and a limited energy source is adopted. Thus an extra degree of freedom is introduced in the problem. The equations of motion are obtained via a Lagrangian approach with some assumed characteristic torque curves. Next, extensive numerical integration is used to detect some interesting geometrical aspects of regular and irregular motions of the system response.
Gear rattling; nonlinear dynamics; nonideal systems
Analysis of Regular and Irregular Dynamics of a Non Ideal Gear Rattling Problem
S. L. T. de Souza
Instituto de Física da Universidade de São Paulo. Brazil
I. L. Caldas
Instituto de Física da Universidade de São Paulo. Brazil
J. M. Balthazar
Departamento de Estatística, Matemática Aplicada e Computação
Universidade Estadual Paulista,
Campus de Rio Claro. Brazil
R. M. L. R. F. Brasil
Departamento de Estruturas e Fundações da Escola Politécnica da Universidade de São Paulo. Brazil
This paper presents a study on the dynamics of the rattling problem in gearboxes under nonideal excitation. The subject has being analyzed by a number of authors such as Karagiannis and Pfeiffer (1991), for the ideal excitation case. An interesting model of the same problem by Moon (1992) has been recently used by Souza and Caldas (1999) to detect chaotic behavior. We consider two spur gears with different diameters and gaps between the teeth. Suppose the motion of one gear to be given while the motion of the other is governed by its dynamics. In the ideal case, the driving wheel is supposed to undergo a sinusoidal motion with given constant amplitude and frequency. In this paper, we consider the motion to be a function of the system response and a limited energy source is adopted. Thus an extra degree of freedom is introduced in the problem. The equations of motion are obtained via a Lagrangian approach with some assumed characteristic torque curves. Next, extensive numerical integration is used to detect some interesting geometrical aspects of regular and irregular motions of the system response.
Keywords: Gear rattling; nonlinear dynamics, nonideal systems
Introduction
Rattling in changeover gears of automobiles is an unwanted comfort problem. It is excited by the torsional vibrations of the drive train system at the entrance of the gearbox, where these torsional vibrations themselves are generated by imbalances of the engine. All gearwheel not under load rattle due to backlashes in the meshes of the gears.
In recent years, general models have been developed to analyze these rattling phenomena, mainly with the goal to find some means to reduce them by parameter variation. They are founded on the procedure based on impact theory. The rattling vibrations posses typical nonlinear behavior leading to periodic and chaotic regimes. The subject has been analyzed by a number of authors, such as Karagiannis and Pfeiffer (1991), for the ideal excitation case. An interesting model of this same problem by Moon (1992) has been recently used by Souza and Caldas (1999) to detect chaotic behavior.
We confine our considerations to single stage rattling. We consider two spur gears with different diameters and gaps between the teeth. Suppose the motion of one gear to be given while the motion ot the other is governed by its dynamics. In the ideal case, the driving wheel is supposed to undergo a sinusoidal motion with given constant amplitude and frequency. In this paper, we consider this motion to be a function of the system response and a limited energy source is adopted( non ideal case). Thus, an extra degree of freedom is introduced in the problem. The equations of motion are obtained via a Lagrangian approach with some assumed characteristic torque curves. Next, extensive numerical integration is used to detect some interesting geometrical aspects of regular and irregular motions of the system response.
Nomenclature
c = viscous damping coeficient
D = motor resisting torque
d = gap width
E_{1} = motor constant
E_{2} = motor constant
G = auxialiary variable
J = rotor moment of inertia
k_{1} = constant of linear part of spring stifness
k_{2} = constant of nonlinear part of spring stifness
L =motor active torque
M = cart mass
m_{1} = ecentric point mass
m_{2} = free point mass
r = eccentricity
S = displacement of the free mass within gap
T = kinetic energy
V = potential energy
x =cart displacement
Y = Lagrangian function
Greek Symbols
a = coefficient
e = small parameter
f = rotor angular displacement
m = parameter related to viscous damping
t = time related variable
Mathematical Model
Our model is presented in Fig. 1. We consider a cart of mass M connected to a inertial reference by a spring k (whose stiffness has a linear part k_{1} and a non linear part k_{2}) and a linear viscous damper c. Its displacement is denoted by x. In the upper part of the cart, a d wide gap is carved. Within the boundaries of this gap a point mass m_{2} (whose displacement is denoted by S) is free to move and eventually impact against them. The motion of the cart is induced by an inboard non ideal motor driving an unbalanced rotor, whose angular displacement is j and whose moment of inertia is J. This situation is modeled by a small point mass m_{1} at a r eccentricity.
We now derive the equations of motion of the cart via Lagrange's equations:
The kinetic energy is:
Thus, we obtain the following set of first order equations of motion for the cart:(5)
where:
E_{1} and E_{2} are constants of the motor.
These equations will render movable boundaries to the motion of the point mass m_{2}. The motion of this point mass is simply given by the solution of the homogeneous equation
Initial conditions are given by each impact with the movable boundaries.
In the next section, motions of both cart and free point mass are obtained via numerical integration.
Numerical Simulations
First, a parameter study of the effect of the motor constant E_{1} upon the motion of the cart is performed. Results displayed in Figures 2 and 3 show that the parameters effect both the frequency
The motion of the cart imply movable boundaries to the displacement of the point mass that is free to move inside the gap. Next, it is interesting to observe Figures 4 and 5. Heavy lines represent the position of the boundaries of the gap at each time t and the thin lines the position of the free point mass. Figure 4 prompts a condition of regular motions while Fig. 5 depicts on e of chaotic motions.
For a better understanding of the phenomenon, a bifurcation diagram is prompted in Fig. 6, showing the speed of the point mass in the moment just before the impact against the movable boundaries, as function of the motor parameter E_{1}.
Comments on the Results
A simple mathematical model of the gearrattling phenomenon is proposed allowing for consideration of limited energy sources such as those liable to occur in practice.
Our investigations confirm that a rich bifurcation structure and chaotic behavior is present in the adopted model. This could be of interest as a possible explanation of the undesirable rattling observed in practical gearboxes.
Acknowledgements
The authors acknowledge financial support by FAPESP and CNPq, both Brazilian researches funding agencies. They also thank the Department of Structural and Foundations Engineering of the Polytechnic School of the University of São Paulo, the Applied Mathematics Department of the State University of São Paulo at Rio Claro, Brazil and Department of mechanical Design of State University of Campinas, Campinas, São Paulo, Brazil..
Presented at COBEM 99  15th Brazilian Congress of Mechanical Engineering, 2226 November 1999, São Paulo. SP. Brazil. Technical Editor: José Roberto F. Arruda.
 Karagiannis, K., Pfeiffer, F., 1991, "Theoretical and experimental investigations of gear rattling", Nonlinear Dynamics, Vol.2, pp. 367387.
 Moon, F.C., 1992, "Chaotic and Fractal Dynamics", J. Wiley, New York.
 Moon, F., 1992., "Chaotic and fractal dynamicsan introduction for applied scientistis and engineers", J. Wiley , New York
 Souza, S.L.T., Caldas, I.L., 1999, "Nonlinear phenomena in gear rattling model", Proceedings of the Sixth Pan American Congress of Applied Mechanics, January 48, pp. 12751277, Rio de Janeiro, Brazil.
Publication Dates

Publication in this collection
21 Aug 2002 
Date of issue
May 2002