Coupled Belt-Pulley Vibration in Serpentine Drives With Belt Bending Stiffness

2004 ◽  
Vol 71 (1) ◽  
pp. 109-119 ◽  
Author(s):  
Lingyuan Kong ◽  
Robert G. Parker

A method is developed to evaluate the natural frequencies and vibration modes of serpentine belt drives where the belt is modeled as a moving beam with bending stiffness. Inclusion of bending stiffness leads to belt-pulley coupling not captured in moving string models. New dynamic characteristics of the system induced by belt bending stiffness are investigated. The belt-pulley coupling is studied through the evolution of the vibration modes. When the belt-pulley coupling is strong, the dynamic behavior of the system is quite different from that of the string model where there is no such coupling. The effects of major design variables on the system are discussed. The spatial discretization can be used to solve other hybrid continuous/discrete eigenvalue problems.

Author(s):  
Lingyuan Kong ◽  
Robert G. Parker

A method is developed to evaluate the natural frequencies and vibration modes of serpentine belt drives where the belt is modeled as a moving beam with bending stiffness. Inclusion of bending stiffness leads to belt-pulley coupling not captured in moving string models. New dynamic characteristics of the system induced by the belt bending stiffness are investigated. The belt-pulley coupling is studied through the evolution of the vibration modes. When the belt-pulley coupling is strong, the dynamic behavior of the system is quite different from that of the string model where there is no such coupling. The effects of major design variables on the system are discussed. The spatial discretization can be used to solve other hybrid continuous/discrete eigenvalue problems.


2003 ◽  
Vol 70 (5) ◽  
pp. 739-750 ◽  
Author(s):  
L. Kong ◽  
R. G. Parker

Serpentine belt drives with spring-loaded tensioners are now widely used in the automotive industry. Experimental measurements show that linear system vibration coupling exists between the pulley rotations and the transverse span deflections. Former models that treat the belt as a string and neglect the belt bending stiffness cannot explain this coupling phenomenon. In this paper, a new serpentine belt system model incorporating the belt bending stiffness is established. The finite belt bending stiffness causes nontrivial transverse span equilibria, in contrast to string models with straight span equilibria. Nontrivial span equilibria cause linear span-pulley coupling, and the degree of coupling is determined by the equilibrium curvatures. A computational method based on boundary value problem solvers is developed to obtain the numerically exact solution of the nonlinear equilibrium equations. An approximate analytical solution of closed-form is also obtained for the case of small bending stiffness. Based on these solutions, the effects of design variables on the equilibrium deflections and span-pulley coupling are investigated.


2004 ◽  
Vol 127 (5) ◽  
pp. 957-966 ◽  
Author(s):  
Lingyuan Kong ◽  
Robert G. Parker

Steady state analysis is conducted on a multipulley serpentine belt drive with a spring-loaded tensioner assembly. Classical creep theory is extended to incorporate belt bending stiffness as well as the belt stretching and centripetal accelerations. The belt is modeled as an axially moving Euler–Bernoulli beam with nonuniform speed due to belt extensibility and variation of belt tension. The geometry of the belt-pulley contact zones and the corresponding belt tension and friction distributions are the main factors affecting belt slip. Bending stiffness introduces nontrivial span deflections, reduces the wrap angles, and makes the belt-pulley contact points unknown a priori. The free span boundary value problems (BVP) with undetermined boundaries are transformed to a fixed boundary form. A two-loop iteration method, necessitated by the tensioner assembly, is developed to find the system steady state. The effects of system parameters on serpentine drive behavior are explored in the context of an actual automotive belt drive.


Author(s):  
Adebukola Olatunde ◽  
J. W. Zu

The objective of this paper is to optimize the belt tensioning mechanism, known as the Twin Tensioner. The optimized tensioner achieves the minimum magnitude of belt tension in a Belt-driven Integrated Starter-generator (B-ISG) system. The B-ISG is an emerging hybrid transmission that closely resembles conventional serpentine belt drives. The system contains an integrated starter-generator (ISG) unit that performs a start-stop function on the engine. A derivation of the system’s equation of motion is simulated in this paper. A parametric study evaluates the Twin Tensioner’s parameters with respect to their impact on static tensions. Design variables are selected from these parameters for optimization. The optimization uses the genetic algorithm (GA) and the sequential quadratic programming (SQP) searches. Computations for belt tension based on the optimized design variables indicate the optimal system contains spans with static tensions that are significantly lower in magnitude than in the original design.


2006 ◽  
Vol 128 (5) ◽  
pp. 555-567 ◽  
Author(s):  
Andrea Tonoli ◽  
Nicola Amati ◽  
Enrico Zenerino

Multiribbed serpentine belt drive systems are widely adopted in accessory drive automotive applications due to the better performances relative to the flat or V-belt drives. Nevertheless, they can generate unwanted noise and vibration which may affect the correct functionality and the fatigue life of the belt and of the other components of the transmission. The aim of the paper is to analyze the effect of the shear deflection in the rubber layer between the pulley and the belt fibers on the rotational dynamic behavior of the transmission. To this end the Firbank’s model has been extended to cover the case of small amplitude vibrations about mean rotational speeds. The model evidences that the shear deflection can be accounted for by an elastic term reacting to the torsional oscillations in series with a viscous term that dominates at constant speed. In addition, the axial deformation of the belt spans are taken into account. The numerical model has been validated by the comparison with the experimental results obtained on an accessory drive transmission including two pulleys and an automatic tensioner. The results show that the first rotational modes of the system are dominated by the shear deflection of the belt.


Author(s):  
João Pedro Xavier ◽  
Eliana Manuel Pinho

Among the famous dynamic string models conceived by Théodore Olivier (1793-1853) as a primary didactic tool to teach Descriptive Geometry, there are some that were strictly related to classic problems of stereotomy. This is the case of the biais passé, which was both a clear illustration of a special warped ruled surface and an example of how constructors dealt with the problem of building a skew arch, solving structural and practical stone cutting demands. The representation of the biais passé in Olivier's model achieved a perfect correspondence to its épure with Monge's Descriptive Geometry. This follow from the long development of representational tools, since the 13th century sketch of an oblique passage, as well as the improvement of constructive procedures for skew arches. Paradoxically, when Olivier presented his string model, the importance of the biais passé was already declining. Meanwhile other ruled surfaces were appropriated by architecture, some of which acquiring, beyond their inherent structural efficiency, a relevant aesthetic value.


Author(s):  
Tomio Koyama ◽  
Weiming Zhang ◽  
Masanori Kagotani

Abstract The jumping characteristics at the driven pulley of synchronous belt drives are experimentally discussed. The jumping of synchronous belt occurs when the elongation of free span of the belt is over than a certain value. In this paper, an empirical formula for the jumping torque of L type synchronous belt is presented. The following parameters are considered in the empirical formula: tensile stiffness of belt, height of belt tooth, coefficient of traction, belt installation tension, wrapping angle of belt, diameter of pulley pitch circle, bending stiffness of shaft.


1999 ◽  
Vol 121 (3) ◽  
pp. 316-321 ◽  
Author(s):  
Jian Lin ◽  
R. G. Parker

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.


2017 ◽  
Vol 116 ◽  
pp. 162-177 ◽  
Author(s):  
Yue Pan ◽  
Xiandong Liu ◽  
Yingchun Shan ◽  
Gang(Sheng) Chen

1946 ◽  
Vol 39 (6) ◽  
pp. 284-287
Author(s):  
H. von Baravalle

For the demonstration of conic sections in high schools and colleges, we generally use the w ooden model of a cone with detachable parts, showing the four kinds of plane sections, a circle, an ellipse, a parabola, and a hyperbola. The wooden model is naturally limited to fixed positions of the intersecting plane and the conic sections appear as four separate facts. As one of the most stimulating parts of a study of conic sections is the realization of how each one of the four curves changes with the position of the intersecting plane and how one kind of conic section can turn into another, a more flexible medium of demonstration seems desirable. It can be found through applying two changes to the usual demonstrations. The first one is to replace the actual cutting of a solid wooden model by projecting a cut on a larger cone, thus achieving an easy flexibility, as either the projector or the model can be moved during the demonstration. The second change is to replace a solid surface by one formed by strings, which not only makes a larger model considerably lighter, but also allows the conic sections to be seen all around the cone. The alternative to string models, models with transparent surfaces, produce too many disturbing light reflections. Figure 1 shows the string model of a cone and Figure 2 its application in the demonstration of ellipses. The system of parallel ellipses in Figure 2 is produced by parallel light planes which are obtained from an ordinary slide projector with slides showing transparent lines on black background. Instead of glass slides also rectangular pieces of stronger drawing paper can be used into which the lines have been cut. Moving the projector slightly to the side makes the ellipses in Figure 2 go up or down the cone and each ellipse widens or contracts during the motion. If one increases the angle of the planes the form of all the ellipses change until they turn into parabolas and finally hyperbolas.


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