Dynamic responses of an engine front-end accessory belt drive system with pulley eccentricities via two spatial discretization methods

2017 ◽  
Vol 232 (4) ◽  
pp. 482-498 ◽  
Author(s):  
Hao Zhu ◽  
Yumei Hu ◽  
Weidong Zhu ◽  
Haiqiang Long
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bending Stiffness ◽  
Drive System ◽  
Dynamic Responses ◽  
Belt Drive ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Galerkin Discretization ◽  
Front End

In this study, a generic mathematical model for calculating the natural frequencies and the dynamic responses of a typical front-end accessory drive system with any number of pulleys and arbitrary configurations of the tensioner and pulleys is established. The belt bending stiffness and the pulley eccentricities are considered in this model, and their influences on the natural frequency and the dynamic responses of the front-end accessory drive system are examined. A generic spatial discretization method and a Galerkin discretization method, which uses Lagrange multipliers to enhance the boundary conditions, are presented to discretize the continuous belt spans and to transform the governing partial differential equations into ordinary differential equations. The accuracies of the generic spatial discretization method and the Galerkin discretization method are validated by modal tests, and the advantages of the generic spatial discretization method with respect to the efficiency and the convenience of implementation are assessed by comparing the generic spatial discretization method with the Galerkin discretization method and the two-layer iteration approach. The dynamic responses of the typical front-end accessory drive system at different operational velocities are calculated from the governing ordinary differential equations derived from these two methods. It is shown that large vibration amplitudes occur in certain belt spans owing to the resonance conditions or the beat phenomena in certain operational conditions and that the belt bending stiffness has a negligible influence on the vibrations of the belt drive system because its value is small.

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Dynamic Analysis of a Rotating Planar Timoshenko Beam Using an Accurate Global Spatial Discretization Method

2019 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Zhu
Keyword(s):  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Slope Angle ◽  
Dynamic Responses ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Dependent Variables ◽  
Spatial Derivatives ◽  
Shear Strains

Abstract A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.

Dynamic response of a front end accessory drive system and parameter optimization for vibration reduction via a genetic algorithm

2016 ◽  
Vol 24 (11) ◽  
pp. 2201-2220 ◽  
Author(s):  
Hao Zhu ◽  
Yumei Hu ◽  
WD Zhu
Keyword(s):  
Genetic Algorithm ◽  
Dynamic Response ◽  
Flexural Rigidity ◽  
Optimization Method ◽  
Drive System ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Front End ◽  
Space Technique ◽  
External Forces

A typical engine front end accessory drive system (FEADS) is mathematically modeled through Hamilton’s principle and Newton’s second law. In this model, the belt’s flexural rigidity and pulley’s eccentricity are considered. Eccentricities of the pulleys are introduced into governing motion equations of the belt spans through the boundary conditions and then transformed to external forces acting on the belt spans. Vibration modes and natural frequencies of the FEADS are calculated by the state-space technique of the complex mode theory. Dynamic responses of the FEADS at different rotational rates of the crankshaft are calculated by solving the spatially discretized governing equations obtained by Galerkin method. The modeling and solution methods are formulated and programmed in a general purposed code. The study shows that the typical resonance and beat phenomenon happen in a certain portion of the belt spans at a certain rotational rate by the excitations of the pulley’s eccentricity. According to the modal analysis and dynamic response analysis, an optimization method based on a genetic algorithm is proposed. By comparing the vibration amplitudes of belt spans before and after optimization at different rotational rates, this optimization method is verified to be effective in reducing transverse vibrations of the belt spans.

scholarly journals A novel analytical model for the thermal behavior of a fiber-reinforced plastic pulley in a front-end accessory drive

2020 ◽  
Vol 12 (6) ◽  
pp. 168781402092449 ◽  
Author(s):  
Xingchen Liu ◽  
Kamran Behdinan
Keyword(s):  
Temperature Distribution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Model ◽  
Operating Conditions ◽  
Belt Drive ◽  
Thermal Flow ◽  
Reinforced Plastic ◽  
Proposed Model ◽  
High Thermal Resistance

This research proposes an innovative model for calculating the temperature distribution of a composite pulley inside a belt drive. The main advantage of the proposed model is a significant reduction in the costs of calculation resources and time. This model adopts two classical theories to determine the heat generation flux and subsequent thermal flow into the pulley. Then, ordinary differential equations are developed in this model according to the irregular geometric structures of a pulley to describe the thermal flow inside this component. Afterward, analytical solutions of the ordinary differential equations are derived to provide final temperature distributions of the pulleys. Moreover, measurements of thermal properties are conducted to reduce the influence of errors. To improve the reliability of the results, experimental temperature measurements were performed on a composite pulley of a designed belt drive system in an engine dynamometer system to validate this analytical model under various operating conditions. The temperature data measured at multiple locations indicate good agreement with the corresponding analytical results. Therefore, the temperature distribution provided by this model can be utilized for the development of high-thermal resistance composite. It can also be used for thermal fatigue simulations of composites under numerous load cases.

scholarly journals Innovative analytical model for temperature prediction of front-end accessory drive

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xingchen Liu ◽  
Kamran Behdinan
Keyword(s):  
Heat Transfer ◽  
Computational Cost ◽  
Operating Conditions ◽  
Drive System ◽  
Belt Drive ◽  
Static State ◽  
Front End ◽  
Proposed Model ◽  
Wide Range ◽  
Analytical Thermal Model

AbstractThe front-end accessory drive belt drive system is a critical component in the vehicle engine. To avoid thermal deterioration under static state operating conditions, the thermal distribution for the belt drive system at each condition must be determined in an efficient manner. Due to the numerical approach is not feasible to address this concern because of its high computational cost, this paper proposes a reliable and efficient novel analytical thermal model to achieve this goal. This work develops the state-of-the-art heat transfer ordinary differential equations (ODEs) describing the thermal flow and heat dissipations on the complex structures of pulleys. Then it integrates these ODEs with heat transfer governing equations of the belt and heat exchanges to establish an innovative system of equations that can be solved within a few seconds to provide temperature plots. Moreover, experiments were conducted on a dynamometer to verify the accuracy of the proposed model under a wide range of conditions. The results indicate that the measured temperatures are in good agreement with the corresponding analytical results. Owing to its efficiency, the proposed model can be integrated with other mechanical characterizations of the belt drive system in terms of design, optimization, and thermal fatigue analyses.

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