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Symbolic and numerical math solvers used to gain understanding of vibrational behaviour and to develop predictive design tools

The Problem
Like other automotive manufacturers, Ford Motor Company wrestled with a common concern – incessant noise and vibration in chain drive systems. Chain drives have been widely used for power transmission in automotive systems for decades. While chain drives are effective, the undesirable noise and vibrations created have always been a problem. This was particularly the case when Ford detected a severe 1800 – 1900 Hz chain noise in a new transmission prototype. Sound pressure levels were 10 -15 dB over nominal values and the cause was unknown.

At Ford, Jack S.P. Liu, Das Ramnath, and Rajesh Adhikari set out to understand the source of the noise and develop simple, analytical models for quick computation of the chain drive system resonances.

The Process
Earlier experimental research identified chain-sprocket meshing noise as the most significant noise source, and suggested that chain drive system dynamic parameters such as speed, tension, mass, and pitch of the chain, sprocket inertia, and the natural frequencies of the chain sprocket system are closely related to the meshing noise. The Ford team took on the challenge of analytically predicting chain drive system resonance based on the assumption that existence of chain resonances can amplify the radiated chain meshing noise.

The team started with the analysis of the chain noise test data and compared this with the theoretical mathematical model. Their results indicated that three types of chain resonance existed: the transverse strand resonance, the longitudinal chain sprocket coupled resonance, and the longitudinal chain stress wave type resonance.

To help deal with the complex calculations and analysis involved in developing these advanced models, Ford used the mathematical software Maple™. Its extensive symbolic and numeric math solvers were used in modelling the physical system to gain an understanding of the vibrational behaviour. The partial differential equations used in the model were solved quickly and easily using Maple’s world-leading differential equation features. When describing results, such as the eigenfunctions that represent the unique mode shapes of the natural resonant frequencies, Maple’s extensive plotting capabilities were indispensable. In addition, the unique documentation capability of Maple enabled Ford to publish integrated worksheets and reports for easy and convenient dissemination across the organization.

By using Maple, Ford could validate mathematical model predictions against both an ABAQUS CAE model and the experimental test results. Also, Ford created a predictive design tool to develop analytical models and predict chain drive dynamics using Maple’s Embedded Components*, including features such as variable slider inputs to modify design variables. This design tool will enable other technical staff to perform future predictions of chain-drive resonances in a quick and easy manner.

“We were amazed at the power of Maple. Its analytical power and modelling capabilities enabled us to get the accuracy we were aiming for,” said Jack S.P. Liu, a CAE engineer at Ford Motor Company. “I especially appreciate Embedded Components and their role in GUI design. Maple’s symbolic math capability exceeds that of other CAE tools in areas where we used it.”

The Results
The Ford team was able to accurately determine the exact locations of the 1800 Hz noise source and the problematic noise peak. By combining transverse and longitudinal natural frequencies, both the analytical and CAE models predicted the 1800-1900 Hz longitudinal chain resonance as observed in chain test data. The team concluded that a thorough understanding of all types of chain resonances is critical for powertrain engineers to design a quiet and smooth chain drive system. Currently, Ford is planning to develop analytical models for predicting chain drive mechanics using Maple.

* Embedded Components allow you to build easy-to-use custom interfaces to interact with the Maple computation engine. They can include such elements as buttons, sliders, plots, check boxes, and boxes for entering and displaying mathematics.

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