Ford Motor Company: Analytical Prediction of

Chain Drive System Resonance

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 they are very effective, the undesirable noise and vibrations created have always been a problem. This was particularly the case with a new prototype trans-mission at Ford, where the drive train development group detected a severe 1800 – 1900 Hz chain noise. Sound pres-sure levels of this noise were 10-15 dB over nominal values and the root cause was unknown.

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

Earlier experimental research has identified chain-sprocket meshing noise as the most significant noise source, and it has been suggested that the 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. In continuing the research, 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, which showed the existence of three types of chain resonance – the transverse strand resonance; the longitudi-nal chain sprocket coupled resonance; and the longitudinal chain stress wave type resonance.

Symbolic and Numerical Math Solvers Used to Develop ModelsTo help deal with the complex calculations and analysis involved in developing these advanced models, the math-ematical software Maple was used. Its extensive symbolic and numerical math solvers were used in modeling the physical system to gain an understanding of the vibrational behavior. The partial differential equations used in the model were solved quickly and easily using Maple’s world-lead-ing 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 capabilities of Maple enabled the publication of integrated worksheets and reports for easy and convenient dissemination across Ford’s organization.

Using Maple enabled the mathematical model predictions to be validated against both an ABAQUS CAE model and the experimental test results. Furthermore, a predictive design tool to develop analytical models and predict chain drive dy-

namics was created using Maple’s Embedded Components, including features like 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 analyti-cal power and modeling capabilities enabled us to get the accuracy we were aiming for”, said Jack S.P. Liu, CAE engineer, 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.”

Why Maple?

• Symbolic & numerical math solvers for physical system modeling

• Slider capability for design variables

• Integrated worksheet for model & report publication

Integrated Model & Report Publication

Calculating Resonances1. Transverse Chain Natural FrequencyThe first natural frequency is represented by:

where , the transverse wave speed

Chain resonances occur at:

2. Longitudinal Chain Natural FrequencyAssuming rigid support and no damping, the non-dimen-sional linear equations of motion for the longitudinal chain span (around w=0) are given as follows:

Boundary Conditions:

By combining transverse and longitudinal natural frequen-cies, excessive resonance was found to occur at the 28th harmonic.

ResultsThe Ford team was able to accurately determine the exact locations of the 1800 Hz noise source and the problematic noise peak. The objective 1800 Hz noise source seemed to occur at the 28th resonant frequency, and the noise peak between 600-800 rpm seemed to occur in the 22nd-23rd band width. 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 concludes that a thorough understanding of all types of chain resonances is critical for powertrain engineers in designing a quiet and smooth chain drive system. Plans to develop analytical models for predicting chain drive mechanics using Maple are now underway.

© Maplesoft, a division of Waterloo Maple Inc., 2007. Maplesoft, and Maple are trademarks of Waterloo Maple Inc. All other trademarks are the property of their respective owners.

www.maplesoft.com | info@maplesoft.com