A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs

Noise, vibration and harshness analyses are of great interest for the latest developments of the gearboxes of electric vehicles. Gearboxes are now the main source of vibrations, since electric powertrains are much quieter than internal combustion engines. Traditionally, the simulation of the non-lin...

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Main Authors: Giacomo Saletti, Giuseppe Battiato, Stefano Zucca
Format: Article
Language:English
Published: MDPI AG 2025-04-01
Series:Vibration
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Online Access:https://www.mdpi.com/2571-631X/8/2/18
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author Giacomo Saletti
Giuseppe Battiato
Stefano Zucca
author_facet Giacomo Saletti
Giuseppe Battiato
Stefano Zucca
author_sort Giacomo Saletti
collection DOAJ
description Noise, vibration and harshness analyses are of great interest for the latest developments of the gearboxes of electric vehicles. Gearboxes are now the main source of vibrations, since electric powertrains are much quieter than internal combustion engines. Traditionally, the simulation of the non-linear gear dynamics is studied by first performing a series of preliminary static analyses to compute the static transmission error (STE). The STE (i.e., in the form of varying mesh stiffness) is then accepted as the system’s excitation source to compute the dynamic transmission error (DTE). This paper presents a novel approach to analyze the non-linear dynamics of gears which does not require any preliminary static analyses. The method consists of a frequency–domain approach based on the Harmonic Balance Method (HBM) and the Alternating Frequency–Time (AFT) scheme, allowing for much faster simulations when compared to the widely used direct–time integration (DTI). The contact between the teeth is modeled as intermittent and penalty based with a varying gap. The time–varying gap between the teeth is initially approximated to a step function that guarantees the design contact ratio. The methodology introduced is tested on a lumped parameter model of a spur–gear pair already proposed and simulated in the literature. The results obtained with the novel approach are compared with the DTI simulation of the model as a reference. The excellent match between the different approaches validates the reliability of developed methodology.
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series Vibration
spelling doaj-art-15f37ed78eca445984689cd9003cdebc2025-08-20T03:29:43ZengMDPI AGVibration2571-631X2025-04-01821810.3390/vibration8020018A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear PairsGiacomo Saletti0Giuseppe Battiato1Stefano Zucca2Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, 10129 Torino, ItalyDipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, 10129 Torino, ItalyDipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, 10129 Torino, ItalyNoise, vibration and harshness analyses are of great interest for the latest developments of the gearboxes of electric vehicles. Gearboxes are now the main source of vibrations, since electric powertrains are much quieter than internal combustion engines. Traditionally, the simulation of the non-linear gear dynamics is studied by first performing a series of preliminary static analyses to compute the static transmission error (STE). The STE (i.e., in the form of varying mesh stiffness) is then accepted as the system’s excitation source to compute the dynamic transmission error (DTE). This paper presents a novel approach to analyze the non-linear dynamics of gears which does not require any preliminary static analyses. The method consists of a frequency–domain approach based on the Harmonic Balance Method (HBM) and the Alternating Frequency–Time (AFT) scheme, allowing for much faster simulations when compared to the widely used direct–time integration (DTI). The contact between the teeth is modeled as intermittent and penalty based with a varying gap. The time–varying gap between the teeth is initially approximated to a step function that guarantees the design contact ratio. The methodology introduced is tested on a lumped parameter model of a spur–gear pair already proposed and simulated in the literature. The results obtained with the novel approach are compared with the DTI simulation of the model as a reference. The excellent match between the different approaches validates the reliability of developed methodology.https://www.mdpi.com/2571-631X/8/2/18gear dynamicsnon–linear dynamicsharmonic balance methodalternating frequency–timetransmission error
spellingShingle Giacomo Saletti
Giuseppe Battiato
Stefano Zucca
A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
Vibration
gear dynamics
non–linear dynamics
harmonic balance method
alternating frequency–time
transmission error
title A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
title_full A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
title_fullStr A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
title_full_unstemmed A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
title_short A Coupled Harmonic Balance-Based Approach for the Non-Linear Dynamics of Spur-Gear Pairs
title_sort coupled harmonic balance based approach for the non linear dynamics of spur gear pairs
topic gear dynamics
non–linear dynamics
harmonic balance method
alternating frequency–time
transmission error
url https://www.mdpi.com/2571-631X/8/2/18
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