The value of meshing impact is highly recognized due to its effect on vibration and noise in mechanical systems. Meshing impacts were investigated in the past using the impulse method characterized by smaller impact forces and longer impact times. They are calculated using the energy method in the paper to demonstrate that the impact force is over eight times greater than that method, while impact time is reduced to approximately one-sixth of that method. An improved model of meshing impact has been developed based on the kinetic energy theorem and Hertzian contact theory to enhance the accuracy of the results. An improved nonlinear dynamics model of spur gear pairs has been developed based on meshing impacts. The ith tooth pair from meshing-in to meshing-out will experience meshing-in impact (MII) → double tooth-pair meshing zone (DTMZ) → double-to-single switching impact (DSSI) → single tooth-pair meshing zone (STMZ) → single-to-double switching impact (SDSI) → DTMZ → meshing-out impact (MOI). Boundary impacts are observed when the gear teeth arrive at the boundary from disengagement, including driving-tooth-side boundary impact (DBI) and backing-tooth-side boundary impact (BBI). The dynamics of the system both with and without the meshing effects are investigated using bifurcation diagrams, top Lyapunov exponent (TLE) diagrams, Poincaré maps, phase portraits, and force–time diagrams. The results indicate that larger force mutations lead to more pronounced impact phenomena and greater changes in displacement and velocity. This research lays the groundwork for future investigations into the meshing impacts of other gears. @@ The ith teeth from meshing-in to meshing-out will experience meshing-in impact <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{{\text{Mi}}}}^{i} \to t_{{{\text{Mi1}}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mi</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mi1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → double tooth-pair meshing zone <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{{\text{Mi1}}}}^{i} \to t_{{\text{A}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mi1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>A</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → double-to-single switching impact <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{\text{A}}}^{i} \to t_{{{\text{A1}}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>A</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>A1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → single tooth-pair meshing zone <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{{\text{A1}}}}^{i} \to t_{{\text{C}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>A1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>C</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → single-to-double switching impact <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{\text{C}}}^{i} \to t_{{{\text{C1}}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>C</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>C1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → double tooth-pair meshing zone <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{{\text{C1}}}}^{i} \to t_{{{\text{Mo}}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>C1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mo</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> → meshing-out impact <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2025_1963_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$(t_{{{\text{Mo}}}}^{i} \to t_{{{\text{Mo1}}}}^{i} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mo</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>t</mi> <mrow> <mtext>Mo1</mtext> </mrow> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> according to the multi-state meshing and meshing impacts of the spur gear pair. Boundary impacts are observed when a tooth reaches the boundary from disengagement, including driving-tooth-side boundary impact and backing-tooth-side boundary impact. An improved model of meshing impact has been established based on the kinetic energy theorem and Hertzian contact theory. An improved nonlinear dynamics model of spur gear pair is developed based on the multi-state meshing and meshing impacts. The dynamics of the system considering meshing impacts and without meshing impacts are investigated using bifurcation diagrams, top Lyapunov exponent diagrams, Poincaré maps, phase portraits, and force–time diagrams. Graphical abstract

Dynamics of spur gear pairs considering meshing impacts based on energy methods

期刊论文
发表日期：

2025年4月2日

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Lingyun Zhu,
Yanfa Guan,
Xiangfeng Gou

摘要

The value of meshing impact is highly recognized due to its effect on vibration and noise in mechanical systems. Meshing impacts were investigated in the past using the impulse method characterized by smaller impact forces and longer impact times. They are calculated using the energy method in the paper to demonstrate that the impact force is over eight times greater than that method, while impact time is reduced to approximately one-sixth of that method. An improved model of meshing impact has been developed based on the kinetic energy theorem and Hertzian contact theory to enhance the accuracy of the results. An improved nonlinear dynamics model of spur gear pairs has been developed based on meshing impacts. The ith tooth pair from meshing-in to meshing-out will experience meshing-in impact (MII) → double tooth-pair meshing zone (DTMZ) → double-to-single switching impact (DSSI) → single tooth-pair meshing zone (STMZ) → single-to-double switching impact (SDSI) → DTMZ → meshing-out impact (MOI). Boundary impacts are observed when the gear teeth arrive at the boundary from disengagement, including driving-tooth-side boundary impact (DBI) and backing-tooth-side boundary impact (BBI). The dynamics of the system both with and without the meshing effects are investigated using bifurcation diagrams, top Lyapunov exponent (TLE) diagrams, Poincaré maps, phase portraits, and force–time diagrams. The results indicate that larger force mutations lead to more pronounced impact phenomena and greater changes in displacement and velocity. This research lays the groundwork for future investigations into the meshing impacts of other gears. @@ The ith teeth from meshing-in to meshing-out will experience meshing-in impact $(t_{{{\text{Mi}}}}^{i} \to t_{{{\text{Mi1}}}}^{i} )$ $(t_{Mi}^{i} \to t_{Mi1}^{i})$ → double tooth-pair meshing zone $(t_{{{\text{Mi1}}}}^{i} \to t_{{\text{A}}}^{i} )$ $(t_{Mi1}^{i} \to t_{A}^{i})$ → double-to-single switching impact $(t_{{\text{A}}}^{i} \to t_{{{\text{A1}}}}^{i} )$ $(t_{A}^{i} \to t_{A1}^{i})$ → single tooth-pair meshing zone $(t_{{{\text{A1}}}}^{i} \to t_{{\text{C}}}^{i} )$ $(t_{A1}^{i} \to t_{C}^{i})$ → single-to-double switching impact $(t_{{\text{C}}}^{i} \to t_{{{\text{C1}}}}^{i} )$ $(t_{C}^{i} \to t_{C1}^{i})$ → double tooth-pair meshing zone $(t_{{{\text{C1}}}}^{i} \to t_{{{\text{Mo}}}}^{i} )$ $(t_{C1}^{i} \to t_{Mo}^{i})$ → meshing-out impact $(t_{{{\text{Mo}}}}^{i} \to t_{{{\text{Mo1}}}}^{i} )$ $(t_{Mo}^{i} \to t_{Mo1}^{i})$ according to the multi-state meshing and meshing impacts of the spur gear pair. Boundary impacts are observed when a tooth reaches the boundary from disengagement, including driving-tooth-side boundary impact and backing-tooth-side boundary impact. An improved model of meshing impact has been established based on the kinetic energy theorem and Hertzian contact theory. An improved nonlinear dynamics model of spur gear pair is developed based on the multi-state meshing and meshing impacts. The dynamics of the system considering meshing impacts and without meshing impacts are investigated using bifurcation diagrams, top Lyapunov exponent diagrams, Poincaré maps, phase portraits, and force–time diagrams.

Graphical abstract