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This study demonstrates the transient and steady state dynamic loading on teeth within a two-stage gear transmission arising from backlash and geometric manufacturing errors by utilizing a nonlinear multibody dynamics software model. Backlash between gear teeth which is essential to provide better lubrication on tooth surfaces and to eliminate interference is included as a defect and a necessary part of transmission design. Torsional vibration is shown to cause teeth separation and double-sided impacts in unloaded and lightly loaded gearing drives. Vibration and impact force distinctions between backlash and combinations of transmission errors are demonstrated under different initial velocities and load conditions. The backlash and manufacturing errors in the first stage of the gear train are distinct from those of the second stage. By analyzing the signal at a location between the two stages, the mutually affected impact forces are observed from different gear pairs, a phenomenon not observed from single pair of gears. Frequency analysis shows the appearance of side band modulations as well as harmonics of the gear mesh frequency. A joint time-frequency response analysis during startup illustrates the manner in which contact forces increase during acceleration.

Gear trains with different designs play very important roles in automobiles, helicopters, wind turbines, and other modern industries. Excessive loading on the gear teeth may arise due to the combination of gear backlash and teeth defects. Without vibration health monitoring to ensure proper operation performance will degrade.

Dubowsky and Freudenstein [

Two notable review papers that discuss the numerical modeling of gear dynamics are by Özgüven and Houser in 1988 [

Previous research shows that the signal patterns due to the combination of backlash, time-varying gear mesh stiffness, and the involute profile errors are very complicated and highly depend on gear train design and configurations. In other words, the signals from a specific gearing system are difficult to interpret until a series of modeling, testing, and data processing work are carried out. However, it is not realistic to experimentally test each type of gear train for the specific fault patterns. To solve this issue, a virtual experiment method based on multibody dynamics and nonlinear contact mechanics simulation is presented. Ebrahimi and Eberhard [^{3} electric mining shovel. The nonlinear contact mechanics is analyzed to predict the bearing support force variation and gear tooth loading of ideal gears and gears with defects using multibody dynamics software. No gear backlash was considered. In this study, the authors demonstrate the importance of accurate geometric modeling of gear tooth involutes, and realistic center distance separation on the transient response of ideal and defective gears. The highly nonlinear character of loading and geometry requires special attention to Hertzian contact modeling. Once modeled accurately, double-sided tooth impacts and associated loading can be determined as well as superposition of effects at a shaft intermediate to sets of gears. The analysis from frequency domain indicates that an eccentric tooth on a gear installed on the intermediate shaft results in a significant increase in force magnitude components. The amplitude of the spectral line at the first-stage gear mesh frequency increases dramatically.

In order to investigate how the interaction of backlash and manufacturing errors affects the dynamic behavior and contact forces of a multiple stage gearing system, the slider-crank mechanism shown in Figure

Gear design parameters.

Modules (mm/tooth) | |
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Number of teeth | |

Standard pitch circle diameter (mm) | |

Total gear ratio | 13.375 |

Pressure angle | 20° |

Simulation parameters.

Backlashes (mm) | |
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Material properties | |

^{3} | |

Force exponent | 2.2 |

Penetration | 10^{−7} mm |

Stiffness | 2 × 10^{7} N/mm |

Crank-slider mechanism.

Two-stage gear train.

The MSC.ADAMS IMPACT algorithm was chosen as the contact force model because of its robustness in numerical integration. The restitution model is extremely sensitive to the duration of the contact event and is best suited for impulse type simulations. It is not ideal for time histories that include a large number of contact events in which the force vector is not known beforehand. The stiffness parameter is reasonable for this lightly loaded steel gear pair and was determined via a trial and error method. The response of interest occurs over a very short time interval, around one hundred milliseconds. Because the damping force in meshing gears is such a small percentage of

Geometric defects including (a) chipped tooth of Gear A and (b) eccentric tooth of Gear C.

The “perfect” waveform represented in Figure

Free vibration response with input shaft, 167.5 rad/s; intermediate shaft, −73.5 rad/s; output shaft, 12.5 rad/s.

The response of the intermediate shaft is due to a superposition of the impacts in Stage 1 and Stage 2. The angular velocity of the intermediate shaft is shown to be combination of the other two shafts’ angular velocities. Double-sided impacts are evident in the time history of the

For the chipped pinion case, the impact force occurs at an earlier time. This can be explained in part through the reduction in inertia. The mass moment of interia of the perfect input pinion is ^{2}, while the chipped tooth pinion is ^{2}. This is a reduction of 1.76%. The chipped pinion will experience a larger acceleration for a given impact force. The teeth neighboring the chipped tooth will contact the mating gear sooner than it would with the standard inertia.

The force response in Stage 2 is depicted in Figure

Comparison of the force magnitudes of Stage 1 and Stage 2 with initial conditions: input shaft, 167.5 rad/s; intermediate shaft, −73.5 rad/s; output shaft, 12.5 rad/s.

An impact occurs in Stage 2 before Stage 1; at first this may seem counterintuitive. The initial velocities given to each shaft are based upon their rated operating speed. The bearings are modeled as frictionless, constraining all degrees of freedom except for rotation in the normal plane. The relative velocity on the pitch circle of Gear C and Gear D is slightly larger than between Gear A and Gear B. For initial conditions in which the relative velocity between Gear A and Gear B is larger than between Gear C and Gear D, the opposite would occur.

The system is modeled as the interaction of three rigid bodies. The first is the input shaft and Gear A. The second is Gear B, the intermediate shaft, and Gear C. The third is Gear D, the output shaft, and the crank. The third body has an inertial mass at least one order of magnitude larger than the other two bodies. Its velocity changes more slowly than for the other two bodies due to inertia effects. The small delay between responses around 18.5 ms can be attributed to the chipped tooth. Because the tooth is chipped, the force response in Stage 2 is slightly delayed.

Figure

Relative displacement

For the case with a chipped tooth on the input pinion, the entire curve is shifted forward in time. Although the impacts in Stage 1 occur earlier, the overall effect in Stage 2 is delayed. The chipped tooth causes Stage 1 to become more excited, as a result it takes longer for the contact in Stage 2 to occur.

A practical step torque of the form

From Figure

Comparison of relative displacements

From Figure

Comparison of the force magnitudes on Stage 1 and Stage 2 with step input torque 149.123 N-m.

In order to demonstrate how the frequency contents of the contact force evolve over time, a joint time-frequency analysis is presented based on the transient start-up conditions. For this procedure, aliasing issues are prevented by using a large number of integration steps and a long simulation duration of 3 seconds. FFT leakage is reduced by overlapping a sliding time sample of 50 ms by 80% and applying a Hamming window to each sample.

From Figure _{1} = 1653 Hz at the first time slice of _{1}) = 3306 Hz. The lines which originate below 100 Hz are the element spin speeds and their respective superharmonics. The largest spin speed amplitude corresponds to input 1X which has a value of 98 Hz at

Three-dimensional FFT of force magnitude in Stage 1 for prescribed backlash and perfect geometry with the application of exponential step torque 149.6 N-m.

To obtain the frequency domain response of Stage 1, a constant angular speed is applied to the input shaft and a small resistive torque on the intermediate and output shafts, shown in Figure

A comparison of frequency domain components of force magnitude on (a) Stage 1 and (b) Stage 2 with prescribed backlash and perfect geometry.

The frequency components in each mesh include the respective gear mesh frequencies and their super harmonics. The shaft speeds and hunting tooth frequencies do not appear because the mesh geometry contains prescribed backlash without manufacturing errors. The force magnitudes in Stage 1 are larger because the input pinion has the smallest inertia and experiences the largest angular accelerations for a given impact. If the frequency components under one hundred Newtons are considered erroneous noise, then the first five harmonics of the gear mesh frequency in each stage comprise the vast majority total force vectors.

Stage 1 initial conditions are input shaft, 167.5 rad/s (26.7 Hz); intermediate shaft, −5.27 N-m; output shaft, −20.0 N-m; Stage 2 initial conditions are input shaft, −1.49 N-m, intermediate shaft, −5.27 N-m; output shaft, 12.5 rad/s (1.99 Hz). The contact events in Stage 1 and Stage 2 are coupled together. This is evident in the frequency spectrum as sideband modulation. The force vector in Stage 1 is modulated by the output shaft speed of 1.99 Hz creating relatively small sidebands surrounding the gear mesh harmonics, shown in Figure

An eccentric tooth on Gear C results in a significant increase in force magnitude components below the gear mesh frequency, shown in Figure _{1} increases 50.4%. This is a potentially new and important vibration signature of the defected gear train.

Frequency domain components of force magnitude on Stage 2 with an eccentric gear tooth, initial conditions: input shaft, −1.49 N-m, intermediate shaft, −5.27 N-m; output shaft, 12.5 rad/s (1.99 Hz).

A nonlinear multibody dynamic software model has been developed for a two stage slider-crank mechanism to demonstrate the effects of dynamic loading on gear teeth with defects during transient, start-up, and steady-state operation. The stiffness, force exponent, damping, and friction coefficients for the MSC.IMPACT force algorithm are presented. The dynamic behavior of the intermediate shaft of a two stage slider-crank mechanism is shown to be a superposition of the impact forces acting in each mesh. The geometric error of a chipped tooth on the pinion gear of this mechanism causes a delay in the contact forces in the second stage. A joint time-frequency analysis on a realistic driving step torque reveals spectral components which increase in frequency and magnitude as the crank accelerates through its operating speed. Frequency domain results of steady state operation demonstrate that the response is dominated by the gear mesh frequency and its harmonics.

The authors acknowledge the Donald E. Bently Center for Engineering Innovation at California Polytechnic State University San Luis Obispo for support of this work.