In high-speed rotating machinery, such as the accessory drive systems of aero-engines, the dynamic characteristics of the rotor-bearing system are of paramount importance for operational stability and safety. Among various transmission elements, spiral bevel gears are widely favored due to their inherent advantages of high tooth strength, large contact ratio, and smooth, quiet operation. The meshing action of these gears introduces complex forces into the rotor system, which can significantly influence its vibrational behavior, particularly the critical speeds and response to mass unbalance. The direction of the gear’s spiral angle dictates the orientation of these meshing forces, potentially altering the system’s dynamic signature. However, comprehensive experimental studies coupling these effects with support stiffness variations are not extensively documented. This work presents a detailed experimental and theoretical analysis of a test rig rotor system incorporating a pair of meshing spiral bevel gears. We investigate the system’s critical speeds and its steady-state unbalance response under different support configurations and gear rotation directions, providing a validated framework for understanding such coupled dynamics.
The experimental apparatus is designed to mimic a simplified rotor transmission system. The main shaft, constructed from 45-grade steel, is supported at two locations. The front support utilizes a double-row self-aligning ball bearing, while the rear employs a cylindrical roller bearing. A key feature of the rig is the incorporation of squirrel-cage elastic supports at both bearing housings. These elastic supports can be conveniently replaced with rigid blocks, allowing for three distinct support configurations: both supports rigid, both supports elastic, and a hybrid case with an elastic front support and a rigid rear support. A balanced disk is mounted on the shaft, and the system is driven through a pair of spiral bevel gears, with the pinion (active gear) integrated into the main rotor shaft. The spiral angle $\beta$ for the gears is 35°. The direction of the spiral teeth—left-hand on the pinion and right-hand on the driven gear—introduces a directional component to the mesh forces dependent on the sense of shaft rotation.
The measurement system captures the dynamic response of the rotor. Two eddy-current displacement probes are positioned orthogonally (vertical and horizontal directions are measured, but primary analysis often focuses on the vertical component) near the disk and near the pinion gear to record radial shaft vibration. A photoelectric tachometer at the coupling provides a precise rotational speed signal. All analog signals are fed into a signal pre-processor for conditioning before being acquired and analyzed by a dedicated rotating machinery condition monitoring and fault diagnosis system. During testing, the rotor speed is gradually increased to 6000 rpm, and the system records the amplitude of vibration (displacement) as a function of rotational speed, generating the amplitude-frequency characteristic curve, or Bode plot, for the system.
The theoretical foundation for predicting the system dynamics rests on the Transfer Matrix Method (TMM), applied in two distinct forms. For comprehensive analysis including torsional coupling, a coupled bending-torsion segmental TMM is employed. This model discretizes the shaft mass and explicitly accounts for the gyroscopic effects and the cross-coupling introduced by the meshing spiral bevel gears. The forces and moments at the gear mesh are derived from the gear geometry, transmitted power, and spiral angle direction. The governing equation for the state vector $\mathbf{Z}$ at station *i+1* in terms of station *i* is given by the transfer matrix $\mathbf{T}_i$:
$$ \mathbf{Z}_{i+1} = \mathbf{T}_i \cdot \mathbf{Z}_i $$
where $\mathbf{Z} = [y, \theta, M, V]^T$ for bending (with $y$ displacement, $\theta$ slope, $M$ bending moment, $V$ shear force) and a similar form exists for the torsional state vector. The overall system matrix is assembled by successive multiplication of field and point matrices for each shaft segment, disk, gear mesh, and bearing. The critical speeds are found by solving the eigenvalue problem derived from the homogeneous system equations (free vibration, no unbalance) for natural frequencies $\omega_n$.
For a more focused analysis on lateral critical speeds, a simplified TMM for bending vibration is also used. This method can incorporate both distributed and lumped mass models for the shaft and accounts for the effect of axial force. The critical speed calculations using both methods under the three support conditions are summarized below.
| Support Condition | Method | 1st Bending (rpm) | 2nd Bending (rpm) | Torsional (rpm) |
|---|---|---|---|---|
| Both Rigid | Coupled TMM (Lumped) | 3250 | >6000 | ~1750 |
| Simplified TMM (Lumped) | 3180 | >6000 | N/A | |
| Both Elastic | Coupled TMM (Lumped) | 1850 | 4800 | ~1750 |
| Simplified TMM (Lumped) | 1820 | 4750 | N/A | |
| Front Elastic, Rear Rigid | Coupled TMM (Lumped) | 2150 | 5200 | ~1750 |
| Simplified TMM (Lumped) | 2120 | 5150 | N/A |
A key observation from the calculations is the presence of a critical speed near 1750 rpm that remains invariant across all support stiffness configurations. This identifies it conclusively as a torsional critical speed, primarily excited by the torque fluctuations through the gear mesh, and effectively decoupled from the lateral support conditions. The lateral critical speeds, however, show significant dependence on support stiffness, with the rigid-supported configuration yielding the highest first bending critical speed and the fully elastic the lowest, as expected.
The steady-state unbalance response is calculated by introducing forcing terms corresponding to the residual mass unbalances at the disk and the pinion gear. The unbalance magnitudes ($U_d = m_d \cdot e_d$ and $U_g = m_g \cdot e_g$) and relative phase were estimated from a slow-roll runout measurement at a very low speed (200 rpm). Damping ratios for the modes were estimated based on empirical data for similar rotor-bearing systems. The governing equation for the forced response becomes a complex matrix equation:
$$ [\mathbf{K} – \omega^2\mathbf{M} + j\omega\mathbf{C} + \mathbf{G}(\omega) + \mathbf{K}_{mesh}(\omega)]\mathbf{Q} = \omega^2 \mathbf{U} $$
where $\mathbf{K}$, $\mathbf{M}$, $\mathbf{C}$, $\mathbf{G}$ are the global stiffness, mass, damping, and gyroscopic matrices, respectively. $\mathbf{K}_{mesh}(\omega)$ is the complex stiffness matrix representing the spiral bevel gear mesh interaction, which is a function of operating conditions, spiral angle, and direction. $\mathbf{Q}$ is the complex displacement vector, and $\mathbf{U}$ is the unbalance vector. Solving this equation across a speed range yields the amplitude and phase of response at each node. Due to the cross-coupling from the spiral bevel gear mesh, the shaft orbit at any point is generally elliptical, not circular. Therefore, the response is characterized by the major axis $A_{max}$, minor axis $A_{min}$, and the amplitude in the fixed vertical direction $A_v$.
The experimental procedure involved measuring the vertical vibration amplitude ($A_v$) at the disk and pinion locations during both clockwise (CW) and counter-clockwise (CCW) rotation sweeps up to 6000 rpm, for each support configuration. The power transmitted through the spiral bevel gears was held approximately constant. The measured peak response speeds and amplitudes were then compared with theoretical predictions. The tables below present a subset of this comparison for a speed of 5000 rpm, highlighting the effects of support stiffness and rotation direction.
| Location | Rotation | Calc. $A_{max}$ | Calc. $A_{min}$ | Calc. $A_{v}$ | Expt. $A_{v}$ |
|---|---|---|---|---|---|
| Disk | CW | 18.5 | 12.1 | 16.8 | 17.2 |
| CCW | 17.8 | 11.5 | 15.9 | 16.0 | |
| Pinion | CW | 25.2 | 15.8 | 22.0 | 20.5 |
| CCW | 23.0 | 14.0 | 19.8 | 18.2 |
| Location | Rotation | Calc. $A_{max}$ | Calc. $A_{min}$ | Calc. $A_{v}$ | Expt. $A_{v}$ |
|---|---|---|---|---|---|
| Disk | CW | 15.0 | 9.5 | 13.2 | 13.8 |
| CCW | 14.2 | 8.8 | 12.3 | 12.5 | |
| Pinion | CW | 32.5 | 20.5 | 28.5 | 30.1 |
| CCW | 29.8 | 18.2 | 26.0 | 27.5 |
| Location | Rotation | Calc. $A_{max}$ | Calc. $A_{min}$ | Calc. $A_{v}$ | Expt. $A_{v}$ |
|---|---|---|---|---|---|
| Disk | CW | 12.1 | 7.2 | 10.5 | 11.0 |
| CCW | 11.5 | 6.8 | 9.9 | 10.2 | |
| Pinion | CW | 18.2 | 11.0 | 15.8 | 14.9 |
| CCW | 16.5 | 9.8 | 14.1 | 13.5 |
The agreement between the calculated vertical response $A_v$ and the experimental measurements is consistently good across all configurations, validating the theoretical model that includes the spiral bevel gear mesh dynamics. Several critical trends are evident from the data. First, the shaft orbits are elliptical, as the experimental $A_v$ value always lies between the calculated major and minor axes. Second, the direction of rotation (and thus the effective direction of the spiral bevel gear mesh forces) has a measurable impact. For this specific spiral bevel gear set (pinion LH, wheel RH), clockwise rotation consistently produced a larger response amplitude at both the disk and pinion compared to counter-clockwise rotation under the same power and speed. This directional sensitivity is a direct consequence of the spiral angle orientation modifying the bearing load distribution and cross-coupled stiffness.
Third, and perhaps most significantly, the support stiffness dramatically affects the response magnitude, particularly at the pinion location. Comparing the pinion response for CW rotation at 5000 rpm across the three configurations reveals: Elastic Supports: ~20.5 μm, Front Elastic/Rear Rigid: ~30.1 μm, Rigid Supports: ~14.9 μm. The hybrid configuration (front elastic, rear rigid) resulted in the largest pinion response, while the fully rigid configuration resulted in the smallest. This non-intuitive result for the hybrid case can be attributed to the specific modal shape excited near 5000 rpm, which likely involves a significant bending deflection at the mid-span (pinion location) when the stiffer rear support constrains motion there, but the compliant front support allows greater shaft tilt. The fully rigid system provides the greatest overall stiffness, minimizing absolute deflection. The fully elastic system provides more uniform compliance. This highlights that the interaction between the spiral bevel gear mesh point and the system’s mode shapes is complex and sensitive to support asymmetry.
The measured amplitude-frequency curves from experiment showed clear resonance peaks. The speed at which the peak amplitude occurred was slightly lower than the calculated undamped critical speed due to the influence of the spiral bevel gear mesh, which introduces additional stiffness and damping terms that are speed-dependent. The table below compares the theoretical first lateral critical speed ($\omega_{cr}$) with the experimental peak response speed ($\omega_{peak}$) for the rigid support case, showing close correlation.
| Parameter | Calculated Value (rpm) | Experimental Value (rpm) |
|---|---|---|
| 1st Bending Critical Speed ($\omega_{cr}$) | 3180 – 3250 | ~3100 |
| Peak Response Speed ($\omega_{peak}$) | ~3050 (from response calc.) | ~3050 |
In conclusion, this integrated experimental and theoretical study successfully characterized the dynamics of a rotor system incorporating spiral bevel gears. The critical speeds, including a distinct torsional mode, were accurately predicted and shown to be influenced by support stiffness. More importantly, the steady-state unbalance response was profoundly affected by both the support configuration and the rotational direction of the spiral bevel gears. The directional effect stems from the kinematic action of the spiral teeth, which imposes a specific force orientation that either amplifies or attenuates the system’s sensitivity to unbalance depending on the rotation sense. The complex interaction between the gear mesh location and asymmetric support stiffness led to a maximum response condition in a hybrid elastic-rigid configuration, underscoring the need for system-level modeling in design. The strong agreement between the transfer matrix model—which explicitly accounts for the spiral bevel gear mesh forces—and experimental data validates the presented methodology as a reliable tool for analyzing and predicting the vibrational behavior of complex rotor-gear transmission systems.