To investigate the influence of engine torque fluctuations on the response amplitude and transient characteristics of the rotor system, this section conducts a theoretical analysis based on a representative single-degree-of-freedom (SDOF) torsional vibration model. From a mechanistic perspective, the intrinsic interaction between torque fluctuations and system dynamic response is elucidated, thereby providing theoretical support for the subsequent analysis of multi-degree-of-freedom (MDOF) systems.

The torsional equation of motion for a typical SDOF system can be expressed as 1Jθ¨+cθ˙+kθ=Tt, where J denotes the rotational inertia, c the torsional damping coefficient, k the torsional stiffness, and Tt the external input torque.

Considering that the engine output torque under real operating conditions exhibits pronounced nonstationary fluctuation characteristics, its variation process, although complex, generally possesses a finite rate of change that is much smaller than the theoretically infinite rate associated with an ideal step load. Therefore, within a sufficiently small time interval, the torque fluctuation signal can be approximated in a piecewise manner as a linear function with a time-varying slope, i.e., an equivalent ramp-type torque input: T(t)=αt, where α denotes the rate of torque variation.

Applying the Laplace transform to Eq. (1) yields 2Js2Θs+csΘs+kΘs=αs2. After obtaining the transfer function, it can be reformulated into the standard second-order system representation, which is expressed as 3Θs=α/Js2s2+2ζωns+ωn2. Applying the inverse Laplace transform yields the torsional response of the SDOF system as 4θt=αkt-2ζωn+e-ζωntAcos⁡(ωdt)+Bsin⁡(ωdt), where ωn=kJ denotes the natural frequency of the system, ζ=c2Jk is the damping ratio, and ωd=ωn1-ζ2 is the damped natural frequency. The constants A and B are determined by the initial conditions.

According to Eq. (4), the system response can be decomposed into three components:

The linear growth term αtk. This term represents the quasi-static response associated with the time-varying torque input. Its slope is directly proportional to the torque variation rate α.

These results indicate that the torque variation rate α is the key parameter controlling transient response intensity. Larger α produces stronger transient vibrations, whereas smaller α leads to rapid attenuation toward a quasi-static state. Overall, torsional vibration is governed primarily by the torque variation rate rather than its amplitude, providing a theoretical basis for analyzing torque fluctuation effects in subsequent MDOF models.

The PIC, a key component of the transmission system, transfers engine power to the combining gearbox, which drives the main and tail rotors. It comprises the power shaft, first-stage reduction gear pair, elastic shaft, second-stage reduction pinion, and combining gear, interconnected by spline and diaphragm couplings, as well as a strut-type overrunning clutch for engagement control. As the power turbine is directly coupled to the PIC, its torque fluctuations strongly affect the torsional response of the system. The 3D schematic of the PT–PIC system is shown in Fig. 1.

A lumped-parameter approach is adopted to model the PT–PIC system. Each component is represented by rigid disks, massless damping, and torsional elastic elements, considering only torsional degrees of freedom. Nonlinear effects such as gear meshing errors and backlash are neglected to emphasize the dominant influence of torque excitation. The system is thus simplified into an 8-degree-of-freedom torsional model, as shown in Fig. 1, with angular displacements of the equivalent inertia disks as generalized coordinates. Based on this model, the torsional vibration responses under different torque fluctuation excitations can be systematically analyzed.

The rotational inertia of each component is obtained through SolidWorks-based modeling, while the corresponding torsional stiffness values are calculated via finite-element analysis using ANSYS. The resulting parameters of the equivalent disks and the dynamic properties of each component are summarized in Tables 1–2.

Based on Newton's second law, the torsional vibration dynamic equations of the coupled PT–PIC system, incorporating the effects of engine torque fluctuations, are established as follows:

5J1θ¨1+c1θ˙1-θ˙2+k1θ1-θ2=T1J2θ˙2+c1θ˙2-θ˙1+c2θ˙2-θ˙3+k1θ2-θ1+k2θ2-θ3=0J3θ˙3+c2θ˙3-θ˙2+c3θ˙3-θ˙4+k2θ3-θ2+k3θ3-θ4=0J4θ¨4+c3θ˙4-θ˙3+k3θ4-θ3+Td1rb1=0J5θ¨5+c5θ˙5-θ˙6+k5θ5-θ6-Td1rb2=0J6θ¨6+c5θ˙6-θ˙5+c6θ˙6-θ˙7+k5θ6-θ5+k6θ6-θ7=0J7θ¨7+c6θ˙7-θ˙6+k6θ7-θ6+Td2rb3=0J8θ¨8-Td2rb4=-T2Td1=cm1rb1θ˙4-rb2θ˙5+km1rb1θ4-rb2θ5Td2=cm2rb3θ˙7-rb4θ˙8+km2rb3θ7-rb4θ8, where T1 and T2 denote the power turbine torque and load torque, respectively. For each operating condition, T2 is assumed constant and determined by multiplying the mean value of the measured T1 by the overall transmission ratio, representing steady rotor power demand while ensuring that torsional response variations are primarily driven by the time-varying fluctuations of T1. θii=1,2,…,8 denotes the torsional displacement of the equivalent disks. Gears 1–2 represent the first-stage reduction pair, and gears 3–4 represent the second-stage reduction pair. rbii=1,2,3,4 are the base radii of the four gears, and Td1 and Td2 denote the dynamic meshing forces of the first- and second-stage gear pairs, respectively.

By selecting the angular displacement and angular velocity of each equivalent inertia disk as state variables, the state-space representation of the system can be established. Subsequently, the torsional vibration response of the system can be solved using the numerical integration solver Ode15s in MATLAB. The angular displacements of each equivalent inertia disk, θi, are first computed, and the relative torsional angular displacements between adjacent disks are then defined as 6θ12=θ1-θ2,θ23=θ2-θ3,θ34=θ3-θ4,θ45=θ4-i12⋅θ5,θ56=θ5-θ6,θ67=θ6-θ7,θ78=θ7-i34⋅θ8, where i12 and i34 denote the transmission ratios of the first- and second-stage gear pairs, respectively, defined as i12=3 and i34=5. The physical definitions of the relative torsional angles are listed in Table 3.

To validate the proposed lumped-parameter model, the eigenvalue problem of the undamped free torsional vibration system is solved to obtain its natural frequencies. A corresponding PT–PIC dynamic model is also developed in Romax, and modal analysis is performed to extract the first two torsional natural frequencies and mode shapes. The results show that the relative errors of the first two natural frequencies are both within 2.7 % (see Table 4), confirming that the model accurately captures the system's torsional dynamics and is suitable for subsequent response analysis under torque fluctuation excitation.

Because the system belongs to a military helicopter platform, full-scale torsional vibration measurements are not publicly available. In such cases, high-fidelity transmission simulation tools serve as practical engineering benchmarks. The close agreement with Romax therefore provides credible validation, although it is currently limited to modal characteristics rather than response-level experimental data. Future work will incorporate higher-fidelity co-simulation and, when available, experimental measurements to further strengthen validation of torsional vibration responses.

Power turbine torque exhibits pronounced nonstationary behavior that varies with flight condition while remaining statistically consistent within each condition. To characterize the excitation, time-domain and statistical analyses are performed using engine data from a representative mission, focusing on three typical operating conditions: hover, aggressive vertical maneuvers, and high-altitude climb and descent maneuver. These scenarios capture distinct rotor power demands and control actions. Hover corresponds to steady hovering or station-keeping flight, during which the engine operates near a quasi-steady state and the rotor power demand remains nearly constant, resulting in small, low-frequency torque fluctuations. Aggressive vertical maneuvers represent rapid low-altitude vertical repositioning (e.g., quick take-off, obstacle avoidance, or emergency climb/descent within ≈30 m), where frequent collective pitch adjustments generate high-rate, quasi-periodic torque variations and strong transient excitation. In contrast, the high-altitude climb and descent maneuver corresponds to large-altitude transition flight (≈200 m), typical of mission transit and terrain clearance, characterized by sustained power increase during climb followed by gradual power reduction during descent, leading to slower but larger-amplitude torque evolution. The corresponding time histories of power turbine torque and speed are presented in Fig. 2, and the fluctuation amplitude ranges are summarized in Table 5.

The analysis shows that torque excitations under different operating conditions differ notably in amplitude, evolution, and local variation rate. First-order statistics, such as amplitude, cannot fully capture the temporal variation characteristics. To quantify these properties, the autocorrelation time τc is introduced as a measure of the signal's temporal correlation. A smaller τc indicates faster decorrelation and more rapid excitation variations, while a larger τc reflects slower evolution.

The autocorrelation time is determined from the normalized autocorrelation function R(τ), which is defined as follows: 7Rτ=ETt-T‾Tt+τ-T‾ETt-T‾2, where T(t) denotes the instantaneous engine torque and T‾ its mean value. The autocorrelation time is defined as the time lag at which the normalized autocorrelation function decays to 1/e of its initial magnitude.

Although the measured torque signals have a relatively low sampling rate (8.3 Hz), resulting in an excitation bandwidth well below the system's natural frequencies, they still exhibit pronounced temporal correlation. Accordingly, the characteristic autocorrelation time τc is adopted, instead of the conventional frequency-ratio criterion, to classify excitation.

From a system timescale perspective, torque disturbances typically correspond to periods of 0.01–0.1 s. When τc<10 s, the excitation varies rapidly and is regarded as fast-varying; when τc>10 s, it evolves slowly and approaches quasi-static behavior. Statistical analysis of measured torque data reveals clear separation among operating conditions: τc>20 s for hover, 1020 s for high-altitude climb and descent maneuver, and τc<10 s for aggressive vertical maneuvers.

Based on engine torque data collected from multiple operating conditions during a representative flight mission, statistical analysis reveals a clear separation among conditions: τc>20 s for hover, 1020 s for high-altitude climb and descent maneuver, and τc<10 s for aggressive vertical maneuvers. Integrating theoretical timescale separation, operational characteristics, and statistical evidence, τc=10 and 20 s are selected as classification thresholds. The excitations are therefore categorized as short- (τc<10 s), intermediate- (10≤τc≤20 s), and long-correlation time (τc>20 s). Table 6 shows strong agreement with the statistical distributions, confirming the effectiveness of this classification for characterizing torque variation rates and supporting subsequent PIC torsional response analysis.

Combined with the analysis in Sect. 2.1, the autocorrelation time τc is found to characterize the time-domain variation rate of torque and is intrinsically related to the ramp excitation parameter α. Specifically, a smaller τc corresponds to a larger α, which is more likely to induce pronounced transient torsional responses. This provides a theoretical basis for subsequent comparisons of PIC responses under different operating conditions.

To investigate how engine torque fluctuations affect the torsional vibration of the PIC, measured torque excitations under three typical flight conditions were applied as inputs. Based on the autocorrelation time τc, hover, high-altitude climb and descent maneuver, and aggressive vertical maneuvers are classified as slow-, moderate-, and fast-varying excitations, respectively. The lumped-parameter method is then used to compute and compare the system's torsional responses.

Representative nodes were selected to capture the coupling between torque fluctuations and system response, including the relative angle between the turbine-disk equivalent inertia and spline coupling driving end (θ12), across the strut-type overrunning clutch (θ56), and the mesh-level relative torsional angles of the two gear stages (θ45 and θ78). Under the high-altitude climb and descent maneuver (Fig. 3), a clear hierarchical distribution of response amplitudes is observed: θ56 is the largest (5.512 ×10-2 rad), followed by θ12 (2.025 ×10-2 rad), whereas θ78 is the smallest (3.013 ×10-7 rad). The same trend appears in the other conditions. This disparity arises from drivetrain inertia and stiffness distributions. Components near the excitation source (turbine, spline coupling) have high equivalent inertia and low torsional stiffness, enabling large angular deformation and transient energy accumulation. Conversely, gear meshes possess very high meshing stiffness and strong kinematic constraints, limiting relative deformation along the line of action; torsional energy is transmitted as torque rather than stored elastically. Consequently, mesh-level angles are orders of magnitude smaller than clutch-level angles, identifying the clutch and spline coupling as primary torsional compliance sites and the gear stages as high-stiffness transmission paths.

For brevity, only the response under aggressive vertical maneuvers is additionally shown (Fig. 4). Across the three conditions, increasing temporal variability of the torque excitation (i.e., decreasing τc and increasing variation rate) leads to substantial growth in both response amplitude and fluctuation range. Under aggressive vertical maneuvers, the amplitude of θ56 reaches 6.083×10-2 rad, which is 10.3 % higher than that under the high-altitude climb and descent maneuver (5.512×10-2 rad) and 280.0 % higher than that under hover (1.599×10-2 rad). This indicates that rapid torque variations enhance internal energy coupling and produce pronounced dynamic amplification in the PIC torsional response.

Analysis of peak responses under aggressive vertical maneuvers shows that maximum torsional vibrations do not coincide with torque peaks and are not directly correlated with torque amplitude. Instead, the vibration intensity is primarily governed by the temporal variation of the excitation. This is confirmed by the first-order time derivatives of turbine torque and rotational speed (Fig. 5), whose peaks align with those of θ12 and θ56, indicating that strong transient responses arise from rapid torque changes rather than absolute magnitude.

Overall, the temporal correlation of engine torque fluctuations, quantified by the autocorrelation time τc, is the dominant factor controlling torsional vibration intensity. As τc decreases, both response amplitudes and transient peaks increase, particularly at critical components such as the strut-type overrunning clutch and spline coupling. These results demonstrate that the high variation rate of torque excitations amplifies system torsional vibrations and that rapid torque fluctuations under typical operating conditions are the primary cause of intensified responses.

To investigate the differences in torsional responses under typical operating conditions and the effects of distinct excitation characteristics, a comparative analysis is conducted for three representative flight conditions. Response amplitudes and peak-to-peak values at each node are extracted, and their distributions are shown in Fig. 6. The results indicate that, except for gear pair nodes, most nodes exhibit the largest relative torsional displacement amplitudes under aggressive vertical maneuvers due to stronger excitation from rapid torque fluctuations. In contrast, the hover condition shows the smallest amplitudes, indicating better vibration stability. Among all nodes, θ12 and θ56 consistently exhibit the largest fluctuations, significantly exceeding those of the secondary gear pair θ78, suggesting that disturbances originate near the power turbine–coupling region and attenuate along the transmission path. This behavior is governed by the system's inertia distribution and torsional stiffness matching. Furthermore, differences in torque variation rates across operating conditions influence not only the response magnitude but also the dynamic coupling and spatial distribution of vibration amplitudes within the system.

To examine the relationship between system response and excitation characteristics, the torque values at the instant when θ56 reaches its maximum are extracted and compared with the corresponding fluctuation ranges. Under hover, high-altitude climb and descent maneuver, and aggressive vertical maneuvers, these values are 16.2 %, 68.6 %, and 57.8 % of the maximum torque, respectively, none coinciding with the torque peaks. Combined with node response comparisons, this indicates that torsional vibration intensity does not scale linearly with torque amplitude, and the fluctuation range alone is insufficient to predict system responses.

To further clarify the mechanism, the first derivative of the torque signals is computed, and the variation-rate curves are shown in Fig. 7. The aggressive vertical maneuver condition exhibits the largest variation rate, followed by high-altitude climb and descent maneuver, while hover shows the smallest, consistent with the distribution of response amplitudes. This confirms that torsional vibration intensity is primarily governed by the torque variation rate rather than its amplitude. Mechanistically, a high variation rate prevents instantaneous inertial equilibrium, leading to transient energy accumulation and dynamic amplification. Overall, response amplitudes are determined by inertia and stiffness distribution, while higher torque variation rates increase vibration intensity and dynamic loads. Rapid torque fluctuations can significantly amplify relative torsional displacements in critical components, such as the strut-type overrunning clutch and spline coupling, thereby increasing the risk of fatigue and failure.

In summary, comparative analysis under three representative operating conditions confirms that the time-varying characteristics of torque excitation dominate the system's torsional vibration behavior. The torque variation rate is identified as the key parameter governing the dynamic response, with a significantly greater influence than other excitation features. Therefore, in the dynamic design and control of the PIC, special attention should be given to the effects of high-rate torque excitation on system stability and fatigue reliability.