In modern motion control systems, the transition from traditional hydraulic and pneumatic actuators to electromechanical solutions represents a significant technological evolution. Among these, the electric cylinder, a modular device integrating a motor-driven screw assembly to convert rotary into precise linear motion, offers distinct advantages in power density, efficiency, controllability, and environmental friendliness. This study focuses on a critical component for marine stabilization systems: a 90 kN electric cylinder driven by a planetary roller screw assembly. While preliminary design and component-level strength validation have been completed, a comprehensive understanding of the complete assembly’s dynamic characteristics and static performance is essential for reliable operation and further optimization. This work employs Finite Element Analysis (FEA) to conduct a detailed modal analysis of the entire electric cylinder structure across its operational range. Furthermore, a thorough axial static stiffness analysis under maximum design loads is performed. The results provide foundational data for assessing resonance risks, informing vibration damping strategies, and enabling precise position control under variable loading conditions, which are paramount for high-performance applications like ship fin stabilizers.
System Description and Finite Element Modeling
The subject of this analysis is a 90 kN electric cylinder designed for actuating ship stabilizer fins. Its core drive mechanism is a precision planetary roller screw assembly, known for its high load capacity, stiffness, and longevity compared to ball screws. The device operates by transferring rotational power from a servo motor via a timing belt to the nut of the screw mechanism. The converted linear motion is delivered through an output rod connected to the fin’s crank mechanism. Two such cylinders are mounted via side-trunnion bearings on the stabilizer frame. Key performance parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Maximum Working Load (Push/Pull) | 90 kN |
| Maximum Working Speed (Rod) | 230 mm/s (at 3450 rpm motor speed) |
| Total Mass (Mechanical Body + Servo Motor) | 139 kg + 36 kg |
| Mounting Configuration | Side-Trunnion (Pivoting) |
The complete three-dimensional CAD model of the electric cylinder, encompassing the housing, planetary roller screw assembly, rod, barrel, bearing supports, and motor mounting plate, served as the basis for the finite element model. To ensure computational efficiency while maintaining accuracy, minor fillets and chamfers were simplified. The model was discretized using 3D solid elements. The core of the actuation system, the planetary roller screw assembly, was modeled in its engaged state, capturing the essential load path from the nut to the screw thread.
Material properties assigned to the primary components are critical for realistic simulation and are detailed in Table 2. The housing and barrel utilize aluminum alloy for weight reduction, while high-strength alloy steels are used for heavily loaded components like the screw, nut, trunnions, and support rods.
| Component | Material | Elastic Modulus, E (GPa) | Yield Strength (MPa) | Poisson’s Ratio, ν | Density, ρ (kg/m³) |
|---|---|---|---|---|---|
| Housing, Barrel | Aluminum Alloy (2A12/6005) | 62 | 235-255 | 0.32 | 2800 |
| Support Plate, Bearing Housings, Trunnion | 42CrMo Steel | 205 | 930 | 0.30 | 7850 |
| Screw (of the planetary roller screw assembly) | 43CrMo4 Steel | 205 | 800 | 0.30 | 7850 |
| Nut (of the planetary roller screw assembly) | 100Cr6 Bearing Steel | 205 | 1700 | 0.30 | 7850 |
| Output Rod | 16Mn Steel | 205 | 345 | 0.30 | 7850 |
Boundary conditions and contact definitions were applied to replicate the actual assembly and mounting. Internally, all component interfaces (threaded contacts, bearing seats, fits) were defined using surface-to-surface contact algorithms with appropriate friction coefficients. The side-trunnion pins, which interface with the external frame, were constrained using cylindrical supports: radial and axial degrees of freedom were fixed, while rotation about the pin axis (the intended pivot motion) was left free. This accurately represents the hinged mounting condition. The servo motor mass was modeled as a point mass/inertia connected to the motor mounting plate.
Theoretical Basis for Modal and Stiffness Analysis
The dynamic behavior of a mechanical structure is governed by its mass and stiffness distribution. The undamped free-vibration eigenvalues are found by solving the equation:
$$ [K]\{\phi\} = \omega^2[M]\{\phi\} $$
where $[K]$ is the global stiffness matrix, $[M]$ is the global mass matrix, $\omega$ is the natural frequency in radians/second (related to frequency in Hz by $f = \omega / 2\pi$), and $\{\phi\}$ is the corresponding mode shape vector. The FEA solver extracts these eigenvalues and eigenvectors, providing the natural frequencies $f_n$ and the associated deformation patterns of the structure. For the electric cylinder, the primary excitation sources are the rotational unbalance and harmonics from the servo motor and the belt drive. The fundamental excitation frequency $f_{ex}$ is related to the motor speed $N$ (rpm) and the belt reduction ratio $i$:
$$ f_{ex} = \frac{N}{60 \cdot i} $$
With a maximum motor speed of 3450 rpm and a reduction ratio of 1.5, the excitation frequency range is 0 to approximately 38.3 Hz. Resonance risk is high when $f_{ex}$ coincides with any structural natural frequency $f_n$.
Static stiffness $K$ is defined as the ratio of an applied force $F$ to the resulting elastic deformation $\delta$ along the same axis:
$$ K = \frac{F}{\delta} $$
For the electric cylinder, the total axial deformation $\delta_{total}$ under a rod load $F_{rod}$ is the sum of compliances (inverse of stiffness) from several components in series:
$$ \delta_{total} = \delta_{screw} + \delta_{nut} + \delta_{bearings} + \delta_{housing} + \delta_{rod} = \frac{F_{rod}}{K_{screw}} + \frac{F_{rod}}{K_{nut}} + \frac{F_{rod}}{K_{bearings}} + \frac{F_{rod}}{K_{housing}} + \frac{F_{rod}}{K_{rod}} $$
Thus, the overall axial stiffness $K_{total}$ is given by:
$$ \frac{1}{K_{total}} = \frac{1}{K_{screw}} + \frac{1}{K_{nut}} + \frac{1}{K_{bearings}} + \frac{1}{K_{housing}} + \frac{1}{K_{rod}} $$
The stiffness of the threaded components, particularly the planetary roller screw assembly, is complex and depends on the engaged thread length, which varies with rod extension. The screw acts as a tension/compression rod combined with a slender shaft, so its stiffness $K_{screw}$ is:
$$ K_{screw} = \frac{A_{screw} \cdot E_{screw}}{L_{engaged}(x)} $$
where $A_{screw}$ is the screw’s effective load-bearing cross-sectional area, $E_{screw}$ is its modulus of elasticity, and $L_{engaged}(x)$ is the load-bearing length of the screw between the nut and the supporting bearing, a function of rod stroke position $x$. This variable engagement length is a key factor in the system’s position-dependent stiffness.
Modal Analysis: Natural Frequencies and Resonance Assessment
Modal analysis was conducted for two critical stroke positions representing the bounds of operation: Rod Fully Extended and Rod Fully Retracted. This assesses how the changing mass distribution and stiffness, primarily due to the exposed length of the screw within the planetary roller screw assembly, affect the dynamic properties. The first 16 natural frequencies for both configurations are summarized in Table 3.
| Mode | Rod Fully Extended Frequency (Hz) |
Rod Fully Retracted Frequency (Hz) |
Primary Mode Shape Description |
|---|---|---|---|
| 1 | 7.20 | 7.30 | Global lateral bending of the cylinder about the trunnion axis (like a cantilever). |
| 2 | 40.23 | 39.37 | Higher-order lateral bending, often with a node near the housing center. |
| 3-10 | 60.6 – 61.0 | 60.6 – 61.0 | Clustered modes involving complex local deformations: twisting/bending of support rods, housing panel vibrations, and local modes of the motor mount and belt guard. |
| 11-12 | 101.3 – 103.0 | 114.0 – 136.2 | Axial-rotational coupled modes and complex housing deformations. |
| 13-16 | 114.3 – 167.0 | 165.1 – 167.2 | High-frequency local modes, primarily involving components of the planetary roller screw assembly support structure and housing. |
The analysis reveals crucial insights for dynamic performance. The first two modes are global bending modes with frequencies within the motor excitation frequency range (0-38.3 Hz). Specifically:
- Mode 1 (~7.2 Hz): Resonance can occur if the motor operates near 432 rpm (7.2 Hz * 60) or its first harmonic at 864 rpm. This is within the low-speed operating range.
- Mode 2 (~40 Hz): This frequency is very close to the upper limit of the excitation range. Resonance is a significant risk when the motor operates near its maximum speed (e.g., 40 Hz corresponds to 2400 rpm motor speed, which is 3600 rpm / 1.5).
The frequency clusters between 60-61 Hz and all higher modes (≥101 Hz) possess natural frequencies significantly above the maximum fundamental excitation frequency of 38.3 Hz. The resonance avoidance ratios for these modes are substantial:
$$ \text{Avoidance Ratio} = \frac{f_n – f_{ex,max}}{f_{ex,max}} \times 100\% $$
For the 60.6 Hz cluster, the avoidance ratio is approximately $(60.6 – 38.3)/38.3 \approx 58\%$. For modes above 100 Hz, the ratio exceeds 160%. Therefore, these higher-frequency modes are not excited by the primary motor/belt harmonics under normal conditions.
A critical observation from the mode shapes is the pronounced vibration amplitude at the locations of the long, slender support rods that connect the front and rear housing sections. These rods, while excellent for tensile loading, present a local compliance that dominates several of the clustered modes in the 60-61 Hz range. This identifies them as a key area for potential structural damping optimization if vibration reduction is required.
Axial Static Stiffness Analysis and Derivation
The axial static stiffness governs the positional accuracy under load. Four load cases were simulated: 90 kN Tension and 90 kN Compression, each with the rod in the Fully Retracted and Fully Extended positions. The total axial deformation $\delta_{total}$ was extracted from FEA, and the overall stiffness $K_{total}$ was calculated using $K = F / \delta_{total}$. The results are presented in Table 4.
| Load Case | Rod Position | Calculated Axial Stiffness, $K_{total}$ (N/µm) | Equivalent Deformation, $\delta$ (mm @ 90 kN) |
|---|---|---|---|
| Tension (90 kN) | Fully Retracted | 143.35 | 0.628 |
| Tension (90 kN) | Fully Extended | 124.57 | 0.722 |
| Compression (90 kN) | Fully Retracted | 237.47 | 0.379 |
| Compression (90 kN) | Fully Extended | 171.04 | 0.526 |
The results demonstrate clear and mechanically explicable trends:
- Position Dependency: For both tension and compression, stiffness decreases as the rod extends. This is directly attributable to the increased engaged length $L_{engaged}$ of the screw in the planetary roller screw assembly, as predicted by the stiffness formula $K_{screw} \propto 1/L_{engaged}$. The screw is the most compliant element in the load chain besides the housing in tension, so its variable length dictates the system’s stiffness variation.
- Asymmetry (Tension vs. Compression): The system is significantly stiffer in compression than in tension. This asymmetry arises from the different load paths. In compression, the load flows from the rod, through the screw of the planetary roller screw assembly, into the thrust bearing, and directly into the stiff rear support plate and housing structure. In tension, the load path pulls on the front housing and the long aluminum barrel, which, due to its lower modulus of elasticity (62 GPa vs. 205 GPa for steel), introduces greater compliance. The total stiffness can be conceptually broken down as:
$$ \frac{1}{K_{Tension}} \approx \frac{1}{K_{Screw(T)}} + \frac{1}{K_{Barrel}} + \frac{1}{K_{Other}} $$
$$ \frac{1}{K_{Compression}} \approx \frac{1}{K_{Screw(C)}} + \frac{1}{K_{Bearing/Support}} + \frac{1}{K_{Other}} $$
where $K_{Barrel} < K_{Bearing/Support}$, leading to $K_{Tension} < K_{Compression}$. - Magnitude and Implications: The average tensile stiffness is approximately 134 N/µm (0.134 N/µm), and the average compressive stiffness is about 204 N/µm (0.204 N/µm). Under the full 90 kN load, this translates to elastic deflections ranging from 0.38 mm to 0.72 mm. For a precision motion system, this non-negligible, load-dependent, and asymmetric compliance must be accounted for. If the application requires high positioning accuracy under varying external loads (as is the case for a ship stabilizer fin opposing hydrodynamic forces), an open-loop system relying solely on motor encoder feedback will have significant positional error due to this structural deflection. This strongly necessitates the implementation of full-closed-loop control using an external position sensor (e.g., a magnetostrictive linear transducer) on the moving rod to measure and correct the actual output position, effectively compensating for the variable stiffness $K_{total}(x, F)$.
Discussion and Concluding Synthesis
This integrated FEA study of the 90 kN planetary roller screw assembly-based electric cylinder provides a comprehensive characterization of its structural dynamics and static load-deformation behavior. The modal analysis successfully identified the vulnerable frequency bands where resonance with the primary motor drive excitation can occur. The first lateral bending mode (~7.2 Hz) and the second bending mode (~40 Hz) require operational scrutiny or potential control system filtering to avoid sustained operation at the corresponding motor speeds. The identification of the support rods as active participants in higher-frequency mode shapes offers a clear target for adding damping material or considering local stiffening if vibration reduction is a critical design goal.
The axial stiffness analysis quantified a fundamental performance characteristic. The derived stiffness values, their dependence on stroke position, and the tension-compression asymmetry are direct consequences of the design choices, particularly the use of a high-lead planetary roller screw assembly for force transmission and aluminum for lightweight structural members. The calculated compliance is not a flaw but a defined property; the key engineering takeaway is the imperative for appropriate control architecture. The data conclusively supports the need for full-closed-loop position control to achieve the high positioning accuracy demanded by the target application despite the variable structural deflection.
In summary, the finite element methodology applied here has transitioned the design from a component-validated state to a system-understood state. The results form a vital dataset for subsequent stages: they inform vibration test planning, guide control algorithm development to manage resonance and compensate for stiffness, and highlight specific structural elements (support rods, housing-barrel interface) for focused attention in any future weight or performance optimization cycle. The planetary roller screw assembly has been confirmed as the central element governing both the variable stiffness and a portion of the dynamic response, underscoring its critical role in the overall electromechanical actuator performance.