In the field of helicopter engineering, the composite blade root segment is critical for aerodynamic performance and structural integrity. Traditional static testing methods for these segments rely heavily on strain gauges, which involve complex calibration processes and are prone to significant errors due to force coupling and deformation effects. I propose a novel approach that leverages a Stewart platform-based six-axis force sensor to directly measure the six force components at the blade root, simplifying the testing procedure and enhancing accuracy. This method eliminates the need for strain gauge decoupling and reduces cumulative errors, offering a more efficient solution for validating blade root strength under centrifugal, flapping, and lead-lag loads.
The traditional static test method involves applying specific forces at a designated loading section to simulate operational conditions, such as centrifugal force (F_X), flapping force (F_Z), and lead-lag force (F_Y). However, blade deformation under these loads introduces cross-coupling, making it difficult to achieve accurate force measurements at the root section (0-profile). For instance, when a flapping load F_Z is applied, the blade deflects, altering the direction and magnitude of the centrifugal force F_X. This coupling effect complicates the relationship between applied loads and actual root forces, as described by the beam deflection equation:
$$ \omega = \frac{100F}{Eh^3} \left( \frac{x^2}{2} – lx + \frac{l^2}{2} \right) $$
Here, ω represents deflection, F is the applied force, E is the elastic modulus, h is the thickness, x is the distance from the loading point, and l is the length. At the loading section, the deflection ω_B is given by:
$$ \omega_B = \frac{50Fl^2}{Eh^3} $$
This deflection causes the actual centrifugal force F_LX at the root to deviate from the applied F_X, and the flapping moment M_HW to differ from the theoretical value. Traditional methods use strain gauges to monitor these forces, but they require extensive decoupling and calibration, which becomes inaccurate under large deformations where material failures like delamination can occur. My approach addresses these issues by integrating a six-axis force sensor between the blade root and the test fixture, enabling direct measurement of the six force components without strain gauges.
The core of this method is the six-axis force sensor based on the Stewart platform structure, which consists of an upper platform, a lower platform, and six connecting rods acting as two-force members. This design allows the sensor to withstand multi-directional loads while providing high precision. The force equilibrium principle for the Stewart platform is derived from force screw theory, where the six-dimensional force vector F at the upper platform center is related to the axial forces f in the rods through a transformation matrix G:
$$ F = G \cdot f $$
Here, F = [F_x, F_y, F_z, M_x, M_y, M_z]^T represents the forces and moments, and f = [f_1, f_2, f_3, f_4, f_5, f_6]^T is the axial force vector. The matrix G is composed of unit line vectors corresponding to each rod’s orientation and position. For a rod i, the unit line vector $ _i is defined as:
$$ S_i = \frac{b_i – B_i}{|b_i – B_i|} $$
$$ S_{oi} = B_i \times S_i = \frac{B_i \times b_i}{|b_i – B_i|} $$
where b_i and B_i are the position vectors of the upper and lower platform hinge points, respectively. The transformation matrix G is then constructed as:
$$ G = \begin{bmatrix}
S_1 & S_2 & S_3 & S_4 & S_5 & S_6 \\
S_{o1} & S_{o2} & S_{o3} & S_{o4} & S_{o5} & S_{o6}
\end{bmatrix} $$
To implement this, I developed a numerical conversion program in MATLAB that computes the relationship between F and f based on the sensor’s geometric parameters. For a sensor with an upper platform radius of 0.3 m, lower platform radius of 0.6 m, platform separation of 0.4 m, and specific hinge angles (q1 = 30°, q2 = 90°), the connecting rod length l3 is calculated as 0.5461 m. The hinge points are arranged symmetrically, with angles between points ensuring proper force distribution. The resulting transformation matrix G is shown in Table 1.
| Row | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 |
|---|---|---|---|---|---|---|
| 1 | -0.549 | -0.402 | -0.074 | -0.275 | 0.623 | 0.677 |
| 2 | -0.402 | -0.549 | 0.677 | 0.623 | -0.275 | -0.074 |
| 3 | 0.732 | 0.732 | 0.732 | 0.732 | 0.732 | 0.732 |
| 4 | 0.220 | 0.000 | -0.110 | -0.190 | -0.110 | 0.190 |
| 5 | 0.000 | -0.220 | -0.190 | 0.110 | 0.190 | 0.110 |
| 6 | 0.165 | -0.165 | 0.165 | -0.165 | 0.165 | -0.165 |
Using this matrix, the MATLAB program can convert between F and f. For example, under calibration forces such as F_x = 10 kN, F_y = 10 kN, F_z = 10 kN, M_x = 1 kN·m, M_y = 1 kN·m, and M_z = 1 kN·m, the axial forces f in the rods are computed as shown in Table 2. This demonstrates the sensor’s ability to resolve complex load states accurately.
| Rod | F_x = 10 kN | F_y = 10 kN | F_z = 10 kN | M_x = 1 kN·m | M_y = 1 kN·m | M_z = 1 kN·m |
|---|---|---|---|---|---|---|
| f1 (N) | -6067.8 | 0.0 | 2275.4 | 1517.0 | 1110.5 | 1011.3 |
| f2 (N) | 0.0 | -6067.8 | 2275.4 | -1110.5 | -1517.0 | -1011.3 |
| f3 (N) | 3033.9 | 5254.9 | 2275.4 | 203.2 | -1869.0 | 1011.3 |
| f4 (N) | -5254.9 | 3033.9 | 2275.4 | -758.5 | 1720.2 | -1011.3 |
| f5 (N) | 3033.9 | -5254.9 | 2275.4 | -1720.2 | 758.5 | 1011.3 |
| f6 (N) | 5254.9 | 3033.9 | 2275.4 | 1869.0 | -203.2 | -1011.3 |
To validate the numerical model, I created a finite element model in ABAQUS, representing the six-axis force sensor with realistic dimensions. The upper and lower platforms were modeled as rigid bodies with thicknesses of 0.02 m and 0.03 m, respectively, and the connecting rods had a radius of 0.01 m. The material properties were assigned high elastic moduli (20,000 GPa for platforms, 200 GPa for rods) to minimize deformation effects, and pin constraints simulated the hinge connections. The mesh used linear reduced-integration elements (C3D8R) to ensure accuracy without excessive computation. The model was subjected to the same calibration loads as in the MATLAB analysis, and the resulting axial forces were extracted from stress simulations.
The finite element analysis confirmed that each rod behaved as a two-force member, with uniform stress distribution along its length. The computed axial forces from ABAQUS, shown in Table 3, closely matched the MATLAB results, with minor discrepancies due to geometric approximations in the numerical model. For instance, under F_x = 10 kN, the axial force f1 was -6471.7 N in ABAQUS compared to -6067.8 N in MATLAB, demonstrating the consistency of the six-axis force sensor model.
| Rod | F_x = 10 kN | F_y = 10 kN | F_z = 10 kN | M_x = 1 kN·m | M_y = 1 kN·m | M_z = 1 kN·m |
|---|---|---|---|---|---|---|
| f1 (N) | -6471.7 | 336.2 | 2516.4 | 1690.2 | 1225.2 | 1124.7 |
| f2 (N) | -348.7 | -6503.1 | 2541.5 | -1240.9 | -1677.6 | -1124.7 |
| f3 (N) | 2931.1 | 5811.9 | 2547.8 | 217.7 | -2064.0 | 1118.4 |
| f4 (N) | -5435.0 | 3518.6 | 2513.3 | -835.7 | 1900.7 | -1118.4 |
| f5 (N) | 3550.0 | -5466.4 | 2538.4 | -1916.4 | 848.2 | 1124.7 |
| f6 (N) | 5780.5 | 2959.4 | 2510.1 | 2086.0 | -235.0 | -1121.5 |
In practical application, the six-axis force sensor is installed between the blade root segment and the test fixture. During static testing, the sensor measures the axial forces in the rods in real-time, which are then converted to the six-dimensional force F at the sensor’s upper platform using the transformation matrix. By knowing the relative position between the sensor’s center and the blade root’s 0-profile, the actual forces and moments at the root can be derived through coordinate transformation. This eliminates the need for strain gauge decoupling and reduces errors associated with large deformations. The calibration process involves applying known loads to the sensor and recording the corresponding axial forces, establishing a lookup table for inverse calculations.
The advantages of this six-axis force sensor-based method are manifold. It simplifies the test setup by reducing the number of sensors and cables, improves measurement accuracy by directly capturing force components, and enhances reliability by avoiding strain gauge failures. Moreover, the Stewart platform structure offers high stiffness and load capacity, making it suitable for high-force applications like helicopter blade testing. Future work could focus on optimizing the sensor’s geometry for minimal cross-coupling and integrating real-time data processing for automated testing.
In conclusion, the integration of a six-axis force sensor into blade root static testing represents a significant advancement over traditional methods. The numerical and finite element validations confirm the feasibility of this approach, paving the way for more efficient and accurate structural validation in aerospace engineering. By leveraging the principles of parallel kinematics and force screw theory, this six-axis force sensor method addresses the limitations of strain-based techniques, ensuring that blade root segments meet stringent strength requirements with greater confidence.