In material testing machines, axial loading is commonly employed for evaluating mechanical properties of materials. However, due to factors such as machining imperfections, installation errors, and operational wear, these machines often exhibit force misalignment, leading to inaccuracies in test results. Force coaxiality error introduces additional bending moments during testing, which can significantly affect the reliability of data. Traditional methods for assessing force coaxiality include the strain gauge method, where strain gauges are attached at specific positions on a specimen to measure strain, and the dual extensometer method, which involves measuring deformation on both sides of the specimen using two extensometers. While these approaches are widely used, they possess inherent limitations. Specifically, they only capture deformations in predetermined orientations, potentially missing the maximum strain方位, which is critical for accurate coaxiality evaluation. This oversight can result in underestimation of the true force coaxiality error.
To address these limitations, I propose a novel method utilizing a six-axis force sensor for force coaxiality detection. A six-axis force sensor is a composite sensor capable of measuring three orthogonal forces and three orthogonal moments in space, providing a comprehensive description of the force state at a detection point. By leveraging this sensor, the stress distribution across a specimen’s cross-section can be analyzed. Based on material properties and the stress-strain relationship, the strain at any orientation of the specimen can be derived, ensuring that the maximum force coaxiality is accurately captured without omission. This approach not only enhances detection accuracy but also overcomes the drawbacks of traditional methods, such as the difficulty in tracing strain-gauge-equipped specimens and the maintenance challenges associated with them.
The relationship between force coaxiality and specimen stress is fundamental to this method. Under ideal conditions, the tensile axis of the material testing machine aligns perfectly with the specimen’s central axis, resulting in uniform stress distribution. The stress on any cross-section of the specimen can be expressed as: $$ \sigma = \frac{P}{A} $$ where \( A \) is the cross-sectional area of the specimen. However, in practical scenarios, force misalignment occurs due to geometric deviations and angular misalignments. The actual force state can be represented by an equivalent analysis, where the force components include axial force, transverse force, and bending moment. For instance, the maximum and minimum normal stresses at a distance \( L \) from the force application point are given by: $$ \sigma_{\text{max}} = \frac{P \cos \alpha}{A} + \frac{P L \sin \alpha}{W} + \frac{P \delta}{W} $$ and $$ \sigma_{\text{min}} = \frac{P \cos \alpha}{A} – \frac{P L \sin \alpha}{W} – \frac{P \delta}{W} $$ where \( \alpha \) is the angle between the tensile axis and the specimen centerline, \( \delta \) is the geometric coaxiality deviation, and \( W \) is the section modulus. The force coaxiality \( e \) is then calculated as: $$ e = \frac{\Delta L_{\text{max}} – \Delta L_{\text{min}}}{\Delta L} \times 100\% $$ where \( \Delta L_{\text{max}} \) and \( \Delta L_{\text{min}} \) are the maximum and minimum deformations, respectively. Using the stress-strain relationship \( \varepsilon = \frac{\sigma}{E} \) (where \( E \) is the elastic modulus) and the deformation-strain relation \( \varepsilon = \frac{\Delta L}{L} \), the coaxiality can be expressed in terms of stress: $$ e = \frac{\sigma_{\text{max}} – \sigma_{\text{min}}}{\sigma_{\text{max}} + \sigma_{\text{min}}} = \frac{A}{W \cos \alpha} (L \sin \alpha + \delta) $$
To validate this theoretical framework, finite element analysis (FEA) software was employed to simulate the stress and deformation of a coaxiality specimen. The model was based on a 250 kN tensile testing machine, with a specimen featuring a uniform diameter of 10 mm, a gauge length of 100 mm, and a total length of 130 mm, made of 45# steel. The simulation decomposed the force misalignment into two states: zero angle between the tensile axis and specimen centerline (\( \alpha = 0^\circ \)) and zero spacing between the axes (\( \delta = 0 \) mm). The results, including stress data and force coaxiality, were analyzed at key points, such as the center of the uniform section. The following tables summarize the relationships between \( \delta \), \( \alpha \), stress, and force coaxiality, comparing the dual extensometer method with the stress-based calculation using the six-axis force sensor approach.
| \( \delta \) (mm) | \( \sigma_{\text{max}} \) (MPa) | \( \sigma_{\text{min}} \) (MPa) | \( e_1 \) (Dual Extensometer) (%) | \( e_{50\text{mm}} \) (Stress-Based) (%) | \( e_1 – e_{50\text{mm}} \) (%) |
|---|---|---|---|---|---|
| 0.01 | 127.64 | 125.56 | 0.85 | 0.82 | 0.03 |
| 0.02 | 128.61 | 124.53 | 1.63 | 1.61 | 0.02 |
| 0.04 | 130.68 | 122.54 | 3.27 | 3.21 | 0.04 |
| 0.06 | 132.74 | 120.38 | 4.87 | 4.88 | -0.01 |
| 0.08 | 134.69 | 118.47 | 6.46 | 6.40 | 0.06 |
| 0.10 | 136.81 | 116.30 | 8.09 | 8.11 | -0.02 |
| 0.12 | 138.96 | 114.36 | 9.67 | 9.71 | -0.04 |
| 0.14 | 140.98 | 112.16 | 11.28 | 11.39 | -0.11 |
| 0.16 | 142.98 | 110.01 | 12.83 | 13.03 | -0.20 |
| 0.18 | 145.11 | 107.85 | 14.49 | 14.73 | -0.24 |
| 0.20 | 147.08 | 106.08 | 16.12 | 16.20 | -0.08 |
| \( \alpha \) (degrees) | \( \sigma_{\text{max}} \) (MPa) | \( \sigma_{\text{min}} \) (MPa) | \( e_1 \) (Dual Extensometer) (%) | \( e_{50\text{mm}} \) (Stress-Based) (%) | \( e_1 – e_{50\text{mm}} \) (%) |
|---|---|---|---|---|---|
| 0.01 | 129.06 | 125.64 | 1.26 | 1.34 | -0.08 |
| 0.02 | 130.78 | 123.92 | 2.61 | 2.69 | -0.08 |
| 0.03 | 132.51 | 122.21 | 3.95 | 4.04 | -0.09 |
| 0.04 | 134.23 | 120.50 | 5.28 | 5.39 | -0.11 |
| 0.05 | 135.96 | 118.78 | 6.62 | 6.74 | -0.12 |
| 0.06 | 137.68 | 117.07 | 7.97 | 8.09 | -0.12 |
| 0.07 | 139.41 | 115.36 | 9.31 | 9.44 | -0.13 |
| 0.08 | 141.13 | 113.64 | 10.64 | 10.79 | -0.15 |
| 0.09 | 142.86 | 111.93 | 11.98 | 12.14 | -0.16 |
| 0.10 | 144.58 | 110.22 | 13.33 | 13.49 | -0.16 |
| 0.11 | 146.31 | 108.50 | 14.65 | 14.84 | -0.19 |
| 0.12 | 148.03 | 106.79 | 16.00 | 16.19 | -0.19 |
The simulation results demonstrate that the absolute error between the dual extensometer method and the stress-based calculation using the six-axis force sensor is within 0.2%, which is acceptable given that the repeatability of force coaxiality detection in material testing machines is typically around 2%. This error primarily arises from uncertainties in finite element mesh node calculations. Thus, the six-axis force sensor method provides a reliable alternative for accurate coaxiality assessment.
The principle of the six-axis force sensor is central to this method. A six-axis force sensor measures three orthogonal forces and three orthogonal moments in space, effectively describing the force state at a detection point. It can be considered as a group of force sensors with a specific structural distribution, where signals from multiple force sensors are processed to obtain the forces and moments acting on the measured object. For example, the Stewart structure six-axis force sensor consists of six elastic rods connecting platforms of different diameters. By detecting the deformation of these elastic rods, which are equivalent to six force sensors, the forces and moments at the sensor detection point (typically the center of the upper platform) can be determined. The general force and moment equations for a six-axis force sensor can be expressed as: $$ F_x = \sum_{i=1}^{6} k_{xi} \cdot d_i $$ $$ F_y = \sum_{i=1}^{6} k_{yi} \cdot d_i $$ $$ F_z = \sum_{i=1}^{6} k_{zi} \cdot d_i $$ $$ M_x = \sum_{i=1}^{6} k_{mxi} \cdot d_i $$ $$ M_y = \sum_{i=1}^{6} k_{myi} \cdot d_i $$ $$ M_z = \sum_{i=1}^{6} k_{mzi} \cdot d_i $$ where \( F_x, F_y, F_z \) are the forces along the x, y, z axes, \( M_x, M_y, M_z \) are the moments about the x, y, z axes, \( k \) are calibration coefficients, and \( d_i \) are the deformation signals from the six sensors. This configuration allows the six-axis force sensor to capture complex force states, making it ideal for coaxiality detection.
Based on the six-axis force sensor, the force coaxiality detection method involves attaching a fixture to the sensor to serve as a coaxiality specimen. The fixture has a uniform section with a diameter of 10 mm and a length of 50 mm, and the distance from the clamping part to the sensor detection point is 65 mm. This assembly is installed in the material testing machine, which is loaded to the force value specified by relevant regulations. The forces and moments at the sensor detection point are measured and imported into FEA software to obtain the stress distribution on the cross-section of the specimen at the detection point. Using the stress-strain relationship, the force coaxiality is calculated as: $$ e = \frac{\sigma_{\text{max}} – \sigma_{\text{min}}}{\sigma_{\text{max}} + \sigma_{\text{min}}} $$ The measurement is repeated three times, and the maximum value is taken as the force coaxiality of the material testing machine. This method ensures that all orientations are considered, eliminating the risk of missing the maximum strain.
The advantages of using a six-axis force sensor for force coaxiality detection are significant. It is a derivative of the strain gauge method but resolves issues such as the difficulty in tracing strain-gauge-equipped specimens, cumbersome maintenance, and undefined failure conditions. Moreover, it overcomes the limitation of traditional methods that may overlook the maximum force coaxiality. However, challenges remain, such as the large size and heavy weight of high-capacity six-axis force sensors, which affect portability. Additionally, existing six-axis force sensor specifications may not cover all ranges of material testing machines, limiting widespread application. As technology advances, including improvements in structural design and the development of new materials, six-axis force sensors are expected to become more compact, lightweight, and capable of higher capacities. This progress will enable their broader adoption in practical detection scenarios, enhancing the accuracy and efficiency of force coaxiality assessments in material testing.
In conclusion, the six-axis force sensor-based method for force coaxiality detection represents a significant improvement over traditional techniques. By providing a comprehensive analysis of stress distribution and enabling the calculation of strain at any orientation, it ensures accurate and reliable results. Future developments in six-axis force sensor technology will likely address current limitations, making this method a standard in the field. The integration of such advanced sensors into material testing protocols will contribute to higher quality control and more precise mechanical property evaluations.