According to prior research on material properties, most magnetostrictive materials experience pre-compression stress in practical applications, typically ranging from 10 to 30 MPa. This preload must remain consistent throughout the entire operational state—whether during static or dynamic deformation processes. To maintain this stress, springs are almost universally used in design due to their reliability and effectiveness.
Consider a practical scenario where a magnetostrictive rod is employed with a pre-stress of 10 MPa. If the rod has a diameter of 20 mm, the resulting pre-pressure could amount to several thousand Newtons. When multiple magnets are involved in a transducer, requiring simultaneous excitation, the required force can easily reach tens of thousands of Newtons. Therefore, to satisfy the pre-compression requirement, a spring with high stiffness (K) is generally preferred. A low stiffness spring may meet the pressure demand but would result in an overly large volume, which is not ideal for real-world applications. Hence, disc springs are commonly used in energy conversion systems, as they can generate significant force with minimal deformation, making them highly efficient.
The ejector rod and the rare earth (Terfenol) rod are connected in series, as the ejector rod can be approximated as a rigid body, meaning its elastic deformation is negligible. As a result, the longitudinal displacement at the top of the rare earth rod is the same as that of the ejector rod. Meanwhile, the spring is connected in parallel with the rare earth rod, so the longitudinal displacement at the top of the rod matches that of the spring. The magnetostrictive force is relatively small compared to the pre-pressure, so the displacement during the magnetostrictive phase is smaller than during the pre-stressing phase.
If hysteresis effects are ignored, under static magnetostriction conditions, the magnetostrictive force Fm generated by a given rod remains constant under a specific magnetic field H and pre-stress T, usually on the order of hundreds of Newtons. This force acts on the spring, causing it to deform longitudinally by x. At this stage, the spring stiffness is K, and the Young’s modulus of the magnetostrictive material is E(T). Assuming the original length of the magnetostrictive rod is L, we have the equations: Fm = Kx + E(T) * ΔL * x / L, and S = x / L. From these, E(T) can be calculated when K is known. Experimental results show that E(T) quickly stabilizes to a constant value after entering the magnetostrictive phase, and it follows a predictable pattern—higher initial pre-stress leads to a higher constant value. This confirms the linear relationship between the magnetostrictive force Fm and the displacement x of the step section. Thus, the magnetostrictive strain S depends not only on Fm but also on the stiffness. In the extreme case, when K approaches infinity, the magnetostrictive strain S should approach zero, equivalent to connecting the rare earth rod to an infinitely rigid body, where the magnetostrictive force can be precisely determined.
Analysis of the flexible disc spring reveals that previous designers often overlooked the conflict between the pre-stressing phase and the magnetostrictive phase regarding spring stiffness. By using a high stiffness spring, the pre-stress requirements are naturally met, while the effect of the magnetostrictive phase seems minor. However, this contradiction can be resolved through smart spring design without complicating the mechanical structure. Introducing a soft spring allows for high stiffness during the pre-stress phase and low stiffness during the magnetostrictive phase, optimizing performance across both stages.
Disc springs offer a unique advantage: they can be fully adjusted via size parameters to achieve an approximate linear force-displacement (F-x) curve. This is the second notable feature of disc springs—the variable stiffness characteristic. Based on empirical formulas for a single disc spring, the load-deformation relationship can be expressed as F = x * S^3 * A * D^2, where D, d, h, and s represent the outer diameter, inner diameter, inner height, and thickness of the disc spring, respectively. The coefficient A is obtained from a table based on the ratio of D/d, ensuring accurate modeling of the spring's behavior.
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