Hysteresis can be effectively modeled using a mechanical system, typically in series configuration. This is because, in the relationship of inelastic deformation, it is assumed that the inelastic deformation corresponds to the force applied to the elastic part rather than the actual displacement. The model consists of a spring and a damper. When the system is suddenly subjected to a force, the spring responds instantaneously, with its behavior determined solely by the spring constant. However, as the force is sustained, the damper gradually relaxes, leading to an increase in total deformation. Conversely, when the force is removed, the spring releases some of its stored energy, causing partial recovery. This model provides a more realistic representation of the hysteresis observed in elastic materials.
Experiments were conducted on single-leaf springs to minimize friction and geometric nonlinearities. In the absence of hysteresis, geometric nonlinearity, and friction, the load-displacement curve during loading and unloading would follow a straight line OA. If geometric nonlinearity is present, this line becomes curved. The stiffness of the leaf spring is generally constant at k = 9P9W = tanH, but when geometric nonlinearity is involved, the stiffness becomes variable. With hysteresis, the unloading path starts from point B (Wa, Pa), and the displacement is divided into two parts: the elastic displacement Wp and the hysteric displacement Wr. Under a given load P, Wp = P/k, while the total displacement during loading and unloading is W1,2 = Wp + Wr. A specific equation is used to derive the relationship between the hysteric displacement and the load.
Considering the hysteretic deformation Wr in a single-section spring, which is often treated as a cantilever beam, the formula for the elastic strain energy is Er = 2KR²/E. The internal bending moment R is given by M(x) = Px, and the stress is σ = Py/I. Substituting these values leads to E(x,y) = 2KP²x²y²/(EI²). By analyzing a small segment dx of the beam, we find that the neutral layer experiences no stress or strain, while the top and bottom surfaces undergo compression and tension, respectively. Using the plane assumption from beam theory, the deflection dH is calculated as dH = dx / [h/(2Er(x,h/2))]. This leads to the relationship between the hysteresis-induced deflection Wr and the curvature Q of the neutral axis: Wr(1 + Wcr²)³/² = 1/Q. Under small deformation conditions, Wcr ≈ 1, and the deflection at the fixed end is zero. Integrating the expressions for variable cross-section beams still results in Wr ∠P².
Applying the same derivation to the unloading process yields a similar relationship, where Wr = KrP². For a few leaf springs, the hysteresis-related displacement is proportional to the load. This was confirmed through experiments, which showed that the ratio of hysteresis displacement to total displacement is only about 0.025, making it negligible in most cases. As a result, the model can be simplified accordingly.
In small leaf spring systems, the displacement due to hysteresis follows a quadratic relationship with the load. During unloading, the displacement difference under the same load is given by W = W2 - W1 = -2Kr(P - Pa/2)² + KrPa²/2. This shows that the hysteresis-induced displacement is a function of the square of the load. These parameters are unique to the material and structure. Once experimentally determined, they allow accurate prediction of the hysteresis effect.
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