Non-linear random vibration isolators

A study of the influence of non-linear spring stiffness characteristics on the effectiveness of vibration isolators with linear damping, subjected to stationary, random white noise ground acceleration is presented. The probability density function of spring displacement is determined analytically by means of the Fokker-Planck equation and both r.m.s. and mean peak values of spring displacement and mass acceleration are presented for three types of spring non-linearity: (a) cubic hard spring, (b) cubic soft spring, (c) tangent spring. The cubic hard spring and cubic soft spring are characterized by the non-dimensional coefficient σ2s0e where σs0 is the r.m.s. spring displacement for the linear system and e is the ratio k3/k1, where k1, and k3 are the linear and cubic stiffness coefficients, respectively. For the hard spring the r.m.s. and mean peak spring displacement are shown to decrease asymptotically for large e in proportion to e−1/4 and the corresponding accelerations are found to increase in proportion to e1/4. The ratio (mean peak/r.m.s.) displacement was found to have an asymptotic value of 2·966 and the corresponding ratio for acceleration was 9·96 for N = 3000 maxima, in contrast to the value of 4·146 for a linear spring. The usefulness of the cubic soft spring in reducing transmitted vibration is of limited value due to the need to avoid the possibility of snap-through buckling caused by a reduction of stiffness with increasing spring deflection. The resulting reductions in acceleration were found not to exceed about 30% compared to those for the linear spring. The tangent spring was found to lead to large increases in mean peak acceleration for gap ratios d/σs0