A tunable electromagnetic stiffness with bistable, hardening and softening characteristics

Abstract

Nonlinear stiffness elements have acquired wide application in recent years to augment the performance of systems like vibration absorbers, vibration isolators and energy harvesters. Hardening, softening, quasi-zero, bistable and multi-stable stiffness characteristics have been shown to improve system performance in a variety of contexts. Still, the main challenge to the wide use of nonlinear stiffness remains the difficulty in physically realising stiffness mechanisms with the desired load–displacement relationship. Even though a wide variety of physical mechanisms have been proposed, they typically have the disadvantage that in order to change the force amplitude value or the character of load–displacement curve, one or more components of the mechanism have to be changed by dismantling the assembly. Electromagnetic stiffness mechanisms make it easier to tune the force amplitude but are usually limited to a single load–displacement curve. The present article proposes an electromagnetic stiffness mechanism, based on a particular arrangement of permanent magnet and current carrying coil, which can be tuned very easily by varying the polarity and value of current through the coil. Just by varying the current through the coil, the proposed mechanism exhibits linear, hardening, softening and bistable stiffness characteristics. The construction of the mechanism is detailed and different stiffness properties are experimentally demonstrated. A closed form expression for stiffness force, which is in very good agreement with experimental curve, is arrived at, with magnet-coil parameters and current as variables. Analytical expression for threshold current values at which softening to hardening and bistable transition happens are obtained and experimentally validated. The method of multiple scales is used to arrive at an asymptotic solution of the oscillator with the proposed electromagnetic stiffness. This solution is also shown to be in excellent agreement with experimental values. A numerical study of dynamic properties are also carried out and experimentally validated.