Optimization of a Viscoelastic Structure: The Seat-Belt Problem

Optimization of the parameters of elementary linear viscoelastic models is considered for the design of a lap seat belt in automobiles. The vehicle is assumed to stop abruptly on impact. The parameters are optimized to allow the speed of the vehicle before impact to have the largest permissible value consistent with constraints imposed for the safety of the user of the belt. The constraints chosen here are: (a) the maximum displacement of the body after impact is equal to or less than a prescribed critical displacement; (b) the forward speed of the body at the critical displacement does not exceed a prescribed maximum value; (c) the force exerted by the belt on the body during the motion following impact does not exceed a prescribed maximum value. It is found that the optimized Kelvin-Voigt viscoelastic model is nearly 40 percent more effective than the purely elastic material. It is nearly as effective as constant deceleration. An additional and advantageous property is proposed, moreover, for belts of viscoelastic materials. This is that the material should have a relatively low spring rate at relatively small strain rates. The optimized belts for the elementary viscoelastic models are shown to be quite stiff at low strain rates, however.