Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop (1955, “The Treatment of Damping Forces in Vibration Theory,” Aeronaut. J., 59(539), pp. 738–742). We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as t→∞. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid (1956, “Free Vibration and Hysteretic Damping,” Aeronaut. J., 60(544), pp. 283–283), which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.

Nonlinear vibration of knitted spacer fabric under harmonic excitation

Knitted spacer fabrics can be an alternative material to typical rubber sponges and polyurethane foams for the protection of the human body from vibration exposure, such as automotive seat cushions and anti-vibration gloves. To provide a theoretical basis for the understanding of the nonlinear vibration behavior of the mass-spacer fabric system under harmonic excitation, experimental, analytical and numerical methods are used. Different from a linear mass-spring-damper vibration model, this study builds a phenomenological model with the asymmetric elastic force and the fractional derivative damping force to describe the periodic solution of the mass-spacer fabric system under harmonic excitation. Mathematical expression of the harmonic amplitude versus frequency response curve (FRC) is obtained using the harmonic balance method (HBM) to solve the equation of motion of the system. Parameter values in the model are estimated by performing curve fit between the modeled FRC and the experimental data of acceleration transmissibility. Theoretical analysis concerning the influence of varying excitation level on the FRCs is carried out, showing that nonlinear softening resonance turns into nonlinear hardening resonance with the increase of excitation level, due to the quadratic stiffness term and the cubic stiffness term in the model, respectively. The quadratic stiffness term also results in biased vibration response and causes an even order harmonic distortion. Besides, the increase of excitation level also results in elevated peak transmissibility at resonance.

On the Nonlinear Vibration Analysis of Ultra Precision Manufacturing Machines With Mode Coupling

Nonlinear free undamped vibrations are investigated for ultra-precision manufacturing (UPM) machines with quadratic stiffness. The modes of the system are linearly coupled. The non-resonant case and the bounded internal resonance case are considered. The results of the non-resonant case indicate that the behavior of the system is the same as the linear behavior. However, for the internal resonance case, the results show that the amplitudes are coupled. The results also indicate that the nonlinear frequencies and amplitudes depend not only on the initial conditions, but also on the location of the isolators with respect to the center of gravity of the UPM.