Balancing Energy to Estimate Damping in a Forced Oscillator with Compliant Contafct

This work identifies damping parameters from compliantcontact vibration systems using energy dissipation concept. To develop the identification algorithms, the energy loss as registered in the force-displacement relationship of the real system is balanced against that of a theoretical model incorporating with an idealized compliant contact. Two approaches, including one based on the harmonic response assumption and the other directly integrating the system responses, are developed. Numerical investigations are performed to illustrate the reliability of the identification algorithms. INTRODUCTION Friction and damping estimation is of great importance to the design, analysis, control, and stability prediction of machines, cutting tools, vehicles and structures [1], some of which can be very sensitive to the damping model [2-8]. Damping estimation can be done by force measurement [9-11], damping coefficient estimation, or general parameter estimation [12]. Long established vibration properties can be exploited for damping estimation in free-response and forced-response systems with viscous damping or Coulomb damping only. Here, we continue a line of work on extracting Coulomb and viscous friction parameters from free oscillation decrements [13-15], and then forced oscillations by which analytical solutions [16-18] are used to obtain estimation equations [19]. Limitations are that these methods are not applicable for damping which is not “smal,”they rely on analytical solutions of single-degree-of-freedom linear systems, and they do not treat friction models other than Coulomb and viscous (see for example, references [8, 20-26]). As such, energy balancing is potentially more generally applicable [27]. There are ways to extend energy balancing to the estimation of damping in multidegree-of-freedom systems, as well [28-31]. In the previous energy-balance work, energy loss in the real vibration systems was expressed in terms of a theoretical model consisting of, for example, viscous and Coulomb friction, and balanced against the input energy. The identification algorithms were derived either by assuming the system with harmonic input and output motion or by directly integrating the input and output signals. These studies concentrate on the rigidly grounded, or rigid-contact problems, such as Coulomb and viscous [27,29], and viscous and quadratic [30]. In this paper, we apply the energy-dissipation identification idea to compliant-contact problems. Many researchers have reported compliance in friction contacts [2-10, 25]. Contact compliance is sufficiently prevalent that a viable damping estimation tool would be of value. This paper aims to add a method to the set of tools for handling these problems. To derive the identification algorithms, the energy dissipated in the theoretical compliant model is introduced in the next section. OSCILLATOR WITH AN IDEAL COMPLIANT CONTACT Contact compliance could be caused by the asperity deformations at the contact interfaces or by the elastic deformation of the surrounding structures. A schematic diagram which shows a base-excited, dual-damped oscillator with an ideal massless compliant contact is presented in Figure 1, where ) (t x and ) (t y indicate the displacements of sliding mass and base excitation, ) (t z denotes the displacement of the hypothetical contact surface, z K represents the stiffness of the contact joint, and ) (t f models the friction force, which is implicitly time dependent via dependence on ) (t x , ) (t x , ) (t y , and ) (t z . The system shown in Figure 1 can have response that consists of “macroscopic slipping” and“microsticking” phases [23]. A schematic x f   diagram is presented in Figure 2 in which both the macroscopic sliding phase (between points D, A and B, C) and the microsticking phase (between points C, D and A, B) are illustrated. We assume that the friction force is modeled as Coulomb friction with equal static and kinetic friction