In this paper, an experimental verification of antagonistic stiffness planning is presented for a 2-DOF parallel mechanism with four actuators. With 2-DOF force redundancy, the magnitude and direction of the stiffness enhancement can be controlled by the additional actuators, where the internal torques of the mechanism exist on the two-dimensional null space. In the experiments, the passive and active stiffness are measured, respectively, during endowing the external force at the end-effector. Two stiffness assignment methods for a given pathway are suggested and are verified by the experiments.
This paper presents analytical bifurcations analysis of a “Jeffcott” type rigid rotor supported by five-pad tilting pad journal bearings (TPJBs). Numerical techniques such as nonautonomous shooting/arc-length continuation, Floquet theory, and Lyapunov exponents are employed along with direct numerical integration (NI) to analyze nonlinear characteristics of the TPJB-rotor system. A rocker pivot type five-pad TPJB is modeled with finite elements to evaluate the fluid pressure distribution on the pads, and the integrated fluid reaction force and moment are utilized to determine coexistent periodic solutions and bifurcations scenarios. The numerical shooting/continuation algorithms demand significant computational workload when applied to a rotor supported by a finite element bearing model. This bearing model may be significantly more accurate than the simplified infinitely short-/long-bearing approximations. Consequently, the use of efficient computation techniques such as deflation and parallel computing methods is applied to reduce the execution time. Loci of bifurcations of the TPJB-rigid rotor are determined with extensive numerical simulations with respect to both rotor spin speed and unbalance force magnitude. The results show that heavily loaded bearings and/or high unbalance force may induce consecutive transference of response in forms of synchronous to subsynchronous, quasi-periodic responses, and chaotic motions. It is revealed that the coexistent responses and their solution manifolds are obtainable and stretch out with selections of pad preload, pivot offset, and lubricant viscosity so that the periodic doubling bifurcations, saddle node bifurcations, and corresponding local stability are reliably determined by searching parameter sets. In case the system undergoes an aperiodic state, the rate of divergence/convergence of the attractor is examined quantitatively by using the maximum Lyapunov exponent (MLE).
Nonlinear elements found in fluid film journal bearings and their surrounding structures are known to induce sub- and super-synchronous, chaos and thermally-induced instability responses in rotor-bearing systems. The current review summarizes the literature on journal bearing induced nonlinear, rotordynamic forces and responses. Nonlinear, thermo-elasto-hydro-dynamic (TEHD) aspects of journal bearings has become increasingly important in high performance turbomachines. These have significant influence on bearing dynamic performance and thermally-induced, rotordynamic instability problems. Techniques for developing TEHD bearing models are discussed in section 2. Nonlinear solution methodology, including bifurcation determination and time and frequency domain methods such as harmonic balance, shooting and continuation, etc. are presented in section 3. Numerical tools to determine nonlinear vibration responses, including chaos, along with examples of bearing induced nonlinear vibrations are presented in sections 4 and 5, respectively.
The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.
This paper presents a numerical study for nonlinear rotordynamic response with bifurcations of tilting pad journal bearings when pad–pivot friction forces are taken into account. A Stribeck friction model is employed to determine the friction coefficient for the contacts between the pads and the spherical-type pivots. The boundary/mixed/hydrodynamic friction mode is determined for each pad surface based on the instantaneous angular motion of the pads. A Jeffcott type rotor supported on 5-pad tilting pad journal bearings is used for the structural model, and finite element fluid film models are utilized to calculate the reaction forces and moments on the pads. The simulation results show that pad–pivot friction plays an important role in determining the stability of the rotor system. For the autonomous condition, the friction induces a Hopf bifurcation and generates limit cycles at high rotor spin speed (>14 krpm), which were originally stable equilibrium states with a no friction condition. For the nonautonomous condition, the 1× synchronous response becomes subsynchronous/quasiperiodic responses in the high-speed range (>14 krpm) with the appearances of Neimark-Sacker bifurcations. It is shown that the outbreak points and corresponding response types are highly dependent on the state of disk imbalance. A comparison of the linear and nonlinear models clearly illustrates the importance of retaining nonlinear forces to determine potential deleterious vibration.
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