Char-Ming Chin

Perturbation Methods in Nonlinear Dynamics—Applications to Machining Dynamics

The role of perturbation methods and bifurcation theory in predicting the stability and complicated dynamics ofmachining is discussed using a nonlinear single-degree-of-freedom model that accounts for the regenerative effect, linear structural damping, quadratic and cubic nonlinear stiffness of the machine tool, and linear, quadratic, and cubic regenerative terms. Using the width of cut w as a bifurcation parameter, we find, using linear theory, that disturbances decay with time and hence chatter does not occur if w w c . In other words, as w increases past W c , a Hopf bifurcation occurs leading to the birth of a limit cycle. Using the method of multiple scales, we obtained the normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that the bifurcation is supercritical; that is, local disturbances decay for w w c . Using a six-term harmonic-balance solution, we generated a bifurcation diagram describing the variation of the amplitude of the fundamental harmonic with the width of cut. Using a combination of Floquet theory and Hills determinant, we ascertained the stability of the periodic solutions. There are two cyclic-fold bifurcations, resulting in large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability. As the width of cut w increases, the periodic solutions undergo a secondary Hopf bifurcation, leading to a two-period quasiperiodic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the torus doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portraits, Poincare sections, and power spectra. The results indicate the importance of including the nonlinear stiffness terms.

BIFURCATIONS IN A POWER SYSTEM MODEL

Bifurcations are performed for a power system model consisting of two generators feeding a load, which is represented by an induction motor in parallel with a capacitor and a combination of constant power and impedance PQ load. The constant reactive power and the coefficient of the reactive impedance load are used as the control parameters. The response of the system undergoes saddle-node, subcritical and supercritical Hopf, cyclic-fold, and period-doubling bifurcations. The latter culminate in chaos. The chaotic solutions undergo boundary crises. The basin boundaries of the chaotic solutions may consist of the stable manifold of a saddle or an unstable limit-cycle. A nonlinear controller is used to control the subcritical Hopf and the period-doubling bifurcations and hence mitigate voltage collapse.

DYNAMICS OF A CUBIC NONLINEAR VIBRATION ABSORBER

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.

Nonlinear Normal Modes of a Cantilever Beam

Two approaches for determination of the nonlinear planar modes of a cantilever beam are compared. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.

On Nonlinear Normal Modes of Systems With Internal Resonance

A complex-variable invariant-manifold approach is used to construct the normal modes of weakly nonlinear discrete systems with cubic geometric nonlinearities and either a one-to-one or a three-to-one internal resonance. The nonlinear mode shapes are assumed to be slightly curved four-dimensional manifolds tangent to the linear eigenspaces of the two modes involved in the internal resonance at the equilibrium position. The dynamics on these manifolds is governed by three first-order autonomous equations. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifrcations of the calculated nonlinear normal modes are investigated.

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

Three-to-One Internal Resonances in Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.

Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifitrcation. A combination ofa shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Nonlinear Dynamics of a Boom Crane

We study a boom crane modeled as a spherical pendulum undergoing base excitations. We demon strate how instabilities in the payload motion arise due to a combination of a one-to-one internal resonance and a primary (additive) resonance or a parametric (multiplicative) resonance. The method of multiple scales is used to derive four nonlinear ordinary-differential equations describing the amplitudes and phases of the in-plane and out-of-plane modes. The modulation equations are used to study the equilibrium and dynamic solutions and their stability. The response could be a single-mode (planar) or a two-mode (three-dimensional) motion. We also study the limit cycles arising in the response and ascertain their stability Numerical re sults indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, intermittency of type I, and cyclic-fold bifurcations.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.