Noel C. Perkins

Modal interactions in the non-linear response of elastic cables under parametric/external excitation

Abstract A theoretical model is derived which describes the non-linear response of a suspended elastic cable to small tangential oscillations of one support. The support oscillations, in general, result in parametric excitation of out-of-plane motion and simultaneous parametric and external excitation of in-plane motion. Cubic non-linearities due to cable stretching and quadratic nonlinearities due to equilibrium cable curvature couple these motion components in producing full, three-dimensional cable response. In this study, a two-degree-of-freedom approximation of the model is employed to examine a class of in-plane/out-of-plane motions that are coupled through the quadratic non-linearities. A first-order perturbation analysis is utilized to determine the existence and stability of the planar and non-planar periodic motions that result from simultaneous parametric and external resonances. The analysis leads to a bifurcation condition governing planar stability and results highlight how planar stability is reduced and non-planar response is enhanced whenever a “two-to-one” internal resonance condition exists between a pair of in-plane and out-of-plane cable modes. This two-to-one resonant behavior is clearly observed in experimental measurements of cable response which are also in good qualitative agreement with theoretical predictions.

Nonlinear Oscillations of Suspended Cables Containing a Two-to-One Internal Resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.

Three-Dimensional Oscillations of Suspended Cables Involving Simultaneous Internal Resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.

Non-linear Dynamic Intertwining of Rods With Self-Contact

Abstract Twisted marine cables on the sea floor can form highly contorted three-dimensional loops that resemble tangles. Such tangles or ‘hockles’ are topologically equivalent to the plectomenes that form in supercoiled DNA molecules. The dynamic evolution of these intertwined loops is studied herein using a computational rod model that explicitly accounts for dynamic self-contact. Numerical solutions are presented for an illustrative example of a long rod subjected to increasing twist at one end. The solutions reveal the dynamic evolution of the rod from an initially straight state, through a buckled state in the approximate form of a helix, through the dynamic collapse of this helix into a near-planar loop with one site of self-contact, and the subsequent intertwining of this loop with multiple sites of self-contact. This evolution is controlled by the dynamic conversion of torsional strain energy to bending strain energy or, alternatively, by the dynamic conversion of twist (Tw) to writhe (Wr).

SUPERCRITICAL STABILITY OF AN AXIALLY MOVING BEAM PART I: MODEL AND EQUILIBRIUM ANALYSIS

Axially moving material problems consider the dynamic response, vibration and stability of long, slender members which are in a state of translation. This study focuses on the response of axially moving beam-like elements at translation speeds that exceed the classical “critical speed stability limit”. A non-linear model for an axially moving beam is derived that accounts for the initial beam curvature induced by supporting pulleys or wheels. Presently, the model is used to determine steady responses at critical and supercritical translation speeds. The properties of the equilibrium problem are examined using an approximate linear solution and an exact, non-linear solution. The deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly near and above the critical speed. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to multiple and bifurcated equilibrium states. The stability of the equilibria is assessed in Part II.

Supercritical stability of an axially moving beam part II: Vibration and stability analyses

Abstract This paper focuses on the stability of axially moving beam-like materials (e.g., belts, bands, paper and webs) which translate at speeds near to and above the so-called “critical speed stability limit.” In the companion paper, a theoretical model for an axially moving beam was presented which accounted for geometrically non-linear beam deflections and the initial beam curvature generated by supporting wheels and pulleys. In that paper, analysis of steady response revealed that the beam possesses multiple, non-trivial equilibrium states when translating at supercritical speeds. The equations of motion are presently linearized about these equilibria and their stability is predicted from the eigenvalue problem for free response. Asymptotic and numerical solutions to the eigenvalue problem are presented for the respective cases of small and arbitrary equilibrium curvature. The solutions illustrate that the translating beam has multiple stable equilibrium states in the supercritical speed regime. The solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections, such as initial curvature.

THE VIBRATION AND STABILITY OF A FRICTION-GUIDED, TRANSLATING STRING

Abstract Eyelets, capstans and cylindrical surfaces are often used in thread, fiber and paper handling machinery to guide the axially moving element. In addition to providing positional control, these “guides” introduce dry friction forces that alter the vibration and stability characteristics of the system. This paper examines the lateral response of a string that slides through an elastically supported, dry friction guide. Exact expressions are derived for the linear response under free and forced conditions. Solutions for the eigenvalue spectrum exhibit unusual features including multiple divergence instabilities, regions of flutter instability, and regions of curve veering associated with mode localization. A second order perturbation solution is derived to examine the behavior of the eigenvalue spectrum in regions of flutter instability and curve veering. The analysis highlights the common features of these two phenomena and suggests ways to minimize vibration by adjusting various design variables. The analysis also demonstrates that the eigenvalue loci in regions of flutter instability and curve veering are naturally described by a local hyperbolic approximation.

An efficient multibody dynamics model for internal combustion engine systems

The equations of motion for the major components in an internalcombustion engine are developed herein using a recursive formulation.These components include the (rigid) engine block, pistons, connectingrods, (flexible) crankshaft, balance shafts, main bearings, and enginemounts. Relative coordinates are employed that automatically satisfy allconstraints and therefore lead to the minimum set of ordinarydifferential equations of motion. The derivation of the equations ofmotion is automated through the use of computer algebra as the precursorto automatically generating the computational (C or Fortran) subroutinesfor numerical integration. The entire automated procedure forms thebasis for an engine modeling template that may be used to supportthe up-front design of engines for noise and vibration targets.This procedure is demonstrated on an example engine under free(idealized) and firing conditions and the predicted engine responses arecompared with results from an ADAMS model. Results obtained by usingdifferent bearing models, including linear, nonlinear, and hydrodynamicbearing models, are discussed in detail.

Harmonic balance/Galerkin method for non-smooth dynamic systems

Models of non-linear systems frequently introduce forces with bounded continuity resulting in non-smooth (even discontinuous) flow. Examples include systems with clearances, backlash, friction, and impulses. Asymptotic methods require smooth (differentiable) flow and are therefore ill-suited for analyzing non-smooth systems. In these cases, the traditional harmonic balance method may be used to obtain approximate periodic solutions, but the method suffers from extremely slow convergence in general. Generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence. These improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution. In particular, forming an infinite series of such functions results in an expansion in the same function space of the exact solution. This expansion converges pointwise to the exact solution and to all derivatives thereof. These improvements are illustrated by example upon re-evaluating a classical single degree-of-freedom model for friction-induced vibration.

Computational Analysis of Looping of a Large Family of Highly Bent DNA by LacI

Sequence-dependent intrinsic curvature of DNA influences looping by regulatory proteins such as LacI and NtrC. Curvature can enhance stability and control shape, as observed in LacI loops formed with three designed sequences with operators bracketing an A-tract bend. We explore geometric, topological, and energetic effects of curvature with an analysis of a family of highly bent sequences, using the elastic rod model from previous work. A unifying straight-helical-straight representation uses two phasing parameters to describe sequences composed of two straight segments that flank a common helically supercoiled segment. We exercise the rod model over this two-dimensional space of phasing parameters to evaluate looping behaviors. This design space is found to comprise two subspaces that prefer parallel versus anti-parallel binding topologies. The energetic cost of looping varies from 4 to 12 kT. Molecules can be designed to yield distinct binding topologies as well as hyperstable or hypostable loops and potentially loops that can switch conformations. Loop switching could be a mechanism for control of gene expression. Model predictions for linking numbers and sizes of LacI-DNA loops can be tested using multiple experimental approaches, which coupled with theory could address whether proteins or DNA provide the observed flexibility of protein-DNA loops.