A continuum model is presented that describes the three-dimensional response of an elastic cable that supports a single attached mass. Two asymptotic forms of this model are analyzed for the free, linear response of slack suspensions having small equilibrium curvature (sag) and level supports. The first model, which is valid for relatively small attached masses, assumes that the cable stretches quasi-statically and results in uniform dynamic cable tension. The quasi-static stretching assumption is partially relaxed in the second model which accounts for spatially varying dynamic tension in an approximate manner. The eigen-solutions associated with free response are compared for the two models. Results indicate that the “small mass model” provides excellent approximations to the natural frequency spectrum and vibration mode shapes for most cables and modes of technical interest. A simple criterion is presented which governs the range of validity of the small mass model.

This study examines the forced response of a sagged elastic cable supporting an array of discrete masses. Such systems arise, for instance, in ocean engineering applications employing cable hydrophone arrays. The excitation considered is harmonic and normal to the cable and may, for instance, approximate prescribed environmental loading. An asymptotic model is presented that describes the linear forced response of a cable/mass suspension having small equilibrium curvature. Closed-form expressions for the Green’s function to an associated boundary-value problem are obtained using a transfer matrix formulation. The derived Green’s function is utilized to construct integral representations for steady-state response under boundary and/or domain excitation. Solutions obtained for a variety of domain loading distributions demonstrate the utility and efficiency of this solution strategy. The theoretical response predictions are verified through experimental measurements of the natural frequency spectrum and frequency response of laboratory cable/mass suspensions.

A nonlinear model is developed which describes the rotational response of automotive serpentine belt drive systems. Serpentine drives utilize a single (long) belt to drive all engine accessories from the crankshaft. An equilibrium analysis leads to a closed-form procedure for determining steady-state tensions in each, belt span. The equations of motion are linearized about the equilibrium state and rotational mode vibration characteristics are determined from the eigenvalue problem governing free response. Numerical solutions of the nonlinear equations of motion indicate that, under certain engine operating conditions, the dynamic tension fluctuations may be sufficient to cause the belt to slip on particular accessory pulleys. Experimental measurements of dynamic response are in good agreement with theoretical results and confirm theoretical predictions of system vibration, tension fluctuations, and slip.

This study presents an investigation of the coupled longitudinal-transverse waves that propagate along an elastic cable. The coupling considered derives from the equilibrium curvature (sag) of the cable. A mathematical model is presented that describes the three-dimensional nonlinear response of a long elastic cable. An asymptotic form of this model is derived for the linear response of cables having small equilibrium curvature. Linear in-plane response is described by coupled longitudinal-transverse partial differential equations of motion, which are comprehensively evaluated herein. The spectral relation governing propagating waves is derived using transform methods. In the spectral relation, three qualitatively distinct frequency regimes exist that are separated by two cut-off frequencies. This relation is employed in deriving a Green’s function which is then used to construct solutions for in-plane response under arbitrarily distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves which propagate through the cable, one characterizing extension-compressive deformations (rod-type) and the other characterizing transverse deformations (string-type). These waves may propagate or attenuate depending on wave frequency. The propagation and attenuation of both wave types are highlighted through solutions for an infinite cable subjected to a concentrated harmonic excitation source.

This paper analyzes the coupled nonlinear tangential-normal waves that propagate along underwater cable suspensions. Taken with the recently developed linear theory governing the in-plane structural waves (3) and an analysis of nonlinear out-of-plane waves for submerged cables (2), this investigation contributes further understanding toward a nonlinear three-dimensional theory for wave propagation along fluid loaded cables. The nonlinearities present in the in-plane model render the cable/fluid model intractable by exact analytical methods. A numerical solution is pursued in this study using finite difference algorithms. To this end, an infinite cable domain is divided to two sub domains, namely an interior (finite computational) domain and exterior (infinite far field) domain. Closed-form solutions for the approximate linear theory are employed for the far field in constructing nonreflecting boundary conditions for the computational domain. Numerical results highlight the governing role of nonlinear hydrodynamic drag for underwater cable suspentions.

A vehicle track model is developed with the objective of providing new capabilities in modeling track vibration response. Understanding track vibration is essential to evaluating the durability of track components, the vibration energy transmitted to the vehicle, and the acoustic emission from the vehicle. A new element model is derived herein that represents a track span as a continuous elastic member with distributed inertia. This model captures the effects of static track sag. static and dynamic track tension, track translation speed, and the coupling of longitudinal and transverse track vibration. Results from a companion experimental study on a section of track support the use of this continuum approximation. The track element model is extended to describe an entire track circuit for an example military vehicle. An eigenanalysis of this circuit model leads to the system vibration modes that are subsequently employed in a low-order model for forced response. The forced response characteristics resulting from two major excitation sources, roadarm motion and polygonal action, are described. The modal content of the track response is then examined to determine the minimum size model required to describe track vibration. It is concluded that low-order system models may be developed as efficient alternatives to established large degree-of-freedom multi-body track models.

This paper extends the methods and results of a previous paper (Ma and Perkins, 1999) on simulating track-wheel-terrain interaction for tracked vehicle dynamics. A new solution algorithm is described that includes an adaptive finite element method for remeshing the track model during simulation. Doing so produces a track model that more accurately describes the mechanics of a track as the vehicle negotiates rough terrain. The model and solution algorithm are illustrated using a full vehicle model of an M1A1 tank.

Analytical studies of vortex-induced vibration (VTV) of cables during lock-in have considered small amplitude and relatively fast dynamic responses about an equilibrium configuration. However, this equilibrium may change as a result of the significantly increased mean drag created during lock-in. In response to increased drag, the cable may slowly drift downstream causing appreciable changes in cable geometry and tension. The resonance conditions for lock-in may be preserved during this slow drift or they may be disrupted. A nonlinear cable/fluid model is discussed mat captures both fast (small amplitude) motions due to VIV and slow (large amplitude) motions due to drift. These two different time scales of motion are clearly observed in numerical simulations. The significant interplay of the slow drifting motion and the fast VTV is highlighted.

Up-down and rifle aiming maneuvers are common tasks employed by soldiers and athletes. The movements underlying these tasks largely determine performance success, which motivates the need for a noninvasive and portable means for movement quantification. We answer this need by exploiting body-worn and rifle-mounted miniature inertial measurement units (IMUs) for measuring torso and rifle motions during up-down and aiming tasks. The IMUs incorporate MEMS accelerometers and angular rate gyros that measure translational acceleration and angular velocity, respectively. Both sensors enable independent estimates of the orientation of the IMU and thus, the orientation of a subject’s torso and rifle. Herein, we establish the accuracy of a complementary filter which fuses these estimates for tracking torso and rifle orientation by comparing IMU-derived and motion capture-derived (MOCAP) torso pitch angles and rifle elevation and azimuthal angles during four up-down and rifle aiming trials for each of 16 subjects (64 trials total). The up-down trials consist of five maximal effort get-down-get-up cycles (from standing to lying prone back to standing), while the rifle aiming trials consist of rapidly aiming five times at two targets 15 feet from the subject and 180 degrees apart. Results reveal that this filtering technique yields warfighter torso pitch angles that remain within 0.55 degrees of MOCAP estimates and rifle elevation and azimuthal angles that remain within 0.44 and 1.26 degrees on average, respectively, for the 64 trials analyzed. We further examine potential remaining error sources and limitations of this filtering approach. These promising results point to the future use of this technology for quantifying motion in naturalistic environments. Their use may be extended to other applications (e.g., sports training and remote health monitoring) where noninvasive, inexpensive, and accurate methods for reliable orientation estimation are similarly desired.

Newly developed miniature wireless inertial measurement units (IMUs) hold great promise for measuring and analyzing multibody system dynamics. This relatively inexpensive technology enables non-invasive motion tracking in broad applications, including human motion analysis. The second part of this two-part paper advances the use of an array of IMUs to estimate the joint reactions (forces and moments) in multibody systems via inverse dynamic modeling. In particular, this paper reports a benchmark experiment on a double-pendulum that reveals the accuracy of IMU-informed estimates of joint reactions. The estimated reactions are compared to those measured by high precision miniature (6 dof) load cells. Results from ten trials demonstrate that IMU-informed estimates of the three dimensional reaction forces remain within 5.0% RMS of the load cell measurements and with correlation coefficients greater than 0.95 on average. Similarly, the IMU-informed estimates of the three dimensional reaction moments remain within 5.9% RMS of the load cell measurements and with correlation coefficients greater than 0.88 on average. The sensitivity of these estimates to mass center location is discussed. Looking ahead, this benchmarking study supports the promising and broad use of this technology for estimating joint reactions in human motion applications.

The energetics of human motion has been intensely studied using experimental and theoretical methods. Knowing the kinetic energy of the human body, and its decomposition into the kinetic energies of the major body segments, has tremendous value in applications ranging from physical therapy, athlete training, soldier performance, worker health and safety, among other uses. Significant challenges thwart our ability to measure segmental kinetic energy in real (non-laboratory) environments such as in the home or workplace, or on the playing/training field. The aim of this research is to address these challenges by advancing the use of an array of miniaturized body-worn inertial measurement units (IMUs) for estimating segmental kinetic energy. As a step towards this goal, this study reports a benchmark experiment that demonstrates the accuracy of IMU-derived estimates of segmental kinetic energy. The study is conducted on a well-characterized mechanical system, a double pendulum that also serves as an apt model for the lower or upper extremities. A two-node IMU array is used to measure the kinematics of each segment as input to the segmental kinetic energy computations. The segments are also instrumented with two high-precision optical encoders that provide the truth data for kinetic energy. The segmental kinetic energies estimated using the IMU array remain within 3.5% and 3.9% of the kinetic energies measured by the optical encoders for the top and bottom segments, respectively, for the freely decaying pendulum oscillations considered. These promising results support the future development of body-worn IMU arrays for real-time estimates of segmental kinetic energy for health, sports and military applications.

A single DNA molecule is a long and flexible biopolymer that contains the genetic code. Building upon the discovery of the iconic double helix over 50 years ago, subsequent studies have emphasized how its biological function is related to the mechanical properties of the molecule. A remarkable system which high-lights the role of DNA bending and twisting is the packing and ejection of DNA into and from viral capsids. A recent 3D reconstruction of bacteriophage φ29 reveals a novel toroidal structure thought to be 30–40 bp of highly bent/twisted DNA contained in a small cavity below the capsid. Here, we extend an elastic rod model for DNA to enable simulation of the toroid as it is compacted and subsequently ejected from a small volume. We compute biologically-realistic forces required to form the toroid and predict ejection times of several nanoseconds.

DNA is a life-sustaining molecule that enables the storage and retrieval of genetic information. In its role during essential cellular processes, this long flexible molecule is significantly bent and twisted. Previously, we developed an elasto-dynamic rod approximation to study DNA deformed into a loop by a gene regulatory protein (lac repressor) and predicted the energetics and topology of the loops. Although adequate for DNA looping, our model neglected electrostatic interactions which are essential when considering processes that result in highly super-coiled DNA including plectonemes. Herein we extend the rod approximation to account for electrostatic interactions and present strategies that improve computational efficiency. Our calculations for the stability for a circularly bent rod and for an initially straight rod compare favorably to existing equilibrium models. With this new capability, we are now well-positioned to study the dynamics of transcription and other dynamic processes that result in DNA supercoiling.

DNA is a long flexible biopolymer containing genetic information. Proteins often take advantage of DNA’s inherent flexibility to perform their cellular functions. Here we present selected results from our computational studies of the mechanical looping of DNA by the Lactose repressor protein. The Lactose repressor resides in the bacterium E. coli and deforms DNA into a loop as a means of controlling the production of enzymes necessary for digesting lactose. We examine this looping process using a computational rod model [1–3] to understand the strain energy and geometry for the resultant DNA loops. Our model captures the multiple looped conformations of the molecule arising from both multiple boundary conditions and geometric nonlinearities. In addition, the model captures the periodic variation of strain energy with base-pair length as suggested by repression experiments (see, for example, [4, 5]).

The design of suspension systems for high speed railway vehicles involves the simultaneous consideration of those requirements as suspension packaging, ride quality, stability, and cost. A design strategy is presented in this paper that enables an optimal design with respect to these competing requirements. The design strategy consists of four steps including the development of a lumped parameter vehicle model, the determination of vehicle parameters, the formulation of a design objective, and the minimization of the objective to optimize key suspension parameters. The design objective captures vehicle requirements including ride quality, suspension packaging, and wheel/rail holding. Power spectral densities (PSDs) are computed for the vertical vehicle body acceleration, suspension travel and dynamic wheel/rail interaction. The design objective function is calculated based on these PSDs and minimized to yield an optimum. An example suspension design is proposed that improves vehicle ride quality and wheel/rail holding without sacrificing other requirements.

Marine cables under low tension and torsion on the sea floor can undergo a dynamic buckling process during which torsional strain energy is converted to bending strain energy. The resulting threedimensional cable geometries can be highly contorted and include loops and tangles. Similar geometries are known to exist for supercoiled DNA (deoxyribonucleic acid) and these also arise from the conversion of torsional strain energy to bending strain energy or, kinematically, a conversion of twist to writhe . A dynamic form of Kirchhoff rod theory is presented herein that captures these nonlinear dynamic processes. The resulting theory is discretized using the generalized-α method for finite differencing in both space and time. The important kinematics of cross-section rotation are described using an incremental rotation “vector” as opposed to traditional Euler angles or Euler parameters. Numerical solutions are presented for an example system of a cable subjected to increasing twist at one end. The solutions show the dynamic evolution of the cable from an initially straight element, through a buckled element in the approximate form of a helix, and through the dynamic collapse of this helix through a looped form.