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### Research Papers

J. Comput. Nonlinear Dynam. 2011;7(1):011001-011001-10. doi:10.1115/1.4004122.

In this investigation, a new three-dimensional nonlinear train car coupler model that takes into account the geometric nonlinearity due to the coupler and car body displacements is developed. The proposed nonlinear coupler model allows for arbitrary three-dimensional motion of the car bodies and captures kinematic degrees of freedom that are not captured using existing simpler models. The coupler kinematic equations are expressed in terms of the car body coordinates, as well as the relative coordinates of the coupler with respect to the car body. The virtual work is used to obtain expressions for the generalized forces associated with the car body and coupler coordinates. By assuming the inertia of the coupler components negligible compared to the inertia of the car body, the system coordinates are partitioned into two distinct sets: inertial and noninertial coordinates. The inertial coordinates that describe the car motion have inertia forces associated with them. The noninertial coupler coordinates; on the other hand, describe the coupler kinematics and have no inertia forces associated with them. The use of the principle of virtual work leads to a coupled system of differential and algebraic equations expressed in terms of the inertial and noninertial coordinates. The differential equations, which depend on the coupler noninertial coordinates, govern the motion of the train cars; whereas the algebraic force equations are the result of the quasi-static equilibrium conditions of the massless coupler components. Given the inertial coordinates and velocities, the quasi-static coupler algebraic force equations are solved iteratively for the noninertial coordinates using a Newton–Raphson algorithm. This approach leads to significant reduction in the numbers of state equations, system inertial coordinates, and constraint equations; and allows avoiding a system of stiff differential equations that can arise because of the relatively small coupler mass. The use of the concept of the noninertial coordinates and the resulting differential/algebraic equations obtained in this study is demonstrated using the knuckle coupler, which is widely used in North America. Numerical results of simple train models are presented in order to demonstrate the use of the formulation developed in this paper.

J. Comput. Nonlinear Dynam. 2011;7(1):011002-011002-11. doi:10.1115/1.4004263.

In this investigation, the effect of revolute joints’ clearance on the dynamic performance of mechanical systems is reported. A computation algorithm is developed with the aid of SolidWorks/CosmosMotion software package. A slider-crank mechanism with one and two clearance-joints is studied and analyzed when working in vertical and in horizontal planes. The simulation results point out that the presence of such clearance in the joints of the system understudy leads to high peaks in the characteristic curves of its kinematic and dynamic performance. For a multiclearance joints mechanism, the maximum impact force at its joints takes its highest value at the nearest joint to the input link. This study also shows that, when the mechanism works in horizontal plane, the rate of impacts at each clearance-joint increases and consequently the clearance-joints and actuators will deteriorate faster.

J. Comput. Nonlinear Dynam. 2011;7(1):011003-011003-7. doi:10.1115/1.4004309.

The effect of a fast harmonic base displacement and of a fast periodically time varying stiffness on vibroimpact dynamics of a forced single-sided Hertzian contact oscillator is investigated analytically and numerically near sub- and superharmonic resonances of order 2. The study is carried out using averaging procedure over the fast dynamic and applying a perturbation analysis on the slow dynamic. The results show that a fast harmonic base displacement shifts the location of jumps, triggering the vibroimpact response, toward lower frequencies, while a fast periodically time-varying stiffness shifts the jumps toward higher frequencies. This result has been confirmed numerically for both sub- and superharmonic resonances of order 2. It is also demonstrated that the shift toward higher frequencies produced by a fast harmonic parametric stiffness is larger than that induced by a fast base displacement.

J. Comput. Nonlinear Dynam. 2011;7(1):011004-011004-10. doi:10.1115/1.4004468.

A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner’s equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner’s equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning,” J Comput. Nonlinear Dyn., 1 , pp. 205–211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner’s equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained.

J. Comput. Nonlinear Dynam. 2011;7(1):011005-011005-14. doi:10.1115/1.4004376.

Regular and chaotic dynamics of the flexible Timoshenko-type beams is studied using both the standard Fourier (FFT) and the continuous wavelet transform methods. The governing equations of motion for geometrically nonlinear Timoshenko-type beams are reduced to a system of ODEs using both finite element method (FEM) and finite difference method (FDM) to ensure the reliability of numerical results. Scenarios of transition from regular to chaotic vibrations and beam dynamical stability loss are analyzed. Advantages and disadvantages of various wavelet functions are discussed. Application of continuous wavelet transform to the investigation of transitional and chaotic phenomena in nonlinear dynamics is illustrated and discussed.

J. Comput. Nonlinear Dynam. 2011;7(1):011006-011006-9. doi:10.1115/1.4004469.

This research aims to analyze the dynamics of the self-excited vibration of a cleaning blade in a laser printer. First, it is experimentally indicated that that the self-excited vibration is not caused by the negative damping effect based on friction. Next, the excitation mechanism and dynamics of the vibration are theoretically clarified using an essential 2DOF link model, with emphasis placed on the contact between the blade and the photoreceptor. By solving the equations governing the motion of the analytical model, five patterns of static equilibrium states are obtained, and the effect of friction on the static states is discussed. It is shown that one of five patterns corresponds to the shape of the practical cleaning blade, and it is clarified through linear stability analysis that this state becomes dynamically unstable, due to both effects of friction and mode coupling. Furthermore, the amplitude of the vibration in the unstable region is determined through nonlinear analysis. The obtained results show that this unstable vibration is a bifurcation classified as a supercritical Hamiltonian-Hopf bifurcation, and confirms the occurrence of mode-coupled self-excited vibration on a cleaning blade when a constant frictional coefficient is assumed.

J. Comput. Nonlinear Dynam. 2011;7(1):011007-011007-9. doi:10.1115/1.4004547.

The rotations of a parametric pendulum fitted onto a suitable floating support and forced to move vertically under the action of water waves have been studied on the basis of a dedicated wave flume laboratory experiment. An extended experimental campaign has been carried out with the aim of providing insight into the mechanics of the pendulum’s response to the wave forcing and data useful as a benchmark for available theories. A large number of time histories of the pendulum’s angular position have been collected. Rotations have been detected for different values of the frequency and of the amplitude of the excitation, showing the robustness in parameter space, and for different initial conditions, showing the robustness in phase space. This experiment, suggested by the recently developed concept of extracting energy from sea waves, constitutes preliminary experimental proof of that concept’s practical feasibility.

J. Comput. Nonlinear Dynam. 2011;7(1):011008-011008-8. doi:10.1115/1.4004377.

This paper examines the limitations of using B-spline representation as an analysis tool by comparing its geometry with the nonlinear finite element absolute nodal coordinate formulation (ANCF) geometry. It is shown that while both B-spline and ANCF geometries can be used to model nonstructural discontinuities using linear connectivity conditions, there are fundamental differences between B-spline and ANCF geometries. First, while B-spline geometry can always be converted to ANCF geometry, the converse is not true; that is, ANCF geometry cannot always be converted to B-spline geometry. Second, because of the rigid structure of the B-spline recurrence formula, there are restrictions on the order of the parameters and basis functions used in the polynomial interpolation; this in turn can lead to models that have significantly larger number of degrees of freedom as compared to those obtained using ANCF geometry. Third, in addition to the known fact that B-spline does not allow for straightforward modeling of T-junctions, B-spline representation cannot be used in a straightforward manner to model structural discontinuities. It is shown in this investigation that ANCF geometric description can be used to develop new spatial chain models governed by linear connectivity conditions which can be applied at a preprocessing stage allowing for an efficient elimination of the dependent variables. The modes of the deformations at the definition points of the joints that allow for rigid body rotations between ANCF finite elements are discussed. The use of the linear connectivity conditions with ANCF spatial finite elements leads to a constant inertia matrix and zero Coriolis and centrifugal forces. The fully parameterized structural ANCF finite elements used in this study allow for the deformation of the cross section and capture the coupling between this deformation and the stretch and bending. A new chain model that employs different degrees of continuity for different coordinates at the joint definition points is developed in this investigation. In the case of cubic polynomial approximation, $C1$ continuity conditions are used for the coordinate line along the joint axis; while $C0$ continuity conditions are used for the other coordinate lines. This allows for having arbitrary large rigid body rotation about the axis of the joint that connects two flexible links. Numerical examples are presented in order to demonstrate the use of the formulations developed in this paper.

J. Comput. Nonlinear Dynam. 2011;7(1):011009-011009-8. doi:10.1115/1.4004886.

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

J. Comput. Nonlinear Dynam. 2011;7(1):011010-011010-11. doi:10.1115/1.4004951.

J. Comput. Nonlinear Dynam. 2011;7(1):011011-011011-11. doi:10.1115/1.4004808.

In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

J. Comput. Nonlinear Dynam. 2011;7(1):011012-011012-8. doi:10.1115/1.4004885.

The identification of parameters in multibody systems governed by ordinary differential equations, given noisy experimental data for only a subset of the system states, is considered in this work. The underlying optimization problem is solved using a combination of the Gauss–Newton and single-shooting methods. A homotopy transformation motivated by the theory of state observers is proposed to avoid the well-known issue of converging to a local minimum. By ensuring that the response predicted by the mathematical model is very close to the experimental data at every stage of the optimization procedure, the homotopy transformation guides the algorithm toward the global minimum. To demonstrate the efficacy of the algorithm, parameters are identified for pendulum-cart and double-pendulum systems using only one noisy state measurement in each case. The proposed approach is also compared with the linear regression method.

J. Comput. Nonlinear Dynam. 2011;7(1):011013-011013-5. doi:10.1115/1.4004963.

A nonlinear model of inverted pendulum that exhibit unbounded single well $φ6$ potential is described. The complete equation for one-dimensional wind-induced sway is derived. The harmonic balance method along with Melnikov theory are used to seek the effects of aerodynamic drag forces on the amplitude of vibration, on the structure failure, and on the appearance of horseshoes chaos. Numerical simulations have been performed to confirm analytical investigation.

J. Comput. Nonlinear Dynam. 2011;7(1):011014-011014-16. doi:10.1115/1.4004962.

In this paper the nonlinear dynamic responses of a rigid rotor supported by ball bearings due to surface waviness of bearing races are analyzed. A mathematical formulation has been derived with consideration of the nonlinear springs and nonlinear damping at the contact points of rolling elements and races, whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The numerical integration technique Newmark-β with the Newton–Raphson method is used to solve the nonlinear differential equations, iteratively. The effect of bearing running surface waviness on the nonlinear vibrations of rotor bearing system is investigated. The results are mainly presented in time and frequency domains are shown in time-displacement, fast Fourier transformation, and Poincaré maps. The results predict discrete spectrum with specific frequency components for each order of waviness at the inner and outer races, also the excited frequency and waviness order relationships have been set up to prognosis the race defect on these bearing components. Numerical results obtained from the simulation are validated with respect to those of prior researchers.

### Technical Briefs

J. Comput. Nonlinear Dynam. 2011;7(1):014501-014501-4. doi:10.1115/1.4004121.

In this paper, a numerical method is proposed to approximate the solution of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The method is based upon Ritz Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the nonlinear DGRLW equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.