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Research Papers

J. Comput. Nonlinear Dynam. 2017;12(5):051001-051001-7. doi:10.1115/1.4035880.

In this work, a mathematical model is developed for simulating the vibrations of a single flexible cylinder under crossflow. The flexible tube is subjected to an axial load and has loose supports. The equation governing the dynamics of the tube under the influence of fluid forces (modeled using quasi-steady approach) is a partial delay differential equation (PDDE). Using the Galerkin approximation, the PDDE is converted into a finite number of delay differential equations (DDE). The obtained DDEs are used to explore the nonlinear dynamics and stability characteristics of the system. Both analytical and numerical techniques were used for analyzing the problem. The results indicate that, with high axial loads and for flow velocities beyond certain critical values, the system can undergo flutter or buckling instability. Post-flutter instability, the amplitude of vibration grows until it impacts with the loose support. With a further increase in the flow velocity, through a series of period doubling bifurcations the tube motion becomes chaotic. The critical flow velocity is same with and without the loose support. However, the loose support introduces chaos. It was found that when the axial load is large, the linearized analysis overestimates the critical flow velocity. For certain high flow velocities, limit cycles exist for axial loads beyond the critical buckling load.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051002-051002-12. doi:10.1115/1.4035214.

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051003-051003-10. doi:10.1115/1.4035671.

This paper concerns the dynamic simulation of constrained mechanical systems in the context of real-time applications and stable integrators. The goal is to adaptively find a balance between the stability of an over-damped implicit scheme and the energetic consistency of the symplectic, semi-implicit Euler scheme. As a starting point, we investigate in detail the properties of a recently proposed timestepping scheme, which approximates a full nonlinear implicit solution with a single linear system, without compromising stability. This scheme introduces a geometric stiffness term that improves numerical stability up to a certain time-step size, but it does so at the cost of large mechanical dissipation in comparison to the traditional constrained dynamics formulation. Dissipation is sometimes undesirable from a mechanical point of view, especially if the dissipation is not quantified. In this paper, we propose to use an additional control parameter to regulate “how implicit” the Jacobian matrix is, and change the degree to which the geometric stiffness term contributes. For the selection of this parameter, adaptive schemes are proposed based on the monitoring of energy drift. The proposed adaptive method is verified through the simulation of open-chain systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051004-051004-6. doi:10.1115/1.4035896.

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051005-051005-6. doi:10.1115/1.4036195.

Static optimization (SO) has been used extensively to solve the muscle redundancy problem in inverse dynamics (ID). The major advantage of this approach over other techniques is the computational efficiency. This study discusses the possibility of applying SO in forward dynamics (FD) musculoskeletal simulations. The proposed approach, which is entitled forward static optimization (FSO), solves the muscle redundancy problem at each FSO time step while tracking desired kinematic trajectories. Two examples are showcased as proof of concept, for which results of both dynamic optimization (DO) and FSO are presented for comparison. The computational costs are also detailed for comparison. In terms of simulation time and quality of muscle activation prediction, FSO is found to be a suitable method for solving forward dynamic musculoskeletal simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051006-051006-8. doi:10.1115/1.4036116.

This paper describes a very simple beam model, amenable to be used in multibody applications, for cases where the effects of torsion and shear are negligible. This is the case of slender rods connecting different parts of many space mechanisms, models useful in polymer physics, computer animation, etc. The proposed new model follows a lumped parameter method that leads to a rotation-free formulation. Axial stiffness is represented by a standard nonlinear truss model, while bending is modeled with a force potential. Several numerical experiments are carried out in order to assess accuracy, which is usually the main drawback of this type of approach. Results reveal a remarkable accuracy in nonlinear dynamical problems, suggesting that the proposed model is a valid alternative to more sophisticated approaches.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(5):051007-051007-12. doi:10.1115/1.4036118.

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Comput. Nonlinear Dynam. 2017;12(5):054501-054501-5. doi:10.1115/1.4036117.

The control of the motion of nonholonomic systems is of practical importance from the perspective of robotics. In this paper, we consider the dynamics of a cartlike system that is both propelled forward by motion of an internal momentum wheel. This is a modification of the Chaplygin sleigh, a canonical nonholonomic system. For the system considered, the momentum wheel is the sole means of locomotive thrust as well the only control input. We first derive an analytical expression for the change in the heading angle of the sleigh as a function of its initial velocity and angular velocity. We use this solution to design an open-loop control strategy that changes the orientation of sleigh to any desired angle. The algorithm utilizes periodic impulsive torque inputs via the motion of the momentum wheel.

Commentary by Dr. Valentin Fuster

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