ABSTRACT This contribution deals with the equivalence problems for systems of implicit ordinary differential equations. Equivalence means that every solution of the original set of equations is a solution of some normal form, and vice versa. The system is identified with the submanifold in a suitable jet-space, defined by the equations. Therefore, a short introduction to jet-theory is presented, as well as its application to systems of differential equations. We present several results for well-determined and under-determined systems and give formulas that describe the transform to an appropriate normal form. Apart from the theoretical results, we give several sketches of computer algebra-based algorithms necessary to solve these problems efficiently. *Communicated by J. McPhee. †This article was published with errors in Mechanics of Structures and Machines, 30(1), pp. 103–121. The complete article with corrections is printed here.

ABSTRACT Vibration isolation components such as rubber bushings or mounts are usually represented by the Voigt model for consideration of their dynamic characteristics in commercial multi-body dynamic analysis programs. The simple Voigt model, however, has limitations in reflecting their frequency-dependent dynamic characteristics. A method, the so-called transfer function model, which can reflect better the frequency-dependent complex stiffness and yet matches well with commercial programs, is suggested. The performance of the proposed technique in dynamic analysis is illustrated for two four-degree-of-freedom systems by comparing the results with those of the Voigt model. *Communicated by G. Hulbert.

To achieve noise and vibration reduction in planetary gear applications, key design parameters are often varied to avoid resonances, optimize load distribution, and reduce weight. In the plots of natural frequencies versus planetary gear parameters, veering phenomena occur when two eigenvalue loci approach each other, but then abruptly veer away. The importance of veering is manifested in the dramatic changes in the vibration modes of the veering natural frequencies and the consequent impact on response. This work analytically characterizes the rules of eigenvalue veering in planetary gears. The coupling factors between two close eigenvalue loci are approximated by perturbation analysis. Special veering patterns were obtained using the unique properties of planetary gear vibration modes. The results are illustrated by an example. Key design parameters were investigated, and generalized guidance is provided for tuning planetary gear natural frequencies. *Communicated by S. Sinha.

The stability of an axially rotating, standing (or hanging) elastic column with an end mass was studied. The base was elastically restrained in the sense that the base moment was proportional to the base slope. It was found that rotation was always destabilizing; the base spring constant was always stabilizing; and the gravity force of the mass was destabilizing when the column was standing. Formulas and graphs for the stability boundaries showing the mutual interaction of the three parameters were found. *Communicated by P. Pedersen.

Numerous studies about the modelization of multibody systems with flexible parts undergoing large rigid body motions and small elastic deformations have recently been performed [[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]]. Some of them [[10][11][12][13][14][15][16][17]], [[20], [[25][26]], [[28]] focused on the influence of geometric nonlinearities, and others [[18][19]], [[21][22][23][24]], [[30][31]] focused on the terms describing dynamic nonlinearities. In this work [[30][31]], we look at the dynamic nonlinearities. The method usually used to model the motion of such systems consists of separating the motion of each component of the system of rigid body motion and small elastic deformations and keeps only the linear terms connected to them. In some cases of fast motion, the problem of the accuracy of this linearization occurs. Precisely, it has been noted that small deformations do not imply small rates of deformation or small accelerations of deformation [[18]]. The question then raised is to know if this linearization is too premature. After establishing a method to allow the evaluation of the relative importance of nonlinearity terms, it appears that these terms must be kept in some cases. The obtained method is general and does not concern only the dynamic terms. A slider-crank mechanism was used to test the theory. Kane's method [[32]] was used to establish the equations of motion, and the software AUTOLEV (Online Dynamics, Sunnyvale, CA) was used for numerical study. *Communicated by E. Haug.

A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper. *Communicated by D. Tortorelli.

The problem of designing minimum weight thin curved beams with piece-wise constant thickness is studied. The material of the beam is assumed to be rigid-plastic that obeys the yield condition and associated flow law. The influence of geometric changes that occur in the post-yield stage are taken into account. The weight minimization is performed under conditions where the maximum deflection of the beam coincides with the deflection of a reference beam of constant thickness. Necessary optimality conditions are derived with the aid of variational methods from optimal control theory. Numerical results are presented for shallow circular beams with two, three, and four steps in thickness.

A modified Adams-Moulton predictor-corrector method is proposed to solve multibody dynamical systems. The proposed method is obtained by combining an Adams-Bashforth predictor method and an Adams-Moulton corrector method with derived weighting coefficients. With the modification, the accuracy of the proposed method is almost one order of magnitude better than the Adams-Moulton predictor-corrector method with the same step size. Stability limits of the proposed method are also studied. Because the present method has greater stability limits than Adams-Moulton predictor-corrector methods, the proposed method has good robustness during the process of time integration. A crank-slider mechanism is used as an example to investigate the capability of the proposed method in solving multibody dynamic systems.

An analytical study of the effect of hob offset on dynamic tooth strength of spur gears is presented. The study was limited to equal and opposite offset values applied to the pinion and gear to maintain the standard operating center distance. The analysis presented is performed using a new version of the NASA gear dynamics code DANST. The operating speed of a transmission has a significant influence on the amount of hob offset required to equalize dynamic stresses in the pinion and gear. In the transmission studied, at low speeds, the optimum hob offset was found to fluctuate within a range. At higher speeds, the optimum value is constrained by the minimum allowed thickness at the tip of the pinion tooth. For gears that must operate over a range of speeds, an average offset value can be used. Spur gears designed with the procedure presented here can have significant improvements in load capacity.

An overall global imperfection-sensitivity analysis of classical bifurcation models is presented. A modification of the classical imperfection-sensitivity functions of the stable-symmetric, unstable-symmetric, and asymmetric bifurcation problems under the effect of configuration-dependent conservative loading device is analyzed. The deformation-sensitive characteristics of loading are considered an imperfection of the dead loading device. By considering the interaction of several types of imperfection, imperfection-sensitivity surfaces are obtained. Geometric, material, and loading imperfections are considered here and applied to classical bifurcation models. To obtain the imperfection-sensitivity surfaces of each model, exact global graphical solutions and approximate local analytical solutions are presented.