Vibration Analysis of a Two‐Step Sine Movable Tooth Drive

Step-type compound planetary gear sets are widely applied in vehicle systems. Comprehensive dynamic model of step-type compound planetary gear sets, which includes translational, rotational vibrations and static transmission errors, was established. Natural vibration characteristics of the system, such as natural frequencies and vibration modes, were analyzed. Belt shape distribution characteristics of its natural frequencies was researched. According to vibration characteristics of both central components and planets, natural vibration modes of the system are classified into three types: central components translational vibration and planets random vibration, central components rotational vibration and planets identical vibration, central components static and adjacent planets reverse vibration mode.

In this paper, a micro electromagnetic harmonic drive system is proposed. Considering Van der Waals force, dynamics equation of the flexible ring for the micro drive system is deduced and resolved. Using the equations, the effects of the molecule force on the natural frequencies and vibration modes of the drive system are investigated. Results show that considering molecule force, natural frequencies of the flexible ring are reduced and its vibration modes are changed. For lower order modes, smaller clearance between the flexible ring and stator, smaller thickness of the flexible ring and larger radius of the flexible ring, the effects of the molecule force on the natural frequencies and vibration modes are more obvious.

In modern aerospace technology, strapdown inertial navigation systems (SINS) are widely used, using fiber-optic (FOG) or ring laser (CLG) gyroscopes. During the operation of such systems, the sensitivity axes are rotated relative to the basic coordinate system. The resulting angles between the axes of the base coordinate system and the axes of sensitivity of the navigation system (non-orthogonality) are one of the factors leading to an increase in the measurement errors of the device, which affects the measurement accuracy. During operation, the system is affected by vibrations of various nature, the impact of which can contribute to the appearance of non-orthogonality. The purpose of this work is to determine the maximum permissible vibration amplitudes affecting the SINS body according to the permissible values ​​of the deviation of the FOG sensitivity axes for two variants of the SINS layout. An approach to determining the permissible amplitudes of an external harmonic impact on the unit of a strapdown inertial navigation system based on fiber-optic or ring laser gyroscopes is considered. A design scheme, mathematical and finite element models for calculating natural frequencies and forced oscillations of a strapdown inertial navigation system unit have been developed. In various frequency ranges, numerical calculations have determined the boundary values ​​of the amplitudes of the external harmonic impact on the base of specific configurations of the SINS assembly. It has been established that dangerous states take place in the region of the 1st natural frequency of the system, as well as near higher frequencies. Comparison of the results for design options 1 and 2 allows us to conclude that in order to weaken the effect of vibrations on the accuracy of the SINS unit, it is advisable that the lowest natural vibration frequencies for the SINS assembly be as high as possible (more than 1000 Hz).
 Key words: vibration; fiber optic gyroscope; strapdown inertial navigation system; finite element method; natural frequencies and modes of vibration.

Based on the principle of axial active magnetic bearing system with single degree of freedom, the difference between magnetic system vibration mode and the traditional bearing system vibration mode were described. Initial four order natural frequencies and vibration mode were calculated by the module of modal analysis of ANSYS11. Axial displacement and control current influence on natural frequencies and vibration mode were also analyzed. At last they were compared with the results of experimental modal analysis. Analysis shows that there was a certain difference between the results of finite element method and experimental model analysis, but the difference was small. It provided theoretical basis for flexible system design and dynamic design.

There is a difficult technology problem of high-pressure differential multi-layer separate injection, which had given birth to after using the technology of water injection in the process of oil extraction, the electrical control type water distributor has been designed for the solving of this difficult technology problem. The dynamic characteristics for the executive body of electrical control water type distributor had been studied based on the technology of virtual prototyping and the theories of multi-rigid-body kinematics and modal analysis. The three-dimensional model and dynamics model had been established firstly using UG software, then the kinematics analysis and modal analysis of its key components had been simulated using ADAMS software, its kinematics discipline and natural frequencies and vibration modes have been simulated at last. Therefore, the simulation results provide theory data for its structural optimization design.

A schematic description of a stand based on a digital speckle interferometer with a diffuse reference wave is given, which makes it possible to determine the natural frequencies and vibration modes of the blades in real-time. In the frequency range of 100 ... 3000 Hz, an experimental study of the vibration characteristics of a turbine rotor blade of a gas turbine engine was carried out, under free boundary conditions, which were achieved by placing the blade on soft silicone racks. Blade dimensions: height along the trailing edge - 288 mm, a chord in the middle section - 88.5 mm. 7 modes of vibrations have been identified. The technology of creating a solid-state geometric model of a rotor blade based on a faceted body obtained by 3D scanning with an accuracy of 0.01 mm is presented. The finite element method, using the Lagrange variation principle, is used to calculate the values of natural frequencies and vibration modes of a blade based on the developed geometric model. Concerning the frequency range 100 ... 3000 Hz, using the Ansys Workbench software package, a series of calculations of the resonant frequencies of the blade, by the finite element method, in the range of variation of the values of mechanical properties was carried out: Young's modulus E = 200 ... 230 GPa; Poisson's ratio μ = 0.26 ... 0.3. The density of the material: ρ = 7830 kg / m³, determined experimentally, by the method of hydrostatic weighing. The final element used in the calculation is (tetrahedron) SOLID 187; the minimum element size is 0.6 mm. The total number of elements is about 1.5 ∙ 10⁶. Based on the two-dimensional spline interpolation of the calculated data, the dependence of the standard deviation of the calculated and experimental values on the adopted mechanical properties of the material of the rotor blade is built. In the range of values E = 215 ... 217 GPa and μ = 0.295 ... 0.3, the minimum standard deviation of the calculated frequencies from their experimental values is 0.73%. Provided the value of the standard deviation of frequencies is 1%, the range of values of mechanical properties will be: E = 211 ... 220 GPa and μ = 0.26 ... 0.3. It is shown that narrowing the studied range of values of Young's modulus and Poisson's ratio does not significantly affect the results obtained.

Free Vibration Modes Analysis of Femur Bone Fracture Using Varying Boundary Conditions based on FEA

The mode characteristics and the parameter sensitivity for a two-stage NGW spur planetary gear system are studied based on the principles of structure natural dynamical characteristics. Considering the influence of flexible structure including shaft, the planet carrier, and the ring-gear, the coupled lateral-torsional-axial vibration dynamical model of the planetary gear system is established under the generalized coordinate system using the shafting element method. With the model, the natural frequency and vibration mode are solved, and the results indicate that the flexibility of ring-gear has a greater effect on natural frequency. Several distinct types of vibration mode are summarized, such as planet torsional mode, sun-gear shaft axial mode, ring-gear axial mode and so on. However, the translational mode which is one of the modes in the coupled lateral-torsional lumped mass model is not found in this study. Within the scope of the time-varying, mesh stiffness mainly affects the planet torsional mode of corresponding stage. Furthermore, the variation of radial bearing stiffness will also do effect on axial vibration mode, and the variation of bearing stiffness not only affects the vibration modes of adjacent stage of planetary gear train, but also affects the nonadjacent stage. The results demonstrate the coupling characteristics of the system under the free vibration condition.

Natural vibration characteristics of the helical gear of the increasing gear box for wind generator has an important influence on the normal use of wind turbine. In order to improve the smooth and reliability of the driven system and improve the design efficiency of helical gear, it is very necessary to the natural frequency of gear by modal analysis. In this paper, the model of helical gear is established by using three-dimensional modeling software PRO/E, and through data exchange interface, the model was converted into ANSYS, then the model analysis was carried out and the low natural vibration frequencies and vibration mode were obtained. This is the basis for the design and optimization of the gear transmission system.

This work examines the free vibration and stability of a spinning, elastic disk-spindle system. The extended operator formulation is exploited to discretize the system using Galerkin’s method (Parker, 1999). The coupled vibration modes of the system consist of disk modes, in which the disk dominates the system deformation, and spindle modes, in which the spindle dominates the system deformation. Both the natural frequencies and vibration modes are strongly affected by disk flexibility. If the membrane stresses associated with disk rotation are neglected then the system exhibits flutter instabilities, but these instabilities are not present when membrane stresses are modeled. Natural frequency veering between disk and spindle frequencies is prominent at low speeds and substantially affects the spectrum and stability. No veering is observed at high speeds where rotational stress stiffening diminishes disk-spindle coupling and causes the natural frequencies to converge to those of a rotating spindle carrying a rigid disk. Changes to the vibration modes are examined in terms of a strain energy ratio measuring the contribution of the disk strain energy to the total modal strain energy.

The paper discusses the possibility to control coupled bending-twisting vibrations of composite wing by means of the monoclinic structures in the reinforcement of the plating. Decomposing the potential straining energy and kinetic energy of natural vibration modes into interacting and non-interacting parts, it became possible to introduce the two coefficients that integrally consider the effect of geometry and reinforcement structure upon the dynamic response parameters of the wing. The first of these coefficients describes the elastic coupling of the natural vibration modes, the second coefficient describes the inertial one. The paper describes the numerical studies showing how the orientation of considerably anisotropic CRP layers in the plating affects natural frequencies, loss factors, coefficients of elastic and inertial coupling for several lower tones of natural bending-twisting vibrations of the wing. Besides, for each vibration mode, partial values of the above mentioned dynamic response parameters were determined by means of the relationships for orthotropic structures where instead of “free” shearing modulus in the reinforcement plant, “pure” shearing modulus is used. Joint analysis of the obtained results has shown that each pair of bendingtwisting vibration modes has its orientation angle ranges of the reinforcing layers where the inertial coupling caused by asymmetry of the cross-section profile with respect to the main axes of inertia decreases, down to the complete extinction, due to the generation of the elastic coupling in the plating material. These ranges are characterized by the two main features: 1) the difference in the natural frequencies of the investigated pair of bendingtwisting vibration modes is the minimum and 2) natural frequencies of bending-twisting vibrations belong to a stretch restricted by corresponding partial natural frequencies of the investigated pair of vibration modes. This result is of practical importance because it enables approximate analysis of real composite wings with complex geometry in the existing commercial software packages.

The paper reports an algorithm to analytically solve one of the problems in the mechanics of elastic bodies, which is associated with studying the natural vibrations of a composite two-stage plate whose concave part is smoothly aligned with the part of a constant thickness. We have defined patterns for stating the boundary and transitional conditions, which should be taken into account when considering the natural vibrations of a two-stage plate.The ratios have been obtained, which make it possible to study the distribution of deflections and determine the values of amplitudes of the curved vibrations of the plate. It was noted that the modes of vibrations are based on the symmetry and factorization methods that we had developed and refined earlier. Specifically, it has been found that the deflections can be explored through expressions that are derived through the sum of relevant solutions to two linear second-order differential equations with variable coefficients.Based on the proposed approach, a system consisting of eight homogeneous algebraic equations has been defined, which allowed us to build a frequency equation for the plate rigidly fixed along the inner contour and free along the outer contour. We have determined the values for the plate’s natural frequencies for the first three modes of natural vibrations. Moreover, in order to verify and expand a set of plates of different configurations, the plates with two types of concave in their variable part have been considered. The new approaches and the ratios based on them could be useful for the further advancement of methods for solving similar problems in mathematical physics on natural values. A practical implementation is the problems about the vibrations of plates with variable thickness and of different modes

Quadruped animals switch gait patterns with walking frequency for energy-effective walking. This is similar to the phenomenon that excited natural vibration modes are switched with vibration frequency in a MDOFs system. Therefore, in this paper, it is assumed that quadruped animals walk by using the natural vibration of their own musculoskeletal systems. In the simplest rigid-body-link model consisting of a body and four legs, there are the natural vibration modes similar to the gait patterns (trot, pace and gallop) of quadruped animals. However, all the natural frequencies in the model exist near the natural frequency in the free leg, and are accordingly different from the walking frequencies of actual quadruped animals. In the rigid-body-link model added a scapular and a pelvic on observations of quadrupted walking, the natural frequency of the gallop mode corresponding to high speed walking raises greatly and approaches the walking frequency. If the physical characteristics of horses were applied to the rigid-body-link model added the leg's joints, the natural vibration in the model is close to the gait patterns of horses.

1. The dam length limitation (influence of the canyon walls) has a substantial effect on its stress-strain state and dynamic characteristics — natural frequencies and modes of vibration. 2. The data from model and analytical investigations indicate that when the ratio Lc/ Hd⩽6 it is necessary to take into consideration in static and dynamic calculations of earth dams their length limitations. 3. A decrease in the ratio Lc/Hd increases the natural frequency of the system, concentrates the spectrum of their periods, and gives rise to modes of vibrations inherent to three-dimensionally working structures (arch dams). 4. A change in the natural modes of vibration of a structure with consideration of its length limitation leads to a different (in comparison with calculation according to the scheme of the two-dimensional problem) distribution of seismic accelerations. The maximum seismic accelerations occur not on the dam crest but lower and are localized in the central zone, occupying a limited area in the form of an ellipse with large vertical axis. 5. With consideration of the dam length limitation the zone of high seismic accelerations, occupies a sixfold smaller area in the dam than in the case of calculation according to the scheme of the two-dimensional problem. In this case the lower and near-wall zones of the dam are subjected to substantially smaller seismic accelerations. 6. Calculations of the slope stability for dams located in narrow sites (Lc/Hd⩽6) should be made by methods which permit taking into account the variable seismic accelerations over the length and height of the dam and the possibility of sliding of part of the slope along an arbitrary (not necessarily cylindrical) surface.

Coupled effect of in-plane magnetic field and size effect on vibration properties of the completely free double-layered nanoplate system