scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

scholarly journals Vibration analysis of rotating rings with complex support stiffnesses

2018 ◽  
Vol 237 ◽  
pp. 01010
Author(s):  
Fuchun Yang ◽  
Yue Zhang ◽  
Hailong Li
Keyword(s):  
Differential Operators ◽  
Natural Frequencies ◽  
Frequency Separation ◽  
Vibration Modes ◽  
Governing Equations ◽  
Nodal Diameter ◽  
Vibration Properties ◽  
Matrix Differential Operators ◽  
Matrix Differential ◽  
Space Matrix

Vibration characteristics of rotating rings with complex support stiffnesses are studied. The complex stiffnesses of the rotating ring include discrete stiffnesses and partially distributed stiffnesses. The governing equations are established by Hamilton’s principle. The governing equations are cast in matrix differential operators and discretized using Galerkin’s method. The eigenvalue problem is dealt with state space matrix and the natural frequencies and vibration modes are obtained. The properties of natural frequencies and vibration modes of rotating rings are studied. The results illustrate that frequency separation and frequency veering happen with the increase of rotation speed. The vibration modes are not dominated by only one nodal diameter while dominated by several nodal diameters because the discrete and partially distributed stiffnesses disrupt the axisymmetry of rotating rings. The influences of several parameters to vibration properties of rotating rings are also investigated.

Vibration of High-Speed Compliant Gear Pairs

2016 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
System Structure ◽  
Vibration Modes ◽  
Mesh Stiffness ◽  
Nodal Diameter ◽  
Gear Teeth ◽  
Wide Range ◽  
System Coupling

This study analytically investigates the vibration of high-speed, compliant gear pairs using a model consisting of coupled, spinning, elastic rings. The gears are elastically coupled by a space-fixed, discrete stiffness element that represents the contacting gear teeth. Hamilton’s principle is used to derive the nonlinear governing equations of motion and boundary conditions. These equations are linearized for small vibrations about the steady equilibrium due to rotation. The equations are cast in operator form, which exemplifies their gyroscopic system structure. The eigenvalue problem is discretized using Galerkin’s method. The natural frequencies and vibration modes for an example aerospace gear pair are numerically calculated for a wide-range of rotation speeds. The system coupling leads to multiple eigenvalue veering regions as the gear rotation speed varies. Highly coupled vibration modes that have meaningful deflection in the discrete mesh stiffness occur within a set frequency band. The vibration modes within this band have distinct nodal diameter components that evolve with rotation speed.

Vibration of Spinning Cantilever Beams Undergoing Coupled Bending and Torsional Motion

2013 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
Original Problem ◽  
Natural Frequencies ◽  
Rotation Axis ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Cantilever Beams ◽  
Torsional Motion ◽  
Governing Equations ◽  
Helicopter Blade ◽  
Wide Range

This study investigates the vibration of a spinning cantilever beam with a rigid body attached to its free end undergoing coupled bending and torsional motion. The rotation axis is perpendicular to the beam (like a helicopter blade). The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system and Galerkin discretization is readily applied where it is not for the original problem. The natural frequencies and vibration modes are investigated over a wide range of rotation speeds.

scholarly journals Vibration of thin cylindrical shells on an elastic foundation coupled with multiple discrete stiffnesses

2018 ◽  
Vol 237 ◽  
pp. 01011 ◽  
Author(s):  
Fuchun Yang ◽  
Wenlei Qiu
Keyword(s):  
Natural Frequency ◽  
Elastic Foundation ◽  
Natural Vibration ◽  
Cylindrical Shells ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Shear Modes ◽  
External Forces ◽  
Circumferential Wave

Vibration properties of thin cylindrical shells on an elastic foundation coupled with multiple discrete stiffnesses were investigated. The discrete stiffnesses were modelled as external forces. Hamilton’s principle was applied to deduce the governing equations. To study the natural vibration properties, the wave-like solutions were applied. Then the governing equations were discretized in matrix form to obtain the eigenvalues. The properties of natural frequencies and vibration modes of thin cylindrical shells were studied. The results illustrate that each natural frequency of cylindrical shells is mainly dominated by one vibration mode, that is, flexural, longitudinal and shear modes. The natural frequency will get closer with the increase of circumferential wave numbers. The influences of several parameters to vibration properties of cylindrical shells were also investigated.

The Free Vibrations of Stiffened Drill Strings With Static Curvature

1967 ◽  
Vol 89 (1) ◽  
pp. 23-29 ◽  
Author(s):  
D. A. Frohrib ◽  
R. Plunkett
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Drill String ◽  
Bending Stiffness ◽  
Mode Shapes ◽  
Static Tension ◽  
Lateral Vibration ◽  
Governing Equations ◽  
Deflection Curve ◽  
Wide Range

The natural frequencies of lateral vibration of a long drill string in static tension under its own weight are primarily the same as those of the equivalent catenary. These frequencies and the mode shapes are affected to a certain extent by the bending stiffness and to a greater extent by the static deflection curve due to lateral deflection of the bottom end. In this paper, the governing equations are derived and general solutions are given in an asymptotic expansion with the bending stiffness as the parameter. Specific numerical results are given in dimensionless form for the first three natural frequencies for a very wide range of horizontal tension and several appropriate values of bending stiffness for zero vertical static force at the bottom.

scholarly journals OPTIMIZATION OF SOME WEAVING MACHINE PARTS DESIGN. THEORETICAL APPROACH

2021 ◽  
Vol 2021 ◽  
pp. 215-221
Author(s):  
A. Mostafa ◽  
W. Hashima ◽  
S. El-Gholmy ◽  
A. Al-Oufy ◽  
M. Hassan
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Operating Time ◽  
Research Work ◽  
Element Model ◽  
Vibration Modes ◽  
Work Study ◽  
Mechanical Factors ◽  
Machine Parts ◽  
The Cost

The factors of increasing productivity, reducing the cost and the quality improvement are the most important research concerns in weaving machinery. Increasing the effectiveness and productivity of production were achieved by increasing the operating time and efficiency of weaving looms. Thus, the manufacturers of weaving equipment attempt to minimize factors that limit production speed and production conditions. Heald frame is one of the known parts of the weaving machine that causes vibrations and noise which are important factors that influence high-speed development of looms. In this research work, study of mechanical factors (stresses and vibration) has been investigated for heald shaft. Finite element model of the heald frame was constructed to simulate different type of material. Then some important natural frequencies and vibration modes are calculated and the results. Results show a major improvement with the usage of these different material. As well as the failure of heald shaft is mainly due to friction and vibration and not due to the stresses or weight.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Vibration Properties of High-Speed Planetary Gears With Gyroscopic Effects

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Eigenvalue Problems ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
High Speeds ◽  
Wide Range ◽  
Modal Property ◽  
Gyroscopic Effects ◽  
Complex Valued

This study investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speed-dependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

ELECTROMECHANICAL DYNAMICS FOR MICRO BEAMS

2006 ◽  
Vol 06 (02) ◽  
pp. 233-251 ◽  
Author(s):  
LIZHONG XU ◽  
XIAOLI JIA
Keyword(s):  
Balance Equation ◽  
Dynamic Equation ◽  
Natural Frequencies ◽  
Electrostatic Force ◽  
Critical Voltage ◽  
Vibration Modes ◽  
Linear Dynamic ◽  
Forced Responses ◽  
Electromechanical Dynamics ◽  
Electromechanical Coupled

In this paper, an electromechanical coupled dynamic equation of a micro beam under an electrostatic force as well as under an electromechanical coupled force is presented. The linearization of above dynamic equation is made, allowing the equation to be divided into a linear dynamic equation for dynamic displacement and a static balance equation for static displacement. Using the balance equation, the changes of the voltage along with displacement are studied. It is shown that there is a critical voltage at which the micro beam will buckle. From the linear dynamic equation, natural frequencies and vibration modes of the micro beam, and its forced responses to voltage excitation are derived. The results show that the natural frequencies and vibrating magnitudes of the micro beam are affected by mechanical and electric parameters. Smaller beam length and voltage as well as larger beam thickness and clearance should be selected in order to obtain smaller vibrating magnitudes. It is also shown that for higher vibration modes, more positions of the peak dynamic displacements occur.

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