Free Vibration of In-Planar Rotating Ring: An Analytical Solution

2000 ◽  
Author(s):  
Hamid R. Hamidzadeh
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Systematic Approach ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Stress Theory ◽  
Wave Numbers ◽  
Annular Disks ◽  
Circumferential Wave

Abstract An analytical method is presented to consider the vibration characteristics of high speed rotating rings. More specifically, a systematic approach based on an established solution for linear in-plane vibration of spinning annular disks is used to compute natural frequencies and mode shapes of rotating rings. The medium is considered to be homogenous, and elastic isotropic. The developed analytical solution is achieved by implementing the two-dimensional plane stress theory. The modal displacements and stresses at both inner and outer boundaries are determined, and the required boundary conditions are satisfied to obtain the frequency equation. The dimensionless natural frequencies for different modes, rotating speeds, and thickness ratios are computed. In addition, variations of dimensionless critical speeds for several circumferential wave numbers versus radius ratio of the ring are presented. The provided results are for the two different cases of clamped-free and free-free rings.

In-Plane Vibration Mode Shapes for Rotating Disks: Exact Solution

2020 ◽  
Author(s):  
Ehsan Sarfaraz ◽  
Hamid R. Hamidzadeh
Keyword(s):  
High Speed ◽  
Angular Speed ◽  
Free Vibrations ◽  
Polar Coordinates ◽  
Mode Shapes ◽  
Rotating Disks ◽  
Stress Theory ◽  
Annular Disks ◽  
Modal Vibration

Abstract An analytical method is developed for the determination of modal vibration characteristics of high speed rotating annular disks. A systematic approach based on established solutions for the linear in-plane free vibrations of the disks which satisfy the displacement and stresses compatibilities is developed. The disk is considered to be a homogeneous, thin and elastic isotropic, and it is rotating at constant angular speed. The developed analytical solution was obtained by implementing the two-dimensional plane stress theory. In this research, fixed-free and free-free boundary conditions for the annular disks are considered, and natural frequencies, as well as mode shapes of the rotating disks, are computed. The mode shapes are represented by eight functions in polar coordinates. The dimensionless natural frequency parameters are depicted for free vibration of the system for a range of dimensionless rotation speed and radius ratios. Also, the results provide several non-dimensional critical speeds.

Dynamic model for high-speed rotors based on their experimental characterization

2017 ◽  
Vol 2 (4) ◽  
pp. 25
Author(s):  
L. A. Montoya ◽  
E. E. Rodríguez ◽  
H. J. Zúñiga ◽  
I. Mejía
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Experimental Characterization ◽  
Rotating Systems ◽  
Computing Performance ◽  
The Impact ◽  
Stiffness Distribution ◽  
Good Agreement

Rotating systems components such as rotors, have dynamic characteristics that are of great importance to understand because they may cause failure of turbomachinery. Therefore, it is required to study a dynamic model to predict some vibration characteristics, in this case, the natural frequencies and mode shapes (both of free vibration) of a centrifugal compressor shaft. The peculiarity of the dynamic model proposed is that using frequency and displacements values obtained experimentally, it is possible to calculate the mass and stiffness distribution of the shaft, and then use these values to estimate the theoretical modal parameters. The natural frequencies and mode shapes of the shaft were obtained with experimental modal analysis by using the impact test. The results predicted by the model are in good agreement with the experimental test. The model is also flexible with other geometries and has a great time and computing performance, which can be evaluated with respect to other commercial software in the future.

scholarly journals Spatially Resolved Experimental Modal Analysis on High-Speed Composite Rotors Using a Non-Contact, Non-Rotating Sensor

Sensors ◽  
2021 ◽  
Vol 21 (14) ◽  
pp. 4705
Author(s):  
Julian Lich ◽  
Tino Wollmann ◽  
Angelos Filippatos ◽  
Maik Gude ◽  
Juergen Czarske ◽  
...  
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Fiber Reinforced ◽  
Structural Dynamic ◽  
Spatially Resolved ◽  
Glass Fiber Reinforced ◽  
Vibration Responses ◽  
And Performance

Due to their lightweight properties, fiber-reinforced composites are well suited for large and fast rotating structures, such as fan blades in turbomachines. To investigate rotor safety and performance, in situ measurements of the structural dynamic behaviour must be performed during rotating conditions. An approach to measuring spatially resolved vibration responses of a rotating structure with a non-contact, non-rotating sensor is investigated here. The resulting spectra can be assigned to specific locations on the structure and have similar properties to the spectra measured with co-rotating sensors, such as strain gauges. The sampling frequency is increased by performing consecutive measurements with a constant excitation function and varying time delays. The method allows for a paradigm shift to unambiguous identification of natural frequencies and mode shapes with arbitrary rotor shapes and excitation functions without the need for co-rotating sensors. Deflection measurements on a glass fiber-reinforced polymer disk were performed with a diffraction grating-based sensor system at 40 measurement points with an uncertainty below 15 μrad and a commercial triangulation sensor at 200 measurement points at surface speeds up to 300 m/s. A rotation-induced increase of two natural frequencies was measured, and their mode shapes were derived at the corresponding rotational speeds. A strain gauge was used for validation.

Vibrations of Spinning Rings With Radial and Circumferential Displacement Constraints

1991 ◽  
Author(s):  
Shyh-Chin Huang ◽  
Chen-Kai Su
Keyword(s):  
Natural Frequencies ◽  
Point Contact ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Complex Conjugate ◽  
Constrained System ◽  
Form Complex ◽  
Circumferential Displacement ◽  
On The Road ◽  
Displacement Constraints

Abstract The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

Analytical Solution for Vibration of a Rotating Delaminated Composite Beam with End Mass

2016 ◽  
Vol 16 (06) ◽  
pp. 1550013 ◽  
Author(s):  
Ramazan-Ali Jafari-Talookolaei
Keyword(s):  
Analytical Solution ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Mode Shapes ◽  
Material Anisotropy ◽  
Rotating Speed ◽  
Multipliers Method ◽  
Free Mode ◽  
The Difference ◽  
Laminated Composite Beams

In this paper, the free vibration of rotating laminated composite beams (LCBs) with general lay-ups and single through-the-width delamination is analytically investigated. The Hamilton’s principle is used to derive the coupled governing differential equations and boundary conditions for the rotating delaminated beam, considering the effects of shear deformation, rotary inertia, material couplings (bending–tension, bending–twist and tension–twist couplings), and Poisson’s effect. Both the free mode and constrained mode assumptions are adopted. Analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the Lagrange multipliers method. The accuracy is assured by the convergence of the natural frequencies, as well as by comparison with published results. The effects of various factors such as delamination parameter, fiber angle, hub radius, material anisotropy, end mass and rotating speed are studied in detail. The difference between the results based on the free mode and constrained mode assumptions is also investigated.

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.

Effects of staggering and pitch error on the dynamic response of a double-helical gear set

2015 ◽  
Vol 23 (11) ◽  
pp. 1844-1856
Author(s):  
Chen Siyu ◽  
Tang Jinyuan
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Helical Gear ◽  
Mode Shapes ◽  
Gyroscopic Effect ◽  
Gear Teeth ◽  
Gear Backlash ◽  
Pitch Error ◽  
Speed Performance ◽  
Helical Gear Pair

Theoretical analysis is conducted to investigate the nonlinear vibration response in a double-helical gear set under the influence of staggering, single pitch error and cumulative pitch error. A dynamic model of the double-helical gear set is established considering the axial deflection of shaft and gyroscopic effect, and the natural frequencies, mode shapes and critical speeds for the gear set with axial degree and gyroscopic effect are calculated. Moreover, several cases are numerically simulated after consideration of time varying mesh stiffness and gear backlash, and then some general results are deduced about the influences of staggering, single pitch error and cumulative pitch error on vibration intensity and gear teeth impact. Although the pitch error is modeled as a lower frequency excitation, high natural frequencies and modes are excited, especially when the double-helical gear pair operates under high speed condition. Comparisons are listed which are useful for understanding the high speed performance of a high speed gear set.

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