On the Vibration of Coupled Traveling String and Air Bearing Systems

1996 ◽  
Vol 118 (3) ◽  
pp. 398-405 ◽  
Author(s):  
A. V. Lakshmikumaran ◽  
J. A. Wickert
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Nodal Point ◽  
Fold Increase ◽  
Computational Effort ◽  
Air Pressure ◽  
Mode Shapes ◽  
Air Bearing ◽  
Design Variables ◽  
Axially Moving

Air bearings are used to position and guide such axially-moving materials as high speed magnetic tapes, paper sheets, and webs. In each case, vibration of the moving medium couples with the air bearing’s dynamics, and techniques are developed here to reduce the computational effort that is required to predict the natural frequencies, damping ratios, and vibration modes of the prototypical traveling string and self-pressurized air bearing model. Automatic nodal point allocation reduces the number of nonlinear equations that arise in finding the equilibrium string displacement and air pressure, and in subsequent vibration analysis, the response is obtained in closed form by using the Green’s function for the traveling string. Global discretization of the air pressure alone then yields a matrix eigenvalue problem which is simpler than that obtained through previous methods which required discretization of both displacement and pressure. Overall, essentially a five-fold increase in computational speed is achieved, thus facilitating design and parameter studies. Changes in the natural frequencies, damping ratios, and coupled displacement-pressure mode shapes with respect to several design variables are discussed and compared with experiments.

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.

Lateral Vibration of Two Axially Translating Beams Interconnected by a Winkler Foundation

2006 ◽  
Vol 129 (3) ◽  
pp. 380-385 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Müftü
Keyword(s):  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Mode Shapes ◽  
Winkler Foundation ◽  
Axial Tension ◽  
Winkler Elastic Foundation ◽  
Critical Tension ◽  
Divergence Instability ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. The natural frequencies and associated mode shapes are obtained. The natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at a critical velocity and a critical tension; and, divergence and flutter instabilities coexist at postcritical speeds, and divergence instability takes place precritical tensions. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams are presented.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2020 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems, a general algorithm has been developed to compute natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam’s cross section.

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.

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.

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.

Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

2005 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Elastic Foundation ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Beam Theory ◽  
Mode Shapes ◽  
Winkler Elastic Foundation ◽  
Canonical State ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

Experimental Study of the Dynamic Response of Partially Filled Pipes Focused on Natural Frequencies and Mode Shapes

2017 ◽  
Author(s):  
Oscar de la Torre ◽  
Xavier Escaler ◽  
Jamie Goggins
Keyword(s):  
Natural Frequencies ◽  
Dynamic Properties ◽  
Computational Cost ◽  
Computational Effort ◽  
Mode Shapes ◽  
Significant Drop ◽  
Fluid Structure ◽  
Hydraulic Machinery ◽  
Piping Systems ◽  
Pipe Geometry

The presence of air in piping systems is a major concern in the industry. Problems like flow disruption, reduction of hydraulic machinery efficiencies or a significant drop in pipe capacity are many times related to this fact. The present paper aims to find a simple and non-intrusive experimental method to detect air in piping systems. The method, based on the dynamic properties of fluid-structure systems and underpinned by a novel low computational cost numerical simulation, accurately predicts the volume of water present in a pipe. Good agreement between numerical and experimental solutions has been obtained using much less computational effort than traditional fully coupled Fluid Structure Interaction with CFD analysis. From the numerical and experimental data, two different mathematical expressions relating the system natural frequencies, both vertically and horizontally, and the area occupied by the water have been obtained. These expressions account for the pipe geometry which theoretically would make them suitable for other diameter and wall thickness values. The paper is combined with a preliminary study of the system’s mode shapes for the different volumes of water.

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