Dynamic Model and Simulation of Flag Vibrations Modeled as a Membrane

2014 ◽  
Author(s):  
Gary Frey ◽  
Ben Carmichael ◽  
Joshua Kavanaugh ◽  
S. Nima Mahmoodi
Keyword(s):  
Constant Velocity ◽  
Cross Flow ◽  
Vertical Direction ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
System Response ◽  
Wind Flow ◽  
Pressure Function ◽  
Model And Simulation ◽  
Reasonable Model

A flag is modeled as a membrane to investigate the two-dimensional characteristics of the vibration response to an uniform wind flow. Both the affecting tension and pressure functions for the wind flow with constant velocity are introduced and utilized in the modeling. In this case, the tension is caused by the weight of the flag. The pressure function is a function describing the pressure variations caused on the flag when in uniform flow. The pressure function is found by assuming that the air flow is relatively slow and that the flag is wide enough to minimize cross flow at the boundaries. An analysis of the downstream motion of the flag is necessary as well. Hamilton’s principle is employed to derive the partial differential equation of motion. The flag is oriented in the vertical direction to neglect the effect of the flag’s weight on the system’s response. Galerkin’s method is used to solve for the first four mode shapes of the system, and the system response is numerically solved. Simulations reveal a very reasonable model when the flag is modeled as a membrane.

scholarly journals Study on dynamic characteristics of leaf spring system in vibration screen

2021 ◽  
pp. 146134842110229
Author(s):  
Jiacheng Zhou ◽  
Chao Hu ◽  
Ziqiu Wang ◽  
Zhengfa Ren ◽  
Xiaoyu Wang ◽  
...  
Keyword(s):  
Dynamic Characteristics ◽  
Vertical Direction ◽  
Maximum Amplitude ◽  
Exciting Force ◽  
Mode Shapes ◽  
Leaf Spring ◽  
Amplification Factors ◽  
Simulation And Experiment ◽  
Exciting Forces ◽  
Spring System

By studying dynamic characteristics of the leaf spring system, a new elastic component is designed to reduce the working load and to a certain extent to ensure the linearity as well as increase the amplitude in the vertical and horizontal directions in vibration screen. The modal parameters, amplitudes, and amplification factors of the leaf spring system are studied by simulation and experiment. The modal results show that the leaf spring system vibrates in horizontal and vertical directions in first and second mode shapes, respectively. It is conducive to loosening and moving the particles on the vibration screen. In addition, it is found that the maximum amplitude and amplification factor in the horizontal direction appear at 300 r/min (5 Hz) while those in the vertical direction appear at 480 r/min (8 Hz), which are higher than those in the disc spring system. Moreover, the amplitude of the leaf spring system increases proportionally with the increase of exciting force while the amplification factors are basically the same under different exciting forces, indicating the good linearity of the leaf spring system. Furthermore, the minimum exciting force occurs in the leaf spring system under the same amplitude by comparing the exciting force among different elastic components. The above works can provide guidance for the industrial production in vibration screen.

Time Domain Simulations on a Single Point Moored Submerged Sphere of Variable Radius

2005 ◽  
Author(s):  
Joa˜o M. B. P. Cruz ◽  
Anto´nio J. N. A. Sarmento
Keyword(s):  
Wave Propagation ◽  
Time Domain ◽  
Physical Phenomenon ◽  
Single Point ◽  
Vertical Direction ◽  
Equation Of Motion ◽  
Hydrodynamic Coefficients ◽  
Energy Converter ◽  
Submerged Sphere ◽  
The Time Domain

This paper presents a different approach to the work developed by Cruz and Sarmento (2005), where the same problem was studied in the frequency domain. It concerns the same sphere, connected to the seabed by a tension line (single point moored), that oscillates with respect to the vertical direction in the plane of wave propagation. The pulsating nature of the sphere is the basic physical phenomenon that allows the use of this model as a simulation of a floating wave energy converter. The hydrodynamic coefficients and diffraction forces presented in Linton (1991) and Lopes and Sarmento (2002) for a submerged sphere are used. The equation of motion in the angular direction is solved in the time domain without any assumption about its output, allowing comparisons with the previously obtained results.

Straked Riser Design With VIVA

2010 ◽  
Author(s):  
Dhyanjyoti Deka ◽  
Paul R. Hays ◽  
Kamaldev Raghavan ◽  
Mike Campbell
Keyword(s):  
Traveling Waves ◽  
Oil And Gas ◽  
Cross Flow ◽  
Oil And Gas Industry ◽  
Response Prediction ◽  
Design Tool ◽  
Mode Shapes ◽  
Fe Model ◽  
Viv Suppression ◽  
Modal Solution

VIVA is a vortex induced vibration (VIV) analysis software that to date has not been widely used as a design tool in the offshore oil and gas industry. VIVA employs a hydrodynamic database that has been benchmarked and calibrated against test data [1]. It offers relatively few input variables reducing the risk of user induced variability of results [2]. In addition to cross flow current induced standing wave vibration, VIVA has the capability of predicting traveling waves on a subsea riser, or a combination of standing and traveling waves. Riser boundary conditions including fixed, pinned, flex joint or SCR seabed interaction can be modeled using springs and dashpots. VIVA calculates riser natural frequencies and mode shapes and also has the flexibility to import external modal solutions. In this paper, the applicability of VIVA for the design of straked steel catenary risers (SCR) and top tensioned risers (TTR) is explored. The use of linear and rotational springs provided by VIVA to model SCR soil interaction and flex joint articulation is evaluated. Comparisons of the VIV fatigue damage output with internal and external modal solution is presented in this paper. This paper includes validation of the VIVA generated modal solution by comparing the modal frequencies and curvatures against a finite element (FE) model of the risers. Fatigue life is calculated using long term Gulf of Mexico (GoM) currents and is compared against the industry standard software SHEAR7. Three different lift curve selections in SHEAR7 are used for this comparison. The differences in riser response prediction by the two software tools are discussed in detail. The sensitivity of the VIVA predicted riser response to the absence of VIV suppression devices is presented in this paper. The riser VIV response with and without external FE generated modal input is compared and the relative merits of the two modeling approaches are discussed. Finally, the recommended approach for VIVA usage for SCR and TTR design is given.

Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

2019 ◽  
Vol 86 (10) ◽  
Author(s):  
Pol D. Spanos ◽  
Ying Zhang ◽  
Fan Kong
Keyword(s):  
Harmonic Balance ◽  
Equation Of Motion ◽  
Periodic Component ◽  
Statistical Moments ◽  
System Response ◽  
Statistical Linearization ◽  
Ensemble Averaging ◽  
Stochastic Component ◽  
Stochastic Excitations ◽  
Linear Elements

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

Approximate Analytic Method of Piping Systems With Elasto-Plastic Damper

2002 ◽  
Author(s):  
Shigeru Aoki ◽  
Takeshi Watanabe
Keyword(s):  
Simple Procedure ◽  
Continuous System ◽  
Approximate Solutions ◽  
Continuous Model ◽  
Equation Of Motion ◽  
Forced Response ◽  
Mode Shapes ◽  
Piping System ◽  
Restoring Force ◽  
Approximate Methods

An elasto-plastic damper is one of the vibration absorbers in which energy is absorbed by elasto-plastic deformation of the hysteretic type damper. It is used for the piping system. The piping system is continuous system. Since it is difficult to find the analytical solution of the equation of motion for the system with elasto-plastic damper, the equation of motion is treated by various approximate methods in which the system is usually considered as a single- or a multiple-degree-of-freedom system, but not as a continuous system. In order to analyze the response of a nonlinear continuous system, however, it is necessary to consider the system as a continuous system. In this paper, the nonlinear steady-state response of the piping system with elasto-plastic damper is undertaken by approximate solutions, which are easily obtained by a simple procedure and are more practical than the exact solutions. As a continuous model of the piping system, a beam simply supported or clamped at one end, with elasto-plastic damper at the other end is used. The restoring force is modeled as hysteresis loop characteristics in order to consider the energy loss in the damper. In the analysis, the restoring force is expanded into the Fourier series, and only fundamental terms are considered. The resonance curves and mode shapes of the beam are obtained from the approximate solution. And effect of elasto-plastic damper on the forced response of continuous system is examined.

Analysis of Free Nonlinear Vibration Behavior for Curved Embedded Carbon Nanotubes on Elastic Foundation

2010 ◽  
Author(s):  
M. Rezaee ◽  
H. Fekrmandi
Keyword(s):  
Carbon Nanotubes ◽  
Dynamic Behavior ◽  
Continuum Mechanics ◽  
Nonlinear Term ◽  
Nonlinear Oscillations ◽  
Equation Of Motion ◽  
System Response ◽  
Response Curves ◽  
Vibration Behavior ◽  
The Continuum

Carbon nanotubes (CNTs) are expected to have significant impact on several emerging nanoelectromechanical (NEMS) applications. Vigorous understanding of the dynamic behavior of CNTs is essential for designing novel nanodevices. Recent literature show an increased utilization of models based on elastic continuum mechanics theories for studying the vibration behavior of CNTs. The importance of the continuum models stems from two points; (i) continuum simulations consume much less computational effort than the molecular dynamics simulations, and (ii) predicting nanostructures behavior through continuum simulation is much cheaper than studying their behavior through experimental verification. In numerous recent papers, CNTs were assumed to behave as perfectly straight beams or straight cylindrical shells. However, images taken by transmission electron microscopes for CNTs show that these tiny structures are not usually straight, but rather have certain degree of curvature or waviness along the nanotubes length. The curved morphology is due to process-induced waviness during manufacturing processes, in addition to mechanical properties such as low bending stiffness and large aspect ratio. In this study the free nonlinear oscillations of wavy embedded multi-wall carbon nanotubes (MWCNTs) are investigated. The problem is formulated on the basis of the continuum mechanics theory and the waviness of the MWCNTs is modeled as a sinusoidal curve. The governing equation of motion is derived by using the Hamilton’s principle. The Galerkin approach was utilized to reduce the equation of motion to a second order nonlinear differential equation which involves a quadratic nonlinear term due to the curved geometry of the beam, and a cubic nonlinear term due to the stretching effect. The system response has been obtained using the incremental harmonic balanced method (IHBM). Using this method, the iterative relations describing the interaction between the amplitude and the frequency for the single-wall nanotube and double-wall nanotube are obtained. Also, the influence of the waviness, elastic medium and van der Waals forces on frequency-response curves is researched. Results present some useful information to analyze CNT’s nonlinear dynamic behavior.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Second Order ◽  
Mode Shapes ◽  
Voltage Response ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

Nonlinear Oscillations of an Autoparametrical System With Two Coupled Pendulums

2007 ◽  
Author(s):  
Danuta Sado
Keyword(s):  
Nonlinear Oscillations ◽  
Type System ◽  
Vertical Direction ◽  
Free Vibrations ◽  
System Response ◽  
Main Body ◽  
Bifurcation Diagrams ◽  
Damping Force ◽  
Internal Resonances ◽  
Non Linear

The nonlinear dynamics of a three degree of freedom autoparametrical vibration system with two coupled pendulums in the neighborhood internal and external resonances is presented in this work. It was assumed that the main body is suspended by an element characterized by non-linear elasticity and non-linear damping force and is excited harmonicaly in the vertical direction. The two connected by spring pendulums characterized are mounted to the main body. It is assumed, that the motion of the pendulums are damped by nonlinear resistive forces. Solutions for the system response are presented for specific values of the uncoupled normal frequency ratios and the energy transfer between modes of vibrations is observed. Curves of internal resonances for free vibrations and external resonances for exciting force are shown. In this type system one mode of vibration may excite or damp another one, and except different kinds of periodic vibration there may also appear chaotic vibration. Various techniques, including chaos techniques such as bifurcation diagrams and: time histories, phase plane portraits, power spectral densities, Poincare` maps and exponents of Lyapunov, are used in the identification of the responses. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits. The results show that the system can exhibit various types of motion, from periodic to quasi-periodic to chaotic, and is sensitive to small changes of the system parameters.

Simulation of Flow and Heat Transfer in a Moving Bed Reactor With Cross Flow

2008 ◽  
Author(s):  
Marcelo J. S. de Lemos
Keyword(s):  
Heat Transfer ◽  
Relative Velocity ◽  
Solid Phase ◽  
Cross Flow ◽  
Vertical Direction ◽  
Momentum Equation ◽  
Solid Matrix ◽  
Flow And Heat Transfer ◽  
Temperature Distributions ◽  
Solid Phases

Heat transfer in a porous reactor under cross flow is investigated. The reactor is modeled as a porous bed in which the solid phase is moving horizontally and the flow is forced into the bed in a vertical direction. Equations are time-and-volume averaged and the solid phase is considered to have a constant imposed velocity. Additional drag terms appearing the momentum equation are a function of the relative velocity between the fluid and solid phases. Turbulence equations are also affected by the speed of the solid matrix. Results show temperature distributions for several ratios of the solid to fluid speed.

An Improved Component-Mode Synthesis Method to Predict Vibration of Rotating Spindles and Its Application to Position Errors of Hard Disk Drives

2005 ◽  
Vol 128 (5) ◽  
pp. 568-575 ◽  
Author(s):  
Takehiko Eguchi ◽  
Teruhiro Nakamiya
Keyword(s):  
Mathematical Model ◽  
Natural Frequencies ◽  
Hard Disk ◽  
Equation Of Motion ◽  
Component Mode Synthesis ◽  
Mode Shapes ◽  
Air Bearings ◽  
Improved Method ◽  
Stationary Part ◽  
Improved Model

This paper describes an accurate mathematical model that can predict forced vibration of a rotating spindle system with a flexible stationary part. In particular, we demonstrate this new formulation on a hard disk drive (HDD) spindle to predict its position error signal (PES). This improved method is a nontrivial extension of the mathematical model by Shen and his fellow researchers, as the improved method allows the flexible stationary part to comprise multiple substructures. When applied to HDD vibration, the improved model consists not only a rotating hub, multiple rotating disks, a stationary base, and bearings (as in Shen’s model) but also an independent flexible carriage part. Moreover, the carriage part is connected to the stationary base with pivot bearings and to the disks with air bearings at the head sliders mounted on the far end of the carriage. To build the improved mathematical model, we use finite element analysis (FEA) to model the complicated geometry of the rotating hub, the stationary base and the flexible carriage. With the mode shapes, natural frequencies, and modal damping ratios obtained from FEA, we use the principle of virtual work and component-mode synthesis to derive an equation of motion. Naturally, the stiffness and damping matrices of the equation of motion depend on properties of the pivot and air bearings as well as the natural frequencies and mode shapes of the flexible base, the flexible carriage, the hub, and the disks. Under this formulation, we define PES resulting from spindle vibration as the product of the relative displacement between the head element and the disk surface and the error rejection transfer function. To verify the improved model, we measured the frequency response functions using impact hammer tests for a real HDD that had a fluid-dynamic bearing spindle, two disks, and three heads. The experimental results agreed very well with the simulation results not only in natural frequencies but also in gain and phase.

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