Frequency Response of a Viscously Damped Flat Plate

2011 ◽  
Vol 78 (4) ◽  
Author(s):  
Michael James Martin
Keyword(s):  
Frequency Response ◽  
Flat Plate ◽  
Driving Force ◽  
Analytic Solution ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Phase Lag ◽  
Biological Sensing ◽  
Fluid Density ◽  
Stokes Second Problem

The frequency response of a flat plate in a viscous fluid is a problem with applications in microsystems, including gyroscopes, accelerometers, viscometers, and biological sensing. To find the frequency response away from the resonant frequency, the equations of motion are combined with the solution to Stokes’ second problem to produce an analytic solution for the motion of the plate in response to a sinusoidal driving force. These results are used to determine the gain, phase lag, and dynamic stability of the system. The behavior of the system depends on an effective damping ratio ζeff, which depends on the resonator dimensions, and is proportional to the square root of the viscosity times the fluid density.

Frequency Response of Cylindrical Resonators in a Viscous Fluid

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Michael J. Martin ◽  
Brian H. Houston
Keyword(s):  
Resonant Frequency ◽  
Frequency Response ◽  
Quality Factor ◽  
Linear Models ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Offshore Structures ◽  
Crossover Frequency ◽  
Fluid Density ◽  
Cylindrical Resonators

The frequency response of a cylinder in a viscously damped fluid is a problem of fundamental engineering interest, with applications ranging from microsystems to offshore structures. The analytic solution for the drag in a vibrating cylinder in the laminar flow regime is combined with the equations of motion for forced vibration of a cylinder attached to a spring. The resulting model gives an analytic solution for the dynamic response of the system, including the gain, frequency lag, resonant frequency, quality factor, and stability of the system. The results show that the response of the system is nonlinear, with the phase of the system differing from the phase predicted by linear models. The gain, quality factor, resonant frequency, and crossover frequency all increase with the nondimensional natural frequency β and decrease with the ratio of the fluid density to the resonator density.

On the Frequency Response of a Spinning Disk With Large Deformations

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Nonlinear Frequency ◽  
Response Characteristics ◽  
Geometrical Nonlinear ◽  
Oscillation Frequencies ◽  
Multi Mode ◽  
Different Levels

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

The Dynamic Behaviour of Passively Compensated, Hydrostatic Journal Bearings with Various Numbers of Recesses

1976 ◽  
Vol 18 (6) ◽  
pp. 292-302 ◽  
Author(s):  
P. B. Davies
Keyword(s):  
Dynamic Behaviour ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
Flow Equation ◽  
External Excitation ◽  
Signal Flow Graphs ◽  
Flow Compensation ◽  
Circular Functions ◽  
Flow Graphs

A previously established small-perturbation analysis is developed to express the unsteady-state continuity-of-flow equation for an isolated recess in a passively compensated, multirecess, hydrostatic journal bearing in terms of generalized co-ordinates. The concise form of this equation enables motion of the shaft about the concentric position to be described by equations which are derived in closed form for bearings with orifice, capillary or constant flow compensation and any number of recesses. These equations of motion, and hence the expressions for the receptances which describe the response of a bearing to external excitation, are shown to be of exactly the same form for all bearings of the type considered. Furthermore, the damping ratio and natural frequency in any particular case are determined by a single dynamic constant which is shown to be equal to a linear combination of circular functions and a limited number of coefficients which may be found explicitly by routine use of signal flow graphs. The results of the analysis, which is exact within the stated assumptions, are compared with those of other workers and the steady-state solution of the equations of motion is shown to give an expression for static stiffness which is useful for design purposes. Numerical values of the dynamic constant for bearings with between 3 and 20 recesses are given graphically.

Optimal Design of Vibration Absorber Systems Supported by Elastic Base

1991 ◽  
Author(s):  
Hashem Ashrafiuon
Keyword(s):  
Dynamic Models ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Base ◽  
Design Parameters ◽  
Primary System ◽  
Vibration Absorbers ◽  
Equivalent Damping ◽  
System Equations ◽  
Different Levels

Abstract This paper presents the effect of foundation flexibility on the optimum design of vibration absorbers. Flexibility of the base is incorporated into the absorber system equations of motion through an equivalent damping ratio and stiffness value in the direction of motion at the connection point. The optimum values of the uncoupled natural frequency and damping ratio of the absorber are determined over a range of excitation frequencies and the primary system damping ratio. The design parameters are computed and compared for the rigid, static, and dynamic models of the base as well as different levels of base flexibility.

Influence of Material Damping on In-Plane Modal Parameters for Rotating Disks

2012 ◽  
Author(s):  
Hamid R. Hamidzadeh ◽  
Ehsan Sarfaraz
Keyword(s):  
Elastic Moduli ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Stability ◽  
Rotating Disks ◽  
Governing Equations ◽  
Annular Disk ◽  
Stress Theory ◽  
Hysteretic Damping ◽  
Circumferential Wave

The linear in-plane free vibration of a thin, homogeneous, viscoelastic, rotating annular disk is investigated. In the development of an analytical solution, two dimensional elastodynamic theory is employed and the viscoelastic material for the medium is allowed by assuming complex elastic moduli. The general governing equations of motion are derived by implementing plane stress theory. Natural frequencies are computed for several modes at specific radius ratios with fixed-free boundary conditions and modal loss factors for different damping ratios are determined. The computed results were compared to previously established results. It was observed that the effects of rotational speed and hysteretic damping ratio on natural frequency and elastic stability of the rotating disks were related to the mode of vibration and type of circumferential wave occurring.

scholarly journals Seismic Assessment of 10-Story Building Using Tuned Mass Damper

2020 ◽  
Author(s):  
Mohammad Aghajani Delavar
Keyword(s):  
Frequency Tuning ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Tuned Mass Dampers ◽  
Earthquake Excitation ◽  
Optimum Parameters ◽  
Seismic Records ◽  
Structural Responses ◽  
Lateral Displacements ◽  
Shear Building

In this paper, optimum parameters of Tuned Mass Dampers (TMD) are considered to control the responses of 10-story shear building under harmonic loading and 22 set of seismic records of FEMA-P695. The criterion used to obtain the optimum parameters is to select mass ratio, the frequency (tuning) and damping ratio that would result in smallest lateral displacements. State-space equations of motion are presented to compute the structural responses by developing a MATLAB file. A 10-story shear building is presented as a case study to assess the effects of TMDs on the multi-story structures. The results indicate that using TMD can reduce structural responses up to the average 20% under earthquake excitation and up to 90% under harmonic loadings. TMDs are not always effective under any type of ground motion; therefore, being aware of the given location is significant to design TMDs properly.

Mechanical Contact Frequency Response Measurements

2000 ◽  
Vol 122 (4) ◽  
pp. 828-833 ◽  
Author(s):  
S. S. Kupchenko ◽  
D. P. Hess
Keyword(s):  
Frequency Response ◽  
Phase Lag ◽  
Lithium Grease ◽  
Mineral Oils ◽  
Vibration Data ◽  
Response Data ◽  
Lubricated Contacts ◽  
Wide Range ◽  
Contact Frequency ◽  
Dry Conditions

This paper presents friction frequency response measurements taken from a planar steel contact subjected to controlled random broadband normal vibration. Data are included from both dry and various lubricated contact conditions under different vibration input levels and different sliding velocities. Frequency response data for dry contacts are found to have nearly steady magnitude and negligible phase lag over a relatively wide range of frequencies. This suggests a coefficient of friction, independent of frequency but dependent on levels of normal acceleration and sliding velocity, may adequately define the dry contact frequency response. The frequency response data for lubricated contacts are mixed. For example, with MoS2 grease the frequency response may adequately be defined by a constant, as with dry conditions. However, frequency response data for contacts with pure mineral oils, mineral oils with additives, and lithium grease are found to be dependent on frequency. [S0742-4787(11)00101-9]

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

Magnetic Field-Induced Nonlinear Vibration of an Unbalanced Rotor

2003 ◽  
Author(s):  
Yuefang Wang ◽  
Ganyun Sun ◽  
Lihua Huang
Keyword(s):  
Magnetic Field ◽  
Potential Energy ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Oscillatory System ◽  
Forced Vibrations ◽  
Electric Motors ◽  
Jump Phenomenon ◽  
Mass Unbalance ◽  
Unbalanced Rotor

The free and forced flexural vibrations are investigated for rotors of electric motors operating in unsymmetrical magnetic fields. The magnetic potential energy reserved in the air-gap is analytically derived and the unbalanced magnetic pull is obtained through the law of energy conservation. With this excitation, the equations of motion of the unbalanced rotor are developed for nonlinear displacements response. For small dynamic eccentricities, the equations of motion are simplified and the rotor is compared to a free Duffing oscillatory system. The analytic solution for forced vibrations subject to residual mass-unbalance excitations is also obtained. Jump phenomenon in the solution is pointed out, and the effects of initial eccentricity and flux density on the natural frequency are also investigated.

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