Passage of a Rotor Through a Critical Speed

1982 ◽  
Vol 104 (2) ◽  
pp. 370-374 ◽  
Author(s):  
K. Tsuchiya
Keyword(s):  
Computer Simulation ◽  
Asymptotic Expansion ◽  
Digital Computer ◽  
Critical Speed ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Peak Amplitude ◽  
Matched Asymptotic Expansion ◽  
Nonstationary Oscillation ◽  
Pass Through

This paper deals with a nonstationary oscillation of a rotor passing through a critical speed. The analysis is based on the method of multiple scales and the method of matched asymptotic expansion. The peak amplitude of the response and the criteria for the onset of the stalling (inability to pass through the critical speed) are derived. These results are compared with those of digital computer simulation.

scholarly journals Response of a Parametrically Excited System to a Nonstationary Excitation

1995 ◽  
Vol 1 (1) ◽  
pp. 57-73 ◽  
Author(s):  
Harold L. Neal ◽  
Ali H. Nayfeh
Keyword(s):  
Computer Simulations ◽  
Governing Equation ◽  
Digital Computer ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Saddle Points ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Analog Computer ◽  
Modulation Equations

The response of a single-degree-of-freedom system to a nonstationary excitation is investigated by using the method of multiple scales as well as analog- and digital-computer simulations. The unexcited system has one focus and two saddle points. The system can be used to model rolling of ships in head or follower seas. The method of multiple scales is used to derive equations governing the modulation of the amplitude and phase of the response. The modulation equations are used to find the stationary solutions and their stability. The response to nonstationary excitations is found by integrating the original governing equation as well as the modulation equations. There is good agreement between the results of both approaches. For some frequency and amplitude sweeps, the nonstationary response found from integrating the original governing equation exhibits behaviors that are analogous to symmetry-breaking bifurcations, period-doubling bi furcations, chaos, and unboundedness present in the stationary case. The maximum response amplitude and the excitation amplitude or frequency at which the response becomes unbounded are found as functions of the sweep rate. The results of the digital-computer simulations are verified with an analog computer.

scholarly journals Response of a Parametrically-Excited System to a Nonstationary Excitation

1993 ◽  
Author(s):  
H. L. Neal ◽  
A. H. Nayfeh
Keyword(s):  
Computer Simulations ◽  
Governing Equation ◽  
Digital Computer ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Saddle Points ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Analog Computer ◽  
Modulation Equations

Abstract The response of a single-degree-of-freedom system to a nonstationary excitation is investigated by using the method of multiple scales as well as analog- and digital-computer simulations. The unexcited system has one focus and two saddle points. The system can be used to model rolling of ships in head or follower seas. The method of multiple scales is used to derive equations governing the modulation of the amplitude and phase of the response. The modulation equations are used to find the stationary solutions and their stability. The response to nonstationary excitations is found by integrating the original governing equation as well as the modulation equations. There is good agreement between the results of both approaches. For some frequency and amplitude sweeps, the nonstationary response found from integrating the original governing equation exhibits behaviors that are analogous to symmetry-breaking bifurcations, period-doubling bifurcations, chaos, and unboundedness present in the stationary case. The maximum response amplitude and the excitation amplitude or frequency at which the response becomes unbounded are found as functions of the sweep rate. The results of the digital-computer simulations are verified with an analog computer.

Study of plate vibrating in fluids having part-through crack at random angles and locations

2021 ◽  
pp. 095440622110127
Author(s):  
Ruqia Ikram ◽  
Asif Israr
Keyword(s):  
Frequency Response ◽  
Multiple Scales ◽  
Peak Amplitude ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
Crack Location ◽  
Cracked Plate ◽  
Nonlinear Frequency Response ◽  
Angle Crack ◽  
Crack Angle

This study presents the vibration characteristics of plate with part-through crack at random angles and locations in fluid. An experimental setup was designed and a series of tests were performed for plates submerged in fluid having cracks at selected angles and locations. However, it was not possible to study these characteristics for all possible crack angles and crack locations throughout the plate dimensions at any fluid level. Therefore, an analytical study is also carried out for plate having horizontal cracks submerged in fluid by adding the influence of crack angle and crack location. The effect of crack angle is incorporated into plate equation by adding bending and twisting moments, and in-plane forces that are applied due to antisymmetric loading, while the influence of crack location is also added in terms of compliance coefficients. Galerkin’s method is applied to get time dependent modal coordinate system. The method of multiple scales is used to find the frequency response and peak amplitude of submerged cracked plate. The analytical model is validated from literature for the horizontally cracked plate submerged in fluid as according to the best of the authors’ knowledge, literature lacks in results for plate with crack at random angle and location in the presence of fluid following validation with experimental results. The combined effect of crack angle, crack location and fluid on the natural frequencies and peak amplitude are investigated in detail. Phenomenon of bending hardening or softening is also observed for different boundary conditions using nonlinear frequency response curves.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Dynamic analysis of a parametrically excited golden Muga silk embedded pneumatic artificial muscle

2018 ◽  
Vol 211 ◽  
pp. 02008 ◽  
Author(s):  
Bhaben Kalita ◽  
S. K. Dwivedy
Keyword(s):  
Dynamic Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Quadratic Function ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Time Response ◽  
Phase Portraits ◽  
Pneumatic Artificial Muscle ◽  
Pneumatic Muscle

In this work a novel pneumatic artificial muscle is fabricated using golden muga silk and silicon rubber. It is assumed that the muscle force is a quadratic function of pressure. Here a single degree of freedom system is considered where a mass is supported by a spring-damper-and pneumatically actuated muscle. While the spring-mass damper is a passive system, the addition of pneumatic muscle makes the system active. The dynamic analysis of this system is carried out by developing the equation of motion which contains multi-frequency excitations with both forced and parametric excitations. Using method of multiple scales the reduced equations are developed for simple and principal parametric resonance conditions. The time response obtained using method of multiple scales have been compared with those obtained by solving the original equation of motion numerically. Using both time response and phase portraits, variation of few systems parameters have been carried out. This work may find application in developing wearable device and robotic device for rehabilitation purpose.

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