Large Amplitude Free Vibration Analysis of a Rotating Beam with Non-linear Spring and Mass System

2005 ◽  
Vol 11 (12) ◽  
pp. 1511-1533 ◽  
Author(s):  
S. K. Das ◽  
P. C. Ray ◽  
G. Pohit
Keyword(s):  
Free Vibration Analysis ◽  
Linear Constraint ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Rotating Beam ◽  
Mass System ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Out Of Plane ◽  
Linear Spring

The free, out-of-plane vibration of a rotating beam with a non-linear spring-mass system has been investigated. The non-linear constraint appears in the boundary condition. The solution is obtained by applying the method of multiple time-scales directly to the non-linear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in the literature. Subsequent non-linear study indicates that there is a pronounced effect of the spring and its mass. The influence of the spring-mass location on frequencies is also investigated for the non-linear frequencies of the rotating beam.

Large Amplitude Free Vibration of a Rotating Nonhomogeneous Beam With Nonlinear Spring and Mass System

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
A. Chakrabarti ◽  
P. C. Ray ◽  
Rasajit Kumar Bera
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Mass Effect ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Governing Equations ◽  
Mass System ◽  
Nonlinear Spring ◽  
Reduced Frequency ◽  
Out Of Plane

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam (β=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied.

Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

On the non-linear Lamb-Taylor instability

1969 ◽  
Vol 38 (3) ◽  
pp. 619-631 ◽  
Author(s):  
Ali Hasan Nayfeh
Keyword(s):  
Layer Thickness ◽  
Travelling Waves ◽  
Finite Thickness ◽  
Standing Waves ◽  
Linear Case ◽  
Wave Number ◽  
Multiple Time Scales ◽  
Solid Boundary ◽  
Multiple Time ◽  
Non Linear

A non-linear analysis of the inviscid stability of the common surface of two superposed fluids is presented. One of the fluids is a liquid layer with finite thickness having one surface adjacent to a solid boundary whereas the second surface is in contact with a semi-infinite gas of negligible density. The system is accelerated by a force normal to the interface and directed from the liquid to the gas. A second-order expansion is obtained using the method of multiple time scales. It is found that standing as well as travelling disturbances with wave-numbers greater than$K^{\prime}_c = k_c[1+\frac{3}{8}a^2k^2_c + \frac{51}{512}a^4k^4_c]^{\frac{1}{2}}$where a is the disturbance amplitude and kc is the linear cut-off wave-number, oscillate and are stable. However, the frequency in the case of standing waves and the wave velocity in the case of travelling waves are amplitude dependent. Below this cut-off wave-number disturbances grow in amplitude. The cut-off wave-number is independent of the layer thickness although decreasing the layer thickness decreases the growth rate. Although standing waves can be obtained by the superposition of travelling waves in the linear case, this is not true in the non-linear case because the amplitude dependences of the wave speed and frequency are different. A mechanism is proposed to explain the overstability behaviour observed by Emmons, Chang & Watson (1960).

Non-Linear Spring and Damping Forces Estimation During Free Vibration

1995 ◽  
Author(s):  
Michael Feldman ◽  
Simon Braun
Keyword(s):  
Free Vibration ◽  
Hilbert Transform ◽  
Time Domain ◽  
Free Vibration Analysis ◽  
Single Degree Of Freedom ◽  
Non Linear ◽  
Linear Spring ◽  
The Time Domain ◽  
The Hilbert Transform ◽  
Vibrating Systems

Abstract A method for dynamic analysis of sophisticated nonlinear single-degree-of-freedom systems, based on the Hilbert transform in the time domain is described. Using the Hilbert transform together with the proposed method for system identification, we obtain both instantaneous modal parameters together with non-linear force characteristics during free vibration analysis under impulse excitation without long resonance testing. Using the Hilbert transform in the time domain is a new method of studying linear and non-linear vibrating systems exposed to impulse or shock inputs.

Non-Linear Constraints and Stiffness Representations in Simple Flexible-Body Dynamic Beam Formulations

2003 ◽  
Vol 9 (8) ◽  
pp. 911-929 ◽  
Author(s):  
William Haering
Keyword(s):  
Linear Constraint ◽  
Linear Constraints ◽  
Model Verification ◽  
Lumped Parameter Model ◽  
Flexible Body ◽  
Flexible Beam ◽  
Lumped Parameter ◽  
Non Linear ◽  
Linear Spring ◽  
Insight Into

Abstract: A simple discrete four-degree-of-freedom (lumped parameter) model representing a flexible beam undergoing large overall planar prescribed motion has been developed. It serves as a simple tool to investigate two previously studied problems involving flexible-body beam dynamics, namely those involving bending and membrane stiffness dominated behavior. The tool is used to investigate the requirements to accurately solve these problems using non-linear constraints and a non-linear spring representation. The validity of this model is demonstrated by comparing results to those previously published for continuous flexible-body beam formulations. One of these continuous representations is modified to include a non-linear tether spring representation. This allows additional model verification as well as insight into the non-linear constraint and stiffness representations. Taken in its entirety, this investigation demonstrates the utility of these simple lumped parameter models, by showing their ability to provide rapid insight into the behavior of the more complicated continuous models, as well as the system in general.

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