Non-Linear Free Vibrations of a Rotating Beam Carrying a Tip Mass With Rotary Inertia

2002 ◽  
Author(s):  
Ahmad A. Al-Qaisia
Keyword(s):  
Transformation Method ◽  
Free Vibrations ◽  
Rotary Inertia ◽  
Inertia Effects ◽  
Assumed Mode Method ◽  
Non Linear ◽  
Mode Method ◽  
Geometric Stiffening ◽  
Tip Mass ◽  
Root Attachment

The non-linear, for each of the first three modes, of planar, large amplitude flexural free vibrations of a beam clamped with an angle to a rigid rotating hub and carrying a tip mass with rotary inertia are investigated. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian dynamics in conjunction with the assumed mode method, assuming constant hub rotation speed, is utilized in deriving the non-linear unimodal temporal problem. The time transformation method is employed to obtain an approximate solution to the frequency-amplitude relation of the beam-mass free vibration, since the order of the nonlinear terms is not small which includes static and inertial geometric stiffening as well as inertial softening terms. Results in non-dimensional form are presented graphically, for the effect beam root-attachment angle, hub radius and the attached inertia element ratio on the variation of the natural frequency with vibration amplitude.

Dynamic Impact Analysis of a Rotating Beam Having a Tip Mass

2006 ◽  
Vol 321-323 ◽  
pp. 1649-1653 ◽  
Author(s):  
Hong Seok Lim ◽  
Hong Hee Yoo
Keyword(s):  
Impact Analysis ◽  
Flexible Structures ◽  
Angular Speed ◽  
Flexible Beam ◽  
Transient Responses ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Stiffness Variation ◽  
Tip Mass ◽  
The Impact

Flexible structures undertaking impact while undergoing overall motion can be found in several industrial products these days. Transient motion and stress induced by impact should be considered elaborately to extend the life of the products. In the present study, a modeling method for a flexible beam with a tip mass that undertakes impact while undergoes large overall motion is presented. The tip mass takes the impact force and the transient responses of the beam are calculated by employing the assumed mode method. The stiffness variation caused by the large overall motion is considered in this modeling. The effects of the tip mass and the angular speed of the beam on the transient responses are investigated.

Effect of Fluid Mass on Non-Linear Natural Frequencies of a Rotating Beam

2003 ◽  
Author(s):  
Ahmad A. Al-Qaisia
Keyword(s):  
Harmonic Balance ◽  
Natural Frequencies ◽  
Harmonic Balance Method ◽  
Continuous System ◽  
Assumed Mode Method ◽  
Non Linear ◽  
Mode Method ◽  
The Stability ◽  
The Mathematical Model ◽  
Analytical Expressions

The non-linear natural frequencies of the first three modes of a beam clamped to a rigid rotating hub and carrying a distributed fluid along its span are investigated. The mathematical model is derived using the Lagrangian method and the continuous system is discretized using the assumed mode method. The resulted unimodal nonlinear equation of motion was solved using two methods; harmonic balance (HB) and time transformation (TT), to obtain approximate analytical expressions for the nonlinear natural frequencies. Results have shown that the two terms harmonic balance method (2THB) is more accurate than the time TT method. Results for the effect and type of distribution, i.e. uniform or linearly distributed, on the variation of the nonlinear natural frequency with the rotational speed of the system and how they affect the stability are studied and presented in non-dimensional form.

scholarly journals On the Steady State Response of a Cantilever Beam Partially Immersed in a Fluid and Carrying an Intermediate Mass

2000 ◽  
Vol 7 (4) ◽  
pp. 179-194 ◽  
Author(s):  
A.A. Al-Qaisia ◽  
M.N. Hamdan ◽  
B.O. Al-Bedoor
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Equation Of Motion ◽  
Steady State Response ◽  
Intermediate Mass ◽  
Assumed Mode Method ◽  
State Response ◽  
Non Linear ◽  
Mode Method ◽  
Assumed Mode

This paper presents a study on the nonlinear steady state response of a slender beam partially immersed in a fluid and carrying an intermediate mass. The model is developed based on the large deformation theory with the constraint of inextensible beam, which is valid for most engineering structures. The Lagrangian dynamics in conjunction with the assumed mode method is utilized in deriving the non-linear unimodal temporal equation of motion. The distributed and concentrated sinusoidal loads are accounted for in a consistent manner using the assumed mode method. The non-linear equation of motion is, analytically, solved using the single term harmonic balance (SHB) and the two terms harmonic balance (2HB) methods. The stability of the system, under various loading conditions, is investigated. The results are presented, discussed and some conclusions on the partially immersed beam nonlinear dynamics are extracted.

Non-linear Frequency Response and Stability Analysis of Piezoelectric Nanoresonator Subjected to Electrostatic Excitation

2019 ◽  
Vol 20 (5) ◽  
pp. 601-621 ◽  
Author(s):  
Sayyid H. Hashemi Kachapi ◽  
Morteza Dardel ◽  
Hamidreza Mohamadi daniali ◽  
Alireza Fathi
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Frequency Response ◽  
Surface Elasticity ◽  
Assumed Mode Method ◽  
Linear Frequency ◽  
Non Linear ◽  
Mode Method ◽  
Parametric Studies

AbstractThe effects of surface energy on the non-linear frequency response and stability analysis of piezoelectric cylindrical nano-shell as piezoelectric nanoresonator are investigated in the current paper using Gurtin–Murdoch surface elasticity and von Karman–Donnell’s theory. The nanoresonator is embedded in visco-Pasternak medium and electrostatic excitation. The governing equations and boundary conditions are derived using Hamilton’s principle and also the assumed mode method is used for changing the partial differential equations into ordinary differential equations. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady-state vibrations of the system. The validation of the mentioned system is achieved with excellent agreements by comparison with numerical results. The parametric studies such as the effects of geometrical and material properties, different boundary conditions, the ratio of length to radius $L/R$ for different values of the voltages ${V_{{\rm{DC}}}}$ and ${V_{{\rm{AC}}}}$, the gap width of the nanoresonator $b/L$, the effect of the voltages ${V_{{\rm{DC}}}}$ and ${V_{{\rm{AC}}}}$ and also the effect of piezoelectric voltage ${V_p}$ are conducted on the non-linear frequency response and stability analysis of the piezoelectric nanoresonator.

Parameter Sensitivity Analysis of Rotating Beams in Frequency Domain

2009 ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Nader Jalili
Keyword(s):  
Sensitivity Analysis ◽  
Closed Form ◽  
Characteristic Equation ◽  
Frequency Analysis ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Tip Mass ◽  
Assumed Mode

Many mechanical rotating systems can be modeled as a cantilever beam attached to a rotating substrate. Vibratory beam gyroscopes are good examples of such systems. They consist of a rotating beam with a tip mass, attached to a rotating base. Due to the base rotation, the governing partial differential equations of the system are coupled, and hence, the system undergoes coupled torsional-bending vibrations. The coupling effect complicates the frequency analysis of the system, especially in determining the system characteristic equation. Many investigators have chosen to use the assumed mode method in their analysis of such systems instead of extracting the exact mode shapes of the system. In spite of all these difficulties, this paper addresses the exact frequency analysis of such systems and presents a closed-form frequency characteristic equation and evaluates the accurate values of the natural frequencies. The application of the proposed method is not limited to the system at hand, as it can be utilized for analyzing general systems with coupled governing equations of motion. Having analyzed a closed-form frequency equation has two valuable advantages: a) it can serve as the basis for the subsequent time-domain analysis; and b) it can be very essential in developing control strategies. In this study a thorough sensitivity analysis is performed to determine the effects of different parameters on the natural frequencies of the coupled vibrating system. The proposed method reveals some interesting findings in the systems which were difficult, if not impossible, to be revealed by the assumed mode method commonly utilized in many research work reported recently in literature.

A Method to Improve the Modal Convergence for Structures With External Forcing

1987 ◽  
Vol 54 (4) ◽  
pp. 904-909 ◽  
Author(s):  
Keqin Gu ◽  
Benson H. Tongue
Keyword(s):  
Spatial Distribution ◽  
Free Vibration ◽  
Convergence Rate ◽  
Traditional Approach ◽  
Vibration Modes ◽  
External Forcing ◽  
Slow Convergence ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The traditional approach of using free vibration modes in the assumed mode method often leads to an extremely slow convergence rate, especially when discete interactive forces are involved. By introducing a number of forced modes, significant improvements can be achieved. These forced modes are intrinsic to the structure and the spatial distribution of forces. The motion of the structure can be described exactly by these forced modes and a few free vibration modes provided that certain conditions are satisfied. The forced modes can be viewed as an extension of static modes. The development of a forced mode formulation is outlined and a numerical example is presented.

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