Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Nader Jalili
Keyword(s):  
Frequency Analysis ◽  
Time Domain ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Bending Vibrations ◽  
Resonant Condition ◽  
Mass System ◽  
Coupled Equations ◽  
Tip Mass ◽  
Gyroscopic Systems

An exact frequency analysis of a rotating beam with an attached tip mass is addressed in this paper while the beam undergoes coupled torsional-bending vibrations. The governing coupled equations of motion and the corresponding boundary condition are derived in detail using the extended Hamilton principle. It has been shown that the source of coupling in the equations of motion is the rotation and that the equations are linked through the angular velocity of the base. Since the beam-tip-mass system at hand serves as the building block of many vibrating gyroscopic systems, which require high precision, a closed-form frequency equation of the system should be derived to determine its natural frequencies. The frequency analysis is the basis of the time domain analysis, and hence, the exact frequency derivation would lead to accurate time domain results, too. Control strategies of the aforementioned gyroscopic systems are mostly based on their resonant condition, and hence, acquiring knowledge about their exact natural frequencies could lead to a better control of the system. The parameter sensitivity analysis has been carried out to determine the effects of various system parameters on the natural frequencies. It has been shown that even the undamped systems undergoing base rotation will have complex eigenvalues, which demonstrate a damping-type behavior.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

2017 ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Large Mass ◽  
Mode Shapes ◽  
Resonance Excitation ◽  
Response Curves ◽  
Mass System ◽  
Spring System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

scholarly journals The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Minimum Problem ◽  
Infinite Class ◽  
Coupled Equations ◽  
Geometrical Problem ◽  
Maximum Minimum

A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.

scholarly journals Natural Frequncies of Coupled Blade-Bending and Shaft-Torsional Vibrations

2007 ◽  
Vol 14 (1) ◽  
pp. 65-80 ◽  
Author(s):  
B.O. Al-Bedoor
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Coupled Model ◽  
Small Deformation ◽  
Torsional Vibrations ◽  
Reduced Order ◽  
Rotating Speed ◽  
Coupled Equations ◽  
Stiffening Effect

In this study, the coupled shaft-torsional and blade-bending natural frequencies are investigated using a reduced order mathematical model. The system-coupled model is developed using the Lagrangian approach in conjunction with the assumed modes method to discretize the blade bending deflection. The model accounts for the blade stagger (setting) angle, the system rotating speed and its induced stiffening effect. The coupled equations of motion are linearized based on the small deformation theory for the blade bending and shaft torsional deformation to enable calculation of the system natural frequencies for various combinations of system parameters. The obtained coupled eignvalue system is ready for use as a reference for comparison for larger size finite element simulations and for the use as a fast check on natural frequencies for the coupled blade bending and shaft torsional vibrations in the design and diagnostics processes. Some results on the predicted natural frequencies are graphically presented and discussed pertinent to the coupling controlling factors and their effects. In addition, the predicted coupled natural frequencies are validated using the Finite Element Commercial Package (Pro-Mechanica) where good agreements are found.

Tuning of a Piezoelectric Harvester by Means of a Cantilever Spring-Mass System

2018 ◽  
Author(s):  
Alberto Doria ◽  
Cristian Medè ◽  
Giulio Fanti ◽  
Daniele Desideri ◽  
Alvise Maschio ◽  
...  
Keyword(s):  
High Frequency ◽  
Natural Frequencies ◽  
Coupled System ◽  
Vibration Absorber ◽  
Main Resonance ◽  
Dynamic Vibration Absorber ◽  
Mass System ◽  
High Frequency Resonance ◽  
Tip Mass ◽  
Piezoelectric Harvester

The possibility of improving the performance of a piezoelectric harvester by means of a cantilever dynamic vibration absorber (CDVA) is investigated. The CDVA cancels the original mode of vibration of the harvester and generates two new modes. Some prototypes are developed using a mathematical model for predicting the natural frequencies of the coupled system. Impulsive tests were performed on prototypes. Experimental results show that a small CDVA can lower the main resonance frequency of an harvester of the same extent as a larger tip mass. The measured voltage shows also an high frequency resonance peak, which can be exploited for collecting energy. A multi-physics numerical model is developed for performing modal analysis and stress analysis. Numerical results show that the stress inside the piezoelectric material of the harvester with CDVA results smaller than the stress inside the harvester with a tip mass tuned to the same frequency.

Planar Motion of a Flexible Beam With a Tip Mass Driven by Two Kinematic Rotational Degrees of Freedom

1998 ◽  
Vol 120 (1) ◽  
pp. 206-213
Author(s):  
D. C. Winfield ◽  
B. C. Soriano
Keyword(s):  
Finite Element Method ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Planar Motion ◽  
Flexible Beam ◽  
The Finite Element Method ◽  
Rotational Degrees Of Freedom ◽  
Tip Mass

The objective was to model planar motion of a flexible beam with a tip mass that is driven by two kinematic rotational degrees of freedom which are (1) at the center of the hub and (2) at the point the beam is attached to the hub. The equations of motion were derived using Lagrange’s equations and were solved using the finite element method. The results for the natural frequencies of the beam especially at high tip masses and high rotational velocities of the hub were calculated and compared to results obtained using the Raleigh-Ritz method. The dynamic response of the beam due to a specified hub rotation was calculated for two cases.

On the Free Vibration Analysis of a Sandwich Beam With Tip Mass

2018 ◽  
Author(s):  
E. F. Joubaneh ◽  
O. R. Barry
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Free Vibration Analysis ◽  
Design Parameters ◽  
Governing Equations ◽  
Tip Mass

This paper presents the free vibration analysis of a sandwich beam with a tip mass using higher order sandwich panel theory (HSAPT). The governing equations of motion and boundary conditions are obtained using Hamilton’s principle. General Differential Quadrature (GDQ) is employed to solve the system governing equations of motion. The natural frequencies and mode shapes of the system are presented and Ansys simulation is performed to validate the results. Various boundary conditions are also employed to examine the natural frequencies of the sandwich beam without tip mass and the results are compared with those found in the literature. Parametric studies are conducted to examine the effect of key design parameters on the natural frequencies of the sandwich beam with and without tip mass.

Small-amplitude free vibration analysis of active multistable shells under static multifield loading

2020 ◽  
pp. 107754632092393
Author(s):  
Dimitris Varelis
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Small Amplitude ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Shell Element ◽  
Curvilinear Coordinates ◽  
Coupled Equations ◽  
Modal Response

This study considers the small-amplitude free vibrational response performed on top of the quasi-static snap through buckling, which is accompanied by large displacements and rotations of shallow doubly curved laminated piezoelectric shells under multifield loading. The mechanics incorporate coupling between mechanical, electric, and thermal fields and encompass geometric nonlinearity effects due to large quasi-static displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. Based on the above mechanics and adopting the finite element methodology, an eight-node nonlinear shell element is developed to yield the linearized discrete coupled small-amplitude dynamic equations of motion. Initially, the nonlinear coupled equations are linearized and solved quasi-statically using an extended cylindrical arc-length method in combination with the Newton–Raphson iterative technique, and subsequently the free vibration analysis is performed at each solution point. Validation and evaluation cases on laminated cylindrical shells demonstrate the accuracy of the present method and its robust capability to predict the modal response on top of the nonlinear quasi-static response of active multistable shells subject to combined thermo–piezo–electromechanical loads. Numerical cases show the feasibility to develop smart shell structures to detect, via the monitoring of natural frequencies, the onset of snap-through instability. The capability of smart shells to actively modify its natural frequencies such as to promote or mitigate snap-through instabilities is quantified. Additional results quantify the effect of thermomechanical loads on actuation capability. The influence of geometric parameters (curvature and thickness) on the modal response is finally investigated.

The Natural Frequencies and Modes of Vibration of a Rotating Beam

1954 ◽  
Vol 58 (525) ◽  
pp. 652-654
Author(s):  
H. S. Liner
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Matrix Form ◽  
Rotating Beam ◽  
Bending Vibrations ◽  
Natural Frequencies And Modes ◽  
Modes Of Vibration ◽  
Frequency Diagram

This note presents in matrix form the equations of motion of a rotating and vibrating beam. The natural frequencies of the system are obtained by plotting an impedance-frequency diagram and noting the frequencies at which the impedance vanishes. Uncoupled bending vibrations are considered but the analysis can easily be extended to include coupled bending-torsion vibrations. The arrangement of the method for solution on a digital computor is quite straight-forward.

Flexural Vibrations of Piezolaminated Slender Beams: A Balanced Model

2002 ◽  
Vol 8 (8) ◽  
pp. 1105-1121 ◽  
Author(s):  
H. Abramovich ◽  
A. Livshits
Keyword(s):  
Natural Frequencies ◽  
Piezoelectric Effect ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
First Order ◽  
Various Boundary Conditions ◽  
Coupled Equations ◽  
Direct Piezoelectric Effect ◽  
Slender Beams ◽  
Balanced Model

A balanced model describing the behavior of piezolaminated composite beams was developed based on a first order shear deformation theory. The model takes into account the additional induced stiffness due to the direct piezoelectric effect for the case of “closed electrodes”, for symmetric and non-symmetric lay-ups. To comply with the condition of “closed electrodes” two cases were considered, E x = 0 or D x = 0. The three-coupled equations of motion were solved to yield eigenfrequencies and eigenmodes for various boundary conditions, materials and lay-up configurations. For the cases studied, the influence of the induced stiffness due to the direct piezoelectric effect on the natural frequencies was found to increase the natural frequencies by a few percent.

Analytical and Experimental Investigation of the Coupled Bladed Disk/Shaft Whirl of a Cantilevered Turbofan

1986 ◽  
Vol 108 (4) ◽  
pp. 567-575 ◽  
Author(s):  
E. F. Crawley ◽  
E. H. Ducharme ◽  
D. R. Mokadam
Keyword(s):  
Analytical Model ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Disk System ◽  
Structural Dynamic ◽  
Nodal Diameter ◽  
Coupled Equations ◽  
Bladed Disk ◽  
Shaft System ◽  
The One

The structural dynamics of a rotating flexible blade-rigid disk-flexible cantilevered shaft system is analytically and experimentally investigated. A simple analytical model yields the equations of motion expressed in the rotating frame, which show that the blade one nodal diameter modes dynamically couple to the rigid body whirling motion of the shaft-disk system. The blade modes higher than one nodal diameter are uncoupled from the shaft-disk dynamics. Nondimensionalization of the coupled equations of motion yield the criteria for the propensity and magnitude of the interaction between the bladed disk and shaft-disk modes. The analytical model was then correlated with the results of a structural dynamic experiment performed on the MIT Aeroelastic Rotor, a fan similar in design to a modern high bypass ratio shroudless turbofan. A special whirl excitation apparatus was used to excite both forward and backward asynchronous whirl, in order to determine the natural frequencies of the system. The agreement between the predicted and experimental natural frequencies is good and indicates the possibility of significant interaction of the one nodal diameter blade modes with the shaft-disk modes.

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