Nonlinear Resonances in a Flexible Cantilever Beam

1994 ◽  
Vol 116 (4) ◽  
pp. 480-484 ◽  
Author(s):  
T. J. Anderson ◽  
B. Balachandran ◽  
A. H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Cantilever Beam ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Continuous Systems ◽  
Modal Interactions ◽  
The Third ◽  
First Case ◽  
Second Mode ◽  
Band Limited

An experimental investigation into the response of a nonlinear continuous systems with many natural frequencies in the range of interest is presented. The system is a flexible cantilever beam whose first four natural frequencies are 0.65 Hz, 5.65 Hz, 16.19 Hz, and 31.91 Hz, respectively. The four natural frequencies correspond to the first four flexural modes. The fourth natural frequency is about fifty times the first natural frequency. Three cases were considered with this beam. For the first case, the beam was excited with a periodic base motion along its axis. The excitation frequency fe was near twice the third natural frequency f3, which for a uniform isotropic beam corresponds to approximately the fourth natural frequency f4. Thus the third mode was excited by a principal parametric resonance (i.e., fe ≈ 2f3) and the fourth mode was excited by an external resonance (i.e., fe ≈ f4) due to a slight curvature in the beam. Modal interactions were observed involving the first, third, and fourth modes. For the second case, the beam was excited with a band-limited random base motion transverse to the axis of the beam. The first and second modes were excited through nonlinear interactions. For the third case, the beam was excited with a base excitation along the axis of the beam at 138 Hz. The corresponding response was dominated by the second mode. The tools used to analyze the motions include Fourier spectra, Poincare´ sections, and dimension calculations.

Nonlinear Dynamics of a Transversely Excited Flexible Cantilever Beam

1995 ◽  
Author(s):  
Mahmood Tabaddor ◽  
Ali H. Nayfeh
Keyword(s):  
Nonlinear Dynamics ◽  
Natural Frequency ◽  
Cantilever Beam ◽  
Experimental Results ◽  
Modal Interactions ◽  
Frequency Sweep ◽  
Second Mode ◽  
Fourth Mode

Abstract Some experimental results concerning the nonlinear dynamics of a transversely excited beam are presented. The excitation is harmonic. A frequency sweep around the fourth natural frequency of the beam reveals some interesting modal interactions. The first phenomenon is the transfer of energy from the fourth mode, approximately 33.10 Hz, to the first mode, approximately 0.70 Hz. This interaction involves modulation of the amplitude and phase of the fourth mode. The second interaction involves the participation of the fourth mode and the second mode, approximately 5.80 Hz. The mechanism by which the second mode is activated as yet remains unidentified.

scholarly journals Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

2021 ◽  
Vol 13 (11) ◽  
pp. 168781402110609
Author(s):  
Hossein Talebi Rostami ◽  
Maryam Fallah Najafabadi ◽  
Davood Domiri Ganji
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Variable Properties ◽  
The Third ◽  
Koch Snowflake

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Natural frequency modulation of a kinematically redundant planar parallel robot

2021 ◽  
pp. 095440622110524
Author(s):  
Jiawei Gu ◽  
Zhijiang Xie ◽  
Jian Zhang ◽  
Yangjun Pi
Keyword(s):  
Natural Frequency ◽  
Frequency Modulation ◽  
Natural Frequencies ◽  
Vibration Suppression ◽  
Excitation Frequency ◽  
Parallel Robot ◽  
Modulation Method ◽  
Kinematic Redundancy ◽  
Resonance Amplitude ◽  
Searching Method

When a parallel robot is equipped with kinematic redundancy, it has sufficient capabilities of natural frequency modulation through adjusting geometric configuration. To reduce resonance of a mechanism, this paper investigates the natural frequency modulation of a kinematically redundant planar parallel robot. A double-threshold searching method is proposed for controlling the inverse kinematics solution and keeping the natural frequencies away from the excitation frequency. The effectiveness of modulating the natural frequencies is demonstrated by comparing it with a non-modulation method. The simulation results indicate that, in all directions, the responses are coupled, and every order should be taken into consideration during natural frequency modulation. Compared to the non-modulation method, the proposed method can reduce the resonance amplitude to a certain extent, and the effect of vibration suppression is remarkable.

Parameter Identification of a Cantilever Beam Immersed in Viscous Fluids With Potential Applications to the Probe Calibration of Atomic Force Microscopes

2009 ◽  
Author(s):  
Wenlung Li ◽  
S. P. Tseng
Keyword(s):  
Parameter Identification ◽  
Natural Frequency ◽  
Cantilever Beam ◽  
Present Method ◽  
Present Report ◽  
Excitation Frequency ◽  
Spring Constant ◽  
Target System ◽  
Phase Lag ◽  
Identification Method

The main objective of the report is to present a new identification method has been derived for single-degree, base-excited systems. The system is actually to mimic a probe of cantilever type of AFMs. In fact, the idea of the present report was initiated by needs for in situ spring constant calibration for such probe systems. Calibration processes can be treated as parameter identification for the stiffness of the probe before it is used. However, sine a real probe is too small to be seen by bare eyes and too costly to verify, a cantilever beam was adopted to replace it during the study. The present method starts with giving a chirp excitation to the target system, and to lock the damped natural frequency. Once the damped natural frequency is obtained, it is possible to locate the frequency at which the phase lag is equal to π/2. From which, the excitation frequency is then purposely changed to that frequency and the corresponding steady-state responses are measured. In the meantime, the system dissipative energy or power may also need to be stored. In fact, the present identification formulation is to express the spring constant of the target systems in terms of two measurable parameters: the phase angle and the system damping. The former can be computed from the damped natural frequency while the latter can be identified along with measuring the input power. The novel formulation is then numerically simulated using the Simulink toolbox of MATLAB. The simulation results clearly showed the current identification method can work with good accuracy. Following the numerical simulation, experimental measurements were also carried out by a cantilever beam that its free end was immersed to viscid fluids. The fluids of different viscosity were used to mimic the environments of a probe in use. The experimental results again substantiated the correctness of the present method. Thus it is accordingly concluded that the new recognition algorithm can be applied with confidence.

Control Research on Construction Excitation Frequency of Resonant Rubblization Machine

2011 ◽  
Vol 305 ◽  
pp. 394-397
Author(s):  
Xin Qiu ◽  
Qing Yang ◽  
Lan Yun Chen
Keyword(s):  
Natural Frequency ◽  
Elastic Foundation ◽  
Thin Plate ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Research Results ◽  
Vibration Theory ◽  
Fundamental Natural Frequency ◽  
Control Research

Based on the assumption of thin plate of elastic foundation and vibration theory, a method for calculating the fundamental natural frequency of cement slab is presented and the influence of slab dimension and foundation reaction modulus on the fundamental natural frequency of cement slab is discussed. As well, according to the analysis results of fundamental natural frequencies of typical cement pavements of China, the selected proposals of the excitation frequency of the resonant rubblization machine are presented. The research results provide a theory support to popularize resonant rubblization technology in overlaying and rebuilding engineering of the existed cement pavements in China.

Approximations for Natural Frequencies of Interconnected Walls and Frames

1975 ◽  
Vol 2 (1) ◽  
pp. 116-119 ◽  
Author(s):  
A. Rutenberg ◽  
A. C. Heidebrecht
Keyword(s):  
Entire Range ◽  
Natural Frequencies ◽  
Maximum Error ◽  
High Accuracy ◽  
Stiffness Parameter ◽  
The Third ◽  
Second Mode ◽  
Beam Approximation ◽  
Flexural Beam

Several approximate formulae for the determination of natural frequencies of interconnected walls and frames with uniform mass and stiffness throughout their height are compared and the range of their applicability discussed. It is found that none of the approximations give high accuracy for the entire range of the stiffness parameter αH encountered in practice. It is recommended that for the fundamental frequency the flexural beam approximation be used for the lower end of the range (αH < 7) and the large αH asymptotic formula be used elsewhere. This ensures that the error never exceeds 5%. For the higher modes there is a choice between three approximations, the maximum error associated with each being practically identical: 10% in the second mode and 5% in the third.

Experimental Investigation of Sloshing Dynamics Coupled With Barge Responses

2009 ◽  
Author(s):  
T. Nasar ◽  
S. A. Sannasiraj ◽  
V. Sundar
Keyword(s):  
Experimental Investigation ◽  
Wave Height ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Wave Excitation ◽  
The Third ◽  
Sloshing Frequency ◽  
Sloshing Dynamics ◽  
Liquid Depth ◽  
Cover Up

An experimental work has been carried out to study the phenomena of sloshing of liquid in a partially filled tank with aspect ratio (hs/l, where hs is the static liquid depth and l is the tank length) of 0.585. The sloshing tank was rigidly fixed in to a barge and was exposed to regular beam waves. The wave excitation frequencies (fw) ranging from 0.70Hz to 1.54Hz that cover up to the third mode natural sloshing frequency (f3) are considered. The incident wave height (Hi) is 0.10m. The effects of wave excitation frequency and wave height on the sloshing oscillation are studied. Attempts are made to evaluate the harmonics present in the sloshing oscillation and compare with the results of earlier studies. The barge responses such as sway, heave and roll are measured and it is found that the barge responses at their natural frequencies are insensitive to induce sloshing oscillation inside the tank.

Modal Overlap and Dissipation Effects of a Cantilever Beam with Multiple Attached Oscillators

2000 ◽  
Vol 123 (2) ◽  
pp. 181-187 ◽  
Author(s):  
M. V. Drexel ◽  
J. H. Ginsberg
Keyword(s):  
Transfer Function ◽  
Natural Frequency ◽  
Cantilever Beam ◽  
Impulse Response ◽  
Critical Issue ◽  
Harmonic Excitation ◽  
Vibration Absorber ◽  
Continuous Systems ◽  
Coupled Equations ◽  
Eigenmode Analysis

This work was prompted by a study performed by Strasberg [7] in which numerous small spring-mass-damper systems are attached to a large suspended mass representing the master structure. The isolated natural frequency of each attached system was selected to match in average the natural frequency of the isolated master structure. Strasberg found that the critical issue when an impulse excitation is applied to the master structure is the bandwidth of the isolated attached systems in comparison to the spacing between the natural frequencies of the system. Modal overlap, which corresponds to bandwidths that exceed the spacing of those frequencies, was shown to greatly influence the response of the master structure. Light damping, for which there is little or no modal overlap, corresponds to an impulse response that consists of a sequence of nearly periodic exponentially decaying pulses, and the transfer function for harmonic excitation of the master structure indicates that the substructure acts as a vibration absorber for the master structure. Increased damping leads to modal overlap, with the result that the impulse response consists of a single decaying pulse. The frequency domain transfer function indicates that the vibration absorber effect is enhanced. The present work explores these issues for continuous systems by replacing the one degree of freedom master structure with a cantilever beam. The system parameters are selected to match Strasberg’s model, with the suspended oscillators placed randomly along the beam. The beam displacement is represented as a Ritz series whose basis functions are the cantilever beam modes. The coupled equations are solved by a state-space eigenmode analysis that yields a closed form representation of the response in terms of the complex eigenmode properties. The continuous fuzzy structure is shown not to display the transfer of energy between the master structure and the substructure that was exhibited by the discrete fuzzy structure, apparently because of the asynchronous motion of the attachment points resulting from the spatial variability of the beam’s motion. The vibration absorber effect for harmonic excitation is only obtained for the heavy damping in the case of a beam.

Mechanics Mechanism Analysis of Concrete Pavement Resonant Rubblization

2015 ◽  
Vol 667 ◽  
pp. 365-369
Author(s):  
Peng Chen ◽  
Xin Qiu ◽  
Qing Zhu ◽  
Chan Chan Ouyang
Keyword(s):  
Boundary Condition ◽  
Natural Frequency ◽  
Elastic Foundation ◽  
Thin Plate ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Concrete Pavement ◽  
Mechanism Analysis ◽  
Vibration Theory ◽  
Fundamental Natural Frequency

Based on the assumption of thin plate of elastic foundation and vibration theory, a method for calculating the fundamental natural frequency of cement slab is presented and the certain relationship between the fundamental natural frequency of cement slab and cement slab boundary condition is discussed. As well, according to the analysis results of fundamental natural frequencies of the typical cement pavements of China, the selected proposals of the excitation frequency of the resonant rubblization machine are presented .The research results provide a theory support to popularize resonant rubblization technology in overlaying and rebuilding engineering of the existed cement pavements in China.

An Addition to Myklestad’s Method Giving Convergence to a Natural Frequency

1975 ◽  
Vol 19 (02) ◽  
pp. 130-132
Author(s):  
B. Dawson ◽  
M. Davies
Keyword(s):  
Natural Frequency ◽  
Cantilever Beam ◽  
Digital Computer ◽  
Quadratic Convergence ◽  
Natural Frequencies ◽  
Limited Range ◽  
Previous Calculation

This paper presents an extension of Myklestad's method that within a limited range either side of a natural frequency enables an improved frequency value to be determined from the results of a previous calculation. The method has quadratic convergence within the vicinity of a natural frequency and is well suited for solution on a digital computer. The method is illustrated by determining the first five natural frequencies of a cantilever beam, and the rates and range of convergence of the method are shown.

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