Fourth Natural Frequency Research Articles

МОДАЛЬНЫЙ АНАЛИЗ ШПИНДЕЛЬНОГО УЗЛА НА ОПОРАХ С АНИЗОТРОПНЫМИ УПРУГИМИ ХАРАКТЕРИСТИКАМИ

The article deals with issues related to the influence of options for the design of the spindle assembly (SA), the anisotropy of the radial rigidity of the bearing supports, as well as the spindle speed on the dynamic quality of the machine. The problem statement is formulated as follows: it is necessary to determine the degree of influence of each of the above parameters on the eigen frequencies and vibration modes of the SA. To identify patterns, numerical finite element modeling was carried out in the ANSYS Workbench environment. Variants of the spindle assembly were considered on the example of four models of the CNC lathe model 16B16T1, which differ in the number of structural elements and their location along the length of the spindle. When modeling, the spindle bearings were represented as solid rings with certain elastic properties, varying which the anisotropy of the stiffness of the bearings was realized. During the numerical experiment, the spindle speed was taken into account in the range from 0 to 20,000 rpm. When analyzing the results, the values of six natural frequencies and modes of vibration were considered for each of the models with a change in the rigidity of the front support; Campbell diagrams were obtained, illustrating the dependence of the dynamic characteristics of the spindle on the rotational speed in the specified range. As a result of the research, it was found that design factors have a significant impact on the vibration modes and natural frequencies of the spindle assembly. The simulation showed that when machining in the centers, the fourth natural frequency and modes of vibration will affect the accuracy, and when using a chuck, the first one. Taking into account the fact that the difference between them is more than 2.5 times, then machining in the centers can be carried out at significantly higher cutting speeds. Changing the stiffness of the front support results in a range of natural frequencies for each waveform, which can be up to 200 Hz. Within the specified range, natural frequencies change depending on the spindle speed.

Transient responses in Piezoceramic Multilayer Actuators Taking into Account External Viscoelastic Layer

The work develops a generalized approach to the study of thickness (radial) vibrations arising in the piezoceramic plates, cylinders, spheres under electrical loads. The state of the problem and the main approaches, used in the problems of studying the oscillations of electroelastic bodies, are described. The use of multilayer elements with electroded interface surfaces and variable direction of polarization of the layers increases the conversion efficiency of electrical energy into mechanical energy, so multilayer piezoceramic plates, cylinders, spheres with changing polarization directions with electroded interfaces are considered. Because of piezoelectric elements are often embedded in the housing and supplemented with matching layers to protect against mechanical damage, it is necessary to study their effect on the oscillations of the element. The proposed approach makes it possible to study the vibrations of plane, cylindrical and spherical bodies with layers made of various electroelastic and elastic materials. Numerical implementation is carried out using finite differences.
 Nonstationary oscillations of PZT-4 ceramic elements at zero initial conditions are investigated. Oscillations of multilayer plates, cylinders and spheres with and without an external elastic or viscoelastic reinforcing layer under impulse and harmonic unsteady loads are investigated and compared. There are found own frequencies for 5-layer bodies of different geometry with and without an external layer. The first natural frequency for cylinder and sphere corresponds to the radial mode of oscillations, while the second natural frequency for cylinders and spheres and the first for flat bodies are almost equal and correspond to thickness mode. The transient processes in the elements under impulse loads and the influence of the outer elastic layer (housing or matching layer) are studied, taking into account the Rayleigh attenuation. It is established that for a flat layer the outer layer increases the amplitude and the period of free vibrations after removing the load, and for cylinders and spheres it decreases. The presence of an elastic layer enhances the third and dampens the fourth natural frequency of the transducer, thereby expanding the frequency range of its operation.

Analysis of the Leakage-Flow-Induced Vibration of a Slip Joint in a Jet Pump of a Boiling Water Reactor

The purpose of this analysis is to evaluate the structural integrity of the jet pump assembly of a BWR during the performance of its operational and safety functions. The natural frequencies and vibration modes of the jet pump assembly immersed in water were determined. It was observed that the fourth mode shape was torsional, and its associated resonance frequency was 41.82 Hz. Also, the vibration induced by the flow in the leakage of the slip joint was analyzed with an axisymmetric model. The gap of the slip joint was varied from 0.2 mm until 0.65bmm. A gap between 0.6 and 0.64, would cause flow-induced vibration because this excitation frequency matches with the fourth natural frequency of the jet pump assembly. The above was carried out using computational fluid dynamics, as well as the finite element method, with ANSYS Structural and ANSYS Fluent codes.

Analysis of lateral vibrations of the axially loaded helical spring

It is shown in this paper that the proposed concept by Kruzelecki and Życzkowski (1990) of the equivalent rod can be applied in calculations of natural lateral vibrations of springs and that the obtained results will be nearer FEM results than the standard model based on the Timoshenko equivalent beam. The model, created on this base, allows one to calculate natural frequencies of the clamped-clamped spring. It is also shown that models based on the equivalent beam concept, which are easier to apply than the models treating the spring as the spatially curved rod, have only a slightly smaller accuracy. It is also indicated that in the most common practice making of manufacturing end coils of springs, the natural frequencies differ significantly from the frequencies calculated by means of all tested methods. The performed simulations show that differences between the first and the second as well as the third and the fourth natural frequency of the spring are small and, therefore, the axially symmetrical equivalent beam model can be used without a large error. The diagram allowing one to determine whether the desired frequencies are lower or higher than the cut-off frequency is developed for the presented model.

Experimental and finite element analysis to identify the source of vibration of a coach

This paper presents an investigation on the cause of severe vibration problem of a coach with four-cylinder engine running at an idle state using vibration and impact hammer modal experiments to obtain the main vibration frequency components and the natural characteristics of the coach. The vibration results indicate that the main vibration component comes from the vibration transmitted from the engine to the chassis frame, which is closely related with the engine idle speed. Based on structural simulation analysis of the coach’s chassis frame and comparison with modal testing, the coach severe vibration problem was due to coupling resonance between the engine idle frequency and the fourth natural frequency of the chassis frame. The solution to eliminate the vibration problem is provided by changing the local structure stiffness of the chassis frame. The contribution of this paper lies in providing a solution to solve similar problems.

Vertical Vibration of Moving Strip in Rolling Process Based on Beam Theory

The shape and thickness qualities of strip are influenced by the vibration of rolling mill. At present, the researches on the vibration of rolling mill are mainly the vertical vibration and torsional vibration of single stand mill, the study on the vibration of tandem rolling mill is rare. For the vibration of tandem rolling mill, the key problem is the vibration of the moving strip between stands. In this paper, considering the dynamic of moving strip and rolling theory, the vertical vibration of moving strip in the rolling process was proposed. Take the moving strip between the two mills of tandem rolling mill in the rolling process as subject investigated, according to the theory of moving beam, the vertical vibration model of moving strip in the rolling process was established. The partial differential equation was discretized by Galerkin truncation method. The natural frequency and stability of the moving strip were investigated and the numerical simulation in time domain was made. Simulation results show that, the natural frequency was strongly influenced by the rolling velocity and tension. With increasing of the rolling velocity, the first three natural frequencies decrease, the fourth natural frequency increases; with increasing of the unit tension, when the rolling velocity is high and low, respectively, the low order dimensionless natural frequency gradually decreases and increases, respectively. According to the stability of moving strip, the critical speed was determined, and the matching relationship of the tension and rolling velocity was also determined. This model can be used to study the stability of moving strip, improve the quality of strip and develop new rolling technology from the aspect of dynamics.

Dynamics for an electromechanical integrated electrostatic harmonic drive

In this paper, an electromechanically coupled dynamic equation for an integrated electrostatic harmonic drive is presented. The dynamic equation is transformed into a balance equation for static displacement and a dynamic equation for dynamic displacement. The electromechanical-coupled force is considered as well. The operating principle for the drive system is analyzed. By defining the electromechanically coupled forces in Fourier series form, the static displacements of the ring are obtained. Changes in the voltage, along with ring displacement, are discussed. Using the same dynamic equation, the natural frequencies and mode functions of the drive system are given for a different electric field. The first natural frequency is influenced by the phase number of the electric field, but not by the pole pair number. Similarly, the second, third, and fourth natural frequencies are influenced both by pole pair number and by phase number. As radial displacement of the ring increases, the voltage will arrive at an extreme value where the flexible ring becomes buckled. For modes having higher natural frequencies, more positions of the dynamic peak displacements occur, and the time periods of the mode functions become shorter. Also, the pole pair number and phase number of the electric field exhibit an obvious influence on the positions of the dynamic peak displacements and time periods of the mode functions.

Non-linear structural dynamics characterization using a scanning laser vibrometer

This paper presents the use of a scanning laser vibrometer and a signal decomposition method to characterize non-linear dynamics of highly flexible structures. A Polytec PI PSV-200 scanning laser vibrometer is used to measure transverse velocities of points on a structure subjected to a harmonic excitation. Velocity profiles at different times are constructed using the measured velocities, and then each velocity profile is decomposed using the first four linear mode shapes and a least-squares curve-fitting method. From the variations of the obtained modal velocities with time we search for possible non-linear phenomena. A cantilevered titanium alloy beam subjected to harmonic base-excitations around the second, third, and fourth natural frequencies are examined in detail. Influences of the fixture mass, gravity, mass centers of mode shapes, and non-linearities are evaluated. Geometrically exact equations governing the planar, harmonic large-amplitude vibrations of beams are solved for operational deflection shapes using the multiple shooting method. Experimental results show the existence of 1:3 and 1:2:3 external and internal resonances, energy transfer from high-frequency modes to the first mode, and amplitude- and phase-modulation among several modes. Moreover, the existence of non-linear normal modes is found to be questionable.

An Experimental Investigation of Multimode Responses in a Cantilever Beam

An experimental investigation into the planar, multimode response of a cantilever metallic beam to a transverse harmonic excitation is presented. The cantilever beam was tested in both the horizontal and vertical directions. In the vertical configuration, the beam was excited near its fourth natural frequency and energy was transferred from the directly excited fourth mode to a low-frequency mode through both resonant and nonresonant modal interactions. In the latter case, the response contained contributions from the first and fourth modes, whereas in the former case, the response contained contributions from the fourth, second, and fifth modes. The mechanism responsible for the resonant interaction involving the three modes is a subcombination internal resonance of the additive type; that is, Ω ≈ ω4 ≈ 1/2(ω2 + ω5), where the ωi are the natural frequencies of the beam. This type of resonance occurs in systems with a dominant cubic nonlinearity. In the horizontal configuration, the beam was excited near its fourth and sixth natural frequencies. Again, energy was transferred to a low-frequency mode, but in this case only through a resonant interaction due to a subcombination internal resonance of the additive type. Power spectra, time series, and frequency- and amplitude-response curves were obtained for characterization of the dynamic multimode responses.

Nonlinear Resonances in a Flexible Cantilever Beam

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.