Dynamic Stiffness Matrix of a Long Rubber Bush Mounting

2002 ◽  
Vol 75 (4) ◽  
pp. 747-770 ◽  
Author(s):  
L. Kari
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Low Frequency ◽  
Material Model ◽  
Incompressible Material ◽  
Noise Abatement ◽  
Arbitrary Boundary ◽  
Dynamic Stiffness Matrix ◽  
Minimum Number ◽  
Standard Linear Solid

Abstract The complete blocked dynamic stiffness matrix of a long rubber bush mounting of particular interest for noise abatement is examined by an analytical model, where influences of audible frequencies, material properties, bush mounting length, and radius, are investigated. The model is based on the dispersion relation for an infinite, thick-walled cylinder with arbitrary boundary conditions at the radial inner and outer surfaces; yielding the sought stiffness matrix, including axial, torsional, radial, and tilting stiffness. A nearly incompressible material model is adopted, being elastic in dilatation while displaying viscoelasticity in deviation. The applied deviatoric Mittag-Leffler relaxation function is based on a fractional standard linear solid, the main advantage being the minimum number of parameters required to successfully model the rubber properties over a broad structure-borne sound frequency domain. The dynamic stiffness components display a strong frequency dependence at audible frequencies, resulting in acoustical resonance phenomena, such as stiffness peaks and troughs. The low-frequency stiffness asymptotes of the presented model are shown to agree with those of static theories.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

2021 ◽  
pp. 146442072098817
Author(s):  
Md. Imran Ali ◽  
Mohammad Sikandar Azam
Keyword(s):  
Power Law ◽  
Functionally Graded Material ◽  
Stiffness Matrix ◽  
Natural Vibration ◽  
Dynamic Stiffness ◽  
Functionally Graded ◽  
Neutral Surface ◽  
Functionally Graded Plate ◽  
Dynamic Stiffness Matrix ◽  
Graded Material

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

scholarly journals Using Waveguides to Model the Dynamic Stiffness of Pre-Compressed Natural Rubber Vibration Isolators

Polymers ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 1703
Author(s):  
Michael Coja ◽  
Leif Kari
Keyword(s):  
Natural Rubber ◽  
Dynamic Stiffness ◽  
Low Frequency ◽  
Complex Problem ◽  
Wide Frequency Range ◽  
Vibration Isolators ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Frequency Magnitude ◽  
Good Agreement

A waveguide model for a pre-compressed cylindrical natural rubber vibration isolator is developed within a wide frequency range—20 to 2000 Hz—and for a wide pre-compression domain—from vanishing to the maximum in service, that is 20%. The problems of simultaneously modeling the pre-compression and frequency dependence are solved by applying a transformation of the pre-compressed isolator into a globally equivalent linearized, homogeneous, and isotropic form, thereby reducing the original, mathematically arduous, and complex problem into a vastly simpler assignment while using a straightforward waveguide approach to satisfy the boundary conditions by mode-matching. A fractional standard linear solid is applied as the visco-elastic natural rubber model while using a Mittag–Leffler function as the stress relaxation function. The dynamic stiffness is found to depend strongly on the frequency and pre-compression. The former is resulting in resonance phenomena such as peaks and troughs, while the latter exhibits a low-frequency magnitude stiffness increase in addition to peak and trough shifts with increased pre-compressions. Good agreement with nonlinear finite element results is obtained for the considered frequency and pre-compression range in contrast to the results of standard waveguide approaches.

Steady-State Unbalance Response of Continuous Rotors on Anisotropic Supports

1993 ◽  
Author(s):  
Graziano Curti ◽  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Steady State ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Analytical Investigation ◽  
Dynamic Stiffness Matrix ◽  
Rotor Systems ◽  
Beam Elements ◽  
Unbalance Response ◽  
Constant Coefficients ◽  
Linearized Model

Abstract An analytical investigation of the steady-state unbalance response of axisymmetric rotor systems with anisotropic, flexible and damped bearings is presented. According to the exact approach of the dynamic stiffness method, the rotor is modelled by means of continuous beam elements. In this work, the expression of the 8 × 8 dynamic stiffness matrix of a rotating Timoshenko beam is derived and it is shown that it is related, by means of a simple law, to the previously published 4 × 4 dynamic stiffness matrix, which holds for the isotropic bearings case. The effects of concentrated disks and bearings are included into the formulation; in particular, each bearing is described by eight constant coefficients, according to the well-known linearized model of the bearing forces. The unbalance response of a typical rotor system taken from the literature is analyzed. A comparison is presented with the finite element results reported by other authors.

Vibration Modeling of Machine Tool Spindles: A Calibrated Dynamic Stiffness Matrix Method

2013 ◽  
Vol 651 ◽  
pp. 710-716 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Spring Element ◽  
Linear Spring ◽  
Beam Bending ◽  
The Stability ◽  
Spinning Speed ◽  
Vibration Modeling

The effects of spindles vibrational behavior on the stability lobes and the Chatter behavior of machine tools have been established. The service life has been observed to reducethe system natural frequencies. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation, exact within the limits of the Euler-Bernoulli beam bending theory, is developed. The system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model usingboth rigid, simply supported, frictionless pins and flexible linear spring elements. The linear spring element stiffness is then calibrated so that the fundamental frequency of the system matches the nominal value.

scholarly journals Seismic Analysis of the Transportation Portal by the Combined Asymptotic Method

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Alexander Tyapin
Keyword(s):  
Asymptotic Method ◽  
Stiffness Matrix ◽  
Transfer Functions ◽  
Seismic Analysis ◽  
Dynamic Stiffness ◽  
Free Field ◽  
Structural Dynamic ◽  
Dynamic Stiffness Matrix ◽  
Double Support ◽  
Soil Dynamic

The author extends the previously proposed combined asymptotic method (CAM) of seismic SSI analysis for the multi-support systems and applies it to the transportation portal as a double-support system (together with the reactor building). The key issue is the development of the structural dynamic stiffness matrix condensed to the supports by the modal approach. Then the condensed structural matrix is combined with the soil dynamic stiffness matrix also condensed to the rigid basements. As a result, a very simple linear system is solved in the frequency domain. This gives the transfer functions from the free-field motion to the motion of the basements. The only important limitations are the linearity of the soil’s and structure’s properties and the rigidity of the basements. The results for the sample system are checked against the full SASSI solution. The results can be used to justify the further simplification of the system.

Updating of Dynamic Finite Element Models Based on Experimental Receptances and the Reduced Analytical Dynamic Stiffness Matrix

1995 ◽  
Author(s):  
Stefan Lammens ◽  
Marc Brughmans ◽  
Jan Leuridan ◽  
Ward Heylen ◽  
Paul Sas
Keyword(s):  
Stiffness Matrix ◽  
Model Updating ◽  
Dynamic Stiffness ◽  
Computational Cost ◽  
Computational Effort ◽  
Reduction Scheme ◽  
Dynamic Stiffness Matrix ◽  
Simple Strategy ◽  
A Minor ◽  
Analytical Dynamic

Abstract This paper presents a model updating method based on experimental receptances. The presented method minimises the so called ‘indirect receptance difference’. First, the reduced analytical dynamic stiffness matrix is expressed as an approximate, linearised function of the updating parameters. In a numerically stable, iterative procedure, this reduced analytical dynamic stiffness matrix is changed in such a way that the analytical receptances match the experimental receptances at the updating frequencies. The updating frequencies are a set of selected frequency points in the frequency range of interest. Some considerations about an optimal selection of the updating frequencies are given. Finally, a mixed static-dynamic reduction scheme is discussed. Dynamic reduction of the analytical dynamic stiffness matrix at each updating frequency is physically exact, but it involves a great computational effort. The presented mixed static-dynamic reduction scheme is a simple strategy to reduce the computational cost with a minor loss of accuracy.

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