An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

2005 ◽  
Vol 286 (3) ◽  
pp. 549-568 ◽  
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Moment Of Inertia ◽  
Mode Shapes ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Elastically Restrained ◽  
Tip Mass

Natural Frequencies of Nonuniform Beams on Multiple Elastic Supports

1958 ◽  
Vol 25 (1) ◽  
pp. 57-63
Author(s):  
R. A. Di Taranto
Keyword(s):  
Natural Frequencies ◽  
Forced Vibrations ◽  
Moment Of Inertia ◽  
Elastic Supports ◽  
Mass Moment ◽  
End Conditions ◽  
Mass Moment Of Inertia ◽  
Gyroscopic Effects

Abstract A method is presented for the determination of the natural frequencies of nonuniform beams on two or more torsionally and linearly elastic supports, including the effect of rotary mass moment of inertia. The method employed is an extension of the Myklestad method. The cases of two supports with varied end conditions and three supports with a torsional and linear restraint at each support are formulated. It is indicated how this method may be used for problems concerning forced vibrations of beams on multiple elastic supports and for the determination of critical rotor speeds including gyroscopic effects.

Nonlinear Vibrations of a Beam With a Tip Mass Attached to a Rotating Hub

2005 ◽  
Author(s):  
Jerzy Warminski ◽  
Jose´ Manoel Balthazar
Keyword(s):  
Light Beam ◽  
Nonlinear Vibrations ◽  
Small Mass ◽  
Moment Of Inertia ◽  
Beam Model ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Dynamical Boundary Conditions ◽  
Tip Mass ◽  
Galerkin's Procedure

A model of a light beam mounted to a rigid hub and forced by an external torque has been analysed in the paper. An additional loading caused by a small mass attached to the end of the beam and the influence of mass moment of inertia of the hub have been also taken into account. The motion of a flexible light beam is a composition of slewing motion and undesired vibrations, which can be crucial if the stiffness of the structure is not high, comparing to the external dynamical load. The nonlinear model of the beam proposed in [2], [3] have been applied to explain behaviour of the system. That model has taken into account bending, tension and a non-linear curvature, which differs the system from the classical approach [4] or the approach presented in [5] for large displacement of a beam model. The influence of the mass moment of inertia of the hub and the tip-mass is investigated in the paper. Differential equation of motion and dynamical boundary conditions are derived by applying the Hamilton’s principle whilst the reduced model have been obtained by virtue of the Galerkin’s procedure.

scholarly journals Determination of the dimensions of the cylindrical weight at the end of the rod

2019 ◽  
Vol 14 (3) ◽  
pp. 214-217
Author(s):  
A.A. Aitbaeva
Keyword(s):  
Differential Equation ◽  
Spectral Problem ◽  
Natural Frequencies ◽  
Moment Of Inertia ◽  
Unknown Parameters ◽  
Transverse Vibrations ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
The Right

This article discusses free transverse vibrations of a homogeneous rod. The left end of the rod is clamped, and a cylindrical weight is concentrated at the right end. The eigenfrequencies of the rod vibration are known. The purpose of this work is to determine the parameters of the end cylindrical weight of the rod (mass, moment of inertia, length and radius) by the natural frequencies of the rod vibrations. We use a partial differential equation derivative of the fourth order to solve this problem. This equation and boundary conditions are reduced to a spectral problem. To find the mass and moment of inertia of the weight, the «Method of an additional unknown» was applied. In the characteristic determinant of the spectral problem, there are terms that contain products of unknown coefficients. The essence of the «Method of an additional unknown» is that some of these products are proposed to be considered new additional unknowns, through which the rest can be expressed. It is shown that the mass and moment of inertia of the weight can be found using the three natural frequencies of the rod vibrations. Formulas for finding the length and radius of a cylindrical weight are obtained, and corresponding examples of finding unknown parameters are considered.

Modeling and validation of crankshaft speed fluctuations of a single-cylinder four-stroke diesel engine

2021 ◽  
pp. 095440702110262
Author(s):  
Mustafa Babagiray ◽  
Hamit Solmaz ◽  
Duygu İpci ◽  
Fatih Aksoy
Keyword(s):  
Diesel Engine ◽  
Dynamic Model ◽  
Moment Of Inertia ◽  
Mass Motion ◽  
Cylinder Pressure ◽  
Motion Equations ◽  
Single Cylinder ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Speed Fluctuations

In this study, a dynamic model of a single-cylinder four-stroke diesel engine has been created, and the crankshaft speed fluctuations have been simulated and validated. The dynamic model of the engine consists of the motion equations of the piston, conrod, and crankshaft. Conrod motion was modeled by two translational and one angular motion equations, by considering the kinetic energy resulted from the mass moment of inertia and conrod mass. Motion equations involve in-cylinder gas pressure forces, hydrodynamic and dry friction, mass inertia moments of moving parts, starter moment, and external load moment. The In-cylinder pressure profile used in the model was obtained experimentally to increase the accuracy of the model. Pressure profiles were expressed mathematically using the Fourier series. The motion equations were solved by using the Taylor series method. The solution of the mathematical model was performed by coding in the MATLAB interface. Cyclic speed fluctuations obtained from the model were compared with experimental results and found compitable. A validated model was used to analyze the effects of in-cylinder pressure, mass moment of inertia of crankshaft and connecting rod, friction, and piston mass. In experiments for 1500, 1800, 2400, and 2700 rpm engine speeds, crankshaft speed fluctuations were observed as 12.84%, 8.04%, 5.02%, and 4.44%, respectively. In simulations performed for the same speeds, crankshaft speed fluctuations were calculated as 10.45%, 7.56%, 4.49%, and 3.65%. Besides, it was observed that the speed fluctuations decreased as the average crankshaft speed value increased. In the simulation for 157.07, 188.49, 219.91, 251.32, and 282.74 rad/s crankshaft speeds, crankshaft speed fluctuations occurred at rates of 10.45%, 7.56%, 5.84%, 4.49%, and 3.65%, respectively. The effective engine power was achieved as 5.25 kW at an average crankshaft angular speed of 219.91 rad/s. The power of friction loss in the engine was determined as 0.68 kW.

scholarly journals Pengaruh Variasi Radius-Dalam Rim Terhadap Pengurangan Massa dan Momen Inersia Massa Dengan Studi Kasus Benda Putar Berdiameter 10 cm

2018 ◽  
Vol 1 (1) ◽  
Author(s):  
Eka Taufiq Firmansjah
Keyword(s):  
Outer Radius ◽  
Moment Of Inertia ◽  
Mass Reduction ◽  
Inner Radius ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Machine Elements ◽  
The Moment ◽  
The Masses

ABSTRAK Mesin terdiri dari sekumpulan elemen mesin yang diam dan bergerak. Elemen mesin yang bergerak dengan gerakan berputar disebut benda putar. Pada beberapa kasus seringkali diinginkan pengurangan massa dari benda putar tersebut untuk alasan ekonomis, biasanya untuk elemen mesin yag diproduksi massal. Namun pengurangan massa berakibat pada pengurangan momen inersia massa benda putar bersangkutan. Jika tuntutan perancangan tidak mempermasalahkan perubahan tersebut, maka pengurangan massa tidak menjadi masalah. Namun jika momen inersia massa tidak boleh terlalu rendah, maka harus dicari kompromi dimana pengurangan massa sebesar-besarnya namun penurunan momen inersia massa sekecil-kecilnya. Pada penelitian ini dilakukan studi kasus terhadap benda putar berjari- jari 10 cm jari-jari dalam hub 2 cm dan jari-jari luar hub 4 cm. Jumlah jari-jari ada 4 dengan lebar 1 cm dan tebal benda putar 0,5 cm. Variasi pengurangan massa dilakukan dengan memvariasikan jari-jari- dalam rim. Untuk tiap variasi, dilakukan perhitungan untuk mendapatkan jumlah massa yang dapat dikurangi dan momen inersia massa dari benda putar. Ternyata pada nilai jari-jari dalam tertentu, dapat diperoleh nilai kompromi dari permasalahan diatas. Kata kunci: benda putar, penghematan bahan, momen inersia massa.  ABSTRACT Machine consists of a set of machine elements that still and moving. Machine elements that move in a circular motion called rotary object. In some cases it is often desirable reduction in the mass of the rotating object for economic reasons, usually for a mass production of machine elements. But the mass reduction results in a reduction in moment of inertia of the mass. If the demands of the design allow this decrease of moment of inertia, mass reduction is not a problem. But if the moment of inertia of the masses should not be too low, it must find a compromise in which a mass reduction profusely but the decrease in the mass moment of inertia of the smallest. In this research conducted a case study of rotating element radius of 10 cm, radius of the hub 2 cm and outer radius hub 4 cm. The number of spoke are 4 with a width of 1 cm and uniform thickness 0.5 cm all over rotating element. Variations mass reduction is done by varying the inner radius of the rim. For each variation, calculation is performed to obtain the amount of mass that can be reduced and the mass moment of inertia of the rotating object. It turned out that in the certain value of inner radius of the rim in particular, can compromise the values obtained from the above problem. Keywords: rotating element, reducing material, mass moment of inertia.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

An Analysis of the Simplified Two-Mass System of the Connecting Rod in Spark Ignition Engines

2002 ◽  
Author(s):  
Richard Stanley
Keyword(s):  
Statistical Study ◽  
Cylinder Liner ◽  
Moment Of Inertia ◽  
Average Error ◽  
Spark Ignition Engines ◽  
Connecting Rod ◽  
Mass System ◽  
Rolling Moment ◽  
Mass Moment ◽  
Mass Moment Of Inertia

Replacing the connecting rod with a lumped two-mass system causes an error, which influences the inertia rolling moment, the thrust force between the piston and the cylinder liner, and the loading on the main bearings. Dimensionless relationships have been found that relate the inertia error due to the connecting rod simplification (the inertia error) to the errors of the forces and moments that are created by it. Additionally, the results of a statistical study of 19 SI connecting rods indicate that the mass moment of inertia of the two mass system is −2.65% to 22% higher than that the experimentally measured moment of inertia of the connecting rod, with an average error value of 9.65%.

scholarly journals Quantification of vibrissal mechanical properties across the rat mystacial pad

2019 ◽  
Vol 121 (5) ◽  
pp. 1879-1895 ◽  
Author(s):  
Anne En-Tzu Yang ◽  
Hayley M. Belli ◽  
Mitra J. Z. Hartmann
Keyword(s):  
Mechanical Properties ◽  
Center Of Mass ◽  
Distal Region ◽  
Density Variation ◽  
Average Density ◽  
Proximal Region ◽  
Moment Of Inertia ◽  
Radius Of Gyration ◽  
Mass Moment ◽  
Mass Moment Of Inertia

Recent work has quantified the geometric parameters of individual rat vibrissae (whiskers) and developed equations that describe how these parameters vary as a function of row and column position across the array. This characterization included a detailed quantification of whisker base diameter and arc length as well as the geometry of the whisker medulla. The present study now uses these equations for whisker geometry to quantify several properties of the whisker that govern its mechanical behavior. We first show that the average density of a whisker is lower in its proximal region than in its distal region. This density variation appears to be largely attributable to the presence of the whisker cuticle rather than the medulla. The density variation has very little effect on the center of mass of the whisker. We next show that the presence of the medulla decreases the deflection of the whisker under its own weight and also decreases its mass moment of inertia while sacrificing <1% stiffness at the whisker base compared with a solid whisker. Finally, we quantify two dimensionless parameters across the array. First, the deflection-to-length ratio decreases from caudal to rostral: caudal whiskers are longer but deflect more under their own weight. Second, the nondimensionalized radius of gyration is approximately constant across the array, which may simplify control of whisking by the intrinsic muscles. We anticipate that future work will exploit the mechanical properties computed in the present study to improve simulations of the mechanosensory signals associated with vibrissotactile exploratory behavior. NEW & NOTEWORTHY The mechanical signals transmitted by a whisker depend critically on its geometry. We used measurements of whisker geometry and mass to quantify the center of mass, mass moment of inertia, radius of gyration, and deflection under gravity of the whisker. We describe how variations in these quantities across the array could enhance sensing behaviors while reducing energy costs and simplifying whisking control. Most importantly, we provide derivations for these quantities for use in future simulation work.

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