Metin Gurgoze

Istanbul Technical University · Department of Mechanical Engineering

Hamilton’s principle is one of the most important milestones of analytical mechanics. This principle is valid for fixed mass and discrete material systems in its original form. The seminal work of McIver published in 1973 (McIver in J Eng Math 7(3):249–261, 1973) made this principle applicable to variable mass systems and has been constantly refere...

Makina Mühendisliği alanında, hareketleri periyodik katsayılı diferansiyel denklemlerle belirlenen sistemler büyük öneme sahiptirler, zira bu sistemlerde, hangi giriş açısal hızlarında, sistemde rezonansların veya yerine göre rezonans bölgelerinin ortaya çıkma olasılıklarının söz konusu olduğunun önceden kestirilebilmesi, hayati öneme sahiptir. Bu...

In this paper, a new dynamic model for the vibration analysis of an inward-oriented rotating cantilever beam with extra distributed mass was presented. The derived differential equation of motion was solved using the meshless methods of generalized Multiquadric Radial Basis Function (RBF) and the eigenfrequencies of the system were determined. The...

Analitik Mekaniğe Giriş kitabı, yazarın İTÜ Fen Bilimleri Enstitüsü bünyesinde "Makina Dinamiği, Titreşim ve Akustik" Yüksek Lisans Programı'nda 20 yıla yakın süre vermiş olduğu "Mekanik Sistemler Dinamiği" dersinde anlatmış olduğu konuların geliştirilmesi, sınavlarında sorulmuş problemlerin ve verilmiş bazı ödev problemlerinin çözümlerinin derlenm...

The first author has been teaching the postgraduate course, “The Dynamics of Mechanical Systems” in The ITU Faculty of Mechanical Engineering for nearly 20 years. He has observed that students frequently have problems in obtaining the equations of motion of the vibrating systems which were placed on moving bases. Starting from this observation, he...

Many vibrating mechanical systems from the real life are modeled as combined dynamical systems consisting of beams to which spring-mass secondary systems are attached. In most of the publications on this topic, masses of the helical springs are neglected. In a paper (Cha et al. 2008) published recently, the eigencharacteristics of an arbitrary supp...

In a recent study, the eigenvalue sensitivities of a combined system consisting of a linear structure with several lumped attachments were analyzed. An efficient approach was developed for the eigenvalue sensitivities with respect to the parameters of the attachments and their locations by using the implicit function theorem. In the study mentioned...

The present study deals with the dynamics of the flapwise (out-of-plane) vibrations of a rotating, internally damped (Kelvin-Voigt model) tapered Bernoulli-Euler beam carrying a heavy tip mass. The centroid of the tip mass is offset from the free end of the beam and is located along its extended axis. The equation of motion and the corresponding bo...

First author, who has been teaching on subjects related to advanced dynamics and vibrations for many years, observed that his students have always found it difficult to understand concepts on " dyads " and " dyadics ". Based on his observation, in this article the authors present information and formulae gathered from different resources, on dyads...

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an ext...

The common point about many systems modeled as Bernoulli–Euler beams with attachments is that the own masses of the helical springs are neglected. Some researchers accounted for the masses of the springs during free vibrations of those systems. Further to these studies, the present study deals with the investigation of the effect of not taking into...

The current study deals with in-class observations made during the teaching of a graduate-level mechanical engineering course entitled ‘Dynamics of Mechanical Systems’. The specific discussion is about the students' various approaches to the application of Lagrange equations in the context of a homework assignment. The analytical solution requested...

The present study deals with the determination of the frequency equation of a Bernoulli-Euler beam simply supported at both ends, to which is attached in-span a longitudinally vibrating elastic rod with a tip mass, representing a helical spring-mass system with mass of the helical spring considered. The principal aim is to underline once more the i...

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin—Voigt model) Bernoulli—Euler beam carrying a tip mass, which can be thought of as a simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem set up is solved by using the Frobenius metho...

This study is concerned with the establishment of the characteristic equation of a combined system consisting of a cantilever beam with a tip mass and an in-span visco-elastic helical spring-mass, considering the mass of the helical spring. After obtaining the "exact" characteristic equation of the combined system, by making use of a boundary value...

Recently, Hu [A note on the frequency of nonlinear conservative oscillators, Journal of Sound and Vibration 286 (2005) 653–662] presented a superposition method for the approximate determination of frequencies of conservative oscillators when the nonlinear restoring force consists of a superposition of several individual characteristics. In this co...

The present note is concerned with the derivation of the characteristic equation of a cantilevered, visco-elastic bending beam (Kelvin–Voigt model), carrying a tip mass. Further, it is attempted to represent this continuous system by an “equivalent” spring-damper-mass system. Then, the “first” eigenvalues of these systems are calculated and tabulat...

A chain structured mass-spring vibration system with N degrees of freedom is considered where massesMj = mjm = m/j and spring stiffnesses kj = cjk = (N + 1 - j)k, j = 1, …, N, have been applied. The conjecture of Mikota [1] on the natural frequencies is discussed showing how difficult it is to find an explicit solution of the related eigenvalue pro...

This paper deals with the dynamical analysis of a cantilevered Bernoulli–Euler beam subjected to distributed external viscous damping in-span and with a viscous end condition by a single damper. In order to evaluate the vibration characteristics of the system, a procedure is presented where overdamped and underdamped ‘‘modes’’ are investigated simu...

This study is concerned with the establishment of the frequency equation of a combined system consisting of a simply supported beam and an in-span helical spring–mass, considering the mass of the helical spring. After obtaining the “exact” frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental...

Lagrange's equations are a well known tool for establishing equations of motion of discrete systems. Kinetic energy, which plays a central role in their use, has to be formulated with respect to an inertial system. Lagrange's equations can be formulated favorably with respect to a moving coordinate system as well (e.g. for mechanical systems which...

The present note deals with the derivation of the characteristic equation of an axially vibrating viscoelastic rod (Kelvin–Voigt model), carrying a tip mass. Further, it is attempted to represent this continuous system by an “equivalent” spring-damper-mass system. Then, the “first” eigenvalues of these systems are calculated and tabulated for a wid...

This communication is concerned with torsional vibrations of an elastic bar of given length and torsional rigidity to which several discs are attached. For fixed–free and fixed–fixed cases, formulas for the sums of the squared reciprocal eigenfrequencies of the vibrational system are established.

The state-space method is frequently used to obtain the eigenvalues of a viscously damped linear mechanical system. Differences in the definition of the state vector and auxiliary matrices found in the literature lead to differences in the formulation of the eigenvalue problems and this in turn can cause difficulties for students on mechanical vibr...

The eigenfrequencies of a cantilever beam carrying a spring-mass system were analyzed. A helical spring with mass was modeled as a longuitudinally vibrating rod which was attatched to the tip of the cantilever beam together with an additional mass. The frequency equation of this combined system was derived. It was found that neglecting the mass can...

A research study on the effect of the spring mass on the frequency spectrum of a combined system consisting of a cantilever to which a spring-mass system is attached at the free end was discussed. It was found that when the mass of the beam mL was small in comparison to the tip mass, the restoring effect of the beam on the tip mass can be represent...

A method of obtaining the exact solution for the forced vibrations of elastic rods coupled by distributed springs and dampers was presented. The method was based on the change of variables to decouple the set of two second order partial differential equations, and then the solutions were obtained by means of modal analysis. The two restrictions of...

Investigation on the possibility of using springs to preserve the fundamental frequency of a thin rectangular plate carrying any number of point masses was carried out. The problems on plates carrying concentrated masses were encountered in the design of electronic systems. The receptance matrix of the modified plate were obtained by calculating th...

The relationship between the fundamental matrices of a linear mechanical system with n degrees of freedom, and the state vector of which is usually defined in two different forms, was discussed. The free vibrations if a discrete linear mechanical system was governed in the physical space by a matrix differential equation of order two. The different...

Two methods were employed for computing the eigencharacteristics of a continuous beam, carrying a tip mass, consisting of several parts having different physical parameters, and subjected to external viscous damping. Both methods used the separation of variables approach and differed in the solution of the corresponding ordinary differential equati...

The article discusses a determinant formula used for the derivation of frequency equations of combined systems. It enables one to obtain the determinant of the sum of a regular matrix and several dyadic products. The present formula is much easier to prove although it is more general than the recently developed one which can be employed only for a...

A novel formulation of the receptance matrix of non-proportionally damped dynamic systems was presented. An iterative method was developed for the calculation of the receptance matrix when the damping matrix was decomposed into the sum of dyadic products. It was shown that it is possible to express the receptance matrix of the damped system in term...

The study on the establishment of a method to compute the eigenvalues and eigenfunctions of a continuous, viscously damped rod is presented. It consists of two parts having different physical parameters. It is found that both the overdamped and the underdamped eigenvalues and corresponding eigenfunctions are computed for two different sets of param...

A study published some years ago (1) investigated how linear damping affected the eigenvalues of proportionally damped systems, and the results were presented graphically. The necessary conditions for the system to be overdamped or underdamped were given with the help of simple formulae. After a brief summary of the previous work, these methods are...

This study is concerned with the determination of the eigenvalues and "eigenfunctions" of an axially vibrating, viscously damped elastic rod, carrying a tip mass and consisting of two parts having different physical parameters. The eigencharacteristics of the rod are determined via an original application of the separation of the variables method....

Continuous and discrete models for a longitudinally vibrating elastic rod fixed at both ends were proposed. The rod was damped viscously by a single damper in-span. The expressions were derived and solved for the eigenfrequencies of the models. A single explicit formula for both complex frequencies and the damping ratio was derived using an asympto...

This paper deals with the determination of the frequency response function of a cantilevered Bernoulli–Euler beam which is viscously damped by a single damper. The beam is simply supported in-span and carries a tip mass. The frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear s...

Vibrations of beams coupled by several double spring-mass systems were investigated. The combined system consisted of two clamped-free beams carrying tip masses to which a double spring-mass system was attached in span. The frequency equation of the system mass's was established using Green's function method. The results were compared with those ob...

Vibrations of beams coupled by double spring-mass systems were investigated. It consisted of two clamped-free laterally vibrating Bernoulli-Euler beams carrying tip masses as the primary system to which a double spring-mass secondary system was attached across the span. The complicated frequency equation of the combined system was formulated and th...

Following a recently developed methodology, the location of the nodes and anti-nodes of a complicated structure can be determined by attaching a virtual element (a lumped mass or grounded spring) to the system of interest and analysing the free vibration of this combined system as a function of the location of the virtual element. The system of an...

A study was carried out to examine the problem of determining the stiffness coefficient of the spring to be placed at a specified position so that the fundamental frequency of the bending beam subject to various supporting conditions does not change despite the addition of a mass at a predefined position. The values of the spring coefficients calcu...

This study addresses a viscously damped linear discrete mechanical system which is excited harmonically. The coordinates of the system are assumed to be subject to several linear constraint equations. The aim is the establishment of the receptance matrix of the so constrained system in terms of the acceptance matrix of the unconstrained system and...

This study addresses a linear discrete mechanical system which is damped by several viscous dampers. The co-ordinates of the system are assumed to be subject to several linear constraint equations. The aim is to establish the characteristics equation of the so constrained system.

This study addresses the natural vibration problem of a mechanical system consisting of a fixed-free, axially vibrating rod which carries an added distributed mass in-span. The frequency equation of the system is derived first. Following this, the mode shapes are given and finally, the numerical results are given in the form of various curves.

In this paper, the vibrations of an axially moving flexible beam sliding through an arbitrarily driven prismatic joint, restricted to move on a horizontal plane, are investigated. Upon considering the assumption of an Euler–Bernoulli beam in addition to the effects of rotary inertia, end-mass and axial force in association with axial foreshortening...

This study is concerned with a linear discrete mechanical system, which is damped by a single viscous damper. The coordinates of the system are assumed to be subject to a linear constraint equation. A simple analytical expression of the characteristic equation of the constrained system is developed.

This note is concerned with the natural vibration problem of a mechanical system, consisting of a fixed-free axially vibrating elastic rod which is restrained by a linear spring in-span. The frequency equation of the system is derived first. Then the mode shapes are given and finally a sensitivity formula is established

This paper deals with the investigation of the sensitivity of the eigenvalues of a special mechanical system. It consists of a clamped–free Bernoulli–Euler beam carrying a tip mass. The vibrations of the beam are damped by a viscous damper which is attached to it within the span. The main concern lies in the determination of the sensitivities with...

The present paper is concerned with the determination of the frequency equation and sensitivity of the eigenfrequencies of a fixed–free longitudinally vibrating rod carrying a tip mass to which a spring–mass system is attached in-span. First, the exact frequency equation is established, and then an approximate formula is given for the fundamental f...

The present study is concerned with the investigation of the eigencharacteristics of a special system consisting of a viscously damped, clamped free Bernouilli Euler beam carrying a tip mass. The exact characteristic equation is established via a boundary value problem formulation

In the first step, the characteristic equation of the continuous system is derived. In the second, the rod is modelled as a uniform oscillator consisting of n equal masses and springs. The main purpose of the work was to study the dependence of the convergence of the uniform oscillator model eigencharacteristics towards those obtained from the cont...

The is carrying a tip mass at the free end. On the basis of Euler Bernouilli beam theory, the exact frequency equation of the system is derived. Then, the frequency equation is solved numerically for some selected values of the system parameters and the corresponding mode shapes are obtained

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The structural design sensitivity problem of a discrete mechanical system with a single viscous damper is considered. The sensitivity of the eigenvalues and the eigenvectors with respect to the damping parameter is evaluated. Exact analytical expressions for the eigenvalues and the eigenvectors of the modified system are obtained in addition to app...

The aim of this paper is to investigate the flexural vibrations of an axially moving robotic arm sliding through a prismatic joint while the joint is undergoing both vertical translation and rotary motion. Considering not only Euler-Bernoulli beam assumption but also the effects of gravitation, rotary inertia and axial force, the equations of motio...

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli—Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is g...

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This note is concerned with a slightly damped uniform n-mass oscillator where two cases are considered regarding the damping: only one mass is damped, all masses are damped simultaneously. For both cases, approximate analytical expressions are established for the eigenvalues which yield quite accurate approximations to the exact eigenvalues.

This paper deals with the derivation of the frequency equation of a special combined dynamic system. It consists of a clamped-free Bernoulli-Euler beam with a tip mass where a spring-mass system is attached to it. The derivation of the frequency equation is essentially carried out by means of the Lagrange multipliers method. Frequency equations of...

Two primary systems are considered: An n-mass oscillator and a one-dimensional continuous structure the displacements of which are discretized by its first n eigenfunctions. If to each of these systems the same spring-mass system is attached for example at their free ends, it may be expected intuitively that the frequency equations of the resulting...

The present study addresses essentially to the investigation of the bending eigenfrequencies of a cantilevered Bernoulli-Euler beam with a tip mass which has an in-span support. First, an exact frequency equation is established via a boundary value formulation and then an approximate frequency equation is derived using the Lagrange's multipliers me...

The vibrations of a linear discrete mechanical system of n degrees of freedom are governed in physical space by a matrix differential equation of nth order. This means, in general, the solution of an eigenvalue problem of the dimension n. for n ≥3, the eigenvalue problems can generally be solved only numerically, by means of a computer. Only in spe...

The initial energy of a conservative mechanical system is distributed between its vibratory modes. It is a known fact that if the system is displaced according to a certain mode shape and released with zero velocity then it vibrates only in that mode. In this study it is investigated whether it is possible, through appropriate selection of initial...

Optimal positioning of dampers is an important aspect regarding the damping of vibrations of multibody systems. This study deals with the problem of finding the optimal damping constants and the optimal positions of viscous dampers for general linear mechanical systems with f degrees-of-freedom on the basis of an energy criterion. A program package...

We discuss small oscillations of an elastic beam clamped radially to the interior of a rotating ring. Using a Ritz Galerkin procedure, the model equation is reduced to a nonlinear ordinary differential equation of second order

An important aspect of the passive and/or active damping of vibrations of multi-body systems is the optimal positioning of the dampers, actuators and sensors. This study is concerned with the problem of finding the optimal positioning of a viscous damper for a linear conservative mechanical system on the basis of an energy criterion. The results ob...