THE EFFECT OF ROTATION UPON THE NATURAL FREQUENCIES OF A MASS–SPRING SYSTEM

2002 ◽  
Vol 250 (5) ◽  
pp. 849-875
Author(s):  
N.S. CLARKE
Keyword(s):  
Natural Frequencies ◽  
Spring System ◽  
Mass Spring

Vibration of Heavy String Tethered to Mass–Spring System

2017 ◽  
Vol 17 (04) ◽  
pp. 1771002 ◽  
Author(s):  
C. Y. Wang ◽  
C. M. Wang ◽  
R. Freund
Keyword(s):  
Natural Frequencies ◽  
Equation Of Motion ◽  
Gantry Crane ◽  
Vibration Absorber ◽  
Dynamic Vibration Absorber ◽  
Tower Crane ◽  
Spring System ◽  
Mass Spring ◽  
First Time ◽  
Top Mass

This paper is concerned with the vibration of a heavy string tethered to a mass–spring system which is a model for tower crane or gantry crane. The governing equation of motion of the aforementioned problem is solved analytically and exact closed-form characteristic solutions for the natural frequencies are presented. These exact solutions would serve as benchmarks for models which may be more complex than the basic vibration problem considered. Moreover, we investigate the effect of the string-mass pendulum as a flexible dynamic vibration absorber. Such a problem is important in vibration control, and is investigated analytically for the first time. It is found that the effect of the bottom mass is to decrease the amplitude of the string in general. The effect of increased forcing frequency increases the number of nodal points on the string. In most cases, the amplitude of the top mass is reduced.

On Transmission Zeros of Mass-Dashpot-Spring Systems

1999 ◽  
Vol 121 (2) ◽  
pp. 179-183 ◽  
Author(s):  
Jong-Lick Lin
Keyword(s):  
Real Axis ◽  
Imaginary Axis ◽  
Natural Frequencies ◽  
Negative Real Axis ◽  
Transmission Zeros ◽  
Novel Approach ◽  
Spring System ◽  
Mass Spring ◽  
Insight Into ◽  
Interlacing Property

For a noncollocated mass-dashpot-spring system with B=CTΓ, a novel approach is proposed to gain a better insight into the fact that none of its transmission zeros lie in the open right-half of the complex plane. In addition, the transmission zeros have physical meanings and will simply be the natural frequencies of a substructure constrained in the equivalently transformed system. Moreover, it is also shown that transmission zeros interlace with poles along the imaginary axis for a mass-spring system with B=CTΓ. They also interlace with poles along the negative real axis for a mass-dashpot system with B=CTΓ. Finally, two examples are used to illustrate the interlacing property.

Simulation on Vertical Vibration of Six-Roller Cold Rolling Mill

2013 ◽  
Vol 441 ◽  
pp. 623-626
Author(s):  
Xu Ma ◽  
Xing Hui Zhu ◽  
Jian Wei Li ◽  
Jing Hong Yang ◽  
Quan Lu Yang ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Vibration Mode ◽  
Vertical Vibration ◽  
Prototype Model ◽  
Cold Rolling Mill ◽  
Spring System ◽  
Tandem Cold Mill ◽  
Mass Spring ◽  
Modal Coordinates ◽  
Cold Mill

According to the chatter phenomenon of 3rd stand of a steelworks six-roller tandem cold mill (TCM), vertical vibration of tandem cold mill was simplified to eight degrees of asymmetric freedom mass-spring system. Virtual prototype model on vertical vibration of tandem cold mill was established and natural frequencies and main vibration mode of mill were solved by using automatic dynamic analysis of mechanical systems in software ADAMS. The modal coordinates of main vibration mode of the mill were referred firstly and obtained; the maximal mode of vibration system was found and finally discovered the fifth-octave-mode chatter was the probable vibration of the mill.

Graphene-Based Resonant Sensors for Detection of Ultra-Fine Nanoparticles: Molecular Dynamics and Nonlocal Elasticity Investigations

NANO ◽  
2015 ◽  
Vol 10 (02) ◽  
pp. 1550024 ◽  
Author(s):  
S. Kamal Jalali ◽  
M. Hassan Naei ◽  
Nicola Maria Pugno
Keyword(s):  
Molecular Dynamics ◽  
Nonlocal Elasticity ◽  
Md Simulations ◽  
Small Scale ◽  
Frequency Shifts ◽  
Resonant Sensors ◽  
Embedded Atom ◽  
Metal Metal ◽  
Spring System ◽  
Mass Spring

Application of single layered graphene sheets (SLGSs) as resonant sensors in detection of ultra-fine nanoparticles (NPs) is investigated via molecular dynamics (MD) and nonlocal elasticity approaches. To take into consideration the effect of geometric nonlinearity, nonlocality and atomic interactions between SLGSs and NPs, a nonlinear nonlocal plate model carrying an attached mass-spring system is introduced and a combination of pseudo-spectral (PS) and integral quadrature (IQ) methods is proposed to numerically determine the frequency shifts caused by the attached metal NPs. In MD simulations, interactions between carbon–carbon, metal–metal and metal–carbon atoms are described by adaptive intermolecular reactive empirical bond order (AIREBO) potential, embedded atom method (EAM), and Lennard–Jones (L–J) potential, respectively. Nonlocal small-scale parameter is calibrated by matching frequency shifts obtained by nonlocal and MD simulation approaches with same vibration amplitude. The influence of nonlinearity, nonlocality and distribution of attached NPs on frequency shifts and sensitivity of the SLGS sensors are discussed in detail.

scholarly journals Reduced Mass-Weighted Proper Decomposition for Modal Analysis

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Venkata K. Yadalam ◽  
B. F. Feeny
Keyword(s):  
Modal Analysis ◽  
Mass Distribution ◽  
Mass Matrix ◽  
Linear Interpolation ◽  
Modal Identification ◽  
Distributed Parameter ◽  
Arbitrary Mass ◽  
Spring System ◽  
Mass Spring ◽  
Proper Orthogonal

A method of modal analysis by a mass-weighted proper orthogonal decomposition for multi-degree-of-freedom and distributed-parameter systems of arbitrary mass distribution is outlined. The method involves reduced-order modeling of the system mass distribution so that the discretized mass matrix dimension matches the number of sensed quantities, and hence the dimension of the response ensemble and correlation matrix. In this case, the linear interpolation of unsensed displacements is used to reduce the size of the mass matrix. The idea is applied to the modal identification of a mass-spring system and an exponential rod.

scholarly journals Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

2013 ◽  
Vol 14 (5) ◽  
pp. 1228-1251 ◽  
Author(s):  
Yan Li ◽  
I-Liang Chern ◽  
Joung-Dong Kim ◽  
Xiaolin Li
Keyword(s):  
Upper Bound ◽  
Mesh Size ◽  
Front Tracking ◽  
Time Step ◽  
Mass System ◽  
Ode System ◽  
Spring System ◽  
Fabric Surface ◽  
Mass Spring ◽  
Eigen Frequency

AbstractWe use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

scholarly journals El conocimiento semántico en la representación de problemas de ecuaciones diferenciales como modelos matemáticos. [Semantic knowledge in the representation of problems of differential equations as mathematical models]

2018 ◽  
Vol 7 (3) ◽  
pp. 31
Author(s):  
Rosa Virginia Hernández ◽  
Luis Fernando Mariño ◽  
Mawency Vergel
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Equilibrium Point ◽  
Semantic Knowledge ◽  
External Representation ◽  
Damping Force ◽  
Mathematical Problems ◽  
Spring System ◽  
Linear Differential ◽  
Mass Spring

En este artículo se presenta la caracterización del conocimiento semántico evidenciado por un grupo de estudiantes en la representación externa a problemas de ecuaciones diferenciales lineales de segundo orden como modelos matemáticos. El trabajo fue cuantitativo de tipo exploratorio y descriptivo utilizando un cuestionario en la recolección de información. El soporte teórico que dio sentido al estudio fue el modelo de dos etapas propuesto por Mayer R. para la resolución de problemas matemáticos, el ciclo de modelación bajo la perspectiva cognitiva según Borromeo Ferri y la teoría de las representaciones de Goldin y Kaput. La investigación se centró específicamente en la fase de representación del modelo. Entre los principales hallazgos se destaca que cada participante hace su propia representación externa a conceptos como: sistema masa-resorte, peso, masa, punto de equilibrio, constante de elasticidad, punto de equilibrio, ley de Hooke, fuerza amortiguadora, fuerza externa, ley de Newton, entre otros. Se evidencian también dificultades en el tránsito del lenguaje natural al lenguaje matemático y la representación externa de cada una de los signos, símbolos o expresiones matemáticas inmersas en el problema de palabra, debido a que el resolutor tiene que construir un modelo mental de la situación real y plasmarlo en un modelo matemático. Lo anterior pone de manifiesto la importancia que tiene el conocimiento semántico en la etapa de traducción cuando se intentan resolver problemas como situaciones reales a modelar.Palabras clave: resolución de problemas, ciclos de modelación, problemas de palabra, representaciones externas, conocimiento extra matemático, modelación matemática. AbstractThis article presents the characterization of the semantic knowledge evidenced by a group of students in the external representation to problems of second order linear differential equations as mathematical models. The work was quantitative exploratory and descriptive using a questionnaire in the collection of information. The theoretical support that gave meaning to the study was the two-stage model proposed by Mayer R. for solving mathematical problems, the modeling cycle under the cognitive perspective according to Borromeo Ferri and the theory of representations of Goldin and Kaput. The research focused specifically on the representation phase of the model. Among the main findings is that each participant makes his own external representation to concepts such as: mass-spring system, weight, mass, equilibrium point, constant of elasticity, equilibrium point, Hooke's law, damping force, external force, law of Newton, among others. Difficulties are also evident in the transition from natural language to mathematical language and the external representation of each of the signs, symbols or mathematical expressions involved in the word problem, because the resolver has to construct a mental model of the real situation and translate it into a mathematical model. This demonstrates the importance of semantic knowledge in the translation stage when trying to solve problems as real situations to be modeledKeywords: problem solving, modeling cycles, word problems, external representations, extra mathematical knowledge, mathematical modeling.ResumoEste artigo apresenta a caracterização do conhecimento semântico evidenciado por um grupo de estudantes na representação externa a problemas de equações diferenciais lineares de segunda ordem como modelos matemáticos. O trabalho foi quantitativo exploratório e descritivo usando um questionário na coleta de informações. O suporte teórico que deu sentido ao estudo foi o modelo de dois estágios proposto por Mayer R. para resolver problemas matemáticos, o ciclo de modelagem sob a perspectiva cognitiva de acordo com Borromeo Ferri e a teoria das representações de Goldin e Kaput. A pesquisa focalizou especificamente a fase de representação do modelo. Entre os principais achados, cada participante faz sua própria representação externa para conceitos como: sistema de massa-mola, peso, massa, ponto de equilíbrio, constante de elasticidade, ponto de equilíbrio, lei de Hooke, força de amortecimento, força externa, lei de Newton, entre outros. As dificuldades também são evidentes na transição da linguagem natural para a linguagem matemática e a representação externa de cada um dos signos, símbolos ou expressões matemáticas envolvidas na palavra problema, porque o resolvedor tem que construir um modelo mental da situação real e traduzi-lo para um modelo matemático. Isso demonstra a importância do conhecimento semântico na fase de tradução ao tentar resolver problemas como situações reais a serem modeladas. ______________________________________________________ Palavras-chave: resolução de problemas, ciclos de modelagem, problemas de palavra, representação externa, conhecimento extra matemático, modelagem matemática

scholarly journals Mistuning Effects on the Nonlinear Friction Saturation of Flutter Vibration Amplitude

2020 ◽  
Author(s):  
Carlos Martel ◽  
Salvador Rodríguez
Keyword(s):  
Vibration Amplitude ◽  
Gas Flow ◽  
Travelling Waves ◽  
Computational Cost ◽  
Travelling Wave ◽  
Nonlinear Friction ◽  
Blade Vibration ◽  
Bladed Disk ◽  
Spring System ◽  
Mass Spring

Abstract The blade vibration level of an aerodynamically unstable rotor is a quantity of crucial importance to correctly estimate the blade fatigue life. This amplitude is the result of the balance between the energy pumped into the blades by the gas flow, and the nonlinear dissipation at the blade-disk contact interfaces. In a tuned configuration, the blade displacements can be described as a travelling wave consisting of one fundamental nodal diameter and frequency and its higher harmonics, and the problem can be reduced to the computation of a time periodic solution in just one sector. This simplification is no longer valid for a mistuned bladed disk. The resulting nonlinear vibration of the mistuned system is a combination of several travelling waves with different number of nodal diameters, coupled through mistuning. In this case, the complete bladed disk has to be considered, which requires an extremely high computational cost, and, for this reason, reduced order models (ROM) are required to analyze this situation. In this work, we use a 3 DOF/sector mass-spring system to describe the nonlinear friction saturation of the flutter vibration amplitude of a realistic mistuned bladed disk. The convergence of the solution of the mass-spring system is still quite slow because of the presence of many unstable modes with very similar growth rates. In order to speed-up the simulations a simpler asymptotic ROM is derived from the mass-spring model, which allows for much faster integration times. The simulations of the asymptotic ROM are compared with the measurements obtained in the European project FUTURE, where an aerodynamically unstable LPT rotor was tested with different intentional mistuning patterns.

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