Min-Max Solutions for the Linear Mass-Spring System

1957 ◽  
Vol 24 (1) ◽  
pp. 131-136
Author(s):  
Eugene Sevin
Keyword(s):  
Upper And Lower Bounds ◽  
Constant Force ◽  
Linear Mass ◽  
Maximum Displacement ◽  
Natural Period ◽  
Load Duration ◽  
Forcing Function ◽  
Total Impulse ◽  
Spring System ◽  
Mass Spring

Abstract Absolute upper and lower bounds have been determined for the maximum displacement of an undamped linear mass-spring system acted on by a non-negative forcing function characterized only by total impulse and duration. The upper bound is shown to result from applying the total impulse to the mass as an initial blow. The lower bound is shown to depend upon the ratio of load duration to natural period of the system, and this response results from a forcing function consisting of an initial and final impulse and an intermediate constant force. In the latter case, for sufficiently short durations, the forcing function reduces simply to equal initial and final impulses.

Computational Approaches to the Min-Max Response of Dynamic Systems With Incompletely Prescribed Input Functions

1967 ◽  
Vol 34 (1) ◽  
pp. 87-90 ◽  
Author(s):  
Eugene Sevin ◽  
Walter Pilkey
Keyword(s):  
Dynamic Systems ◽  
Initial Conditions ◽  
Restoring Force ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Load Duration ◽  
Analytical Technique ◽  
Single Degree ◽  
Forcing Function ◽  
Total Impulse

Linear, nonlinear, and dynamic programming formulations are developed for the solution of the min-max response of a single-degree-of-freedom dynamic system with incompletely prescribed input functions. The problem is: Given an oscillator whose equation of motion is mx¨ + g(x, x˙) = f(t), subject to stated initial conditions, and acted upon by a forcing function, f(t), which is nonnegative, and of specified finite duration and total impulse, find the particular forces which produce the least possible maximum displacement of the oscillator, and find this bounding value. Previously, Sevin developed an analytical technique for the solution which is inherently dependent upon a linear undamped form for the restoring force g(x, x˙). In the current work, an alternate statement of the problem is presented which lends itself to tractable computational formulations involving less stringent restrictions on g(x, x˙). Results obtained by dynamic and linear programming for specified forms of g(x, x˙) are given as functions of load duration.

scholarly journals Forced Response of a Centrifugal Compressor Stage Due to the Impeller–Diffuser Interaction

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
Edward J. Walton ◽  
Choon S. Tan
Keyword(s):  
Practical Implication ◽  
Forced Response ◽  
Physical Interaction ◽  
Impeller Blade ◽  
Forcing Function ◽  
Modal Coupling ◽  
Spring System ◽  
Unsteady Loading ◽  
Mass Spring ◽  
The Impact

The impact mode coupling between impeller blades and the disk backwall has on the forced response amplitude of impeller blades is assessed. The assessments focus on the forced response of two splitter blade modes to a variety of representative boundary conditions and unsteady loadings. The forcing function is the synchronous unsteady loading generated by the impeller–diffuser interaction at resonance. The results indicate that modal coupling of blade- and disk-dominant modes renders the forced response highly sensitive to small variations in airfoil and disk backwall thickness. As a complement, a reduced-order model based on the forced response of a two mass–spring system is used to elucidate the physical interaction of modal coupling. The practical implication of this finding is that a forced response issue with an impeller blade cannot be addressed adequately by stiffening the structure, such as thickening the blade or disk. Thus, appropriate measures need to be taken to avoid potential blade–disk mode couplings within the manufacturing tolerances of the part.

scholarly journals Forced Response of a Centrifugal Compressor Stage due to the Impeller-Diffuser Interaction

2015 ◽  
Author(s):  
Edward J. Walton ◽  
Choon S. Tan
Keyword(s):  
Practical Implication ◽  
Forced Response ◽  
Physical Interaction ◽  
Impeller Blade ◽  
Forcing Function ◽  
Modal Coupling ◽  
Spring System ◽  
Unsteady Loading ◽  
Mass Spring ◽  
The Impact

The impact mode coupling between impeller blades and the disk backwall has on the forced response amplitude of impeller blades is assessed. The assessments focus on the forced response of two splitter blade modes to a variety of representative boundary conditions and unsteady loadings. The forcing function is the synchronous unsteady loading generated by the impeller-diffuser interaction at resonance. The results indicate that modal coupling of blade and disk dominant modes renders the forced response highly sensitive to small variations in airfoil and disk backwall thickness. As a complement, a reduced-order model based on the forced response of a two mass-spring system is used to elucidate the physical interaction of modal coupling. The practical implication of this finding is that a forced response issue with an impeller blade cannot be addressed adequately by stiffening the structure, such as thickening the blade or disk. Thus appropriate measures need to be taken to avoid potential blade-disk mode couplings within the manufacturing tolerances of the part.

On Running a Machine through its Resonant Frequency

1956 ◽  
Vol 60 (549) ◽  
pp. 620-621 ◽  
Author(s):  
J. P. Ellington ◽  
H. McCallion
Keyword(s):  
High Speed ◽  
Exciting Force ◽  
Linear Mass ◽  
Running Speed ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Constant Rate ◽  
Summation Of Series ◽  
Spring System ◽  
Mass Spring

A solution, in terms of known integrals, is obtained for the motion from rest of a machine, idealised as an undamped linear mass-spring system, when subjected to an exciting force whose frequency varies at a constant rate.In many installations of modern high speed machinery the running speed of the machine is in excess of the resonant or natural frequency of the system, and consequently starting up or stopping the machine could result in vibrations of large amplitude. The problem of assessing the magnitude and duration of these vibrations is very complicated and has been solved analytically only for the case of a single degree of freedom system excited by an oscillating force whose frequency varies linearly with time. However, even this solution is not easy to evaluate, the integrals involved demanding either graphical construction and numerical integration or summation of series.

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

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