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Coupled spring equationsFay, TH, Graham, SD January 2003 (has links)
Coupled spring equations for modelling the motion of two springs with
weights attached, hung in series from the ceiling are described. For the linear
model using Hooke’s Law, the motion of each weight is described by a fourthorder
linear differential equation. A nonlinear model is also described and
damping and external forcing are considered. The model has many features that
permit the meaningful introduction of many concepts including: accuracy of
numerical algorithms, dependence on parameters and initial conditions, phase
and synchronization, periodicity, beats, linear and nonlinear resonance, limit
cycles, harmonic and subharmonic solutions. These solutions produce a wide
variety of interesting motions and the model is suitable for study as a computer
laboratory project in a beginning course on differential equations or as an
individual or a small-groupundergraduate research project.
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Coupled spring equationsFay, TH 01 January 2003 (has links)
Summary
Coupled spring equations for modelling the motion of two springs with
weights attached, hung in series from the ceiling are described. For the linear
model using Hooke’s Law, the motion of each weight is described by a fourthorder
linear differential equation. A nonlinear model is also described and
damping and external forcing are considered. The model has many features that
permit the meaningful introduction of many concepts including: accuracy of
numerical algorithms, dependence on parameters and initial conditions, phase
and synchronization, periodicity, beats, linear and nonlinear resonance, limit
cycles, harmonic and subharmonic solutions. These solutions produce a wide
variety of interesting motions and the model is suitable for study as a computer
laboratory project in a beginning course on differential equations or as an
individual or a small-groupundergraduate research project.
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Modelo pedagógico de resistores elétricos para descrever fraturas em sistemas físicos / Pedagogical model of electrical resistors describe fractures in physical systemsCunha, Airton Modesto da 09 August 2018 (has links)
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Previous issue date: 2018-08-09 / The present dissertation is the result of the educational, “Teaching model of electrical
resistors to describe fractures in physical systems”, product worked in a class of the third
year of high school of the State School Prof. Marcolino de Barros of the municipality of Patos
de Minas-MG. In which, we present a simple pedagogical model with the purpose of studying
the complex process of ruptures in physical systems. For this, we developed an experimental
model together with its theoretical counterpart where we consider a system of electric
resistors network subject to a potential difference and analyze the current that passes
through the system as a function of the random removal of their resistive components. We
use a circuit board, voltage source stabilized at a pre-established potential difference, and a
resistor network with a few ohms units each. After the proposed experiments are executed,
we obtained appropriate experimental data and from these graphs are generated with which
the dependence of the current through the circuit with the interruption of the joints is shown.
Taking into account that this system behaves as an ohmic system, the model is used to study
several relevant physical concepts. As is the case of an elastic solid subject to Hooke’s law-
like deformations, this is then a very broad model. / A presente dissertação é resultado de nosso produto educacional, “Modelo pedagógico de
resistores elétricos para descrever fraturas em sistemas físicos”, trabalhado em uma turma
do terceiro ano do ensino médio da Escola Estadual Prof. Marcolino de Barros do município
de Patos de Minas-MG. No qual, apresentamos um modelo pedagógico simples com a
finalidade de estudar o complexo processo de rupturas em sistemas físicos. Para tanto,
desenvolvemos um modelo experimental munido de sua contraparte teórica onde
consideramos um sistema de resistores elétricos sujeito a uma diferença de potencial e
analisamos a corrente que passa através do sistema em função da remoção aleatória doscomponentes resistivos. Usamos uma placa de circuitos, fonte de tensão estabilizada em uma
diferença de potencial pré -estabelecida e uma malha de resistores com algumas unidades de
ohms cada. Realizadas as experiências propostas obtivemos apropriados dados
experimentais e a partir destes foram gerados gráficos com os quais são mostrados a
dependência da corrente através do circuito com a interrupção das junções. Levando em
consideração que este sistema se comporta como um sistema ôhmico, o modelo é usado para
estudar vários conceitos físicos relevantes. Como é o caso de um solido elástico sujeito a
deformações do tipo lei de Hooke, sendo este então um modelo de caráter bem amplo.
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Žemės plutos horizantaliųjų judesių Ignalinos atominės elektrinės rajone tyrimas geodeziniais metodais / Research of the eartch’s crust horizontal movements in the Ignalina nuclear power plant region by geodetic methodsStanionis, Arminas 17 January 2006 (has links)
A method prepared and created algorithm for computations and evaluation of relation between Earth’s crust horizontal deformations and variations of tectonic stresses. Hooke’s law was used for relation description. Method of evaluation of Earth’s crust horizontal deformations was improved as well as modelling method based on observation data. Elements of deformation tensors were evaluated by applying finite element method. New data of Earth’s crust horizontal movements and their geodynamic interpretation at the Ignalina Nuclear Power Plant region were obtained.
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Linear FEM Analysis of a Commercial Elastomer for Machine FoundationsJakel, Roland 20 June 2024 (has links)
The presentation describes partial results from an industrial project in which a transmission
test bench from ZF Prüfsysteme was decoupled from the foundation in terms of vibration using
commercial PU foam material ('Sylomer' SR220 from Getzner).
The presentation shows how this material was extensively tested by the manufacturer and
characterized in numerous data sheets in order to enable the engineer to perform a simple,
operating point-dependent dynamic design using clear diagrams and the classic equation for a
single-mass oscillator.
However, if a more complex analysis is to be carried out using the finite element method, e.g. to
determine all 6 rigid body shapes and natural frequencies of the dynamically decoupled test rig and
not just the purely vertical natural shape/frequency, the apparent elasticity and shear moduli
specified in the manufacturer's data sheets must be converted into true values for the
corresponding operating points, which can then be used in a linear FE calculation.
For this purpose, FEM models of the elastomer test specimens are generated for different shape
factors. The conversion of the apparent to the true characteristic values is then carried out using
the optimizer available in the PTC software “Creo Simulate” in a so-called 'feasibility study' and
the results are discussed.
It can be seen that the true moduli of elasticity and transverse strain coefficients are only slightly or
no longer dependent on the form factor. Depending on the density of the PU foam, the transverse
strain coefficient is also significantly lower than 0.5. The true shear modulus is practically identical
to the measured shear modulus, as a pure biaxial stress and strain state occurs in the shear
specimens, in which strain restraints due to transverse strain plays no role - quite unlike in the
specimens loaded in the normal (compression) direction, in which triaxial compression stress
states occur due to transverse strain restraints.
Finally, the true material properties determined in this way are used for an exemplary modal
analysis of the entire, idealized test rig on the strip foundation using the finite element method. The
error is evaluated if the apparent modulus of elasticity and a Poisson ratio of zero is used instead,
so that a simple evaluation and error estimation of analysis results is possible in practical
applications. / Der Vortrag beschreibt Teilergebnisse aus einem industriellen Projekt, in dem ein
Getriebeprüfstand der ZF Prüfsysteme schwingungstechnisch über kommerzielles PU-
Schaummaterial („Sylomer“ SR220 der Firma Getzner) vom Fundament abgekoppelt wurde.
Der Vortrag stellt dar, wie dieser Werkstoff vom Hersteller umfangreich getestet und in zahlreichen
Datenblättern charakterisiert wurde, um dem Ingenieur schließlich eine einfache,
betriebspunktabhängige dynamische Auslegung mittels übersichtlicher Diagramme und der
klassischen Gleichung für einen Einmassenschwinger zu ermöglichen.
Soll jedoch eine aufwendigere Analyse mittels der Methode der Finiten Elemente durchgeführt
werden, z.B. um alle 6 Starrkörperformen und Eigenfrequenzen des dynamisch abgekoppelten
Prüfstandes zu bestimmen und nicht nur die rein vertikale Eigenform/Eigenfrequenz, müssen die in
den Herstellerdatenblättern angegeben formzahlabhängigen scheinbaren Elastizitäts- und
Schubmoduli in wahre Werte für die entsprechenden Betriebspunkte umgerechnet werden, die
dann in einer linearen FE-Rechnung verwendet werden können.
Dafür werden FEM-Modelle der Elastomer-Probekörper für verschiedene Formfaktoren erzeugt.
Die Umrechnung der scheinbaren in die wahren Kennwerte wird anschließend mittels des in der
PTC-Software „Creo Simulate“ vorhandenen Optimierers in einer sogenannten
„Machbarkeitsstudie“ durchgeführt und die Ergebnisse diskutiert.
Es zeigt sich, dass die wahren E-Moduli und Querdehnzahlen nur noch gering bzw. nicht mehr
vom Formfaktor abhängen. Je nach Dichte des PU-Schaums stellt sich auch eine Querdehnzahl
von deutlich kleiner als 0,5 ein. Der wahre Schubmodul ist praktisch identisch wie der gemessene
Schubmodul, da in den Schubproben ein reiner zweiachsiger Spannungs- und Dehnungszustand
auftritt, in dem Dehnungsbehinderung durch Querdehnung keine Rolle spielt – ganz anders als in
den in Normalenrichtung (Druck-) belasteten Proben, in denen durch die Querdehnungs-
behinderung dreiachsige Spannungszustände auftreten.
Schließlich werden die so bestimmten wahren Werkstoffkennwerte für eine exemplarische
Modalanalyse des gesamten, idealisierten Prüfstandes auf den Streifenfundamenten mittels der
Methode der Finiten Elemente verwendet. Der Fehler wird bewertet, wenn man stattdessen den
scheinbaren E-Modul und eine Querdehnzahl von Null verwendet, so dass in der
Anwendungspraxis eine einfache Bewertung und Fehlerabschätzung von Analyseergebnissen
möglich ist.
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