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.

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.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-12T10:47:41Z No. of bitstreams: 2 Dissertação - Airton Modesto da Cunha - 2018.pdf: 36005906 bytes, checksum: 5f095bd3c548d6f6e958d83f126908ef (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-12T10:50:15Z (GMT) No. of bitstreams: 2 Dissertação - Airton Modesto da Cunha - 2018.pdf: 36005906 bytes, checksum: 5f095bd3c548d6f6e958d83f126908ef (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-12T10:50:15Z (GMT). No. of bitstreams: 2 Dissertação - Airton Modesto da Cunha - 2018.pdf: 36005906 bytes, checksum: 5f095bd3c548d6f6e958d83f126908ef (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) 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.

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.

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.