Luiz Fernando Soares Camargo Mestre em Engenharia Mecânica pela UFSC
curso de MECANISMOS
Universidade Federal de Santa Maria Departamento de Fabricação e Projeto de Màquinas -1995-
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PREFÁCIO
Este trabalho ter por objetivo principal reunir, em um único texto, todo o conteúdo da disciplina de MECANISMOS A do Curso de Engenharia Mecânica da Universidade Federal de Santa Maria. Para que tal objetivo fosse alcançado, buscou-se na literatura clássica os conteúdos condizentes com o programa didático da referida disciplina. Ele deve servir como um texto de apoio aos alunos, nunca devendo substituir os livros editados por autores de renome. A literatura consultada se encontra listada no final deste polígrafo. Como é um primeiro trabalho, gostaríamos de receber de nossos alunos sugestões que nos permita melhorar o conteúdo deste polígrafo.
Prof. Luiz Fernando S. Camargo, M.Sc. Engenheiro Mecânico
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ÍNDICE
ÍNDICE
Prefácio Capítulo 1. INTRODUÇÃO ................................................................................. 1.1. Breve história da cinemática cinemática ............................................................ Capítulo 2. CONCEITOS RELATIVOS AO ESTUDO DOS MECANISMOS ................... 2.1. Ciência dos mecanismos mecanismos .................................................................. 2.2. Máquina ................................................................. ....................................................................................... ...................... 2.3. Mecanismo ................................................................................... 2.4. Classificação Classificação dos mecanismos mecanismos ......................................................... 2.5. Corpo rígido ................................................................................. 2.6. Movimento de um corpo rígido ........................................................ 2.7. Graus de liberdade liberdade ........................................................................ 2.8. Pares cinemáticos ......................................................................... 2.9. Ponto morto de um mecanismo mecanismo ....................................................... 2.10. Inversão ...................................................................................... Capítulo 3. MECANISMOS CARACTERÍSTICOS .................................................. 3.1. Mecanimsos de quatro barras .......................................................... 3.2. Sistema biela-manivela biela-manivela ................................................................... 3.3. Garfo escocês .............................................................................. 3.4. Mecanismos Mecanismos de retorno rápido ........................................................ 3.5. Junta de Oldham ........................................................................... 3.6. Mecanismos Mecanismos geradores geradores de reta ....................................................... 3.7. Pantógrafo ..................................................................... .................................................................................. ............. 3.8. Roda de Geneva ......................................................................... 3.9. Juntas Universais ......................................................................... Capítulo 4. ANÁLISE CINEMÁTICA DOS MECANISMOS COM MOVIMENTO PLANO . 4.1. Introdução .................................................................................. 4.2. Pontos coincide
01 01 04 04 05 05 05 07 08 10 11 16 17 18 18 19 20 21 23 23 24 25 25 31 31 31
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ÍNDICE
4.5. 4.6. 4.7. 4.8.
Movimento relativo ....................................................................... Centros instantâneos instantâneos de rotação ..................................................... Teorema de Kennedy ..................................................................... Métodos de determinação determinação da velocidades velocidades em mecanismos .................. 4.8.1. Polígono de velocidade velocidade ...................................................... 4.8.2. Equações Equações vetoriais na forma complexa ................................ 4.8.3. Imagem da velocidade velocidade ...................................................... 4.8.4. Método da linha de centros .............................................. 4.8.5. Método barra por barra ..................................................... 4.8.6. Análise da velocidade velocidade pelas componentes componentes ........................... 4.9. Mecanismo Mecanismo com contato direto ....................................................... 4.10. Relação de velocidades angulares angulares ................................................... 4.11. Aceleração Aceleração relativa de partículas em mecanismos mecanismos ............................. 4.11.1. Aceleração relativa de partículas em uma mesma peça ........... ..... ........ 4.12. Caso geral de aceleração ................................................................ Capítulo 5. SINTESE DE MECANISMOS MECANISMOS ARTICULADOS ..................................... 5.1. Método de Rosenauer Rosenauer .................................................................... 5.2. Método de Freudenstein Freudenstein ............................................................... Capítulo 6. CAME .......................................................................................... 6.1. Introdução ................................................................................... 6.2. Classificação Classificação das cames e seguidores seguidores .............................................. 6.3. Geometria da came radial radial ................................................................ 6.4. Diagramas de deslocamento deslocamento .......................................................... 6.5. Desenvolvimento Desenvolvimento gráfico dos perfis da came ..................................... 6.6. Ângulo de pressão ........................................................................ 6.7. Raio de curvatura mínimo .............................................................. Apêndice A. NOTAÇÃO VETORIAL .................................................................. EXERCÍCIOS PROPOSTOS ............................................................................... REFERÊNCIAS BIBLIOGRÁFICAS .......................................................................
37 38 40 41 42 43 44 45 47 48 49 51 52 53 54 59 59 62 65 65 65 66 68 73 75 78 80 85 106
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INTRODUÇÃO
CAPÍTULO
1 INTRODUÇÃO
1.1 - BREVE HISTÓRIA DA CINEMÁTICA: Máquinas e Mecanismos tem sido inventados por pessoas desde o início da história da humanidade. Os antigos Egípcios inventaram máquinas primitivas para construírem as pirâmides e outros monumentos. Embora a roda e a polia não serem conhecidas em todo o reino Egípcio, eles utilizavam a alavanca, o princípio do plano inclinado e provavelmente usavam rolos de madeira. A origem da roda e do eixo não é conhecida. Mas provavelmente as primeiras tenham sido utilizadas na Mesopotâmia em torno de 3.000 a 4.000 anos A.C. Uma quantidade muito grande de esforços foram gastos na tentativa de inventar melhores mecanismos para serem utilizados nos relógios. E muitos inventos de máquinas primeiramente foram direcionado para as aplicações militares (catapulta, aparatos de escalar muralhas, etc..). Inclusive o termo Engenharia Civil foi criado para diferenciar as aplicações com finalidade civis das militares. A Engenharia Mecânica só começou na revolução industrial, quando surgiu a necessidade de projetar(inventar) máquinas que requeriam cada vez mais soluções complicadas e sofisticadas para o controle dos movimentos. James Watt (1736-1819) merece o título do primeiro estudioso da cinemática, pela construção de um mecanismo gerador de retas (ver fig. 3.8), utilizado para guiar um pistão em um curso muito longo, de um nova máquina à vapor. Watt foi o primeiro a reconhecer o valor do movimento em sistemas articulados nos mecanismos de quatro barras. Oliver Evans (1755-1819) foi o primeiro inventor Americano, tendo desenvolvido sistemas articulados para geração de linhas retas para máquinas à vapor. Euler (1707-1783) foi um contemporâneo da Watt, apesar de aparentemente nunca terem se encontrado. Euler apresentou um tratamento analítico para mecanismos na Mechanica sive Motus Scienta Analytice Exposita (1736-1742), que incluía o conceito que o movimento plano é a composição de duas componentes independentes, isto
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INTRODUÇÃO
d'Alembert e Kant, Kant, também propuseram similar idéia. Essa é a origem da nossa divisão em cinemática e dinâmica. No princípio de 1800, L'Ecole Polytechnic de Paris, era a responsável pelos mais brilhantes pesquisadores da época. Lagrange e Fourier são alguns destes pesquisadores. Um de seus fundadores é Gaspard Monge (1746-1818) inventor da Geometria Descritiva. Monge criou um curso de elementos de máquinas e dedicou-se a tarefa de classificar todos os mecanismos e máquinas conhecidas pelos homens na época. Seu colega, Hachett, Hachett, completou este trabalho em 1806, e, publicou em 1811, o que provavelmente tenha sido o primeiro texto sobre mecanismo. Andre Marie Ampere (1775-1836), também pesquisador da escola politécnica, realizou a tarefa de classificar "todos os conhecimentos humanos" em seu Essai sur la Philosophie des Science, usando pela primeira vez o termo cinematique. cinematique. Ele descreve o estudo do movimento sem levar em consideração o efeito das forças e sugere que "a ciência deve incluir todos os pensamentos a respeito do estudo do movimento em suas diferentes parte, independentemente das forças que as produzem". Este termo mais tarde em inglês tornou-se "Kinematics" e em alemão "Kinematik". Robert Willis (1800-1875) escreveu o texto Principles of Mechanism em 1841, enquanto professor da University of Cambridge, Inglaterra. Ele tentou sistematizar os problemas de sintese de mecanismos, enumerando cinco caminhos de obtenção de movimento relativo entre a entrada e a saída de uma ligação: contato por rolamento, contato por escorregamento, contato por articulação, conecção por envoltório (correia), conecção por cabo ou corrente. Franz Reuleaux (1829-1905) publicou Theoretische Kinematik em 1875. Muitas de suas idéias são ainda correntemente utilizadas. Alexander Kennedy (1847-1928) traduziu Reuleaux para o inglês em 1876. Este texto tornou-se o início da moderna cinemática e ainda é publicado. Ele nos proporcionou os conceitos de pares cinemático (juntas), cuja forma e interação definem o tipo de movimento que é transmitido entre os elementos de um mecanismo. Reuleaux classificou os seis componente mecânicos básicos para mecanismos: parafusos, barras, rodas, cames, catracas e órgãos de tração-compressão. Ele também definiu par cinemático inferior e superior. Reuleaux é considerado, normalmente, o Pai da Cinemática Moderna e é responsável pela notação simbólica, genericamente usada em todos os textos modernos de cinemática.
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INTRODUÇÃO
1940, quando A.E.R. DeJonge escreve What Is Wrong with "Kinematics" e "Mechanisms", o qual chamou a atenção da Europa sobre a educação em Engenharia Mecânica no Estados Unidos. Desde, então, muitos trabalhos tem sido desenvolvidos, especialmente sobre os problemas de sintese, por pesquisadores Americanos e Europeus, por autores como: J. Denavit, A. Erdman, F. Freundentein, A.S. Hall, R. Hartenberg, R. Kaufman, B. Roth, G. Sandor e A. Soni, Soni, (todos dos Estados Unidos) e K.Hain (Alemanha). Muitos destes pesquisadores tem utilizado o computador para solucionar rapidamente problemas de difícil tratamento. Mas, tanto na área da análise como na da sintese, utilizam, ainda, muitas das teoria de seus antecessores.
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CONCEITOS
CAPÍTULO
2
CONCEITOS RELATIVOS AO ESTUDO DOS MECANISMOS 2.1 - CIÊNCIA DOS MECANISMOS: Ao projetar uma máquina ou quando se analisa outra já construída, se apresentam dois problemas distintos, mas altamente relacionados entre si: primeiro, as dimensões dos elementos da máquina e suas ligações devem permitir que cada um destes elementos tenha seu próprio e determinado movimento; segundo, os elementos devem ser capazes de resistir aos esforços que atuam sobre eles. Como a natureza dos movimentos não depende das dimensões das máquinas nem da resistência da partes móveis, pode-se estudar força e movimento em separado, por isto, a ciência dos mecanismos é dividido em dois ramos: - Cinemática (a ciência do mecanismos puros): puros) : se ocupa do movimento (deslocamento, velocidade e aceleração) dos elementos das máquinas, de suas formas e da maneira como eles são guiados, sem ter em conta as suas forças. - Construção dos mecanismos: mecanismos : compreende o cálculo das forças que atuam sobre cada parte da máquina, a seleção dos materiais mais adequados por sua resistência, duração e demais propriedades físicas necessárias para suportar os esforços, e, estuda
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CONCEITOS
2.2 - MÁQUINA: Uma máquina é uma combinação de corpos resistentes, dispostos de tal forma que as forças da natureza atuam para produzir algum efeito ou trabalho útil mediante certos e determinados movimentos. Em geral pode-se dizer que uma máquina é um conjunto de peças interpostas entre a fonte de energia e o trabalho que se deseja realizar, com fim de adaptar o primeiro ao segundo. Cada uma das peças de uma máquina se move ou serve de guia para outra peça móvel. As máquinas podem ser divididas em: - MÁQUINAS MOTRIZES: São as máquinas que recebem energia natural ( combustível, eólica, eletricidade, etc.. ) e a transformam em movimento ou energia cinética de caracteristicas conhecidas. Exs: Motor de combustão, turbinas, cataventos, etc.. - MÁQUINAS OPERATRIZES: OPERATRIZES: São as máquinas que recebem energia cinéticas das máquinas motrizes (motores elétricos, motor de combustão, turbinas, etc..) e a aplicam em qualquer trabalho útil. Exs: Máquinas-Ferramentas, prensas, etc..
2.3 - MECANISMO: Recebe este nome toda a combinação de corpos rígidos dispostos de forma que o movimento de um obrigue que se movam todos os demais, de acordo com leis que dependem da natureza da combinação. Pode-se dizer, então, que mecanismo é um conjunto de peças interligadas capazes de transmitir, modificar ou ambos simultaneamente, um determinado movimento. Se o mecanismo produzir força de considerável magnitude, é considerado uma
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CONCEITOS
A classificação mais utilizada é do estudioso alemão de cinemática Franz Reuleaux, que dividiu todos os mecanismos conhecidos em seis tipos básicos, que são: - mecanismos de parafuso - mecanismos de barras - mecanismos de roda, incluindo engrenagens - mecanismos de cames - mecanismos de catraca - órgãos de tração-compressão, ou seja, partes contendo rigidez em um único sentido como correias, correntes e circuitos hidráulicos. A partir destas partes é que os mecanismos e máquinas são construidos. Pode-se, ainda, separar os mecanismos em dois grupos no que concerne aos movimentos transmitidos: os que transmitem movimentos uniformes, que é o caso das engrenagens circulares, correntes, correias e similares e os que transmitem movimento nãouniformes, que é o caso de engrenagens não-circulares, cames, catracas e mecanismos articulados planos e espaciais. Os mecanismos podem, também, ser classificados em planos, esféricos e espaciais (Fig 2.1).
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CONCEITOS
Em um mecanismo plano, todas as partes movem-se em planos que permanecem paralelos entre si, ocorrendo o mesmo com os eixos de todos os pivôs rotativos. No mecanismo esférico, os eixos dos pivôs se interceptam em um ponto. Os movimentos das partes ocorrem sobre as superfícies de esferas concêntricas cujo centro é o ponto de intersecção dos eixos dos pivôs. Nos mecanismos espaciais, algumas das partes ou mesmo todas elas podem se deslocar livremente na três dimensões.
2.5 - CORPO RÍGIDO: Na análise cinemática, é costume estabelecer uma premissa bastante importante no estudo do movimento dos corpos. Devemos admitir que, quaisquer duas partículas em um corpo permanece sempre à mesma distância, independentemente da magnitude das forças que possam atuar para mudar seu espaçamento. Isto é equivalente a dizer que o corpo é incapaz de experimentar qualquer deformação, ou que ele é absolutamente rígido.
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CONCEITOS
rígido" é no corpo movimenta-se da posição 1 para 2, ele também se movimentará. O termo " rígido" sentido matemático, pois sempre haverá uma pequena deformação em todos os corpos, mesmo os mais rígidos.
2.6 - MOVIMENTO DE UM CORPO RÍGIDO: Tratando-se do estudo dos mecanismos, é necessário definir os vários tipos de movimentos produzidos por estes. Consideremos o mecanismo de biela e dupla manivela da figura 2.3. Este é composto das manivelas 2 e 4, das conectoras 3 e 5 e da corrediça 6. Todos esses membros podem ser tratados como corpos rígidos. Se observarmos a corrediça 6 veremos que durante a operação do mecanismo, todas as partículas de 6 tem exatamente o mesmo movimento. A isto chama-se TRANSLAÇÃO. A translação de um corpo rígido ocorre quando cada partícula do corpo tem exatamente o mesmo movimento que qualquer outra partícula de que ele é composto. Consideremos agora a conectora 3. Os pontos A e B sobre ela movem-se em círculos que tem os mesmos raios. A medida que a conectora se move, ela é obrigada a
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A conectora 5 tem um movimento que consiste em rotação e translação, e é mais complexo. A trajetória de B 5 é um círculo, mas a de C 5 é uma linha reta. Cada partícula da barra entre os pontos mencionados descreve uma trajetória diferente durante seu movimento. Diz-se, então, que um corpo animado de rotação-translação, tem um movimento plano. plano.
Figura 2.3 - Mecanismo de biela e dupla manivela. Movimento é o nome chave que descreve um mecanismo. Para haver movimento, um elemento do mecanismo deve ser o condutor (elemento motor) e outro elemento deve ser o seguidor (elemento movido). O movimento é transmitido do condutor para o seguidor por uma das seguintes maneiras: Movimento de rolamento Transmissão direta MOVIMENTO
a
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2.7 - GRAUS DE LIBERDADE: O termo graus de liberdade, na descrição dos sistemas mecânicos exprime o número de dimensões, ou coordenadas necessárias para especificar a posição de todas as partes do sistema: isto é, se forem necessárias três coordenadas para descrever a posição de um sistema mecânico, diz-se que o sistema tem três graus de liberdade. Como exemplo de graus de liberdade, consideremos o mecanismo bielamanivela. Se o ângulo da manivela, θ, for especificado, a posição de cada parte pode ser achada. Assim, apenas uma coordenada, θ, neste caso, é necessária para definir a posição de todos os elementos do mecanismo. Desta forma, diz-se que o mecanismo tem um grau de liberdade (Fig. 2.5).
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CONCEITOS
Figura 2.6 - Sistema com seis graus de liberdade. Um sistema flexível, f lexível, tal como uma correia ou cabo, tem um número infinitos de graus de liberdade porque há um número infinito de partículas, cada uma podendo ter um movimento diferente.
2.8 - PARES CINEMÁTICOS:
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Figura 2.7 - Pares cinemáticos inferiores e superiores.
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CONCEITOS
A distinção entre os tipos de par inferior e superior baseia-se no número de graus de liberdade ("f") do par. O movimento no espaço de um corpo rígido, como já foi visto, possui seis graus de liberdade, ou seja, para especificar o deslocamento de um corpo rígido são necessárias seis variáveis, isto é, três deslocamentos lineares nas direções dos eixos lineares e três rotações em torno destes eixos. Cada uma destas variáveis de movimento está associada com um grau de liberdade, sendo o movimento de translação ou rotação. Para que se possa formar uma idéia clara disso tudo, vai-se examinar em detalhes os movimentos possíveis da barra (2) em relação a barra (1), na figura 2.8, observando a natureza da ligação:
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CONCEITOS
movimento HELICOIDAL relativo das duas barras. O símbolo para o par helicoidal é S L e o grau de liberdade é f=1.
4. Se for permitido rotação e translação ao longo do eixo w, duas variáveis independentes, uma para a translação e outra para a rotação, serão necessárias para CILÍNDRICA; é o caso descrever o movimento entre as barras. Tal ligação é denominada de CILÍNDRICA; de um eixo e o seu mancal de deslizamento em que não há restrição axial. A sua notação simbólica é C e o grau de liberdade é f=2.
5. Supondo z1 = 0, ou seja, barra (2) diretamente sobre o plano x-y de (1). Então detectar-se-iam três movimentos possíveis -duas translações e uma rotação em torno de w. Tal conexão PLANA (par plano), que ocorre raramente, tem o símbolo F e o seu grau de liberdade é f=3.
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CONCEITOS
Fgura 2.10 - Ligações cilíndrica, esférica e plana.
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CONCEITOS
sendo os pares superiores não seguem uma classificação simples como nos pares inferiores nem são descritos convenientemente por meio de símbolos. Ligações por pares superiores podem ocasionalmente ser substituídas por uma combinação de pares inferiores. Considere a construção da figura 2.11.
Figura 2.11 - Substituição de um par superior por dois pares inferiores.
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CONCEITOS
2.10 - INVERSÃO: Utilizando novamente o mecanismo biela-manvivela como exemplo prático (Fig. 2.13.a), pode-se ver que este sistema mantém fixo o elemento (eixo) 1. Com isto, a manivela 2 tem um movimento de rotação e o cursor 4 tem um movimento linear alternado. Agora se mudarmos e fixarmos a manivela 2, o mecanismo terá na biela 3 uma rotação contínua, mas com velocidade angular variável e na barra 1 haverá uma velocidade angular constante, sendo a 1 a condutora. Este novo mecanismo pode ser utilizado em máquinas-ferramentas como mecanismo de retorno rápido (fig. 2.13.b). Mudando novamente, e, fixando-se a biela 3, ter-seá um mecanismo com outras características que é utilizado em máquinas a vapor e em bombas Fig. 2.13.c). E, finalmente, fixando-se o cursor 4, tem-se um novo mecanismo que é
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3 - MECANISMOS CARACTERÍSTICOS
CAPÍTULO 3
MECANISMOS CARACTERÍSTICOS
-1-
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3 - MECANISMOS CARACTERÍSTICOS
Na fig.3.1b as peças 2 e 4 têm o mesmo comprimento e, portanto, sempre permanecem paralelas: tendo, assim, movimento de rotação. A fig.3.1c mostra outro arranjo no qual a peça motriz e a conduzida giram continuamente. Esta forma de quadrilátero articulado é base para o mecanismo de manivela dupla e corrediça, que será abordada no item relativo a
-2-
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-3-
3 - MECANISMOS CARACTERÍSTICOS
A partir da posição angular da manivela pode-se calcular a posição, velocidade e aceleração do cursor, pelas seguintes equações: θ
R
2 2
θ
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3 - MECANISMOS CARACTERÍSTICOS
3.4 - MECANISMO DE RETORNO RÁPIDO: Estes mecanismos são utilizados em Máquinas-Ferramentas para dar lhes um curso de corte lento e um curso de retorno rápido para uma velocidade angular constante da
-4-
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3 - MECANISMOS CARACTERÍSTICOS
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ANÁLISE CINEMÁTICA
2
2
r
AP =
d x P 2
+ j.
A B =
2
dt
dt 2
2
r
d y P
d x B 2
dt
+ j.
d y B 2
dt
2 2 d 2 x′ d 2 y′ d y ′ d y ′ cos cos . θ .sen θ + j. .sen θ + . θ AP 2 = 2 2 2 2 dt dt dt dt r
AC = - 2. r
′ d θ ′ d θ d x d y . .sen θ - 2. . .cos θ dt dt dt dt
′ d θ ′ d θ d y d x . .cos θ - 2. . .sen θ .j dt dt dt dt
+ 2.
Na equação "f", A B é a aceleração absoluta do ponto B, o qual é fixado ao corpo 2 e está momentaneamente coincidente com P. O segundo termo, A P2 é a aceleração de P em relação ao corpo 2. O terceiro termo é a aceleração de Coriolis A C de grandeza:
AC = 2.V P 2.
A velocidade relativa V P2 é a velocidade de P em relação ao corpo 2. E w é a velocidade angular do corpo 2. A aceleração de Coriolis tem como direção, a direção da velocidade relativa do ponto P em relação ao corpo, girada sobre a origem de 90 no sentido da velocidade angular do corpo 2.