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IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Mecanismos
Mecanismos e Máquinas:
Grau de Liberdade
Prof. Paulo Boni
Aula II
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Conteúdo
• Definição de cadeia cinemática
• Cálculo do grau de liberdade de
um mecanismo
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Junta ou Par cinemático
Par de superfícies mantidas permanentemente em contato
de tal forma a produzir um movimento relativo bem
definido entre elas.
Pode ser:
• inferior: o contato é uma superfície
• superior: o contato é uma reta ou um ponto
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Junta ou Par cinemático
Par cinemático inferior: o contato é uma superfície
Rotativo Deslizante
Superfície 1
(externa do
pino)
Superfície 2
(interna do furo)
Superfície 1
(interna do furo)
Superfície 2
(externa do prisma)
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Junta ou Par cinemático
Par cinemático superior: o contato é um ponto ou
uma linha
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 6
Junta ou
Par
cinemático
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Mobilidade
Representa a soma dos movimentos possíveis em
cada “direção”.
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Mobilidade (Grau de Liberdade)
Representa a soma dos movimentos possíveis em
cada “direção”.
No plano: M=3 No espaço tridimensional: M=6
x
y
xy
z
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22/03/2013 9
Elo (link)
Corpo rígido com dois ou mais pares cinemáticos,
através dos quais se pode conectar outros corpos
Binário Ternário Quaternário
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Cadeia cinemática
Sistema resultante da conexão de vários elos formando um
circuito ou malha e que ainda pode produzir movimento.
Cadeia cinemática aberta Cadeia cinemática fechada
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Mecanismo de cadeia cinemática
simples
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Mecanismo de cadeia cinemática
composta
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Grafo de uma cadeia cinemática
Polígono composto por vértices e arestas, que representa a
conectividade da cadeia cinemática.
Vértices Elos - Números
Arestas Pares cinemáticos - Letras
1
2
3 4
5
d
c
b
a
1
6
e f
g
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplo 1
Bomba de petróleo
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplo 1
1-Identificação dos elos
números
1
2
3
4
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplo 1
1-Identificação dos elos
2-Identificação dos pares cinemáticos
letras
d
1
2
3
4
a
b
c
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplo 1
3
4 1
2
1-Identificação dos elos
2-Identificação dos pares cinemáticos
3-Definição dos vértices
elos
d
1
2
3
4
a
b
c
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplo 1
3
4 1
2
d
a
b
c
1-Identificação dos elos
2-Identificação dos pares cinemáticos
3-Definição dos vértices
4-Definição das arestas
juntas (pares cinemáticos)
d
1
2
3
4
a
b
c
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 19
Inversão
Fixando-se cada um dos elos tem-se um mecanismo diferente (inversão)
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 20
Inversão
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Inversão
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22
Graus de liberdade (F) de um mecanismo
é o número de parâmetros independentes que são
necessários para se definir a posição de um corpo
noespaço em qualquer instante.
Critério de Gruebler e Kutzbach: M = 3.(N - 1) – 2P1 – P2
M – número de graus de liberdade do mecanismo
N – número total de elos do mecanismo (incluindo o solo)
P1 – número de pares cinemáticos com 1 g.l.
P2 – número de pares cinemáticos com 2 g.l.
Casos:
a) M = 0 Estrutura
b) M > 0 Mecanismo
c) M < 0 Sistema Hiperestático
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
M=1 M=2
Graus de liberdade
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Example
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Meia Junta
Junta Múltipla –Três - Elos Ligados P1 = 2
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
2.7 Paradoxes
• Greubler criterion does not include geometry, so it
can give wrong prediction
• We must use inspection
E-quintet
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
2.5 Mechanisms and Structures
• Mechanism: DOF>0
• Structure: DOF=0
• Preloaded Structure –
DOF<0, may require
force to assemble
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exemplos
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22/03/2013 37
Graus de liberdade (F) de um mecanismo
Exemplo 2:
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Graus de liberdade (F) de um mecanismo
Exemplo 3:
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Graus de liberdade (F) de um mecanismo
Exemplo 4:
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Graus de liberdade (F) de um mecanismo
Exemplo 5:
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exercícios
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22/03/2013 42
Exercício 1
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exercício 4
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Exercício 5
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22/03/2013 45
Exercício 7
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 46
Exercício 8
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 47
Exercício 9
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni 22/03/2013 48
Exercício 10
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
IFSC Paulo Boni IFSC Paulo
Boni IFSC Paulo Boni IFSC
Paulo Boni IFSC Paulo Boni
Comparar
Exercício 11