Aula%15%(%Estudode Torçãoe Ângulode Torção.

1 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Aula 15 - Estudo de Tor o e ngulo de Tor o. Prof. Wanderson S. Paris, Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS ngulo de Tor o O projeto de um eixo depende de limita es na quan5dade de rota o ou tor o ocorrida quando o eixo subme5do ao torque, desse modo, o ngulo de tor o importante quando se analisam as rea es em eixos esta5camente indeterminados. Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS ngulo de Tor o = ngulo de tor o de uma extremidade do eixo em rela o outra. T(x) = Torque interno na posi o arbitr ria x.

2 J(x) = Momento de in rcia polar do eixo expresso em f(x). G = M dulo de elas5cidade ao cisalhamento do material. Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS C lculo para rea e Torque Constantes Normalmente, o material homog neo, de modo que G constante, bem como, a rea da se o transversal e o torque aplicado tamb m s o constantes, portanto, a equa o que determina o ngulo de tor o pode ser expressa do seguinte modo: Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS C lculo para rea e Torque Constantes Se o eixo es5ver sujeito a diversos torques diferentes, ou a rea da se o transversal e o m dulo de elas5cidade mudarem abruptamente de uma regi o para outra, o ngulo de tor o pode ser determinado a par5r da adi o dos ngulos de tor o para cada segmento do eixo, assim: Prof.

3 Wanderson S. Paris - MEC NICA DOS S LIDOS Conven o de Sinais A dire o e o sen5do do torque aplicado definido a par5r da aplica o da regra da m o direita. Torque e ngulo ser o posi5vos se a dire o indicada pelo polegar for no sen5do de afastar- se do eixo. Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS EIXO SUJEITO A DIVERSOS TORQUES Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Exerc cio 1 1) As engrenagens acopladas ao eixo de a o com uma das extremidades fixa est o sujeitas aos torques mostrados na figura. Supondo que o m dulo de elas5cidade de cisalhamento seja G = 80 GPa e o eixo tenha di metro de 14 mm, determinar o deslocamento do dente P da engrenagem A.

4 O eixo gira livremente no mancal em B. Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Solu o do Exerc cio 1 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Solu o do Exerc cio 1 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Exerc cio 2 O Parafuso de a o A- 36 com 8 mm de di metro est parafusado firmemente ao bloco em A. Determine as for as conjugadas F que devem ser aplicadas chave de torque de modo que a tens o de cisalhamento m xima no parafuso seja de 18 Mpa. Calcule tamb m o deslocamento correspondente de cada for a F necess rios para causar essa tens o. Considere que a chave de torque seja r gida.

5 2008 by Hibbeler. Published by Pearson Prentice Hall,Pearson Education,Inc.,Upper Saddle River,NJ. All rights reserved. This material is protected under allcopyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the 9/19/07 8:16 PM Page 186 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Solu o do Exerc cio 2 2008 by Hibbeler. Published by Pearson Prentice Hall,Pearson Education,Inc.,Upper Saddle River,NJ. All rights reserved. This material is protected under allcopyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the 9/19/07 8:16 PM Page 186 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Solu o do Exerc cio 2 2008 by Hibbeler.

6 Published by Pearson Prentice Hall,Pearson Education,Inc.,Upper Saddle River,NJ. All rights reserved. This material is protected under allcopyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the 9/19/07 8:16 PM Page 186 Prof. Wanderson S. Paris - MEC NICA DOS S LIDOS Exerc cios Propostos [P68] O eixo de a o A- 36 composto pelos tubos AB e CD e uma se o maci a BC. Est apoiado em mancais lisos que permitem que ele gire livremente. Se as engrenagens, presas s extremidades dos eixos, forem subme5das a torques de 85 Nm, determine o ngulo de tor o da engrenagem A em rela o engrenagem D.

7 Os tubos t m di metro externo de 30 mm e di metro interno de 20 mm. A se o maci a tem di metro de 40 mm. OFTWIST20955 propellers of a ship are connected to a A-36steel shaft that is 60 m long and has an outer diameter of340 mm and inner diameter of 260 mm. If the power output MW when the shaft rotates atdetermine themaximum torsional stress in the shaft and its angle of twist.*5 shaft is subjected to a torque T. Compare theeffectiveness of using the tube shown in the figure with thatof a solid section of radius do this, compute the percentincrease in torsional stress and angle of twist per unit lengthfor the tube versus the solid rad>s,5 hydrofoil boat has an A-36 steel propellershaft that is 100 ft long. It is connected to an in-line dieselengine that delivers a maximum power of 2500 hp andcauses the shaft to rotate at 1700 rpm.

8 If the outerdiameter of the shaft is 8 in. and the wall thickness is determine the maximum shear stress developed in theshaft. Also, what is the wind up, or angle of twist in theshaft at full power?38 in.,PROBLEMSTTccc2 Prob. 5 48400 mm400 mm250 mm85 N m!85 N m!ABCDProb. 5 49 5 A-36 steel axle is made from tubes ABandCDand a solid section is supported on smooth bearingsthat allow it to rotate freely. If the gears, fixed to its ends, aresubjected to torques, determine the angle of twistof gear Arelative to gear tubes have an outerdiameter of 30 mm and an inner diameter of 20 mm. Thesolid section has a diameter of 40 #m100 ftProb. 5 50 ABProbs. 5 51/525 engine of the helicopter is delivering 600 hpto the rotor shaft ABwhen the blade is rotating at1200 Determine to the nearest the diameterof the shaft ABif the allowable shear stress is and the vibrations limit the angle of twist of the shaft rad.

9 The shaft is 2 ft long and made from L2 steel.*5 engine of the helicopter is delivering 600 hpto the rotor shaft AB when the blade is rotating at1200 Determine to the nearest the diameter ofthe shaft ABif the allowable shear stress is and the vibrations limit the angle of twist of the shaft rad. The shaft is 2 ft long and made from L2 ksi18 > ksi18 > Wanderson S. Paris - MEC NICA DOS S LIDOS Exerc cios Propostos [P69] As extremidades estriadas e as engrenagens acopladas ao eixo de a o A- 36 75x109 Pa est o subme5das aos torques mostrados. Determinar o ngulo de tor o da extremidade B em rela o extremidade A. O eixo tem di metro de 40 mm. 210 CHAPTER5 TORSION 5 20-mm-diameter A-36 steel shaft is subjectedto the torques shown.

10 Determine the angle of twist of theend assembly is made of A-36 steel and consists of asolid rod 20 mm in diameter fixed to the inside of a tubeusing a rigid disk at B. Determine the angle of twist at tube has an outer diameter of 40 mm and wall thicknessof 5 assembly is made of A-36 steel and consists of asolid rod 20 mm in diameter fixed to the inside of a tubeusing a rigid disk at B. Determine the angle of twist at tube has an outer diameter of 40 mm and wall thicknessof 5 N!m20 N!m30 N!mBDC800 mm600 mm200 m60 N!m150 N! 54/55*5 splined ends and gears attached to the A-36steel shaft are subjected to the torques shown. Determinethe angle of twist of end Bwith respect to end shafthas a diameter of 40 N!mA200 N!m500 N!m300 mm400 mm500 mm400 N! 566 57/58 5 motor delivers 40 hp to the 304 stainless steelshaft while it rotates at 20 Hz.