A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= -F A

Transcription

1 5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (= - ) Since the foces ae equal and oppositely diected, the esultant foce is zeo. ut the displacement of the foce couple (d) does ceate a couple moment. The moment, M, about some abitay point O can be calculated. d = + + O ( F ) If point O is placed on the line of action of one of the foces, say, then that foce causes no otation (o tendency towad otation) and the calculation of the moment is simplified. O This is a significant esult: The couple moment, M, depends only on the position vecto between foces and. The couple moment does not have to be detemined elative to the location of a point o an axis.

2 Example : Moment fom a Lage Hand Wheel The stem on a valve has two hand wheels: a small wheel (30 cm diamete) used to spin the valve quickly as it is opened and closed, and a lage wheel (80 cm diamete) that may be used to fee a stuck valve, o seat the valve tightly when it is fully closed. 80 cm 30 cm If the opeato can impose a foce of 150 N on each side of the lage wheel (a foce couple), what moment is imposed on the valve stem? Solution 1 s dawn, both the foce and position vectos have x- and y-components. The vectos may be defined as:» F_ = [ ]; Newtons» _ = [ ]; centimetes» F_ = [ ] Newtons» _ = [ ]; centimetes The moment of the couple can be calculated using the coss poduct opeato on the Matix toolba.» coss(_,f_) + coss(_,f_); Newton centimetes» M/ The esult is a couple moment of 120 N m diected in the z diection (into the page). Solution 2 Pehaps a moe easonable positioning of the axes fo this poblem might look like this:

3 x y In this case, the vecto definitions become:» F_ = [ ]; Newtons» = [ ]; centimetes Since the position vecto oiginates fom the line of action of foce, does not contibute to the moment. The moment is then calculated as» coss(,f_); Newton centimetes» M/ The esult is, of couse, the same no matte how the axes ae situated. Solution 3: Using Scalas Finally, the poblem can also be solved using a scala fomulation. The pependicula distance between the foces is 80 cm. With axes established as in Solution 2, the moment can be calculated as» F = 150; Newtons» d = 80; centimetes» M= d * F; Newton centimetes» M/ The diection must be detemined using the ight-hand ule.

4 nnotated MTL Scipt Solution Solution 1 Define the vectos F_ = [ ]; Newtons _ = [ ]; centimetes F_ = [ ]; Newtons _ = [ ]; centimetes Compute the moment using MTL's coss poduct function coss(_,f_) + coss(_,f_); Newton centimetes fpintf('the esulting couple moment is = [ 1.2f 1.2f 1.2f ] (Nm)\n', M) Solution 2 Define the vectos F_ = [ ]; Newtons = [ ]; centimetes Compute the moment coss(,f_); Newton centimetes fpintf('the esulting couple moment is = [ 1.2f 1.2f 1.2f ] (Nm)\n', M) Solution 3 Define the applied foce and the pependicula distance between the foces F = 150; Newtons d = 80; centimetes Compute the moment M= d * F; Newton centimetes fpintf('the esulting couple moment is = 1.2f (Nm)\n', M) fpintf(' The diection must be detemined using the ight-hand ule.\n\n') Example : Moment fom the Small Hand Wheel The moment esulting fom applying the same foces to the smalle hand wheel can be detemined using any of the thee solution pocedues outlined above. Solution 2 is shown hee. y x

5 Example : Moment fom the Small Hand Wheel The moment esulting fom applying the same foces to the smalle hand wheel can be detemined using any of the thee solution pocedues above. Solution 2 is shown hee. Define the vectos F_ = [ ]; Newtons = [ ]; centimetes Compute the moments coss(,f_); Newton centimetes fpintf('the esulting couple moment fo the smalle wheel is = [ 1.2f 1.2f 1.2f ] (Nm)\n', M)