Lecture 4 part II

Lecture 4 part II - PART II CONTENTS VECTOR REPRESENTATION OF A MOMENT MOMENT OF A FORCE ABOUT A LINE(AXIS COUPLES 15 VECTOR REPRESENTATION OF A

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15 PART II CONTENTS: VECTOR REPRESENTATION OF A MOMENT MOMENT OF A FORCE ABOUT A LINE (AXIS) COUPLES
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16 VECTOR REPRESENTATION OF A MOMENT In Fig. 4-1, the moment of the force F about point O can be represented by the expression O = × Mr F (9) where r is a position vector from point O to a point A on the line of action of the force F . The sign × means the cross product of two vectors, and is defined as sin sin O α = × = ⋅⋅ = F r F e F r e Disp where is the angle (0 180 ) °° ≤≤ between the two vectors r and F , e is a unit vector perpendicular to the plane determined by vectors r and F as shown in Fig. 4-3 a . It can be seen that the term ||s i n r denotes the perpendicular distance d from the moment center O to the line of action. The distance d is independent of the position A along the line of action of the force since A 3 A 2 A 1
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17 11 2 2 33 sin sin sin d ααα ⋅= rr r (11) Also, from this chart, OO dF dM == = MF e e e (12) where the direction of unit vector e is determined by using right-hand rule : figures of the right hand curl from positive r to positive F and the thumb points in the direction of positive O M . The physical meaning of the moment O M represent the tendency of the force F to rotate the body about an axis through point O that is perpendicular to a plane containing force F and position vector r . It is important to note that the sequence of × rF must be maintained in calculating moment since the sequence otherwise will produce the moment in the opposite direction. a) Right hand rule for positive moment
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18 b) Cartesian vector form of positive vector r c ) Position vector r A/B in terms of vectors r A and r B Fig. 4-3 Moment as the cross product of vector of r and F The vector r from the point about which a moment is to be determined to any point on the line of action of a force F can be expressed in terms of unit vectors i , j , and k , and the coordinates ( x A , y A , z A ) and ( x B , y B , z B ) of points A and B , respectively. The position vectors r A and r B shown in Fig. 4-3 b can be written as AA A A x yz =++ ri j k BB B B x j k (13) As shown in Fig. 4-3 c , / ABA B = + rrr (14) Then the vector from B to A is / AB A B == = rr r r r ( ) ( ) AAA BBB x xyz =+ + −+ + i j ki j k ( ) ( ) ( ) xx yy zz =− +− i j k (15)
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19 Up to now, we need to clarify some potential confusion in the sudcripts we are using for the moment arm e.g., / BA d , and the position vector e.g., / AB r . For / d , we are focusing on the distance from point B to the line of action of force through point A . However, for / r , we are defining a position vector starting from point B to point A . See the following figures for the illustration.
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This note was uploaded on 04/20/2009 for the course CVEN 221 taught by Professor - during the Summer '08 term at Texas A&M.

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Lecture 4 part II - PART II CONTENTS VECTOR REPRESENTATION OF A MOMENT MOMENT OF A FORCE ABOUT A LINE(AXIS COUPLES 15 VECTOR REPRESENTATION OF A

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