3 - Moments, Resultants, and Couples in 3D Flashcards
What is the moment-arm rule?
When the resultant couple vector M is parallel to the resultant forces R, the resultant is called a wrench. By definition, a wrench is positive if the couple and force vectors point in the same direction and negative if they point in opposite directions.
What method is best for determining the magnitude of moments in 3D? What about 2D?
3D: Vector approach with cross-product multiplication
2D: Scalar multiplication with dot product and moment-arm rule
What is a moment of F about a point O?
The moment M_o of F about an axis through O normal to the plane with magnitude M_o = Fd, where d is the perpendicular distance from O to the line of F.
What is the expression of a moment of F about an axis through a point O?
M_o = r x F
Are r x F = F x r ?
No, they give equal but opposite values.
What is the moment about an arbitrary axis?
Take an arbitrary direction lambda. n is a unit vector in this direction, use the dot-product expression to obtain Mo * n. Use the formula:
M_lambda = (r x F*n)n
What is a triple scalar product?
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
What is Varignon’s theorem in three dimensions?
r x F1 + r x F2 + … = r x (F1 + F2 + …)
= r x Sigma F
Mo = Sigma (r x F) = r x R
What is the expression used to find a couple in three dimensions?
M = r x F
How do you find the resultant of a system of concurrent forces?
When forces are concurrent at a point, only the below expression is needed:
R = F1 + F2 + F3 + … = Sigma F
As there are no moments about the point of concurrency.
How do you find the resultant of a system of parallel forces?
For a system of parallel forces not all in the same plane, the magnitude of the parallel resultant force R is simply the magnitude of the algebraic sum of the given forces.
The position of its line of action is obtained from the principle of moments by requiring that r x R = Mo. Here r is a position vector extending from the force-couple reference point O to the final line of action of R, and Mo is the sum of the moments of the individual forces about O.
What are coplanar forces?
When all forces are acting in the same plane, they are called coplanar whereas when forces act at the same time and at the same point, they are called concurrent/forces. When three or more coplanar forces are acting at a point and the vector diagram closes, there is no resultant.
