理论力学教案3

2.7 Couples

1. Definition:

Two parallel, noncollinear forces that are equal in magnitude and opposite in direction are known as a couple.

2. Moment of a couple about a point：

(a). Scalar calculation

M O = F(a+d) –F(a) = Fd

Characteristics:

· A couple has no resultant force (ΣF = 0).

· The moment of a couple is the same about

any point in the plane of the couple.

(b). Vector calculation

The moment of the couple about point O is given by:

M O= r OA ×F + r OB × (–F)

= (r OA – r OB) × F = r BA × F

Conclusion:

· The moment of a couple is the same about every point.

· So, the moment of a couple is a free vector.

· But the moment of a force about a point is a fixed vector！

3. Equivalent couples：

The following four operations can be performed on a couple to produce its equivalent couples.

(a) Changing the magnitude F of each force and the perpendicular distance d while keeping the product Fd constant;

(b) Rotating the couple in its plane;

(c) Moving the couple to a parallel position in its plane;

(d) Moving the couple to a parallel plane.

4. The addition and resolution of couples

(1) The addition of couples

(a) By the usual rule of vector addition.

(b) Bing free vectors, concurrency is not required.

(c) To minimize the possibility of confusion, we use M to denote moment of forces and reserve C for couples.

(2) The resolution of couples

The resolution of couples is the same as the resolution of moments of forces. For example, the moment of a couple about an axis AB can be written as

M AB= C·λ

Sample Problem 2.7

For the couple shown in the figure, determine (1) the corresponding couple-vector and (2) the moment of

the couple about the axis GH.

Solution: (1) The magnitude of the couple is:

C = Fd = 100 × 0.6 = 60 kN·m

The sense of the couple is counterclockwise.

Let λ be the unit vector along the direction of the couple.

Then it can be written as λ = (3j + 4k)/5

Therefore the couple-vector is C = Cλ = 60λ = 36j + 48k kN·m

= C·

λ

GH

λGH So, GH

GH

2.8 Changing the Line of Action of a Force

Step 1: Introduce two equal and opposite forces of magnitude F at A

Step 2: Identify the two forces that form a couple and the magnitude of the couple is C T = Fd . The couple

C T = Fd is called as the couple of transfer.

Conclusion: The couple of transfer is equal to the moment of the original force (acting at B) about the transfer pint A .

· Vector terminology:

C T = r × F