Think about a pinwheel. By blowing on the pinwheel, you're actually applying a force to it. Unlike the examples in previous chapters, the pinwheel doesn't move in the direction of the force because it's tied to a stick. But it does spin or rotate. The pinwheel stays in place, but it spins. The harder you blow (the more force you apply) to a pinwheel, the faster it spins.

Rotational behaviors can also occur in objects that are translating. Rolling is a combination of both linear motion (or translation) and rotation. The old tumblin' tumbleweeds that you see in classic Westerns are a great example. These dried plants move along the dusty countryside (translation), turning under the force of the wind (rotation).

The general physics definition of a moment is always "force times distance." This simple concept doesn't account for several other considerations that come into play, but it's efficient enough to illustrate the units of a moment: Newton-meters (N-m) for SI units and pound-feet (lb-ft) for U.S. customary units.

Because a moment is often a direct result of the action of a force vector, a moment is also a vector and must also have similar characteristics. Like any vector, a moment vector has a magnitude, point of application, and sense (see Chapter 4 for more on these basic characteristics).

The magnitude of the moment is a measure of the intensity of the rotational effect. Instead of having a unique point of application (location in space) or a defined line of action (line in space on which the vector is acting) as forces do, the moment actually rotates around an axis called the axis of rotation. The sense is the direction of rotation about its axis of rotation. You usually represent it as a clockwise (negative) or counterclockwise (positive) behavior. An axis of rotation can either be within the object, which results in a rotational behavior known as pivoting, or outside an object, which results in a rotational behavior known as orbiting.

Figure 12-1 shows you the similarities between force vectors and moment vectors. You display a force with a single-headed arrow, as I describe in Chapter 4. The figure also shows you two ways for depicting a rotation — one with a circular arrow, and another with a double-headed arrow. See the similarities between moment and force sketches? Don't worry, I discuss these methods for drawing moments later in this chapter.

Figure 12.1. Rotation about an axis.

Developing rotational behaviors: Meeting couples and concentrated moments

You can develop a rotational behavior in several different ways. Some require forces, some require distances, and some require neither of those. In statics, rotational behaviors are created by one of three principal methods: one force and a distance, two parallel forces separated by a distance (or couples), and a concentrated rotational effect (or concentrated moment), which I cover in the following sections.

One force and one distance

Consider the behavior of an open door when you push on it as shown in Figure 12-2.

Figure 12.2. Rotation with a single force and single distance.

In a door assembly, doors typically hang from several hinges that are aligned along a single vertical axis of rotation. When you push on the door (apply a force), the resulting action is that the door begins to move. Regardless of where you apply the force on the door (at the top, on the handle, or by your foot on the bottom of the door), the same resulting behavior (a moment) occurs along the axis of rotation. This moment is what creates the rotation that results in the door swinging open or closed.

Two parallel forces and a distance: Couples

A couple is a type of moment produced by two parallel forces of the same magnitude acting in opposite directions and separated by a distance that result in a rotational behavior on the object. One example of a couple is the forces from your fingers on a doorknob. As you turn the knob, one finger is pushing up on the side of the doorknob and the other is pushing down. These two forces together cause the knob to rotate. Actually opening or closing the door requires that you push on the door after you have turned the knob, which is the scenario that I discuss the preceding section. Opening a door actually requires two moments (mechanically speaking, that is).

As another example, imagine you're driving your car with your hands at "ten and two" in proper driving fashion. As you're traveling down the road, a ball bounces out into the lane in front of you. Out of instinct, you quickly turn the wheel clockwise to swerve around the obstacle by pushing up on the left edge of the wheel with your left hand while pulling down on the right edge of the wheel with your right hand as shown in Figure 12-3.

Figure 12.3. Turning a steering wheel creates a couple.

The behavior of your hands on the wheel when you turn clockwise is actually caused by two separate forces applied to the wheel. The force on the left side of the steering wheel is acting upward, while the force on the right side of the wheel is acting downward. In this example, the resultant of these two forces is zero (F + (–F) = 0) or balanced. The wheel isn't actually translating (or moving) in any direction, but it still experiences a rotational behavior — a couple, from these two forces despite having no net force (or zero resultant forces) acting on it.

No distance? Concentrated moments

Another scenario that can cause an object to rotate is the application of a force or moment to another connected object. This resulting and transmitted moment is known as a concentrated moment. In fact, these mechanisms are very common in mechanical shaft design. The purpose of a shaft is to transmit a force or moment from one location in an object or mechanism to another through the action of the shaft. Consider Figure 12-4, which shows a single force F acting on an L-shaped bar.

Figure 12.4. Concentrated moment.

In this example, the force causes the shaft to rotate about its axis of rotation, which in turn creates a resulting moment on the plate at the location where the shaft and plate are connected.

This transmission of moments can occur through multiple objects as well. An example of this type of situation occurs in the drive train of your automobile. The engine of your car causes the transmission to rotate, which in turn causes the axles to rotate, which in turn cause the wheels to rotate. Now obviously, moving a car down the road requires many more factors, but the overall concept of the transmitted moment is still valid.

Taking on torque and bending: Types of concentrated moments

You can create different types of rotational effects depending on which axis a concentrated moment is causing a rotation about. In statics, two of the most common effects are bending and torque (or torsion), shown in Figure 12-5.

• Torque: A torque is a torsional moment, or one that causes rotation about a longitudinal axis of an object that causes a twisting action.

• Bending: A bending moment is a moment that is applied about an axis that is perpendicular to a member's longitudinal axis, or applied in the plane of the cross section (or a slice through the member — see the shaded region in Figure 12-5). That is, if it isn't a torsional moment, it has to be a bending moment.

Figure 12.5. Torque and bending.

Note

Because moments are also vectors, a resultant moment may have components that produce torque and multiple bending effects simultaneously. I talk more about computing these effects in Chapter 20.

Getting a handle on the right-hand rule for moments of force

In statics, you can find several different variations of right-hand rules that prove to be very useful as you start working problems. One of the most useful versions helps you determine the sense of a moment about its axis of rotation.

You can determine the sign (or sense) for moments by making an L-shape with your right thumb and forefinger. Align your thumb with the positive x-axis of your Cartesian coordinate system and then line up your forefinger with the positive y-axis at the same time. (This orientation may feel a bit awkward at first, but it does work!). Bend your middle finger naturally so that it's perpendicular to your thumb and forefinger and pointing outward from your palm. Your middle finger represents the direction of the positive z-axis (see Figure 12-6).

Figure 12.6. Right-hand rule for moments of force.

Warning

The right-hand rule only works with the right hand! If you use your left hand by mistake, the direction of your z-axis will be backward.

After aligning your fingers with the Cartesian system, if your moment is about one of the Cartesian x-, y-, or z-axes, you can determine the sense of the moment by looking at the end of the finger that is parallel to the axis of rotation of the moment. In three dimensions, a moment is positive about the x-axis if it's acting counterclockwise when you're looking at the tip of your thumb. The same applies to the y-axis and your forefinger and to the z-axis and your middle finger.

To calculate the magnitude of a moment of a force, you need to include two pieces of information in your computations: the force and the distance. In general, you can calculate the magnitude of the moment from the following equation:

This equation represents the scalar form of the moment calculation. When you use this formula, you're actually only calculating the magnitude of the moment — you haven't actually considered the sense of the line of action required to fully define it as a vector, nor have you defined the axis of rotation. In the scalar equation, the distance term is the distance from the axis of rotation.

In two dimensions, the x and y Cartesian axes are usually in the plane of the page, which results in the third axis, the z-axis, being out of the page because it must be perpendicular to the two-dimensional axes. A two-dimensional moment of a force located in the xy Cartesian plane is always about the z-axis (the axis of rotation is parallel to the z-axis). In vector terms, this fact means that the moment in the xy plane has a unit vector direction of k (either positive or negative depending on the sense of the moment).

Planar rotation about a point

In this section, I show you how to perform the calculation of the moment after you know the location of the point in space. I actually explain how to choose the necessary moment locations for your calculations when I discuss the various techniques of Part VI.

Note

The major drawback of the scalar method of computing moments is that you have to assign the sense of the vector based on logic.

Suppose you want to calculate the moment of the force in Figure 12-7a (which shows a force with a magnitude of 300 pounds acting at an angle of 60 degrees above the negative x-axis) about Point A. Just follow these steps:

Figure 12.7. Calculating a scalar moment.

1. Break the vector into components in the x- and y-direction.

   The first step is to compute the x- and y-components by using the basic trigonometry principles that I discuss in Chapter 8.

   

   After you've computed the components, you can apply them to the original object one at a time, as shown in Figures 12-7b and 12-7c.

2. Calculate the moment contribution of each component that you calculated in Step 1 about the point of interest.

   For the vertical force Py shown in Figure 12-7b, you can calculate the moment by multiplying by the perpendicular distance.

   Tip

   If the dimensions of the force measured to the point are in directions parallel to the Cartesian x- and y-axes, you want to break the force into components parallel to those axes as well.

   In this case, because the force P_y of Figure 12-7b is vertical, you need to use the horizontal distance when you calculate the moment at Point A (M_A1).

   

   The direction clockwise is determined by considering which direction the force would rotate the object if A were pinned in its current position. This clockwise statement is an indication of the sense of the vector. As I discuss in "Getting a handle on the right-hand rule for moments of force" earlier in the chapter, a moment about the z-axis is considered positive if it's acting counterclockwise about the axis of rotation; therefore, the clockwise moment of the force is actually a negative moment.

   Similarly, you can calculate the moment for the horizontal force P_x of Figure 12-7c. In this case, you need to use the vertical distance, which is perpendicular to the line of action of the horizontal force.

   

3. Compute the net effect of the moments of the component forces about the location of interest.

   

   Because the net magnitude is positive, you know that the net moment about Point A is acting in a positive (or counterclockwise) direction with a magnitude of 201 lb-in.

Determining the magnitude and sense of a two-dimensional couple

You can treat a couple as either two separate forces or as a pair of forces separated by a distance. Both come out to the same magnitude value. You compute the moment couple of a pair of forces by relying on the same general principle of the force times distance relationship. Consider the couple shown in Figure 12-8; it's created by a pair of 200-Newton forces separated by a distance of 2 meters.

Figure 12.8. Computing the magnitude and sense of a two dimensional couple.

To calculate the magnitude of the force couple, you use the formula from the preceding section:

where the force is the magnitude of one of the forces in the couple and the distance is the perpendicular distance between the lines of action of the forces. To determine the sense of the couple, you need to choose both a point on one of the forces' lines of action as well as the other force itself. For this example, if you choose Point 1 on the line of action #1 as your reference point, you'd choose the force on the line of action #2. Now, to determine sense of the moment, you consider the direction of rotation of your selected force about your selected point. In this case, the selected force wants to rotate about Point 1 in a clockwise direction (which indicates a negative sense), so you can say the sense of this couple is negative. Your final solution may look something like the following:

Notice that if you choose the other point (Point 2) as your reference point and the force on the line of action #1, you end up with the same rotational sense.

Moment magnitudes are pretty simple to calculate by using scalar information (and the preceding section), and a little logic helps you determine the sense, particularly when you can compute (or already know) the required perpendicular distance. For two-dimensional problems, determining a perpendicular distance may not be all that difficult, especially if the axis of rotation is aligned to one of the Cartesian axes.

But what happens when you have a three-dimensional force that's creating a moment about some random point in space? Finding that perpendicular distance can be a little rough, especially when the axis of rotation isn't aligned conveniently. You almost always have to fall back to using the following vector form to compute the moment vector M.

where r is a position vector from the axis of rotation to any point on the line of action of the force vector F. Warning: The strange × in the equation isn't the multiplication symbol but rather is called the cross product, an operation performed on two different vectors that produces a third vector that's perpendicular to each of the original vectors. You read this equation as "r cross F." I show you how to actually perform this calculation a little later in this section.

The major advantage of using the vector form over the scalar form when calculating a moment is that you don't have to worry about calculating those pesky perpendicular distances because they're already handled by the vector mathematics contained within the cross product calculation. In fact, the vector solution forms will always work, although for two-dimensional problems, the scalar math calculations are often a lot simpler than performing a cross product.

Completing the cross product

The most difficult part of creating a moment vector is actually the computation of the cross product. Although it's not mathematically difficult, it can be a somewhat lengthy process (as you can see in the equation I show you in Chapter 6). Figure 12-9 shows a three-dimensional vector and a center of rotation at Point 1.

Figure 12.9. Vector formulation for a moment.

To compute the moment, you need two pieces of information. The first is the Cartesian vector formulation for the force that is creating the moment. You can use any of the techniques that I describe in Part I to help you create the force vector. The second piece of information you need is the position vector that starts at Point 1 at the center of rotation (or point of interest) in Figure 12-9 and connects to Point 2, which is a point at any location along the line of action of the force. It doesn't matter where you place the second point as long as it's somewhere along the force's line of action. Normally, you pick that second point as the point of application of the force (because you often know those coordinates) or a place on the geometry where the dimensions are already defined for you.

After you have this information, you can substitute the scalar component values into the cross product formula to compute the magnitude of the moment. As I cover in Chapter 6, one technique for solving a cross product calculation is using a determinant. The determinant form is shown in the following equation.

In the vector formulation, you include the unit vectors for the Cartesian axes on line 1. The position vector information goes on line 2, where you include the scalar magnitudes of the components of the position vector. On line 3, you input the force vector information, which includes the component magnitudes of the force. For lines 2 and 3, you place the x-component information for both the position and force vectors in the column below the x-direction unit vector (i), the y-component information below the y-direction unit vector (j), and the z-component information below the z-direction unit vector (k). If your position vector or force vector doesn't have a particular value for an x-, y-, or z-component, you simply put a zero value in that location. The final answer from this calculation is a vector representation of the moment of the force about the center of rotation.

Using unit vectors to create moment vectors

In some cases, you may know the magnitude of an applied moment about an axis of rotation, particularly if you've used scalar computations to compute the moment. You can create a unit vector defining the direction of the moment by creating a different unit vector that describes the direction of the axis of rotation.

Note

You can always relate any vector to its magnitude and direction. For moments, you use something like the following:

In this equation, u_M is a unit vector in the direction of the sense of the moment (or the direction of your thumb if you've used the right-hand rule for moments, discussed earlier in the chapter).

Rule of Sarrus

A fairly easy shortcut known as the rule of Sarrus can actually eliminate some of the troubles with signs that may pop up when you compute the determinant in your moment vector calculations. The following procedure simplifies the computation:

1. Enter the values into the determinant as described in the nearby section "Completing the cross product."

2. Augment the matrix from Step 1 by copying the i and j columns to the positions shown in the following equation.

   Your calculation now looks like this:

   

3. Starting at the upper left i value, multiply everything on the diagonal from upper left to lower right; repeat this process for both the j and k values and then add these three multiples.

   You get something like the following expression:

   

4. Starting at the lower left F_x value, multiply everything on the diagonal from lower left to upper right; repeat this process for both the F_y and F_z values and then add these three multiples.

   Here's what that expression looks like:

   

5. Subtract the results of Step 4 from the results of Step 3.

   This value is the vector formulation of the moment and the mathematical solution of your determinant.

   

The results of this calculation are identical to those from the cross product methodology I describe in Chapter 6.

Consider the example shown in Figure 12-10.

Figure 12.10. Creating vectors by using scalar moment magnitude.

The unit vector describing the sense and direction of the moment is actually the same unit vector that describes the direction of the axis of rotation.

Tip

In Chapter 5, I show you three different techniques for creating unit vectors for forces; if you can create a unit vector describing a force's line of action, you can use the same techniques to define an axis of rotation, as the following list shows.

• Using position vectors: If you know two points on the axis of rotation, you can use the position vector method to create a unit vector by dividing a position vector by its magnitude. In Figure 12-10, the position vector goes from Point 1 to Point 2 (both of which lie on the axis of rotation) because it must be in the same direction as the unit vector uM. The denominator of this equation is just the magnitude of the position vector in equation form.

  

• Using direction cosines: If you happen to know the angles between each of the Cartesian axes and the axis of rotation, you can use the direction cosine formulation (which follows) to create your unit vector for the moment.

  

In Chapter 4, I show you how to draw a double-headed vector. These two-headed monsters are actually extremely useful when you start working with moments. One of the major points of confusion with moments has to do with the concept of rotation. How do you accurately depict a rotation behavior about a point or axis?

In two dimensions, you can easily illustrate a circular arrow depicting the direction of the moment because the circular arrow almost always acts around the z-axis. However, three-dimensional cases, where the moment can act around any axis in space, are a little harder to illustrate. And when you're computing components from the rotational depiction, such illustration becomes next to impossible.

For this reason, I like to use the double-headed vector notation as shown in Figure 12-11 (though I only use it when I'm working with moments) which lets you handle moments easily and effectively in the same manner as you would treat a force vector.

When performing your basic statics calculations, you often find relocating a force from one point to another convenient. By creating an equivalent force couple, you can move a force vector to a new location by simply relocating the force and creating a new moment at the new location.

Figure 12-13 shows a rigid body with two different points, A and B. In the first picture, a force vector F acts at Point A. Notice that the force F at A is eccentric to (or not acting through) Point B.

Figure 12.13. Equivalent force couples.

To produce the same translational effect on this rigid body (a body not deformed by the force), you simply need to relocate the force at Point A to its new position at Point B. However, after you move the force from Point A to Point B, the rotational behavior of the object changes. In order to capture the rotational effects of the force at Point A with respect to Point B, you have to include an additional rotational effect, or a moment, which you can compute with the following formula:

where rBA is a position vector from the new point (B) to the original point (A). The methods for computing the position vectors and force vectors remain unchanged. I show you more about the uses and implementation of this idea in Part VI.