Ronald J. Anderson - Introduction to Mechanical Vibrations

As an example, consider Figure 1.1. The figure shows a small bead with mass, , sliding on a frictionless semicircular wire that rotates about a vertical axis with a constant angular velocity, . The wire has radius . Gravity acts to pull the mass to the bottom of the semicircle while centripetal effects try to move it to the top. The single angular degree of freedom, , is sufficient to describe the motion of the bead on the wire.

Figure 1.1 A bead on a wire.

Using the formal vector approach, the first step in the kinematic analysis is to choose a coordinate system (i.e. a set of unit vectors) that is convenient for expressing the vectors that will be used. The coordinate system may be fixed or rotating with some known angular velocity. In this case, we will use the ( , , ) system shown in Figure 1.1. This is a rotating system fixed in the wire so that and stay in the plane of the wire and is perpendicular to the plane. Furthermore, and remain horizontal and is always vertical. The angular velocity of the coordinate system is .

We use the general approach to differentiating vectors, as follows, where can be a position vector, a velocity vector, an angular momentum vector, or any other vector.

(1.1)

It is important to understand that the angular velocity vector, , is the absolute angular velocity of the coordinate system in which the vector, , is expressed. There is a danger that the rate of change of direction terms will be included twice if the angular velocity of the vector relative to the coordinate system in which it is measured is used instead.

We start the kinematic analysis by locating a fixed point, in this case point , and writing an expression for the position vector that locates with respect to .

(1.2)

The absolute velocity of is

(1.3)

Then, using Equation 1.1and recognizing that since is a fixed point and that since the radius of a semicircle is constant,

(1.4)

which can be simplified to

(1.5)

The absolute acceleration of is then

(1.6)

which simplifies to

(1.7)

Once an expression for the absolute acceleration has been found, the kinematic analysis is complete and we move on to drawing a Free Body Diagram (FBD). For this example, the FBD is shown in Figure 1.2.

Figure 1.2 Free Body Diagram of a bead on a wire.

Constraints are taken into account when showing the forces acting on the bead. The forces shown and the rationale behind them are: