Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 15

Here we consider a two‐dimensional view of the system as shown in Figure 1.3 and work out the kinematic expressions for the accelerations from our knowledge of kinematics. There are three acceleration terms shown. They are a tangential acceleration, , a normal acceleration, , and a centripetal acceleration, . Both and are due to the rates of change of the angle . Since the wire has constant radius, , we can immediately write and in the directions shown. Centripetal accelerations are of the form and arises from the rotation of the wire with constant angular velocity . The relevant radius here is as shown. Therefore in the direction shown.

Figure 1.3 A 2D representation of the bead on a wire.

The inset in Figure 1.3 shows a FBD of the bead with the gravitational force and radial normal force being visible in this plane. There is another normal force perpendicular to the plane that can't be seen in this view. It is in Figure 1.2 and was shown to be equal to in Equation 1.9. The acceleration in this expression is a Coriolis acceleration. One needs quite a lot of experience with kinematic analysis to get the correct form of this term using an informal approach. Thankfully, it is perpendicular to the plane in which the bead moves relative to the wire, so it never appears in the equation of motion¹.

Summing forces in the vertical and horizontal directions gives

(1.12)

(1.13)

To eliminate the constraining normal force from these two equations, we multiply Equation 1.12 by and Equation 1.13 by and subtract the resulting expressions. The result is

(1.14)

where it is clear that both and are multiplied by zero and disappear from further consideration whereas is multiplied by a trigonometric identity equal to 1. Simplifying and substituting the derived kinematic expressions for and gives

(1.15)

which is the same nonlinear equation of motion (Equation 1.11) found in Subsection 1.1.1.