Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 14

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 1.1 and 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:

  = the weight of the body acting vertically downward. This is the effect of gravity.

  = one component of the normal force that the wire transmits to the mass. Since is perpendicular to the plane of the wire, there can be a normal force in that direction.

  = the other component of the normal force. We let it have an unknown magnitude and align it with the radial direction since that direction is normal to the wire.

 Note that there is no friction force because the system is frictionless. If there were, we would need to show a friction force acting in the direction that is tangential to the wire.

Once the FBD is complete, we can proceed to write Newton's Equations of Motion by simply summing forces in the positive coordinate directions and letting them equal the mass multiplied by the absolute acceleration in that direction. The result is three scalar equations as follows

(1.8)

(1.9)

(1.10)

At this point in the majority of undergraduate Dynamics courses we would count the number of unknowns that we have in the three equations to see if there is sufficient information to solve the problem. We would find five unknowns

and say that we are unable to solve this without further information since we have only three equations. A typical textbook problem would say, for example, that the mass is released from rest (i.e. ) at a specified angle, , thereby removing two of the unknowns and letting you solve for , and .

This solution gives an instantaneous look at the system that really doesn't point out the value of the equations derived. Equations do not have five unknowns. They have two unknown constraint forces, and , and a group of variables (, , ) that are related by differentiation. Rather than counting five unknowns as we did earlier, we should say that there are three unknowns

and three equations.

We can combine the three equations to eliminate and and we will be left with a single differential equation containing , , and . This nonlinear, ordinary differential equation is the equation of motion for the system. Given initial conditions for and , we can solve the equation of motion as a function of time and predict the angle, its derivatives, and the two normal forces at any time. The solution of nonlinear differential equations is not a trivial exercise but can be handled fairly easily using numerical techniques.

The equation of motion for this system can be found by multiplying Equation 1.8 by and adding the result to Equation 1.10 multiplied by , giving

(1.11)

Equation 1.9 is useful only for determining during the motion. An expression for can be found by multiplying Equation 1.8 by and subtracting it from Equation 1.10 multiplied by . As a result, we could solve the differential equation of motion (Equation 1.11) numerically and always have the ability to predict the two constraint forces. These forces provide useful design information that is difficult to get from the methods considered next.