Introduction to Mechanical Vibrations. Ronald J. Anderson

Introduction to Mechanical Vibrations

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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. images ) at a specified angle, images , thereby removing two of the unknowns and letting you solve for images , images and images .

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, images and images , and a group of variables ( images , images , images ) 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 images and images and we will be left with a single differential equation containing images , images , and images . This nonlinear, ordinary differential equation is the equation of motion for the system. Given initial conditions for images and images , 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 images and adding the result to Equation 1.10 multiplied by images , giving

(1.11) equation

Equation 1.9 is useful only for determining images during the motion. An expression for images can be found by multiplying Equation 1.8 by images and subtracting it from Equation 1.10 multiplied by images . 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.

1.1.2 Informal Vector Approach using Newton's Laws

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, images , a normal acceleration, images , and a centripetal acceleration, images . Both images and images are due to the rates of change of the angle images . Since the wire has constant radius, images , we can immediately write images and images in the directions shown. Centripetal accelerations are of the form images and images arises from the rotation of the wire with constant angular velocity images . The relevant radius here is images as shown. Therefore images in the direction shown.

A two-dimensional representation of the bead on a wire, working out the kinematic expressions for three acceleration terms and the inset of a free body diagram.

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 images in Figure 1.2 and was shown to be equal to Скачать книгу