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The Transition from Dynamics to Vibrations

Introductory undergraduate courses on dynamics typically consider large scale motions of systems of particles and/or rigid bodies and instantaneous solutions to their nonlinear, governing equations. You may recall working on dynamics problems where a system of bodies starts from rest at a prescribed position and your task was to determine, for example, the angular acceleration of a body or the forces acting on some part of the system. Solutions like this, while having some utility, provide only part of the understanding of the system that is required for a successful design. In most cases, the derived governing equations are complete enough but the "snapshot" solutions don't help much with the design process.

There are, in fact, many things that can be done with the equations governing the dynamic motion of the system. Briefly, they can be used to

1. Find where the bodies in the system would be if the system were at rest. These are the Equilibrium States.
2. Determine whether the equilibrium states are stable or unstable.
3. Determine how the system behaves for small motions away from a stable equilibrium state.
4. Determine the response of the system in the time domain through the use of numerical simulations. This is the most complex type of analysis and, perhaps surprisingly, gives the least information to the designer until the design has reached the fine tuning phase. The simulations are the analog of "cut and try" experiments where an unsuccessful result gives little information on what to change in order to improve the design.

While going through the material presented in this book, you will be concentrating on very small motions of systems about stable equilibrium states. In doing so, you will see connections to topics you may have covered in courses on statics, on dynamics, and on control systems. You will become very familiar with the linearized, differential, equations of motion for dynamic systems moving around stable equilibrium states and methods for deriving and solving them. This is the essence of Vibrations.

To get started and as a review of sorts we begin with the dynamic analysis to a relatively simple system - a bead sliding on a rotating semicircular wire.

1.1 Bead on a Wire: The Nonlinear Equations of Motion

First courses on the subject of Dynamics, whether for particles or rigid bodies, are primarily concerned with teaching the basics of kinematics, free body diagrams, and applications of Newton's Laws of Motion. Applying these three concepts sequentially will lead to a set of simultaneous force and moment balance equations that take account of kinematic constraints.

There are different ways of approaching these problems. One can use a formal vector-based approach and we will start with that here because it gives a complete set of governing equations including solutions for all constraint forces that are required to enforce kinematic constraints on the motion. A shorthand version of this approach which may be called an "informal vector approach" is often used in practice and that will be the second method addressed here. It typically works with two-dimensional views and leads to the governing equations of motion without necessarily solving for all constraint forces. The third approach will see the equations of motion derived using Lagrange's Equations. This is a work/energy approach that leads to the nonlinear differential equation of motion with minimal effort on the part of the analyst. The kinematic constraint forces are automatically eliminated as the governing equations are derived, leaving a designer with no information about forces acting on elements of the system unless extra work is done to find them. Lagrange's Equations are not typically introduced to undergraduate engineers as often as Newton's Laws are, so extra effort is made in this chapter to introduce the procedures for applying Lagrange's Equations to mechanical systems.

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.

1.1.1 Formal Vector Approach using Newton's Laws

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.

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 .

The absolute velocity of is

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

which can be simplified to

The absolute acceleration of is then

which simplifies to

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

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

Equation 1.9 is...