EN4: Dynamics and Vibrations
                 

 

  Division of Engineering
  Brown University

 

2.6.2 Examples using normal and tangential coordinates.

We proceed to illustrate the use of normal and tangential coordinates with a few representative examples. The first two problems are just concerned with relating velocity to acceleration.

 

Example I: A powerboat being tested for maneuverability is started from rest and driven in a circular path of 40ft radius. The magnitude of its velocity is increased at a constant rate of .

(a) Determine the velocity of the boat as a function of time, in terms of normal and tangential coordinates.

 

Idealize the boat as a particle. The picture above shows a plan view of its path. Let s denote the distance traveled by the boat. Define a basis as shown in the picture. We must take to be in the direction of increasing s, and must point towards the center of the circle. Then, from the discussion in the preceding section, we know that the velocity of the boat may be expressed as

The magnitude of the velocity is therefore . We are told that

Integrate

Hence

(b) Determine the acceleration of the boat as a function of time.

Here, we can use the general result for motion around a circular path

We may now substitute numerical values to conclude that

 

 

Example II: At time t=0, a car starts from rest a point A. It moves toward the right, and the tangential component of its acceleration is . If R=50m and d=200m, what is the magnitude of the car’s acceleration when it reaches point B?

Let s denote the total distance traveled by the car. Introduce a basis as shown. The general results tell us that

Here, denotes the instantaneous radius of curvature of the car’s path. At A, , while at B, .

We are given the tangential component (along ) of the acceleration. Therefore

Since the acceleration and speed of the car increase with time, we need to find the time at which the car reaches point B. The total distance traveled from A to B is

Hence, the car reaches B at time

Finally, we may compute the acceleration vector

Substituting numerical values shows that

 

 

Next, we will show how to apply Newton’s laws in a moving basis. In the next two problems, we will compute normal and tangential acceleration components and use them to deduce forces acting on objects.

Example III: Small parts on a conveyer belt moving with constant speed are allowed to drop into a bin. Show that the angle at which the parts start sliding on the belt satisfies

where is the coefficient of friction between the parts and the belt.

 

A free body diagram for a part on the curved portion of the belt is shown below

T and N are the components of the reaction force tangential and normal to the belt.

Assume that the part is not slipping on the belt. It therefore moves around a circular path of radius R at the same speed as the belt. We can deduce its acceleration from the general results in the preceding section. Define a basis as shown in the picture; note that it is essential to take pointing towards the center of curvature of the path of the particle. Then, since v is constant

Now express the resultant force acting on the particle as components in the basis

Newton’s second law (F=ma) requires that

Comparing components reveals that

When the part just begins to slip

Substitute for N and T and rearrange to see that

 

 

 

 

Vehicle Dynamics. If you drive your car along a straight road, your speed is virtually unlimited (The speed of your car is limited by the police and by the power of its engine. Both these problems have easy engineering solutions. A car’s speed along a straight track is actually limited by aerodynamics – if the car goes too fast it starts to behave like an airplane, which is not a good thing.) On the other hand, you know you must slow down when you drive around a corner. The sharper the corner, the slower you must drive. Why is this?

Consider a car that travels around a curve of radius R, as shown below. Suppose the roadway is banked at angle . Assume that the car’s instantaneous speed is v and its instantaneous tangential acceleration is a. Denote the coefficient of friction between the car’s wheels and the road by . Find the conditions necessary for the car to start skidding on the road.

The picture below shows the forces acting on the car

There is a normal reaction force from the roadway, a lateral force T , a longitudinal force L and a gravitational force acting on the car. The resultant may be expressed as

Our general results tell us that the acceleration of the car is

Finally, Newton’s second law shows that

These equations are readily solved for T and N

The car is just on the point of slipping when

In terms of a and v, we find that:

For a given tangential acceleration a, curve radius R and bank angle , we may compute the maximum possible speed of the car v.

The equation is a bit too complicated as it stands to see exactly what it is telling us. We can use it to look at simpler cases, however.

Suppose for simplicity that the roadway is level, so that . Our equation reduces to

The relationship between a and v is a circle, radius , as shown below

 

We can interpret this picture as follows. Suppose you need to drive your car around a curve of given radius R. You may vary your speed v, and you may vary your tangential acceleration a (by stepping on the gas or applying the brakes). Your acceleration and speed must be such that the point falls within the red circle, otherwise your car will skid.

Vehicle designers refer to this limiting boundary as a car’s `traction budget.’ In practice, a car’s traction budget is not a perfect circle as we have predicted with this simple analysis – the actual shape is determined by the way forces are shared between the four wheels during cornering. Nevertheless, our estimate is quite accurate.

We can learn a great deal from this result. For example:

If you want to take a corner at greatest speed (maximize v), drive at constant speed (a=0): don’t step on the gas or apply the brakes.

Changing a car’s mass has little effect on its ability to take corners

If you want to drive around a bend in the shortest time, you should hug the inside curb (see if you can work this out for yourself).

You may have heard of the `racing line.’ This is the path that minimizes the lap time for a racetrack, and the path that racing drivers try to follow. We might expect the racing line to follow the inside curb of all corners. This is almost right, but not quite. To minimize lap time, drivers want to hit the straight parts of the track at the maximum possible speed. Therefore, they start to straighten the path of the car as they leave a curve, and apply the gas at the same time. The picture below shows a typical racing line, and shows how the driver is using the car’s traction budget at various points around the curve.

Let’s now look at our predictions from the point of view of a highway designer. Our equation allows us to compute the smallest radius for a curved roadway required to carry traffic at a given speed. For example, you might like to compute the minimum allowable curve radius for an interstate.

We also predict that banking the roadway will allow cars to corner safely at much greater speeds. For example, if we take a=0¸ we may show that the maximum allowable speed is

Note that if we take

cars may apparently safely corner at infinite speed (under these conditions, the car remains on the road if its speed exceeds a critical value – if the speed is too low, the car slides off the road. You might like to fill in the details to show this). For typical values of friction coefficient (), this critical bank angle exceeds 45 degrees, however, and most drivers would probably be too scared to drive on the road.