Analysis: static and dynamic loads

We’ll analyse the experiments we just completed. First, the low deflection chart. Here’s the load deflection chart we created in the experiment that used heavy loads. Notice that we have plotted extension along with deflection axis rather than overall length so the curve goes through the origin. It’s not a straight line. It is non-linear. If it had been linear, we could have described it with a single quantity– it’s stiffness. We often use the symbol, k, for stiffness where k equals change in load divided by change in length. It’s the slope of the line. Units in our case are washers per millimeter but more usually it is newtons per millimeter.

A softer spring requires less load to extend it by a given amount, say 200 millimeters. A stiffer spring requires more.

For a non-linear spring, you can find the stiffness at any point on the chart by finding the slope of the tangent there, like this. We can describe our chart using these terms. At low loads, it is a softening spring. The stiffness reduces as the load is increased. At higher loads, it is a stiffening spring. The stiffness increases as the load is increased. Linear springs find many uses. We’ll see one next week when we look at a force transducer. Softening and stiffening springs have uses too. For example, the suspensions in railway freight bogies often are arranged to be softer when lightly loaded. That was statics. Acceleration is not present. Next, we’ll look at dynamics where accelerations are important.

Bounce is an example of a dynamic situation. In the experiments, we tested the effect of dropping the load pan from the unloaded height. First, we found the static deflection.

Then, we dropped the pan from the unstretched height. For a linear spring, we expected the maximum dynamic drop to be twice the deflection we found for the static loaded condition.

Dropping from the low load condition didn’t quite match the prediction for a linear spring. We expected the weight pan to touch the table. It didn’t. Perhaps, it’s not surprising. It’s not a linear spring, and it is not undamped either. If it were undamped, it would have gone on bouncing forever and it didn’t. Depending upon its effectiveness, damping can make a big difference. And the non-linear spring can make a difference too. As the deflection increases, the load might increase fast enough to stop the motion sooner. What about quasi-static? This applies where there are accelerations, but they are so small that you can neglect them, such as when we loaded up the pan.

You’ve learned about spring stiffness of a linear spring, spring stiffness at a given extension or load, linear softening and stiffening springs, the difference between static and dynamic conditions.

What do you think of the general agreement between theory and experiment for this week’s experiments? Post your thoughts on the discussion.