Lab 9

Springs or 2$^{nd}$ order Linear ODE's

According to Newton's second law of motion MATH while Hooke's law states that the restoring force exerted by a spring is opposite to but proportional to the distance the spring is distorted from rest, i.e. $Force=-k\ast x,$ where $x$ is the distance the spring is stretched (or compressed) and $k>0.$ Therefore in the case of the spring we have the undamped situation MATH

I. Examine the graph of the undamped spring with an attached mass of 1 and restoring factor of 16 ($k=16).$ Use the initial conditions of $x(0)=10,$ $x^{\prime }(0)=1.$

To do this on ODE Architect you must first write the ODE is system format.

Now solve the ODE and look at how the solution and the graph compare. What is the limit as MATH ?

Damping is caused by something creating a drag on whatever the spring is trying to move (like moving in water). The effect of damping is that of a force proportional to the velocity but opposite to the direction of movement, i.e. MATH $\ \beta >0.$ Therefore the ODE controlling the movement of the spring is MATH There are 3 types of damping: underdamped, critically damped, and overdamped.

For the next 3 problems follow the instructions for I.

II. The underdamped spring: $m=1,$ $k=25,$ $\beta =6,$ $x(0)=10,$ $\ x\prime (0)=1.$

III. The critically damped spring: $m=1,$ $k=9,$ $\beta =6,$ $x(0)=10,$ $\ x\prime (0)=1.$

III. The over damped spring: $m=1,$ $k=4,$ $\beta =5,$ $x(0)=10,$ $\ x\prime (0)=1.$

Now it's your turn. I'll give you an undamped situation and you are to add the amount of damping needed to get the various situations. Be sure to show the graphs of your results.

IV. Problem: If a spring is stretched 6 inches by a 2 lb weight find $k. $ Hint look at Hooke's law. Now use this result to find the ODE of the undamped motion if a mass $\ m=1$ is attached to the spring (in place of the 2 lb weight) and the spring is stretched so that $x(0)=8$ inches, MATH ft./sec. Graph this situation. BE CAREFUL ABOUT UNITS.

Your turn: Pick a damping to put on the spring in #IV that will result in each of the following V. critically damped, VI. overdamped, VII. underdamped and print on the same graph.

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