If the characteristic equation of a body of single degree of freedom spring mass system are equal,
Answers
Free vibration of conservative, single degree of freedom, linear systems.
First, we will explain what is meant by the title of this section.
* Recall that a system is conservative if energy is conserved, i.e. potential energy + kinetic energy = constant during motion.
* Free vibration means that no time varying external forces act on the system.
* A system has one degree of freedom if its motion can be completely described by a single scalar variable. We’ll discuss this in a bit more detail later.
* A system is said to be linear if its equation of motion is linear. We will see what this means shortly.
It turns out that all 1DOF, linear conservative systems behave in exactly the same way. By analyzing the motion of one representative system, we can learn about all others.
We will follow standard procedure, and use a spring-mass system as our representative example.
Problem: The figure shows a spring mass system. The spring has stiffness k and unstretched length . The mass is released with velocity from position at time . Find .
There is a standard approach to solving problems like this
(i) Get a differential equation for s using F=ma (or other methods to be discussed)
(ii) Solve the differential equation.
The picture shows a free body diagram for the mass.
Newton’s law of motion states that
The spring force is related to the length of the spring by . The i component of the equation of motion and this equation then shows that
This is our equation of motion for s.
Now, we need to solve this equation. We could, of course, use Mathematica to do this in fact here is the Mathematica solution.
This is the correct solution but Mathematica gives the result in a rather more complicated form than necessary. For nearly all simple vibration problems, it is actually simpler just to write down the solution. We will discuss the general procedure you should follow to do this in the next section.
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