Introduction
In this post I will give an
introduction to free vibrations of one degree of freedom systems. The name
already tells you that we are dealing with free vibrations which mean that the
object of mechanical system oscillates about system’s equilibrium position in
the absence of an external excitation. Free vibrations are a result of a
kinetic energy imparted to the system or of a displacement from the equilibrium
position that leads to a difference in potential energy from the system’s
equilibrium position.
Ok now let’s consider a
mechanical system that is shown in the next figure.
Figure 1.1 - System consisting of body and the spring
From the previous figure we
can see that mechanical system consists of an object with mass m, spring with
stiffness k. Also you can see that a spring is a connection between the object
and the fixture. When block/object is displaced at distance x0 from it’s
equilibrium position then the potential energy kx0^2/2 is developed in the
spring. When the system is released from equilibrium, the spring force draws
the block toward the system’s equilibrium position with the potential energy
being converted to kinetic energy. When the block reaches the equilibrium
position, the kinetic energy reaches a maximum and motion continues. The
kinetic energy is converted to potential energy until the spring is compressed
a distance x0. This process of transfer of potential energy to kinetic energy
and vice versa is continual in the absence of non-conservative forces. In a
physical system such perpetual motion is impossible. Dry friction, internal
friction in the spring aerodynamic drag, and other non-conservative mechanisms
eventually dissipate the energy.
Another example of free
vibrations systems with one degree of freedom are: oscillations of the pendulum
about a vertical equilibrium position, the motion of a recoil mechanism of a
firearm once it has been fired, and the motion of the vehicle suspension system
after the vehicle encounters a pothole.
The oscillations of these
systems can be described by the second-order ordinary differential equation. In
this case the dependent variable is time, while the dependent variable is
chosen generalized coordinate. The chosen generalized coordinate represent
displacement of a particle in the system or an angular displacement and is
measured from the system’s equilibrium position.
There exist two possible
methods that can be used to derive the differential equation governing motion
of a one-degree-of-freedom system and they are:
1)
Free body diagram method
2)
Equivalent system method
And as we discussed earlier,
if the system is nonlinear, a linearizing assumption will be made.
The generalized solution of
the second. Order differential equation is a liner combination of two linearly
independent solutions. The arbitrary constants, called constants of
integration, are uniquely determined on application of two initial conditions.
The necessary initial conditions are values of the generalized coordinate and
its first time derivative at a specific time, usually t=0.
The form of the solution for
the differential equation depends on system parameters. A mathematical form of
the solution for an un-damped system is different from the solution of a system
with viscous damping.
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