Mass Spring system
Let us start with the following
The characteristics of this
system are the following:
m=30 with a charge q=1000 to allow the application
of an external force.
- k=3000 (spring constant)
- w=sqrt(k/m) = 10 (theoretical value), leading
to a Period of T=0.2p.
- T(measured in no-loss
- External Force (amplitude)F 0 = 20
a 0 = F 0 /m = 666.6
- loss of velocity =0.01 per simulation step
The frequency of the external
force has been selected equal to the natural frequency of the system,
that it will be just in resonance. The mass has an initial zero
and zero displacement in respect to the equilibrium position. As soon
you click "animation" the mass will start vibrating with a gradually
amplitude. As the system has losses (0.01 loss of velocity per
step), the oscillations will reach a stable amplitude.
In the next section a brief
review of the theory of forced oscillations is presented. There it is
that the amplitude of the oscillation for the steady state is given by:
So that, in resonance (w = w
0 ) the amplitude should be 2bwa 0 .
To measure this steady state
amplitude allow for time being in the order of 10 (for this system) and
use the step by step execution until the value of Vx change the sign.
will observe an amplitude (value of x) around 52.
HINT: remember that disabling
monitor in the particle inspector window allows a faster execution. To
detect the change in sign of Vx you should activate monitor again.
Another important parameter
is the phase delay, d, between the position, x, and the acting external
force. In the section on "Theory of Forced oscillations", this delay is
shown to be 90º when the system is at resonance.
Checking the results
For this experiment we have
measured a steady state amplitude of
The theoretical value for
A, at resonance is a 0 /(2bw). The value for b in our case is
section on "Theory of Forced oscillations")
2b = xyzVLoss/xyzTimeStep =
0.01/(1/128) = 1.28
In addition, a 0 = 666.66
and w = 10.00. Substituting in the expression for A we get the value
very close to the measured
The phase delay, d, in this
case is easily measured to be 90º: When the external force is zero
the x value is maximum or minimum.
In coincidence with the predicted
Change the loss factor and:
Check that the time needed to reach the steady
state is in the order of 10/(2b) (do not change the time step!)
- Measure and compute the new amplitude
- Change the period of the external force and
observe that the amplitude decreases in respect to the value it has at