# Mass Spring system

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)
• rest-length=1000
• w=sqrt(k/m) = 10 (theoretical value), leading to a Period of T=0.2p.
• T(measured in no-loss condition)=62812/10=0.62812
• External Force (amplitude)F 0 = 20 000. a 0 = F 0 /m = 666.6
• loss of velocity =0.01 per simulation step (1/128 = 7.8125e-3)
The frequency of the external force has been selected equal to the natural frequency of the system, so that it will be just in resonance. The mass has an initial zero velocity and zero displacement in respect to the equilibrium position. As soon as you click "animation" the mass will start vibrating with a gradually increasing amplitude. As the system has losses (0.01 loss of velocity per simulation 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 shown 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. You 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
• A (measured) = 52.08
The theoretical value for A, at resonance is a 0 /(2bw). The value for b in our case is (see 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

• A (theoret.) = 52.083
very close to the measured one.

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.

• d (measured) = 90º
In coincidence with the predicted theoretical value.

#### Assignments

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 resonance.