Tools

Types of Oscillators: Simple Harmonic, Damped and Forced Oscillators

Procedure for the experiment is as follows:

1. To study Simple Harmonic Motion

  • Step 1: Set the damping factor (b) = 0.

  • Step 2: Set v0 = 0 m/s.

  • Step 3: Drag the ball to the maximum starting position, say x0 = 10.0 m, and click the play animation button.

  • Step 4: Observe the position graph as shown in Fig. 1 and wait until the animation ends.

  • Step 5: To get the position and velocity graph as shown in Fig. 2, tick the velocity checkbox (V).

Fig. 1: Graph representing S.H.M. motion (position vs time)

SHM position graph

Fig. 2: Graph representing S.H.M. motion (position and velocity vs time)

SHM position and velocity graph


2. To study Damped Oscillatory Motion

  • Step 1: Reset the simulation.

  • Step 2: Set v0 = 0 m/sec.

  • Step 3: Change the damping factor (b) (for example, b = 0.5, 2.83 or 3).

  • Step 4: Drag the ball to the maximum initial position, say x0 = 10.0 m and click the play button.

  • Step 5: Perform the experiment for three different values of the damping factor (b = 0.5, 2.83 and 3), resetting the simulation for each case.

  • Step 6: Repeat the entire procedure as done previously.

Note: After selecting any damping checkbox (Under, Over, or Critical), the value of b is automatically set — no need to manually enter it.

  • Case 1: Under-Damping

    • Set the value of b < 2√2 (say b = 0.5)

    Fig. 3: Graph representing under-damped oscillatory motion (position vs time)

     Under-damped position graph

    Fig. 4: Graph representing under-damped oscillatory motion (position and velocity vs time)

     Under-damped position and velocity graph

  • Case 2: Critical-Damping

    • Set the value of b = 2√2 ≈ 2.83

    Fig. 5: Graph representing critically-damped oscillatory motion (position vs time)

     Critical-damped position graph

    Fig. 6: Graph representing critically-damped oscillatory motion (position and velocity vs time)

     Critical-damped position and velocity graph

  • Case 3: Over-Damping

    • Set the value of b > 2√2 (say b = 3)

    Fig. 7: Graph representing over-damped oscillatory motion (position vs time)

     Over-damped position graph

    Fig. 8: Graph representing over-damped oscillatory motion (position and velocity vs time)

     Over-damped position and velocity graph

In all the above cases, observe the graph and wait until the animation ends.


3. To study Forced Oscillatory Motion

  • Step 1: Reset the simulation again.

  • Step 2: Set the damping coefficient (b) > 2√2 (say b = 3).

  • Step 3: Change the value of initial velocity v0 = 2 m/sec.

  • Step 4: Set value of driven angular frequency (ω) = 1.41 rad/s.

  • Step 5: Set some value of driven amplitude F0 which will be the force for oscillation.

  • Step 6: Drag the ball to the maximum starting position, say x0 = 10.0 m and press the play button.

  • Step 7: Observe the graph till the animation ends.

Fig. 9: Graph representing forced-oscillatory motion (position vs time)

Forced oscillatory position graph

Fig. 10: Graph representing forced-oscillatory motion (position and velocity vs time)

 Forced oscillatory position and velocity graph


Perform the following experiments

Experiment 1:

Perform an experiment to calculate the value of k (spring constant) for the cases described below. In each case, record the observations in the table and compute the value of k. Take the value of m = 1 kg in each case.

Case 1: Free oscillator (Simple Harmonic Motion) when b = 0

  1. Set different values for x0.

  2. Keep other parameters = 0.

  3. Record the value of time period (T) in each case.

  4. Calculate the corresponding angular frequency ω = 2π/T.

  5. Then compute k (spring constant) using the formula:

    omega = sqrt(k/m)
    ω = √(k/m) ⇒ k = mω²

Table 1: Test values for Free Oscillator (b = 0, v0 = 0, m = 1 kg)

S.No x0 (m) T (s) f (Hz) ω (rad/s)
1 1.0 4.44 0.225 1.414
2 2.0 4.44 0.225 1.414
3 4.0 4.44 0.225 1.414
4 6.0 4.44 0.225 1.414
5 8.0 4.44 0.225 1.414
6 10.0 4.44 0.225 1.414

The mean ω = 1.414 rad/s

Computed value of k = mω² = 1 × (1.414)² = 2.0 N/m


Case 2: Under-Damped Oscillator when b < 2√2 (say b = 0.5)

  1. Set different values for x0.

  2. Set b = 0.5 Ns/m and keep other parameters = 0.

  3. Record the value of time period (T) in each case.

  4. Calculate the corresponding damped angular frequency:

    damped angular frequency
    ωd = √(k/m − b²/(4m²))

Value of b used in the experiment: 0.5 Ns/m

Table 2: Test values for Under-Damped Oscillator (b = 0.5, v0 = 0, m = 1 kg)

S.No x0 (m) T (s) f (Hz) ωd (rad/s)
1 1.0 4.51 0.222 1.392
2 2.0 4.51 0.222 1.392
3 4.0 4.51 0.222 1.392
4 6.0 4.51 0.222 1.392
5 8.0 4.51 0.222 1.392
6 10.0 4.51 0.222 1.392

The mean ωd = 1.392 rad/s

Computed value of k = m(ωd² + b²/(4m²)) = 1 × (1.9375 + 0.0625) = 2.0 N/m


Case 3: Forced oscillator when b > 2√2 (say b = 3)

  1. Set different values for F0.

  2. Set driven angular frequency ω = 1.41 rad/s.

  3. Obtain the value of steady-state amplitude (A) by observing the graph each time.

  4. Obtain the value of natural frequency ω0 using the relation:

    forced oscillation amplitude formula
    A = F0 / √((k − mω²)² + b²ω²)

  5. Then compute k (spring constant) from the calculated mean value of ω0.

Parameters: b = 3 Ns/m, ωdrive = 1.41 rad/s, x0 = 10 m, v0 = 2 m/s

Table 3: Test values for Forced Oscillator (b = 3, ω = 1.41 rad/s, m = 1 kg)

S.No F0 (N) A (m) ω0 (rad/s)
1 1.0 0.236 1.414
2 2.0 0.473 1.414
3 3.0 0.709 1.414
4 5.0 1.182 1.414
5 8.0 1.891 1.414
6 10.0 2.364 1.414

The mean ω0 = 1.414 rad/s

Computed value of k = mω0² = 1 × (1.414)² = 2.0 N/m