Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Forced Oscillation and Resonance

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Forced Oscillation and Resonance

The decaying energy of a damped oscillation could be restored by applying a driving force. This type of the oscillation is known as forced or driven oscillation. This force could be periodic and has the following form:

Fd=Focos⁡(ωdt)\boxed{F_d=F_o\cos\left(\omega_dt\right)}Fd​=Fo​cos(ωd​t)​

Here ωd\omega_dωd​is the driven angular frequency which is the frequency at which the force is exerted. 

Example: A swinging baby pushed by her father.

Watch Out!
Note: In driven or forced oscillation, the system oscillates with the driven angular frequency.

The amplitude of a driven oscillation is found to be proportional to the following expression: 

A∝1(ωo2−ωd2)2+(bωdm)2A\propto\frac{1}{\sqrt{\left(\omega_o^2-\omega_d^2\right)^2+\big(\dfrac{b\omega_d}{m}\big)^2}}A∝(ωo2​−ωd2​)2+(mbωd​​)2​1​

If ωd\omega_dωd​is close to ωo=km\omega_o=\sqrt{\frac{k}{m}}ωo​=mk​​, the amplitude becomes very large.

Wize Concept
The condition in which ωo=ωd\omega_o=\omega_dωo​=ωd​ is called resonance and it is where the amplitude gets its maximum value.

Exam Tip
Pay attention to keywords which describe resonance such as largest amplitude or largest angle in the problem. 
We can usually solve problems about resonance condition by putting natural frequency equal to the driven frequency.

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Example: Forced Oscillation in a Pendulum

A simple pendulum with the length of 20 cm20\ cm20 cmundergoes a damped oscillation with b=2 kgsb=2\ \frac{kg}{s}b=2 skg​. How many times per second you need to tap the pendulum at its left most position in order to get the maximum amplitude of the oscillation? (Hint: The natural frequency of a pendulum is equal to ω0=g/l\omega_0 =\sqrt{g/l}ω0​=g/l​ where lllis the length of the pendulum.)

Solution:

Note that, by tapping, we exert force on the pendulum and force it to oscillate. Thus, we have a forced oscillation. In addition, since we are looking for "maximum amplitude", we are talking about resonance condition. Hence:

ωd=ω0=gl=9.80.2=7 rads\omega_d=\omega_0=\sqrt{\frac{g}{l}}=\sqrt{\frac{9.8}{0.2}}=7\ \frac{rad}{s}ωd​=ω0​=lg​​=0.29.8​​=7 srad​   at resonance condition
  To find the number of times to tap the pendulum to get resonance, we need to find the frequency at which force is exerted: 

→fd=ωd2π=72π=1.114 Hz\to f_d=\frac{\omega_d}{2\pi}=\frac{7}{2\pi}=1.114\ Hz→fd​=2πωd​​=2π7​=1.114 Hz

Practice: A Tractor Driver and Forced Oscillation

A tractor driver is driving across a field which has regular bumps 5 m apart from each other. The spring under the driver seat has k=5×104Nmk=5\times10^4\frac{N}{m}k=5×104mN​ and the mass of the seat is 60 kg and the mass of the driver is 70 kg. What is the tractor speed at which the driver feels the largest oscillations on the seat?

Extra Practice

Forced Oscillation and Resonance

A tractor driver is driving across a field which has regular bumps 5 𝑚 apart from each other. The spring under the driver seat has k=5×104 Nmk=5\times10^4\ \frac{N}{m}k=5×104 mN​ and the mass of the seat is 60 𝑘𝑔 and the mass of the driver is 70 𝑘𝑔. What is the speed of the tractor at which the driver feels the highest oscillations on the seat?