To solve the question regarding which type of vibrations has a constant amplitude, we can analyze each option step by step:
### Step-by-Step Solution:
1. **Understanding Free Vibrations:**
- In free vibrations, a system oscillates without any external force acting on it after an initial disturbance. For example, if a mass is attached to a spring and is pulled and released, it will oscillate back and forth.
- In the absence of damping forces (like friction), the amplitude of these vibrations remains constant over time. The motion can be described by the equation:
\[
y(t) = A \sin(\omega t + \phi)
\]
- Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant. Since there are no external forces acting on the system, the amplitude does not change.
**Conclusion:** Amplitude remains constant in free vibrations.
2. **Understanding Damped Vibrations:**
- Damped vibrations occur when there are forces (like friction or air resistance) that oppose the motion of the oscillating system. This damping force causes the amplitude of the oscillation to decrease over time.
- The amplitude decreases exponentially, and eventually, the system may come to rest. Therefore, in damped vibrations, the amplitude does not remain constant.
**Conclusion:** Amplitude does not remain constant in damped vibrations.
3. **Understanding Maintained Vibrations:**
- Maintained vibrations involve an external force that compensates for any damping forces acting on the system. This external force keeps the system oscillating at a constant amplitude.
- An example could be a mass-spring system where an external periodic force is applied to maintain the oscillation against damping. Therefore, the amplitude can remain constant in this case.
**Conclusion:** Amplitude remains constant in maintained vibrations.
4. **Understanding Forced Vibrations:**
- Forced vibrations occur when an external periodic force is applied to the system. This force can maintain the amplitude of the oscillation at a constant value, even in the presence of damping.
- The external force can be expressed as \(F(t) = F_0 \cos(\omega t)\), where \(F_0\) is the amplitude of the external force. This force can counteract the effects of damping, allowing the system to oscillate with a constant amplitude.
**Conclusion:** Amplitude remains constant in forced vibrations.
### Final Answer:
From the analysis:
- **Free vibrations**: Amplitude remains constant.
- **Damped vibrations**: Amplitude does not remain constant.
- **Maintained vibrations**: Amplitude remains constant.
- **Forced vibrations**: Amplitude remains constant.
Thus, the correct options where the amplitude of vibrations remains constant are **(i) free vibrations, (iii) maintained vibrations, and (iv) forced vibrations**.
