Harmonic Vibration क्या है? | Undamped & Damped System Analysis in Hindi

Harmonic Vibration वह motion है जिसमें system periodic या sinusoidal displacement exhibit करता है। यह mechanical vibrations का fundamental type है और both undamped और damped systems में analyze किया जाता है।

1. Undamped Harmonic Vibration

Single Degree of Freedom (SDOF) undamped system में:

m(d²x/dt²) + kx = 0

Solution:

x(t) = A sin(ωₙ t) + B cos(ωₙ t)

• ωₙ = √(k/m) → natural frequency
• Amplitude constant रहती है
• Energy conserved रहती है

2. Damped Harmonic Vibration

Viscously damped SDOF system के लिए:

m(d²x/dt²) + c(dx/dt) + kx = 0

Under-damped solution (0 < ζ < 1):

x(t) = A e^{–ζωₙ t} sin(ω_d t + φ)

• ω_d = ωₙ √(1 – ζ²) → damped natural frequency
• Amplitude धीरे-धीरे decay होती है
• Energy dissipates over time due to damping

3. Parameters of Harmonic Vibration

• Amplitude (A) – Maximum displacement from equilibrium
• Frequency (f) – Number of oscillations per second
• Period (T) – Time for one complete oscillation
• Phase angle (φ) – Initial position of motion at t = 0

4. Applications

• Machine foundation vibration analysis
• Vehicle suspension systems
• Seismic response studies of buildings
• Understanding resonance in mechanical and structural systems

5. Conclusion

Harmonic vibration mechanical vibrations का base concept है। Undamped systems में amplitude constant रहती है, जबकि damped systems में amplitude decay होती है। यह structural और mechanical system analysis में fundamental parameter है।