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Calculate The Response of The Following Underdamped System 3

Reviewed by Calculator Editorial Team

This calculator determines the response of an underdamped second-order system given its natural frequency, damping ratio, and initial conditions. The solution is expressed in terms of the system's natural frequency, damping ratio, and initial displacement and velocity.

Introduction

An underdamped system is a second-order system that oscillates with decreasing amplitude over time. The response of such a system to an initial displacement or velocity can be calculated using the following formula:

x(t) = e-ζωnt [x0 cos(ωdt) + (v0 + ζωnx0) / ωd sin(ωdt)]

Where:

  • x(t) = displacement at time t
  • ζ = damping ratio
  • ωn = natural frequency (rad/s)
  • ωd = damped natural frequency = ωn√(1-ζ²)
  • x0 = initial displacement
  • v0 = initial velocity

The response of an underdamped system is oscillatory with decreasing amplitude. The natural frequency (ωn) determines the frequency of the oscillations, while the damping ratio (ζ) determines how quickly the oscillations die out.

Formula

The complete solution for the displacement of an underdamped system is given by:

x(t) = e-ζωnt [x0 cos(ωdt) + (v0 + ζωnx0) / ωd sin(ωdt)]

Where:

  • ωd = ωn√(1-ζ²) is the damped natural frequency
  • The term e-ζωnt represents the exponential decay
  • The terms involving cos(ωdt) and sin(ωdt) represent the oscillatory component

Note: The damping ratio ζ must be less than 1 for the system to be underdamped. If ζ ≥ 1, the system is overdamped or critically damped.

Example Calculation

Consider an underdamped system with:

  • Natural frequency (ωn) = 10 rad/s
  • Damping ratio (ζ) = 0.2
  • Initial displacement (x0) = 1 m
  • Initial velocity (v0) = 0 m/s

First, calculate the damped natural frequency:

ωd = ωn√(1-ζ²) = 10√(1-0.2²) ≈ 9.6 rad/s

Then, the displacement at time t = 0.5 s is:

x(0.5) = e-0.2×10×0.5 [1 × cos(9.6×0.5) + (0 + 0.2×10×1)/9.6 × sin(9.6×0.5)]

≈ e-1 [cos(4.8) + (2)/9.6 × sin(4.8)]

≈ 0.3679 [0.8776 + 0.2083 × 0.9994]

≈ 0.3679 × 0.8776 ≈ 0.323 m

The system will oscillate with decreasing amplitude, reaching approximately 0.323 meters at t = 0.5 seconds.

Interpreting Results

The response curve of an underdamped system has several key characteristics:

  • The system oscillates with a frequency determined by the damped natural frequency ωd
  • The amplitude of the oscillations decreases exponentially with time
  • The time to reach a specific amplitude can be calculated from the exponential decay term
  • The phase angle between displacement and velocity is determined by the damping ratio

For engineering applications, understanding these characteristics helps in designing control systems that can effectively dampen oscillations while maintaining system stability.

FAQ

What is the difference between natural frequency and damped natural frequency?

The natural frequency (ωn) is the frequency at which the system would oscillate if there were no damping. The damped natural frequency (ωd) is the actual frequency of oscillation when damping is present, which is always less than ωn.

How does the damping ratio affect the system response?

The damping ratio (ζ) determines how quickly the oscillations die out. A lower damping ratio results in slower decay of oscillations, while a higher damping ratio causes the system to return to equilibrium more quickly.

What happens if the damping ratio is greater than 1?

If the damping ratio is greater than 1, the system is overdamped and does not oscillate. The response is a smooth, non-oscillatory return to equilibrium.

Can this calculator be used for real-world engineering problems?

Yes, this calculator provides the theoretical response of an underdamped system. In practice, you would need to consider additional factors such as external forces, nonlinearities, and measurement uncertainties.