In an underdamped system, how many roots does the characteristic equation have?

In an underdamped system, how many roots does the characteristic equation have?

• A. Two identical real roots
• B. One real root
• C. Two complex roots
• D. No roots

Answer: (C)

In an underdamped system, the characteristic equation associated with its differential equation typically has two complex roots. This situation arises in systems where the damping ratio is less than one, indicating that the system experiences oscillations that gradually decay over time. The characteristic equation can be found in second-order linear differential equations, which in standard form is often represented as: s² + 2ζω_ns + ω_n² = 0 Here, ζ is the damping ratio and ω_n is the natural frequency of the system. For an underdamped system, the condition ζ < 1 leads to a discriminant that is negative when solving the characteristic equation, resulting in complex conjugate roots. These roots are of the form: s = -ζω_n ± jω_d where ω_d represents the damped natural frequency. The presence of the imaginary component (jω_d) indicates oscillations in the response of the system, which is characteristic of underdamped behavior. Recognizing this, the correct answer highlights the two complex roots as the defining feature of the characteristic equation for an underdamped system, confirming the nature of the system's response to disturbances.

In an underdamped system, the characteristic equation associated with its differential equation typically has two complex roots. This situation arises in systems where the damping ratio is less than one, indicating that the system experiences oscillations that gradually decay over time.

The characteristic equation can be found in second-order linear differential equations, which in standard form is often represented as:

s² + 2ζω_ns + ω_n² = 0

Here, ζ is the damping ratio and ω_n is the natural frequency of the system. For an underdamped system, the condition ζ < 1 leads to a discriminant that is negative when solving the characteristic equation, resulting in complex conjugate roots. These roots are of the form:

s = -ζω_n ± jω_d

where ω_d represents the damped natural frequency. The presence of the imaginary component (jω_d) indicates oscillations in the response of the system, which is characteristic of underdamped behavior.

Recognizing this, the correct answer highlights the two complex roots as the defining feature of the characteristic equation for an underdamped system, confirming the nature of the system's response to disturbances.