Question

For the given parameters for a forced mass-spring-dashpot system with equation mx

′′
+cx
′
+kx=F
0
​
cosωt. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C(ω) of steady periodic forced oscillations with frequency ω. Sketch the graph of C(ω) and find the practical resonance frequency ω (if any). m=1,c=2,k=2,F
0
​
=50

Answer 1

In a forced mass-spring-dashpot system, the amplitude C of steady periodic forced oscillations with frequency ω can be found using the equation C(ω) = F₀/√((k - mω²)² + (cω)²). The graph of C(ω) can be plotted by varying ω. The practical resonance frequency ω can be found by locating the maximum amplitude C(ω).

Step-by-step explanation:

In the given forced mass-spring-dashpot system with the equation mx″ + cx′ + kx = F₀cos(ωt), we can investigate the possibility of practical resonance. The amplitude C(ω) of steady periodic forced oscillations with frequency ω can be found using the equation:

C(ω) = F₀/√((k - mω²)² + (cω)²)

We can sketch the graph of C(ω) by plotting the amplitude C on the y-axis and the frequency ω on the x-axis. To find the practical resonance frequency ω, we look for the maximum value of C(ω). Plugging in the given values for m, c, k, and F₀, we can find the specific values for C(ω) and ω.

Answer 2

Frequency ω of a forced mass-spring-dashpot system, you need to maximize the amplitude of the steady periodic forced oscillations C(ω). This occurs when the denominator of the amplitude equation is minimized. By taking the derivative of the denominator with respect to ω and setting it equal to zero,

The practical resonance frequency ω. The equation for the amplitude of steady periodic forced oscillations C(ω) is given by:

C(ω) = F0/√((k - mω²)² + (cω)²)

Where:

• F0 is the amplitude of the driving force.
• m is the mass of the system.
• c is the damping coefficient (dashpot).
• k is the spring constant.
• ω is the angular frequency of the external force.

The practical resonance frequency ω, you need to maximize the amplitude C(ω).

This occurs when the denominator is minimized since C(ω) is inversely proportional to the denominator.

The minimum of the denominator D(ω), you can take the derivative of D(ω) with respect to ω and set it equal to zero:

dD/dω = 0

Therefore, the amplitude C(ω) is maximized, representing the practical resonance frequency.