Vibration Theory 2.2: Damped Forced Oscillations

 

4.2 Under Damped System

 

If the vibrational system includes damping, the complementary solution and particular solution are determined as follows.

 

By substituting the particular solution given by equation (4.15) into the vibrational equation (equation 4.13) and simplifying, we obtain the following expression.

 

Here, to simplify the expression, we use 2k/m=ω2 and c/m=2ωξ, and set the sine and cosine terms to 0. The resulting expression is as follows.

 

Using β=p/ω, rearranging once again, the expression becomes as follows.

 

And solving the system of equations for G1 and G2, we obtain the following.

 

Therefore, the particular solution is determined as given in equation (4.16).

 

Here, the phase angle, representing the phase difference between the direction of the external force and the resulting response, is given by equation (4.17).

 

Therefore, the final general solution x(t) is given by equation (4.18).

 

The second term in equation (4.18) represents the oscillatory component determined by the initial conditions of the vibrational system. In a damped vibrational system, this component gradually diminishes over time and is referred to as the transient response. It becomes negligible when the external force acts over an extended period. On the contrary, the first term continuously oscillates in response to the external force and, after a sufficient amount of time, settles into a steady-state oscillation with a constant amplitude.

Here, the first term in equation (4.19), F0/k, represents the static displacement due to the static load F0, and the second term represents the ratio of dynamic displacement to static displacement. Therefore, it is referred to as the dynamic magnification factor or dynamic response factor.

 

Figure 4.2 and Figure 4.3 display the dynamic magnification factor (D) and phase angle (θ).

As evident from the figures, D and θ are functions of the frequency ratio (β) and damping ratio (ξ), defined as the ratio of the excitation frequency to the natural frequency of the vibrational system. The variations of dynamic magnification factor and phase angle with changes in the frequency ratio illustrate the dependence on the damping ratio ξ.

 

Fig- 4.2 Dynamic Magnification Factor based on Frequency Ratio

Fig- 4.3 Variation in Phase Angle based on Frequency Ratio

 

The maximum value of the dynamic magnification factor D occurs when the square root term in equation (4.20) is minimized. Denoting the frequency ratio at this point as βm​, we can express it with equation (4.21).

 

Therefore, the maximum dynamic magnification factor at the frequency ratio given by equation (4.21) is expressed as equation (4.22).

In the case of no damping (ξ=0), the maximum dynamic magnification factor occurs at βm​=1, resulting in the maximum response. For damped vibrational systems, βm​<1, so the frequency ratio at which the dynamic magnification factor is maximized is always less than 1.

Typically, the frequency ratio at which the dynamic magnification factor is maximized is referred to as the resonance frequency. In undamped systems, the resonance frequency occurs when the frequency ratio is 1. If we define the resonance frequency in this way, then substituting β=1 into equation (4.22) results in D=1/(2ξ), which is not the maximum value, but it signifies the resonance magnification factor.

During resonance, the amplitude of steady-state oscillation in the system is larger than the amplitude of the input, but it does not increase indefinitely. Instead, it increases gradually with time and eventually converges. This is because energy from the input is transferred to the vibrational system, and the damping force in the system, if present, consumes energy until equilibrium is reached. If there is no damping, the response would diverge to infinity.

 

Furthermore, the phase difference between the direction of input displacement and the response displacement, as represented by equation (4.17), is illustrated in Figure 4.3.

As evident from the figure, the phase difference during resonance is always 90 degrees, irrespective of the presence or absence of damping.

 

The statement that the phase difference is 0 degrees or 180 degrees has the following physical interpretations:

 

• A phase difference of 0 degrees implies that the direction of the input displacement completely aligns with the direction of the response displacement.

• A phase difference of 180 degrees means that the direction of the input displacement is completely reversed compared to the direction of the response displacement.

 

In other words, in the case of undamped systems, when an external force acts with a frequency below the resonance frequency, the response occurs in the same direction as the force. However, as the frequency of the external force increases and surpasses the resonance frequency, the direction of the force and the response reverses.

For damped systems, this transition occurs gradually.

 

Closing Remarks

 

To those who have followed from free vibrations to forced vibrations, you have truly worked hard.

Now, in dynamics, there are no equations more complex than this, and although it may seem complicated, the mathematics is not difficult at all.

 

Let's revisit the physical meaning of the solution for forced vibrations with damping (Equation 4.18).

Examining the equation, we see that it is composed of a linear combination of the general solution for the case when external force is zero and the particular solution for forced vibrations.

The general solution vibrates with the structure's natural frequency, as studied in free vibrations.

What about the particular solution? The oscillation frequency mentioned in the sine wave function is represented by P, the frequency of the external force.

What does it mean that the mathematically derived response is expressed with different frequencies?

In other words, just as we observed in free vibrations, a structure tends to move with its own natural frequency.

However, no matter how much the structure has its own natural frequency, the external force causes it to vibrate at the frequency of the external force.

And the amplitude increases as the two frequencies match, but decreases as the two frequencies differ, eventually decreasing around the resonance point.

 

In summary, a structure vibrates in response to the frequency of the external force, responding beautifully when the frequency of the external force is close to the structure's own frequency. But if the frequencies don't match, the response remains minimal.

It's similar to the love between a man and a woman, where mutual understanding is crucial for a successful relationship. This reflects the nature of all things.

 

So, what properties does a seismic-resistant structure utilize? This relates to the fact that seismic waves, as a type of external force, have very few low-frequency components.

Therefore, by understanding the frequency characteristics of seismic waves, we use the power of modern science to extend the natural period of a structure to prevent it from resonating with earthquakes.

 

Next, let's explore the meaning of phase angle during resonance.

Many tend to overlook the importance of the phase angle depicted in Figure 4.3, which is as crucial as dynamic magnification.

The figure illustrates the phase angle in the case of no damping, which is often skipped over. The phase angle remains at 0 degrees until the vibration ratio reaches 1. Afterward, it suddenly shifts to 180 degrees, a phenomenon depicted like a hidden picture.

Additionally, as damping ratio increases, the phase angle gradually changes from 0 to 180 degrees.

What does this mean? In the case of no damping, as the frequency of the external force increases until resonance, the direction of the external force aligns with the response direction. However, after resonance, there is an abrupt reversal in the direction of the external force and the response.

You can experience this by placing a toy on a surface and gently shaking it horizontally. While shaking slowly, the toy will move in the same direction. But as you increase the shaking speed, there will be a moment when the toy's movement is felt in the opposite direction, and this is the resonance point.

 

Such resonance phenomena are also utilized in vibration control devices. Imagine hanging a pendulum with a matching natural frequency on the rooftop of a building. During an earthquake, the building vibrates with its natural frequency.

The installed pendulum, having a resonant match with the building, vibrates tremendously. However, due to the 180-degree phase angle during resonance, the additional motion is in the opposite direction to the building, preventing the building from shaking.

This concept is similar to the traditional use of poison as a remedy, a prescription found in traditional Korean medicine.

 

#Under damped systems
#Dynamic magnification factor
#Resonance frequency

 

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