This time, continuing from last time, I will talk about the basics of vibration measurement.
Last time we discussed free vibration, but this time we'll talk about forced vibration. Forced vibration is a phenomenon in which an external force is applied to a vibrating system, causing it to vibrate forcibly. Even if there are damping elements in the vibrating system, the vibration energy is supplied from the outside, so it will continue to vibrate indefinitely. Generally, forced vibration can occur when a force is applied directly to a machine, or when the machine's support system vibrates. In the case of an automobile, the former is the vibration of the engine's explosion, and the latter is forced vibration due to displacement from the road surface, such as on a rough road. Here, we will explain forced vibration caused by an external force.
Similar to the free vibrations discussed previously, let's consider a one-degree-of-freedom vibration system as shown in Figure 1, where a constant external force f(t) is applied to the point mass. In this case, the resultant force of the inertial force, viscous resistance, and restoring force present in this vibration system balances the external force, so the equation of motion is given by equation (1).
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Figure 1: Example of an external force applied to a one-degree-of-freedom vibration system.
Similar to the free vibrations we considered last time, let's consider the case where there is no damping (c = 0) as shown in Figure 2. In this case, the equation of motion is:
It will be.
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Figure 2: Example of 1-degree-of-freedom undamped forced oscillation
Generally, external forces can be considered as complex time signals containing various frequency components, but any waveform can be viewed as a superposition of sine waves. Therefore, we apply a harmonic excitation force f(t) (an excitation force that can be expressed as a sine wave function) and investigate its response displacement x(t). The harmonic excitation force has amplitude F and angular frequency ω;
If we define it as follows and substitute this into equation (2):
It will be.
Solving equation (4):
.................................（5）
(Natural angular frequency)
(Natural frequency)
.................................（6）
.................................（7）
From equation (5), we can see that the response displacement x(t) of the excitation force oscillates with the same angular vibration ω as the harmonic excitation force in equation (3). As the angular frequency ω approaches the natural angular frequency ωn, the amplitude of the displacement increases rapidly, and when the two become equal, the displacement response becomes infinite. This phenomenon is called resonance, and the frequency at this point is called the resonance point. (In reality, there is always damping, so the amplitude will never be infinite...)
In equation (5), if we let / kFXst = (static displacement) and the displacement amplitude be X, then:
.................................（8）
Equation (8) is called the amplitude factor. Furthermore, in equation (5), when the angular frequency ω is greater than the natural angular frequency ωn, the amplitude part becomes negative;
.................................（9）
This can be rewritten as follows. From these, the graph of the amplitude magnification and the phase difference of the response displacement to the excitation force when the angular frequency is swept from near DC to high frequencies is as shown in Figure 3.
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Figure 3 Amplitude and phase in forced oscillation of a one-degree-of-freedom undamped system
From the graph in Figure 3, we can see that at the resonance point, the phase is discontinuously delayed by 180 degrees (π radians), meaning the phase is inverted.
Next, let's consider forced oscillations in damped systems.
Substituting equation (3) into equation (1), the differential equation in this case is:
This is the result. When the damping ratio ζ, which appeared in the previous damped free vibration, is less than 1, solving equation (10) gives:
.................................（11）
Here;
Damping ratio
.................................（12）
Damped Natural Angular Frequency
.................................（13）
Damping rate
The first term on the right-hand side of equation (11) represents the response displacement to forced vibration, and the second term represents a transient vibration that appears at the moment forced vibration is applied in damped free vibration and disappears over time. Figure 4 shows the case where the angular frequency of the excitation force is smaller than the natural angular frequency of the system.
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Figure 4 Transient characteristics of forced vibration in a damped system (when excitation angular frequency < natural angular frequency)
The response displacement in the steady state where free vibrations have disappeared is:
.................................（15）
The amplitude multiplier at this time is given by X st = F/k (static displacement), where X is the displacement amplitude;
.................................（16）
Also, the phase angle φ is:
.................................（17）
From equation (16), even when the excitation force angular frequency matches the natural angular frequency, the amplitude multiplier does not become infinite, but is 1/(2ζ). (Note: This value is not the maximum value.) Also, from equation (17), when the excitation force angular frequency is equal to the natural angular frequency, the phase lag is 90 degrees (π/2 radians), and as it becomes larger, the phase lag approaches 180 degrees (π radians). Figure 5 illustrates the amplitude multiplier from equation (16) and the phase lag from equation (17) (here the lag is considered to be a negative magnitude). Thus, it can be seen that the shape of both graphs depends on the value of the damping ratio ζ.
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Figure 5 Amplitude and phase in forced oscillation of a 1-degree-of-freedom damped system
Figure 6 is a graph showing the amplitude scaling factor on a logarithmic scale.
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Figure 6: Differences in amplitude magnification due to damping ratio ζ
Finally, here's a summary.
(1) In a forced vibration system with one degree of freedom and no damping, when the angular frequency of the harmonic excitation force matches the natural angular frequency, the response displacement becomes infinite, and this state is called resonance.
(2) At the resonance point of a one-degree-of-freedom undamped forced vibration system, the phase of the response displacement to the excitation force is reversed.
(3) The response displacement of a forced vibration system with one degree of freedom damping is a combination of the free-damped vibration and the response displacement vibration of the forced vibration.
(4) The amplitude and phase characteristics of the response displacement in a steady state of a forced vibration system with one degree of freedom damping change depending on the value of the damping ratio ζ.
(5) Near the resonance point of a forced vibration system with one degree of freedom damping, the phase of the response displacement to the excitation force lags from 0 degrees to 180 degrees, and the lag at the resonance point is 90 degrees.
【keyword】
Free vibration, forced vibration, one-degree-of-freedom vibration system, inertial force, viscous resistance force, restoring force, harmonic excitation force, natural angular frequency, natural frequency, resonance, resonance point, amplitude magnification, magnitude factor, damping ratio, damped natural angular frequency, damping rate, transient vibration, steady state
【reference】
- "Introduction to Modal Analysis," by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
- "Technologies and Laws for Pollution Prevention," edited by the Editorial Committee for Technologies and Laws for Pollution Prevention, Japan Industrial Pollution Prevention Association (1990).
(Excerpt from the email newsletter issued on September 17, 2015)
