Walking, traffic vibration, wind, and other factors can cause floors and buildings to sway greatly, and this is often a phenomenon known as “resonance“.
In this article, we will explain this “resonance” in an easy-to-understand manner.
What is natural frequency?
To understand resonance, it is first necessary to explain natural frequencies. Each object has its own unique oscillatory frequency (number of vibrations per second), and this frequency is called the “natural frequency.
When a force such as a tap is applied to an object, the natural frequency of the object starts shaking immediately after the tap. The natural frequency varies depending on the shape and physical properties of the object. For example, the natural frequency of a stringed instrument such as a piano is determined by the length (ℓ) and tension (S) of the string, and when the string is struck, it vibrates at that natural frequency and sounds.
For reference, the natural frequency of a string is expressed by the formula
ｆ = 12ℓ√Sρ
- f: natural frequency Hz (number of vibrations per second)
- ℓ: Length of the string m
- S: Tension N (force to pull the string)
- ρ: wire density kg/m (weight per meter of string)
As can be seen from this relationship, pianos and other instruments change their natural frequencies by adjusting the length and tension of the strings to express the do-re-mi scale.
What is resonance?
Resonance refers to a phenomenon in which an external input of vibration equal to (close to) the natural frequency of an object causes a large increase in sway. For example, on a pedestrian bridge, if the tempo of walking (external input) matches the natural frequency, even a relatively small force can cause a large sway, just as one person walking on a pedestrian bridge may feel a sway.
Incidentally, when sound resonates instead of vibrating, it is called resonance.
Meaning of Resonance Curve
If the natural frequency of an object is 2 Hz (two vibrations per second), the relationship between the frequency of the input vibration (Hz) and the magnitude of the generated vibration oscillation is represented by the resonance curve shown in Figure 1.
As the excitation frequency approaches the natural frequency (2 Hz), the shaking increases, reaching a maximum at the natural frequency. As the excitation frequency moves away from the natural frequency, the shaking becomes smaller.
Figure 1 Resonance curve
How to prevent increased vibration due to resonance
Resonance phenomena, in which even small forces cause large vibrations, can sometimes deteriorate occupant performance (comfort) and become a major problem. There are several ways to prevent this.
Take the case of resonance caused by floor swaying due to walking.
For example, assuming a walking tempo of 2 steps per second (2 Hz), a force with excitation force components at integer multiples of 2 Hz, such as 2, 4, and 6 Hz, is input to the floor (Figure 2). If the natural frequency of the floor coincides with these excitation force components, the floor will resonate and sway significantly.
Figure 2: Force input to the floor by walking
In this case, there are two typical measures to prevent resonance: the first is to avoid matching the excitation frequency with the natural frequency of the floor (i.e., not allowing it to resonate); the second is to reduce the increase in sway due to resonance.
The first specific measure is to raise the natural frequency of the floor to shift it from the excitation frequency of walking to avoid resonance (Figure 3). This is done by increasing the rigidity of the structure and raising the natural frequency by increasing the size of the beams and the number of small beams.
Incidentally, resonance can also be avoided by lowering the natural frequency of the floor. However, lowering the natural frequency of the floor is not a good method because it makes the floor more delicate (easier to sway), and thus the swaying itself may increase even if resonance is avoided.
Figure 3: Measures to prevent resonance by raising the natural frequency
The second measure is to suppress swaying by adding damping to reduce amplification of swaying due to resonance (Figure 4). Specifically, a TMD (Tuned Mass Damper) or AMD (Active Mass Damper), called a damping device, is installed on the floor, and by shaking the weight of the damping device to absorb floor shaking, the increase in shaking caused by resonance can be suppressed (Figure 5).
Figure 4: Measures to suppress increased shaking due to resonance by adding damping
Figure 5: Model of floor-mounted vibration control device