Technical Report on Vibration-Damping Materials and Their Performance Measurement 1

"Vibration suppression" refers to a technology that converts the vibrational energy of solid surfaces into thermal energy, thereby reducing the vibrations of the solid surface. In contrast, "vibration isolation" corresponds to a technology that reduces the vibration transmission rate between the vibration source and the affected object. It is necessary to clearly distinguish between these two. Vibration suppression not only reduces the vibrations of solid surfaces but can also decrease the solid sound radiated from the surface. It is particularly effective in reducing vibrations near the resonance points of the vibrating surface. This vibration suppression technology has been researched for a long time as a vibration countermeasure, and recently, it has gained attention and research has progressed as an application in noise reduction technologies, while the development of suppression materials has become active.

Vibration damping materials are those that consist of viscoelastic materials (which possess both "viscosity," indicating the fluid's flow characteristics, and "elasticity," indicating the solid's restorative characteristics) bonded to substrates such as steel, wood, concrete, and plastic. These materials can be classified into "non-restrained (substrate + damping material)" and "restrained (substrate + damping material + restraining material)" based on the method of bonding. Non-restrained types include standalone damping materials (some of which come with double-sided tape) that are commercially available. A representative of the restrained type is known as vibration damping steel plates, which are also available on the market.

The loss coefficient η is one of the evaluation indicators of the damping characteristics of damping materials. The original meaning of the loss coefficient is that when there is damping in a vibrating response system, the stress-strain diagram (or load-displacement diagram) draws a hysteresis loop as shown in Figure 1. From this hysteresis curve, the force f = K ₁ χ at maximum displacement χ and the force f = K ₂ χ at zero displacement are measured, and the loss coefficient η is derived from η = K ₂ / K ₁. In the case of the representation method shown in Figure 2, where K ₁ is expressed as the real part and K ₂ as the imaginary part, K ′ is called the complex modulus, K ₁ is called the storage modulus, and K ₂ is called the loss modulus.

	z	η	Ｋ	Δ	Ｄ	Δｆ	Ｑ	ψ
z	−	2 z	ω _Rζ	2 pi zeta	aω _R ζ	ω _R ζ / π	1/2 zeta	4 π ζ
η	eta/2	−	ω_R η /2	pi eta	aω _R η /2	ω _R η /2π	1 over eta	2 π η
Ｋ	K/ω _R	2K/ω _R	−	2 π K / ω _R	a K	K /π	ω _R /2K	4 π K / ω _R
Δ	Delta / 2 pi	Δ / π	ω _N Δ /2π	−	aω _N Δ /2π	ω_N Δ /2π	π /(Δ - Δ ²)	2 (Δ - Δ ²)
Ｄ	D/a ω _R	2D/a ω _R	D/a	2 π D /a ω_N	−	D /aπ	a ω _R / 2D	4 π D /a ω _R
Δｆ	π Δｆ/ω _R	2 π Δｆ/ω _R	π Δｆ	2 π ² Δｆ/ω _N	a pi Delta f	−	ω _R 2 π Δｆ	4 π ² Δｆ/ω _R
Ｑ	1/2 Q	1/ Q	ω _R / 2 Q	π / Q	a ω _R / 2 Q	ω _R / 2 π Q	−	2 π / Q
ψ	psi / 4 pi	ψ / 2 π	ω _R ψ / 4 π	ψ/2	a ω _R ψ / 4 π	ω _R ψ / 4 π²	2 π ψ	−

Table 1

ζ: Damping ratio, the ratio of the viscous coefficient C to the critical viscous damping constant = 2 √km
η: Loss coefficient, indicating the loss when the elastic term is represented as k(1+iη) including spring and viscous resistance
K: Damping constant, for vibrational damping
Δ: Logarithmic decay rate, the natural logarithm of the ratio of adjacent amplitudes when the amplitude decays
D: Decay degree, the amount of decay per second in dB displayed for decaying oscillations
Reverberation time (the time it takes for the signal to decay by 60 dB) T = 60 / D
Δf: Half-width, the frequency change depicted near resonance
The width of the frequency that indicates 1 / √2 of the maximum value of the resonance curve
Q: Q factor, a measure of the sharpness of resonance when the oscillating system resonates, calculated as twice the energy of the oscillating system and the energy supplied externally per cycle to sustain that oscillation
ψ: Intrinsic damping capacity, the ratio of the energy lost per cycle in the oscillating system to the energy stored in the system

The most common indicator of vibration damping performance is the loss coefficient η. The representative measurement methods for the loss coefficient include the half-width method, the decay rate method, and the mechanical impedance method. All of these measurement methods are resonance methods measured near the resonant frequency, although there are also non-resonance methods that are not commonly used. Below is an organized summary of the measurement methods.