Technical Report on Vibration-Damping Materials and Their Performance Measurement 2

The frequency response function H(jω) of a one-degree-of-freedom system vibrating under an external force F is defined with its displacement as X, dynamic spring constant as k, viscous damping coefficient as c, and mass as m.

Thus, let the maximum value of the amplitude of this frequency response function be |H| ² _max, and let the angular frequency at this point be ω ₀. If the frequencies at which the amplitude near ω ₀ is 1/2 of |H| ² _max (half the amplitude power) are ω ₁ and ω ₂, the loss factor is given by the following equation.

In actual tests, it is common to read the frequencies f ₁ and f ₂ at points that are 3 dB lower than the peak value of the frequency response function, and to determine the damping performance from the resonance frequency f ₀.

Determine from the formula above (η = K(f ₂ − f ₁)/f ₀) using the half-width of the imaginary part of the apparent mass (force/acceleration) (not 1/√2 but 1/2). Note that the imaginary part of F / α has all "negative" values, input acceleration into FFT's CH A and input into CH B, and if the valley (it seems that measuring from the peak of α/F with the signal inverted might also work, but the imaginary parts of these two are completely different and do not match even if flipped) is denoted as -X, then f ₁ and f ₂ are determined at the positions where the following values occur.

(The correction value is the same as that for the magnitude of the frequency response function, although the magnification factors are different.)

In any of the above cases

Perform calculations using the formula.

is represented. a ₀ is a constant determined by the initial conditions. The rate of decay of this envelope D (dB / sec) is,

It becomes. Therefore,

From this, the loss coefficient can be determined. However, ω ₀ = 2 πf ₀.

In a similar way, measure the time T ₆₀ it takes for the vibration to decay by 60 dB.

It is also possible to determine the loss coefficient. This method is called the reverberation time method. These methods are generally used when the damping is small, and when the vibration damping waveform is displayed through a logarithmic amplifier like a level recorder, a straight line proportional to the envelope is obtained, from which the damping rate can be measured based on the slope of this line. Additionally, a method using the Hilbert transform to measure the damping rate is also effective.

When determining the loss coefficient from the damping waveform, one should not adopt the slope of the waveform immediately after the start of damping or the point where the waveform approaches a constant value in the latter half of the damping. Additionally, if the damping waveform is affected by lower-order vibration modes, resulting in a non-linear appearance, filtering should be performed. The amplitude values of the damped sine wave at a certain time and the next amplitude value are denoted as χ _n and χ _n+1, respectively, and their natural logarithm value is referred to as the logarithmic damping rate, which is commonly used to represent the damping capacity of metallic materials.

When measuring the loss coefficient of materials with significant attenuation, the half-value width method may not yield accurate measurements, and in such cases, the impedance method is considered an effective technique. If we denote the amplitude of the mechanical impedance (F/V) of a one-degree-of-freedom system under forced vibration as |z|, at the resonance point,Thus, |z| = c, and the impedance will represent the viscous damping coefficient. Here, V represents the vibration velocity. Therefore, the loss coefficient can be determined from the following equation.

The mass m can be determined, strictly speaking, from μ∫W _i² dx (where μ is the mass per unit length and W _i is the reference function of the i-th vibration mode). Similarly, the loss coefficient can be obtained from the mechanical compliance (X / F). When a one-degree-of-freedom system is vibrating under a forcing function (f = f ₀ e ^jwt), the compliance can be expressed as g(1 + η j).

Thus, if we denote the real part of the expression as R and the imaginary part as I, then g and η are

can calculate more.

impedance

In practical terms, the following formula is used in the case of the following system.

Calculate here.

f ₀: Resonant frequency (Hz)
M: Sample weight (below the weight) + upper weight (kg)
h: Thickness of the sample (m)
A: Sample area (m ²) (below the weight)
E: Young's modulus (N/m ²)

The loss coefficient is usually calculated using the ratio of accelerometers 1 and 2 (vibration transmission rate) by the half-width method.

The half-width method and the damping ratio method are both used to measure loss coefficients at resonance points or near them, and are also referred to as resonance methods. However, the technique presented here can determine loss coefficients at any frequency.

This testing method involves burying one end of a long beam specimen in sand and vibrating the other end with a vibrator to measure the wavelength λ (m) and vibration attenuation D (dB/m) generated in the beam from standing or traveling waves, and to determine the loss coefficient using the following formula.