Technical Report: Vibration Damping Materials and Their Performance Measurement (Part 2)

The frequency response function H(jω) of a one-degree-of-freedom system vibrating under an external force F is given by the following, where X is the displacement, k is the dynamic spring constant, c is the viscous damping coefficient, and m is the mass:

The maximum value of the amplitude of this frequency response function is |H|. ² Let max be the angular frequency at this point, and let ω0 be the amplitude |H| around ω0. ² The frequency at which the amplitude power is halved (1/2 of the maximum) is ω. ₁, ω ₂ Therefore, the loss coefficient is given by the following equation.

In actual testing, the common method involves reading the frequencies _f1 and _f2 at points 3 dB smaller than the peak value of the frequency response function, and then determining the vibration damping performance from the resonant frequency _f0.

The full width at half maximum (1/2, not 1/√2) of the imaginary part of the apparent mass (force/acceleration) is used to determine the value from the formula (η = K(_f2-_f1)/f0). However, the imaginary part of F/α is always negative. If we input acceleration into CH A and CH B of the FFT and let the value of the trough (it might seem possible to input the signals in reverse and measure from the peak of α/F, but the imaginary parts of these two are completely different and will not match even if reversed) be -X, then _f1 and _f2 are determined at the position where the following values are obtained.

(Although the magnification is different, the correction value is the same as when using the MAG of the frequency response function.)

In any of the above cases

The calculation is performed using the following formula.

It is expressed as follows: _{a 0} is a constant determined by the initial conditions. The decay rate D (dB/sec) of this envelope per second is:

Therefore,

The loss coefficient can then be calculated, where _ω₀ = _2πf₀.

Using a similar approach, we can measure the time T ₆₀ required for the vibration to decay by 60 dB.

Furthermore, the loss coefficient can also be determined. This method is called the reverberation time method. These methods are generally used when the attenuation is small. When the vibration attenuation waveform is displayed through a logarithmic amplifier such as a level recorder, a straight line proportional to the envelope is obtained, and the attenuation rate is measured from the slope of this line. Another effective method for measuring the attenuation rate is to use the Hilbert transform.

When determining the loss coefficient from a damped waveform, the slope at the point where the waveform asymptotically approaches a constant value immediately after the start of damping or in the latter half of damping should not be used. Also, if the damped waveform is affected by low-order vibration modes or other factors and does not form a clean straight line, filtering should be performed. Furthermore, the natural logarithm of the amplitude values of a damped sine wave, χ _n and χ _n+1, is called the logarithmic damping rate and is often used to represent the damping capacity of metallic materials.

When measuring the loss coefficient of a material with high damping, the half-width method may not provide accurate measurements, and in such cases, the impedance method is considered an effective technique. If the amplitude of the mechanical impedance (F/V) of a 1-degree-of-freedom system undergoing forced vibration is |z|, then at the resonance point...Therefore, |z| = c, and the impedance represents the viscous damping coefficient. Here, V represents the vibration velocity. Thus, the loss coefficient can be calculated from the following equation.

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

This is the result. If we let R and I be the real and imaginary parts of the expression, then g and η are

It can be calculated more easily.

Impedance

In practical terms, the following formula is used for the system described below.

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 (m2) (under the weight)
E: Young's modulus (N/m²)

The loss coefficient is typically calculated using the ratio of Accelerometer 1 and 2 (vibration transmission coefficient) and the half-width method.

The half-width method and the attenuation method, among others, both measure the loss coefficient at or near the resonance point and are also called resonance methods. However, the method described here can determine the loss coefficient at any frequency.

This test method involves burying one end of a long beam specimen in sand and exciting the other end with a vibrator. The wavelength λ (m) and vibration damping D (dB/m) are then measured from the standing or traveling waves generated in the beam, and the loss coefficient is calculated using the following formula.