Vibration Analysis -7 "Miscellaneous Stories about Transfer Functions -3" (Transfer Functions and Bode Plots, Frequency Response Functions and Transfer Functions, Meaning of Dynamic Mass and Dynamic Stiffness, Mechanical Impedance, Vibration Modes)

(6) Transfer function and Bode plot

A transfer function describes the relationship between the input and output of a transfer system.
A Bode plot is a method of representing a transfer function as output signal / input signal, using a pair of amplitude ratios and phase differences (relative to the input signal) for each frequency. When performing a shock test with an impulse hammer and Accelerometer, the input signal is the force signal of the impulse hammer (unit: N), the output signal is the acceleration signal of Accelerometer (unit: m/s²), and the transfer system is between the point of impact and the measurement point where Accelerometer is attached. The amplitude ratio (acceleration/force), which is output/input, represents the magnitude of the acceleration at the excited measurement point when a force of 1N is applied. Therefore, to predict the magnitude of the acceleration when a force of any magnitude is applied, it can be obtained simply by multiplying the magnitude of that force by the transfer function. For this reason, it is easier to understand the frequency response function (transfer function) as a filter characteristic (function). The point to note here is that both the force waveform and the acceleration waveform can be Fourier transformed, so they can be thought of as periodic functions, or simply as sine waves of a certain frequency.
For example, if the measured Bode plot shows an amplitude ratio of 0.8 (m/s²/N) at 10 Hz, then the acceleration when a sine wave force of magnitude 2N and 10 Hz is applied will be 0.8 × 2 = 1.6 (m/s²). For this reason, Bode plots, which are represented on the frequency axis, are frequently used.

(7) Frequency response function and transfer function

I have been using the terms frequency response function and transfer function interchangeably, but what is generally measured with an FFT analyzer is called the frequency response function (FRF).

FRF = Input/Output Cross-Spectrum ÷ Input Power Spectrum
(Alternatively, output power spectrum divided by input/output cross spectrum)

It is required.
A transfer function is defined as the Laplace transform of the impulse response being measured, but in practice, the two terms are used almost interchangeably.
The frequency response function obtained using an FFT analyzer has a name determined by which of the force input signal's response (displacement, velocity, or acceleration) is used to represent it, as shown in Table 1.

Table 1

(Note) Compliance is also called acceptance, admission, or dynamic flexibility. Acceleration is also called inertance.

Table 1 shows the relationship between compliance (G(ω)) and other frequency response functions. These relationships stem from the fact that they can be expressed using complex exponential functions. Velocity is obtained by differentiating displacement by 1 degree, so it is the displacement multiplied by jω, and acceleration is obtained by differentiating displacement by 2 degrees, so it is the displacement multiplied by (jω)² = -ω². As can be seen from these relationships, frequency response functions are generally complex numbers, and since j = ejπ/2, each multiplication by jω corresponds to multiplying the amplitude by ω and advancing the phase by 90°. Integration is the opposite of differentiation, dividing by jω, and corresponds to lagging by 90°. The following equations show the method of finding the value by differentiation with respect to displacement x and the method of finding the value by integration with respect to acceleration a.

Displacement = x, Acceleration = a

Speed = jωx Speed = a/(jω)

Acceleration = (jω) ² x Displacement = a/(jω) ²

　　　　　＝−ω ²x　　　　　　　＝−a／ω²

FFT analyzers have this kind of calculus function on the frequency axis, and can display the frequency response function shown in Table 1.

References: Introduction to Modal Analysis, by Akio Nagamatsu, Corona Publishing Co.

(8) Meaning of dynamic mass and dynamic stiffness

What do these frequency response functions represent? Let's consider the physical dimensions of dynamic mass = force/acceleration.

Force (N) = Mass (kg) × Acceleration (m/ ^s²)

Force/acceleration = mass (kg)

This is what it will be.
Similarly, the force/displacement ratio for dynamic stiffness is Force = Spring constant × Displacement, so it becomes the spring constant (stiffness).
The frequency response function is a frequency characteristic, and if we let M be the mass and K be the stiffness,

Frequency = 1/(2π)・√(K/M) ...(1)
(K/M) indicates the route.

Therefore, the frequency response function can be viewed as a change in mass and a change in stiffness.

Let's consider these changes in mass and stiffness a little more.
Experientially, when you touch something that is vibrating, you will notice that the magnitude of the vibration decreases at some points and remains constant at others. This can be thought of as the antinodes (where vibrations are easily altered) and the nodes (where vibrations are less easily altered). In fact, when vibration measurements are taken at each location, it is observed that the vibration is large where it is easily altered and small where it is less easily altered. The frequency response functions at the antinodes and nodes at these points are shown in Figure 1.

Figure 1: Vibration Modes

If we view the diagram from the perspective of a frequency response function, the Y-axis represents a ratio, so assuming a constant force as described above, the antinodes are areas where acceleration is high and therefore the mass at frequency f1 is small and easily moves, while the nodes are areas where mass is large and difficult to move. The fact that the effect of added mass from touching the object by hand is less as the mass increases is the reason why the points supporting the object being measured are taken as nodes. Touching the antinodes changes the overall mass with even a small added mass, causing the resonant frequency f1 to shift downwards. This is why, when measuring light objects, it is necessary to use the lightest possible Accelerometer, and in some cases, non-contact displacement sensors or laser Doppler vibrometers. You intuitively understand that it takes a lot of force to move a heavy object, right?

Now, the frequency when there is an added mass is given by equation (1), where m is the added mass and f2 is the resonant frequency.

　　　f2=1／（2π）・√（K／（M＋m）　　　　　　 ・・・(2)

As stated in section (8), M + m = force/acceleration can be considered a dynamic physical constant determined by the shape of the vibration mode and its frequency f2.
When we consider f2 as a range of frequencies, this is different from static mass, which is measured on a scale; it is called dynamic mass in the context of vibration analysis.
Similarly, K = force/displacement is also called dynamic stiffness.

(9) Mechanical impedance

In addition to mass and spring constant, damping is another vibration element. Force/velocity (N/ms⁻²) represents damping. Mechanical elements and electrical elements are related in their respective ways.

Voltage V = Excitation force F,
Current I=speed u,
Resistance R = Damping R,
Capacitor C = (1 / spring constant) C,
Inductance L = mass m

This can be considered in relation to the above.
The impedance (R, C, L) = V/I is force/velocity, so force/velocity is called mechanical impedance, and, like item (9), it represents dynamic damping.
Damping is a combination of internal damping within the structure and external damping such as viscous damping. This amount of damping is determined by testing, such as by measuring the loss coefficient, but it is fundamentally subject to change depending on temperature and measurement conditions, and this must be taken into consideration during measurement.

(10) Vibration Mode

To investigate the vibration modes, one excitation point is defined, and multiple measurement points are set to represent the shape of the object being measured, and the frequency response function is measured. This can be rephrased as measuring the response to the excitation force (displacement, velocity, acceleration) by measuring the distributed mass and spring constant at each measurement point. Therefore, the vibration of a typical structure is essentially collecting data to create a model of the concentrated mass and spring constant at each measurement point for the vibration of a continuous body.
In reality, many points are not measured, and the object being measured, which is an infinite mass system model, is modeled using a finite number of measurement points. It is important to note that this inherently results in model errors.
To roughly plot 10 Hz vibration modes from a Bode plot, plot the positions of each acceleration measurement point on the X-axis. Read the phase difference and amplitude ratio at each measurement point at 10 Hz. Mark points corresponding to the magnitude of the amplitude ratio, in the positive direction of the Y-axis if the phase difference is positive, and in the negative direction of the Y-axis if the phase difference is negative. Connecting these marked points with lines will plot the 10 Hz vibration modes.

(Excerpt from the email newsletter issued on July 22, 2004)