I've put together a summary of what we discussed last time as a review. It's compiled from quotes in "Introduction to Modal Analysis" by Professor Akio Nagamatsu (published by Corona Publishing).
Free vibrations and natural modes
Free vibration of an object is an inevitable consequence of the absence of external forces, and is solely due to physical and mechanical reasons.
From an energy perspective, free vibration is the movement of energy only within a system isolated from the outside world, and it alternates between the forms of kinetic energy and potential energy. Natural frequencies are the frequencies at which free vibration can occur while satisfying the law of conservation of energy.
When a multi-degree-of-freedom system or continuum undergoes free vibration, it can only vibrate in a form specific to that system, called an eigenmode. The number of eigenmodes is equal to the number of degrees of freedom of the system. Each eigenmode has its own unique frequency, and free vibration does not occur at other frequencies. Furthermore, the free vibration due to each eigenmode has a unique damping ratio.
"Natural modes, natural frequencies, and modal damping ratios" are three fundamental phenomena that govern the dynamic characteristics and overall behavior of not only free vibration, but also forced vibration, transient response, self-excited vibration, and servo analysis, and are collectively referred to as modal parameters.
On the other hand, vibrations in an object are generated and their properties are determined by three types of physical characteristics: mass, stiffness, and damping. The mode characteristics that represent the phenomenon of vibration and the inherent nature of the material that determines the vibration, i.e., the physical characteristics, correspond in a 3:3 ratio, and both are mutually convertible as two sides of the same dynamic characteristic.
Of the three mode characteristics, the natural frequency and mode damping ratio are common to the entire system and are global terms that do not change even if the excitation point or response point changes. In contrast, the natural modes indicate the distribution state of the response within the system and are local terms that change when the vibration point or response point changes.
In identifying mode characteristics, the frequency response function is approximately represented not by all eigenmodes of the system, but by a linear combination of only the eigenmodes included within the target frequency range. To express the influence of eigenmodes outside the target frequency range that were omitted in this process, inertial constraints (coefficients approximating the lower-order eigenmodes, the reciprocal of which is called the extra-power mass and has the dimension of mass) and extra-power compliance (coefficients approximating the higher-order eigenmodes, the reciprocal of which is called the extra-power stiffness and has the dimension of stiffness) are introduced. These are approximations that can be derived from the eigenmodes and are not essential, so they are derived quantities.
Additionally, mode mass, mode stiffness, and mode damping coefficient may be added as mode characteristics, but these are also derived quantities determined from the three mode characteristics.
While physical properties such as mass, stiffness, and damping constitute a physical model, modal properties such as natural modes, natural frequencies, and modal damping ratios constitute a modal model.
Mode Analysis
Experimental identification is the process of identifying the dynamic characteristics of a system based on frequency response functions measured in impact tests, vibration tests, and other methods. Experimental identification can be divided into methods for identifying mode characteristics and methods for identifying physical characteristics. The most commonly performed method is the identification of mode characteristics, and this entire process from experiment to identification is called experimental modal analysis. Experimental identification is also called curve fitting because it determines the mode characteristics to fit the frequency spectral curve or time history response curve.
In theoretical modal analysis, such as the finite element method, a modal model is derived from a mathematical model represented by the equations of motion, consisting of modal characteristics obtained through eigenvalue analysis. The frequency response function and time history response are then used with this modal model. In contrast, experimental modal analysis determines the modal model based on the frequency response function and time history response obtained from vibration tests. Although both theoretical and experimental modal analysis are called modal analysis, it is important to recognize that they follow completely opposite paths.
Natural frequencies and natural modes, when viewed from the perspective of force equilibrium, represent the speed and shape at which an object can vibrate in a free state without external forces, with internal forces balanced in all degrees of freedom. Similarly, from an energy perspective, they represent the speed and shape at which an object can vibrate in an isolated state, with the initial energy flowing into the system conserved. Mathematically, they represent a meaningful (moving) solution where the right-hand side of the equation of motion is zero.
In multi-degree-of-freedom undamped equations of motion
[M]{x“}＋[k]｛x｝＝｛0｝ ・・・（1）
This solution
xi＝Φiexp(jΩt） ・・・ （2）
Let's set this and represent it as a vector.
{x}＝｛Φ｝exp（jΩt)
Furthermore, since xi'' = -Ω^2exp(jΩt), substituting this into equation (1) gives
｛−Ω^2[M]＋[k]}{Φ}＝{0} ・・・ （3 ）
The special condition for having a solution other than stationary ({x}=[0}) is Ω = √k/m for a single-degree-of-freedom system, but for a multi-degree-of-freedom system...
｛−Ω^2[M]＋[k]}＝｛0｝ ・・・ （4）
Since [M] and [k] are known, we can find Ω from equation (4). Ω can be found as many times as there are degrees of freedom, and by substituting this Ω into equation (3) and finding {Φ} as a ratio of Φi, we obtain the solution. The amplitude {x} is not determined as an absolute value, but can be found as a ratio, and it takes a specific vibrational form (mode), and since this form is an intrinsic value determined by M and k, it is called an eigenmode.
Conversely, if we consider that an N-degree-of-freedom system has N eigenmodes, then it would seem that an N-degree-of-freedom system cannot vibrate in any mode other than these N eigenmodes. This appears contradictory, because the vibrations of actual machines and structures, both free and forced, are infinitely varied and can change in countless ways. This can be explained by the following three reasons. Firstly, eigenmodes merely indicate the form of vibration; their magnitude, or absolute quantity, can change infinitely depending on the magnitude of the initial disturbance in the case of free vibration, or the magnitude of the excitation force in the case of forced vibration. Secondly, vibration in a single eigenmode is extremely rare; in most vibrations, multiple eigenmodes are mixed together to form a single phenomenon. And the degree of this mixing can change infinitely depending on the initial disturbance and excitation force. Thirdly, all actual machines and structures are continuums, and therefore have infinite degrees of freedom. Standard modal analysis focuses on the second of these factors, creating a physical model by determining the degrees of freedom of the object through modeling. This model is then converted into a mathematical model using force equilibrium and energy principles. The resulting equations are solved through theoretical analysis and numerical calculations to determine the natural vibrations and their natural modes. Excitation forces are applied to determine the degree of mixing of the natural vibrations, and these are combined to obtain the response.
There are several methods for experimental modal analysis.
The response function of a multi-degree-of-freedom system can be represented as a superposition of the frequency response functions of single-degree-of-freedom systems. When the mode damping ratio is small, the frequency response function of its eigenmode is dominant. The frequency response function near its resonance can be considered as a single-degree-of-freedom system, and the mode characteristics of only that eigenmode can be determined independently. This method is called the single-degree-of-freedom method. One method of this method, which involves determining the response function from the magnitude of the frequency response function, is as follows.
The equations of motion for a 1-degree-of-freedom system are
mx"＋cx'＋kx＝Fexp（jωt)
From this, the frequency response function G(ω) of compliance is
G(ω）＝X/F＝1/k÷（1-β^2＋2jζβ） ・・・ （6）
β＝ω/Ω、Ω^2＝k/m
Using this as a reference, we can express the equation for an N-degree-of-freedom system as follows:
G(ω)＝Σ1/Kr÷(1-βr^2＋2jζrβr) ・・・ (7)
Σ represents the sum of r=1 to n.
Correcting equation (7) with the aforementioned inertia constraint C and excess compliance D yields the following equation.
G(ω)＝Σ1/Kr÷（1-βr^2＋2jζrβr）＋C/ω^2＋D
Here, we will proceed with equation (7).
Since equation (6) is the same if we consider only the r-th order natural mode, we can determine equation (6) by finding the natural frequency Ω and natural damping ζ.
The magnitude of equation (6) is
|G|=1/k÷√{(1-β^2)^2+(2ζβ)^2} ・・・(8)
{ } indicates what is inside the square root.
Since equation (8) is the MAG (amplitude ratio) of the frequency response function, it is maximum at the resonant frequency, and for ζ≪1,
|G|max≒1/（2kζ） ・・・ （9）
Furthermore, the resonant frequency ω0 is ζ≪1
ω0 = Ω√(1-2ζ^2) ≈ Ω ... (10) () indicates what is inside the square root.
Therefore, by measuring the MAG of the self-compliance frequency response function (the point where the excitation point and response point are the same), and using the frequency ω0 of the resonant peak of the r-th order eigenmode,
Ω ≈ ω0 = 2πf0 (f0 = frequency of |G|max) ... (11)
By the half-width method
ζ＝⊿ω/2Ω ・・・（12）
Δω is the difference between 2πfb and 2πfa, where fb and fa are the frequencies of |G|max/√2.
Since |G|max can be read from the measured value, from equation (9)
k＝1/（2ζ|G|max） ・・・（13）
Also, from Ω^2 = k/m
m＝k/Ω^2 ・・・（14）
In this way, the resonant frequency ω0, mode mass m, mode stiffness k, and mode damping ratio ζ can be determined. Similarly, the eigenmodes of other orders can be determined, and an approximate formula can be obtained from their superposition (equation 7).
(Excerpt from the email newsletter issued on January 20, 2005)
