2 degrees of freedom system
As a simple example, let's consider a point mass model of a two-degree-of-freedom system with no damping, N=2, as shown in Figure 3 of the previous issue. The equation of motion can be written as follows. Let the mass, stiffness, and displacement be m1, k1, x1, m2, k2, and x2 respectively.
m1・x1”−k1(x2−x1)=0
m2・x2”＋(k1＋k2)x2−k1・x1=0 (1)
x1'' and x2'' represent the second derivatives of x1 and x2, respectively.
The dot in m1・x1 represents multiplication.
If we group x1'', x2'', x1'', and x2 together, the above expression can be written as follows:
┏ m1 0┓┏ x1”┓ ┏ k1 - k1┓ ┏ x1┓ ┏ 0 ┓
┗ 0 m2┛┗ x2”┛ ＋ ┗ -k1 (k1+k2)┛┗ x2┛ ＝ ┗ 0 ┛ (2)
The sudden shift to matrix notation might be unfamiliar to some since high school, but this matrix representation will be helpful for understanding the material going forward.
Here
M＝┏ m1 0 ┓ K＝┏ k1 - k1 ┓
┗ 0 m2 ┛ ┗ - k1 (k1+k2)┛
E(0) = ┏ 0 ┓
┗ 0 ┛
X”＝┏ x1” ┓ X＝┏ x1 ┓
┗ x2” ┛ ┗ x2 ┛
If we represent this as a matrix, then equation (2) is
M・X”+K・X＝E(0) (3)
This is expressed as an equation of motion using a matrix of springs and point masses.
If you divide this by M
X”＋K／M・X＝E(0) (4)
From this, we can find the "eigenvalues" and multiply them by 1/2π to find the natural frequencies. Furthermore, by finding the "eigenvectors" from the obtained eigenvalues, these eigenvectors represent the shape of the vibration. Finite element analysis (FEM) and other methods generate the above-mentioned mass matrix M and stiffness matrix K from data with defined shapes and perform eigenvalue analysis to calculate the natural frequencies and vibration mode shapes computationally.
Let the eigenvector be Φ. Multiplying both sides of the equation of motion in equation (3) by Φ⁻¹ (the inverse matrix of Φ) from the left and Φ from the right,
Φ^-1・M・X”・Φ＋Φ^-1・K・X・Φ＝E(0)
Organize
Φ^-1・M・Φ・X”＋Φ^-1・K・Φ・X＝E(0)
Φ^-1・M・Φ and Φ^-1・K・Φ are both "normalized diagonal matrices (triangular matrices)". This means that the combined (mutually influencing) equations of motion of a 2-degree-of-freedom system are separated into two 1-degree-of-freedom models (equations formed by combining independent equations of motion). Mathematically, this can be expressed as "a coordinate transformation into two orthogonal degrees of freedom". The same applies to multi-degree-of-freedom systems; conversely, this means that "the equations of motion of an N-degree-of-freedom system that mutually influence each other can be considered as a collection of N independent 1-degree-of-freedom system equations of motion". If we can separate and examine a multi-degree-of-freedom system into individual 1-degree-of-freedom systems, then we can separate and examine each of the multiple power spectral peaks analyzed by FFT.
This application forms the basis of the concept of modal analysis.
The concept of modal analysis
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A 2-degree-of-freedom system model can be decomposed into two 1-degree-of-freedom system components. Similarly, an N-degree-of-freedom system model can be decomposed into N 1-degree-of-freedom system components. This is the concept behind modal analysis.
So, what is this equation of motion used for? For example, if we want to perform vibration control, let's say we add an actuator for vibration control to the object. To be continued...
(Note) Eigenvalues, eigenvectors, standardization, diagonal matrices (triangular matrices), orthogonal, and independent are mathematical terms.
(Excerpt from the email newsletter issued on February 19, 2004)
