For an HBM cyclic analysis, the dynamics equations of a cyclically symmetric structure are transformed to cyclic coordinates (see The General Cyclic Symmetry Analysis Equations) leading to decoupled system matrices.

The structure is subjected to a traveling wave excitation whose fundamental harmonic has engine order (EO), , which is referred to as the fundamental engine order, with higher time harmonics, , having the engine order . This excitation may be represented as [9]:

(1–19)

where the force applied on the sector is given by [9]:

(1–20)

where:

is time

is the time harmonic of the excitation

is the number of time harmonics needed to describe the excitation

is the excitation amplitude for the time harmonic, , applied on the first sector of the physical structure

is the sector index

is the engine order (EO) of the excitation. It may be positive for a backward traveling wave or negative for a forward traveling wave.

is the time frequency of the excitation, which is related to the rotational velocity of the excitation, , by:

(1–21)

Substituting Equation 1–20 into Equation 1–19 we obtain:

(1–22)

where:

is the applied excitation amplitude in cyclic coordinates

(1–23)

where:

is the primitive root of unity.

Note that is equal to some scalar multiple of a column of the complex Fourier matrix. Thus, is the unique spatial harmonic associated with the time harmonic, , of the traveling wave excitation, whose fundamental harmonic has engine order .

In HBM analysis, the displacement solution and the internal forces are assumed to be harmonic with:

(1–24)

(1–25)

Substituting into Equation 1–5 and equating coefficients of , the equation of motion in the frequency domain in physical coordinates for time harmonic, , is obtained:

(1–26)

Because the time harmonic, , is associated with only the spatial harmonic, , it is isolated by pre-multiplying the equation above by :

(1–27)

Due to the block-circulant structure of the matrices of a cyclic symmetric structure [10] and the properties of columns of a Fourier matrix, the nonzero rows of the above equation reduce to:

(1–28)

which is the equation of motion in cyclic coordinates.

is the complex dynamic stiffness matrix in cyclic coordinates associated with time harmonic, , which may be obtained from partitions of real cyclic system matrices [10] and the equivalent direction of the traveling wave component:

(1–29)

(1–30)

where:

Subscripts and represent the DOFs of the base and duplicate sectors

are system matrices obtained by applying cyclic constraints to base and duplicate sectors with the aliased spatial harmonic index :

(1–31)

Note that the direction of the traveling wave component for harmonic, , in the equation above (for and ) is reversed from the direction of the fundamental EO if .

may be further reduced by using reduction techniques such as component mode synthesis.