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where Pk is the kth Tooth ID on Pi. For a given planet Pi
i

and accelerometer Aj, the local rotation angle of the planet gear is given by:
θj Nr )(θij
pi =(1 . + m(2π)) (18)
Np sep

where m is the number of carrier cycles from the initial ori-entation, θc = 0, and the identity:
Nr
θpi =(1 . )θc (19)
Np

is used that relates a planet’s local rotation to that of the carrier angle. Recall, m starts at 1 due to the rejection of the .rst carrier cycle in the data. Once θpij is determined, then:
Ni j .. = .oor(mod(θipi,2π)缸V 2. π 1 ) (20)

Assembly Holding Matrix
To ease the assembly, handle special cases, and assess the individual contributions of each extracted waveform to the overall Vibration Separation Vector , an Assembly Holding Matrix is used. Figure 17 is a schematic of the Assembly Holding Matrix. Each column of the TSA set

Fig. 17. Assembly Holding Matrix (Mv = 5)

is windowed and assigned to a column within the Assem-bly Holding Matrix according to the target gear’s mapping index N.. .
ij

This matrix has length缸V and width Nset . Each column vector of the TSA Set is assigned a position within one of the columns in the Assembly Holding Matrix correspond-Each column of the Assembly Holding Matrix consists ing to the tooth numbers the data represents. This is given of zeros, except for the locations where data is inserted.
by The Vibration Separation Vector is created by summing
RANGE:{N.
ij,N.
ij + 1,N.
ij +缸vs . 1} (21) across the columns.
where N.
ij = Ni j .. . .oor(缸 2 E ) (22) An example output is given in Figure 18.

SGVS-General

Sun Gear Vibration Separation is challenging because it seeks to isolate dynamics associated with a given sun gear tooth through a simultaneously meshing planet gear. In ad-dition, if the hunting tooth ratio between the sun gear and the ring gear is not equal to Ns, only a fraction of the teeth are aligned when data is extracted; thus, a direct tooth-to-tooth comparison is not possible for all teeth. Figure 19 shows the sun gear where the are numbered clockwise.
　
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