312 S.D. Yavuz et al. 2c 1c 1( ) k t 2( ) k t ( ) 1e t ( ) 2e t 1θ 2θ 3θ 1r 2r 3r 1I 2I 3I 1 2b 2 2b Fig. 30.2 Dynamic model of the system where g1 and g2 are nonlinear displacement functions defined mathematically as g1 D 8ˆ ˆ< ˆˆ: hr1 1 .t/ Cr2 2 .t/ Ce1 .t/i b1, hr1 1 .t/ Cr2 2 .t/ Ce1 .t/i >b1 0, ˇ ˇ ˇr1 1 .t/ Cr2 2 .t/ Ce1 .t/ˇ ˇ ˇ b1 hr1 1 .t/ Cr2 2 .t/ Ce1 .t/iCb1, hr1 1 .t/ Cr2 2 .t/ Ce1 .t/i < b1 , (30.2a) g2 D 8ˆ ˆ< ˆˆ: hr2 2 .t/ Cr3 3 .t/ Ce2 .t/i b2, hr2 2 .t/ Cr3 3 .t/ Ce2 .t/i >b2 0, ˇ ˇ ˇr2 2 .t/ Cr3 3 .t/ Ce2 .t/ˇ ˇ ˇ b2 hr2 2 .t/ Cr3 3 .t/ Ce2 .t/iCb2, hr2 2 .t/ Cr3 3 .t/ Ce2 .t/i < b2 . (30.2b) The above three-degrees-of-freedom semi-definite system can be reduced to a two-degrees-of-freedom definite system by defining the following two new coordinates: p1 .t/ Dr1 1 .t/ Cr2 2 .t/ Ce1 .t/, (30.3a) p2 .t/ Dr2 2 .t/ Cr3 3 .t/ Ce2 .t/. (30.3b) These new coordinates represent the relative gear mesh displacements, which are the combinations of the dynamic and static transmission errors. Using Eqs. (30.1)–(30.3), the following new system of equations is obtained Rp1 .t/ Cc1 r2 1 I1 C r2 2 I2 . p1 .t/ Cc2 r2 2 I2 . p2 .t/ Ck1 .t/ r2 1 I1 C r2 2 I2 g1 .t/ Ck2 .t/ r2 2 I2 g2 .t/ D r1 I1 T1 C r2 I2 T2 C Re1 .t/, (30.4a) Rp2 .t/ Cc1 r2 2 I2 . p1 .t/ Cc2 r2 2 I2 C r2 3 I3 . p2 .t/ Ck1 .t/ r2 2 I2 g1 .t/ Ck2 .t/ r2 2 I2 C r2 3 I3 g2 .t/ D r2 I2 T2 C r3 I3 T3 C Re2 .t/. (30.4b) In order to obtain the dimensionless equations of motion, the following transformations are applied: m1 D I1I2 r2 1I2 Cr 2 2I1 , m2 D I2 r2 2 , m3 D I2I3 r2 3I2 Cr 2 2I3 , (30.5a-c) k1 .t/ D k1 .t/ k1m , k2 .t/ D k2 .t/ k2m , (30.5d,e) !2 11 D k1m m1 , !2 12 D k2m m2 , !2 21 D k1m m2 , !2 22 D k2m m3 , (30.5f-i) 11 D c1 2m1!11 , 12 D c2 2m2!12 , 21 D c1 2m2!21 , 22 D c2 2m3!22 , (30.5j-m)
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