24 O. Ozaydin and E. Cigeroglu Bearing Coupling Moving Disk Stationary Disk Actuator Tail Drive Shaft Fig. 3.1 Tail drive line mN Fig. 3.2 Dynamic model of tail drive line 3.2 Mathematical Model of the System The tail drive shaft system shown in Fig. 3.1 consists of a shaft supported by two bearings at both ends and connected to two couplings. A dry friction damper is used in the middle of the shaft in order to decrease vibration amplitudes. The tail drive shaft is modeled by Euler-Bernoulli Beam theory and spring elements are used to represent the bearings and couplings as shown in Fig. 3.2. Equation of motion of an Euler Bernoulli beam can be written as follows A @2w.x; t/ @t2 C EI @4w.x; t/ @y4 C c @w.x; t/ @t D f .t/ı .x L2/ fn .w.x; t//ı .x L1/ ; (3.1) where w(x, t) is transverse displacement, is density, Ais cross sectional area, Eis Young’s Modulus, I is moment of inertia, c is viscous damping coefficient, fn(w(x, t)) is the nonlinear friction force, f (t) is external forcing, L1 is the location of dry friction damper and L2 is location of external forcing. 3.3 Solution Methodology For the solution of the partial differential equation of motion given in Eq. (3.1) Galerkin’s method is used. Utilizing expansion theorem, transverse displacement of the shaft can be expressed as given below; w.x; t/ DX j aj.t/ j.x/; (3.2) where aj(t) and j(x) are the j th generalized coordinate and mass normalized eigenfunction of a beam with springs at the supports are included. Substituting Eq. (3.2) into Eq. (3.1) the following result is obtained.
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