tailieunhanh - Hydrodynamic Lubrication 2011 Part 9

Tham khảo tài liệu 'hydrodynamic lubrication 2011 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sinusoidal Squeeze with a Soft Surface 153 We then solve Eqs. and simultaneously under the boundary condition Pn 1 0 using the Newton-Raphson method. Since Eq. includes h ị the time change of the pressure distribution and that of the fluid film shape cannot be obtained by a single iterative calculation. The equation is calculated from the beginning for every time step and the hi value obtained is used as h for the next time step. High-Frequency Squeeze When the frequency of the sinusoidal squeeze motion becomes high the viscoelasticity of the rubber cannot be ignored. For high frequencies the apparent elastic coefficients increase and the phase difference between stress and strain becomes significant. Consider a square rubber block. For simplicity divide the block into many columns pillars as shown in Fig. and assume that each column deforms in the axial direction only and independently from each other. Also assume that the dynamic characteristics of rubber can be expressed by the spring-dashpot models of three elements four elements and five elements shown in Fig. and that the dynamic behavior of a column can be expressed by the following constitutive equation a a1cr a2a E e b1é b2 é where E is Young s modulus. Fig. . Column model and viscoelastic models of rubber The stress required to deform a column at a constant strain rate can be expressed by the following function of time 154 7 Squeeze Film C1 c2t c3 exp -t r1 c4 exp -t r2 F t F t in the above equation is called the constant-strain-rate modulus. The coefficients ci i 1 2 3 4 and ri i 1 2 of the above equation can be expressed in terms of the coefficients a1 a2 b1 b2 of Eq. and E. Using F the stress an after an arbitrary strain history can be expressed approximately as Xn n F tn - ti-1 i 1 - 3-2 At where ai and ei are the stress and strain respectively at time t ti i At. The strain before t to has been assumed to be zero. Equation