Issue 32

N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02 16 φ is 0.4 and C is a coefficient dependent on the carrier fluid of the MR fluid (C = 1 for hydrocarbons), according to [17] The magnetic B-H relationship of a MR fluid can be defined as [18]:               0 10.97 1.133 0 0 1.91 1  mrf H mrf B e H (2) Therefore, a MR fluid’s relative permeability can be defined as:            0 10.97 1.133 0 0 1.91 10.97 mrf H r mrf dB e dH (3) where, B is in Tesla, H mrf is in A/m, and μ 0 = 1.25x10 -6 H/m is the permeability of free space. Fig. 4 shows the graphs of the B-H relationship of the MRF 140-CG, the values of yield stress τ B and the relative magnetic permeability as a function of the magnetic field intensity H mrf . In order to reach the best performances, the material which composes the magnetic circuit should have high magnetic permeability and high magnetic saturation. A material with such properties is the AISI 1010, which is a low-carbon steel (C% < 0.10). This material though, is hardly available because of is being used for niche applications. Hence the AISI 430 was used. AISI 430 is a ferritic stainless steel with a high relative magnetic permeability, of about 600. (a) (b) (c) Figure 4 : MRF 140 CG [15] properties: B-H relationship (a) , yield stress τ B vs H (b) and relative permeability vs H (c) . Analytical design of the MR damper Fig. 5 shows the forces developed by a magnetorheological damper [19, 20]. Considering the parallel-plate Bingham model, the forces can be decomposed into three contributions [21]. First, the controllable force F τ , Eq. (4), directly correlates with the magnetic field applied through the yield stress τ B .    ( ) B P A D L A F c sign V h (4) where L P is the axial activation length of the piston head, A A is the annular piston’s area, h is the fluid gap and c is a coefficient that depends on the volumetric flow rate, the viscosity and the yield stress. Second, F η , Eq. (5), represents the viscous forces and depends on the length of the orifice, the fluid’s viscosity and flow rate.    3 12 A QLA F k wh (5) where Q is the flow rate, L is the total axial length of the piston head, w is the mean circumference of the damper’s annular flow path and k is a constant depending on the volumetric flow rate and the velocity. Third, F f that stands for the friction forces like those related to the seals system. Moreover we should also account for the force derived from the effect of pressure F P . Hence, the total force will be obtained by adding up all these contributions:       tot f P F F F F F (6) The Dynamic Range D, is also a fundamental parameter which provides an estimate of the influence of the control variable on the system behavior. D can be calculated as the controllable forces divided by the uncontrollable forces (Eq. 7).