The cervical disc is indicated for implantation in neck joints of skeletally mature patients for reconstruction of a degenerated cervical disc following arm pain and/or a neurological deficit, due to a single-level abnormality localized to the disc space. Considering that a commercially available cervical disc is composed of cobalt-chromium (mainly Co, 30 % Cr and 0.8 % Fe) endplates and a central polyethylene core, the broad radius of the polyethylene core allows for the required joint motion (as shown below). Co-Cr Spherical interface [ “Selectively constrained” articulation Polyethylene Cylindrical interface Co-Cr (a) Please propose a manufacturing process flow for producing the cervical disc, excluding surface finishing steps. You may include drawing to describe the shape changes of materials throughout the process. Here, for a simple analysis, we may consider the cervical disc as 3 blocks of materials as shown below. The upper and lower Co-Cr blocks have the same side lengths Loc=20 mm) and thickness Hour (= 2 mm). Surface finish (rough) Lace Lacr Co-Cr 1 Hag Surface finish (smooth) Polyethylene Radaptor Lac Lao Co-Cr 1 Hace Surface finish (smooth) 9 Surface finish (rough) Assuming the weight of head for a 70-kg man (blood volume -5 L) is –5 kg and the neck joint has Wweck (100) times per day of rotations with 0 degree (< 90° or 1/2 radian) everyday, we may consider the wear only between the upper Co-Cr plate and the polyethylene core with a radius of Rare with an equivalent sliding distance of 26R0/3 for each the neck rotation. The allowable core is between 5 mm and 8 mm. The polyethylene core contains a rod structure with a radius of Rudytor (= 5 mm) for adhesion to the lower Co-Cr plate. Further, the ion release rates of iron QR = 0.6 ug/cm2/week), chromium (Qc = 0.03 ug/cm/week) and cobalt Qo = 0.1 mg/cm2/week) should also be considered together with the wear effect in order to ensure all the ion levels are within the ratio fibresbad = 6 % or 0.06) of the maximum allowable intake levels for iron (IFe = 200 ug/L/day), chromium (IG = 35 mg/L/day) and cobalt (Io = 28 mg/L/day). To enable the cell attachment on the uppermost and lowest surfaces (grey surfaces shown in the above figure) of the cervical disc, surface finish should be performed to obtain the surface roughness Oranged between 2 um and 3 um. On the other hand, the inner surfaces (highlighted with dotted shading in the above figure) should be smooth enough (Cosworth $ 200 nm) to minimize the cell attachment. To achieve the required surface roughnesses, we can implement slab milling on each of the 4 surfaces with the path including 2 stages of slab milling as described in the below figure (Here, we only shown the ‘smooth surfaces’ as an example, with an allowance about Dw around the core perimeter or the rod. The ‘rough surfaces’ do not have the circular vacancies.) H. First Stage Second Stage Drill/2 Dwi Dw/2 Surface profile D./2 N Surface profile OrR D/2 . ť D. + Hill Hi! Du/2 HAY The chosen milling cutter has a diameter Dosil = 3 mm) and a height Hai = 2 mm), with the number of teeth on the cutter 1 (= 2). The moving velocity has to be set the same throughout the milling stages. Between the 2 stages, the milling head would change the orientation and move from the end position of stage 1 to start position of stage 2. The maximum moving speed/feed (F) rate of milling head is 50 mm/s. The maximum rotational speed of the milling cutter is 6,000 rpm, yet the suggested rotational speed for this milling operation (N) is only 300 rpm. In addition, the milling machine need other processing time for setting up/finalization the machine process (T. = 2 min), re-orientation of the milling head/working material between the milling stages (Thange = 10 s). For the costs, the machine cost (M) is $0.5 /min with an overhead ratio (OH) 0.1; and the labor rate (W) is $3 /min with an overhead ratio (OH) 0.25. Each the tool (milling cutter) costs Cow = $120; and the tool life (Tow) is ~3 hr cutting time. We assume that the tool is not broken during the process. (b) Now, please propose an optimization statement (including both the objective function and the constraints) for the minimum cost for the surface finish process. (Note: Use symbols only, but not the given values, for this part.) (c) Based on your answer in part (b), please fin the minimized manufacturing cost by considering the above given values provided, with explanations. Further, discuss on how much production cost can be potentially saved by comparing this value with the maximum possible cost of a feasible manufacturing process.

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