Reference no: EM132737458

ENM7005-B Modelling and Optimisation - University of Bradford

Question 1: Machine Tool

A machine tool spindle is supported by two rolling bearings of stiffness k₁ = C kN/μm and stiffness k₂ = D kN/μm (see diagram). Data: spindle radius r = (50 + B) mm, overhang h = (100 + A) mm, Young's modulus for spindle material (steel) is 210 kN/mm².

Distance between bearings is x and α = h/x

Deflection per unit load is given by

δ = h³/3EI (1+1/α) + (1+1/α)²/K₁ + α²/k₂

I = Πr⁴/4 is the moment of inertia of the spindle section. Be careful with units!

(a) Develop an objective with your parameters, identifying design variables and constraints as required.

(b) Select and implement a suitable numerical method to determine spindle spacing ?? for minimum deflection per unit load (max. stiffness) to a tolerance of ±0.1 mm. You should (i) justify your method selection, (ii) provide tabulated and/or graphical information showing solution progress, (iii) state the optimised design value and objective, and (iv) determine the tool-tip deflection Δ.

(c) (i) Explore the analysis for a selection of two or three other suitable spindle materials and briefly report on a comparison of the outcomes with an evidenced recommendation for a preferred material. You will need to seek suitable data.

(ii) Is there any merit in changing the spindle radius? Build a simple 1-factor 3-level model based on numerical results to evidence a response to this question.

Question 2: Multi-stage Compressor

It is required to minimise work done per unit volume on a gas in a multi-stage compressor, from pressure P) via pressures {P₁, ... , P_n-1} to final pressure P* (in atm), given by

w/V = P0/k [∑ⁿ_i=1(P_i/P_i-1)^k - n], k =1 -1/r

(a) For P₀ = 1, P_n = 2A, and r = 2B, develop an objective with your parameters, identifying all design variables and constraints as required.

(b) (i) Numerically investigate the optimal solutions for 2, 3, and 4-stage compressors. In each case (i) identify the feasible design space, (ii) evidence a valid initial design point, and (iii) indicate why any constraints you've identified may be redundant.

Provide tabulated information of solution progress, stating optimised intermediate pressures p to p*+! and objective value accurate to three decimal places.

(ii) Confirm the numerical result obtained for n = 3 by solving the problem analytically.

(c) From your results in (b) discuss the value (or otherwise) of increasing the number of stages in the compressor.

Question 3: Tubular Column

A thin-walled tubular column (internal radius r, wall thickness t) is subject to a compressive load. The column is to be designed for minimum weight, subject to direct stress and a buckling constraints, described by:

Minimise: f(r,t) = rt Subject to: r > 0, t > 0
g₁(r, t) = 1/rt - (150 + A) ≤ 0

g₂ (r, t) = 1 - (6400 + 3C)r³t ≤0

(a) In the design space, plot the objective function contours and the constraint functions to obtain a graphical solution. Discuss the need, or otherwise, for the side constraints given in the description

(b) Obtain a numerical solution using two initial design points, (0.1,0.2) and (0.1,5.0), and discuss your solution.

(c) Repeat part (b) with the additional constraint that the radius must be at least 10 times the tube-wall thickness (you will need to formulate the constraint) and discuss your solution).

Question 4: Renewable Energy Integration

You are working on a renewable energy integration development for a small country that includes competing energy supply systems: hydroelectric, wind, and solar. A Project Team has proposed 13 alternatives for the country's energy solution. Your Technical Team have assessed the benefits expected in each objective under each alternative.

(a) Are any of these alternative proposals sub-optimal? If so, how would you explain the meaning of sub-optimality to the Project Team?

(b) You need to determine and explain the trade-off between the alternatives to the Project Team. Suggest different visualisation plots you might use to illustrate the trade-off.