ASE, Coimbatore 16ED654 Optimization Techniques in Engineering Practice exercise 1 : Problem formulation
1. In the design of a shell-and-tube heat exchanger (Fig. 1), it is decided to have the total length of tubes equal to at least α1. The cost of the tube is α2 per unit length and the cost of the shell is given by α3D2.5L, where D is the diameter and L is the length of the heat exchanger shell. The floor space occupied by the heat exchanger costs α4 per unit area and the cost of pumping cold fluid is α 5L/d 5 N2 per day, where d is the diameter of the tube and N is the number of tubes. The maintenance cost is given by α6 NdL. The thermal energy transferred to the cold fluid is given by α7/N1.2dL1.4 + α8/d 0.2L. Formulate the mathematical programming problem of minimizing the overall cost of the heat exchanger with the constraint that the thermal energy transferred be greater than a specified amount α9. The expected life of the heat exchanger is α10 years. Assume that αi , i = 1, 2, . . . , 10, are known constants, and each tube occupies a cross-sectional square of width and depth equal to d.
Figure 1. Shell and Tube Heat exchanger 2. A beam-column of rectangular cross section is required to carry an axial load of 25 lb and a transverse load of 10 lb, as shown in Fig. 2. It is to be designed to avoid the possibility of yielding and buckling and for minimum weight. Formulate the optimization problem by assuming that the beam-column can bend only in the vertical ( x x y) plane. Assume the material to 3 be steel with a specific weight of o f 0.3 lb/in , Young’s modulus of 30 × 106 psi, and a yield stress of 30,000 psi. The width of the beam is required to be at least 0.5 in. and not greater than twice the depth. Hint: The compressive stress in the beam-column due to Py is Py / bd and that due to Px is
The axial buckling load is given by
Figure 2: Beam – Beam – Column Column
3. A simply supported beam, with a uniform rectangular cross section, is subjected to both distributed and concentrated loads as shown in Fig. 3. It is desired to find the cross section of the beam to minimize the weight of the beam while ensuring that the maximum stress induced in the beam does not exceed the permissible stress (σ0) of the material and the maximum deflection of the beam does not exceed a specified limit (δ0). The data of the problem are P = 105 N, p0 = 106 N/m, L = 1 m, E = 207 GPa, weight density (ρw) = 76.5 kN/m3, σ0 = 220 MPa, and δ0 = 0.02m
Figure 3: Simply Supported beam Formulate the problem as a mathematical programming problem assuming that the crosssectional dimensions of the beam are restricted as x 1 ≤ x2, 0.04m ≤ x1 ≤ 0.12m, and 0.06m ≤ x2 ≤ 0.20 m. 4. It is required to stamp four circular disks of radii R 1,R 2,R 3, and R 4 from a rectangular plate in a fabrication shop (Fig. 4). Formulate the problem as an optimization problem to minimize the scrap. Identify the design variables, objective function, and the constraints.
Figure 4 : Locations of circular disc in a rectangular plate 5. Understand the formulation of a 2D Car suspension system with an objective of minimizing the transmissibility factor (comfort of the passengers) – Deb and Saxena (1997) A two dimensional model of a car suspension system and its dynamic model having 4 degrees of freedom (q 1 to q 4) is show in the Fig.5 and 6 respectively. For simplicity, only 2 wheels (one each in rear and front) are considered. The sprung mass of the car is considered to be supporting on 2 axles by means of coil springs and shock absorber. Each axle contains some unsprung mass which is supported by the tyre.
Figure 5: 2-D model of a car suspension system Step1 : Choosing the design variables
In order to formulate the optimal design problem, first important design variables are identified, which govern the dynamic behaviour of car vibration.
Figure 6: Dynamic car suspension model and important design variables In order to simplify the problem formulation it is considered 4 of the above parameters and other parameters are kept constant as shown below:
l1, l2 are the horizontal distance of the front and rear axles from the CG of the sprung mass. Using the parameters, differential equations governing the vertical motion of the unsprung mass at the front axle (q 1), the sprung mass (q 2) and the unsprung mass at the rear axle (q 4) and angular motion of the sprung mass (q 3) are written (Refer Fig.6)
The parameters d 1,d 2,d 3 and d 4 are the relative deformations in the front tyre, the front spring, the rear tyre and rear spring respectively. The Fig. 6 shows all the 4 dof of the system (q 1 to q 4). The relative deformation in springs and tyres can be written as :
The time varying functions f 1(t) and f 2(t) are road irregularities as functions of time. For example bump can be modeled as f 1(t)=A*sin(pi*t/T), where A is the amplitude of the bump and ‘T’ is the time required to cross the bump. When the car is moving forward, the front wheel experiences the bump first, while the rear wheel experiences the bump later depending upon the speed of the car. The function f 2(t) can be written as f 2(t) =f 1(t- l/v), where ‘l’ is the axle to axle distance and ‘v’ is the speed of the car. f 2(t) can be written as f 2(t)=A*sin(pi*(t-l/v)/T). The coupled differential equations specified in Equations 1.2 to 1.5 can be solved using Runge – Kutta method to obtain pitching and bouncing dynamics of sprung mass ms. The equations can be integrated for time range from zero to tmax. Step 2: Formulating the constraints
For simplicity only one constraint ‘jerk’ is considered (rate of change of vertical accelerations of the sprung mass). Jerk is a major factor concerning the comfort of the riders. Industry standards suggests that the ‘maximum jerk ’ experienced by the passengers should not be more than about 18 m/s2. Mathematically Max q2''' t 18 When 4 coupled equations 1.2 to 1.5 are solved, the above constraint can be computed by numerically differentiating the vertical movement of the sprung mass (q 2) thrice w.r.t ‘time’. Step 3: Formulating the objective function
IN this problem, the primary objective is to Minimize the transmissibility factor which is calculated as the ratio of bouncing amplitude q 2(t) of the sprung mass to the road excitation amplitude ‘A’. The objective function is mathematically represented as
max
Minimize
abs( q2 (t )) A
The above objective function can be calculated from the solution of the 4 differential equations mentioned earlier. Minimum value of the transmissibility factor suggests the minimum transmission of road vibrations to the passengers. The factor is also directly related to the ride characteristics as specified by ISO standards. The optimized design will provide the minimum transmissibility of the road vibration to the passengers with a limited level of jerk. Step 4 : Side constraints
Minimum and maximum limit for design variables cab be set based on previous experience with car suspension design 0 k fs , krs
2
kg / mm
0 f , r 300
kg /(m / s)
Step 5 : Problem statement (Deb and Saxena)
Minimize
max
abs( q2 (t )) A
Subject to 18 max q2 t 0, '''
0 k fs ,
k rs
2,
0 f ,
r
300.
KRK/ 28-12-2017/PE1 ###
