Design Optimization Of Power Screw Jack 3f7231

Reena M Arora* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 9, Issue No. 2, 171 - 177

“Design Optimization of Power Screw Jack” Reena M Arora 1* Mtech student PCEA nagpur india, [email protected] om

M.K. Sonpimple 2 Assist. Professor PCEA Nagpur india, me.mksonpimple@ pce.ltjss.net

Kiran R. Kaware 3 Mtech student PCEA nagpur india kiran.kaware@gma il.com

Dilip L Budhlani 4 Sr.Lecturer GNIET Nagpur India dilipbudhlani@redi ffmail.com

Pankaj M Arora 5 Shift Manager INDORAMA Nagpur India [email protected]

1.Introduction

A power screw (or power screw), also known as a lead screw or translation screw, is a screw designed to translate turning motion into linear motion. Common applications are machine slides (such as in machine tools), vises, presses, and jacks Power screws are manufactured in the same way as other thread forms. A power screw can be used in conjunction with split nut.

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The design of automotive or industrial power screw jack involves many interrelated parameters. It is necessary to understand this interrelationship and the constraints involved to obtain the optimum design of power screw jack. Thus, Optimization play a key role in field of engineering application. In our work power screw is machine component is to be optimized using the graphical & MATLAB software It is essential to determine stresses in local areas and other areas using three dimensional, symmetric and axisymmetric models, the preliminary conclusion is that finite element analysis is an extremely powerful tool for design and optimization of power screw. Depending on the desired solutions, there are different methods that offer faster run times and less error. The recommended methods included symmetric models using shell elements and axisymmetric models using solid elements. Design optimization of power screw concerns with the idea to get optimized design dimensions of power screw to minimize weight under given set of constraint by taking pitch & mean diameter of as design variables and screw should be self locking assume coeffient of friction between screw and nut is 0.16, screw is safe in buckling, Permissible stress should be less than or equal to yield strength/FOS, screw should be safe in shear failure as design constraints. Further the verification of optimized graphical solution for minimum weight is compared with MATLAB software.

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Abstract

Keywords:- power screw jack, optimization, optimization by graphical method & by MATLAB software

Types

Power screws are classified by the geometry of their thread. V-threads are less suitable for Power screws than others such as Acme because they have more friction between the threads. Their threads are designed to induce this friction to keep the fastener from loosening. Power screws, on the other hand, are designed to minimize friction. Therefore, in most commercial and industrial use, V-threads are avoided for Power screw use. Nevertheless, V-threads are sometimes successfully used as Power screws, for example on microlathes and micromills. Square thread Square threads are named after their square geometry. They are the most efficient, having the least friction, so they are often used for screws that carry high power. But they are also the most difficult to machine, and are thus the most expensive. Acme thread

Acme Thread

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Reena M Arora* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 9, Issue No. 2, 171 - 177

2. Optimization by graphical method

Acme threads have a 29° thread angle, which is easier to machine than square threads. They are not as efficient as square threads, due to the increased friction induced by the thread angle.

Optimization is the act of obtaining the best result under given circumstances. Optimization can be defined as the process of finding the condition that give the maximum or minimum value of a function.

Buttress thread Buttress threads are of a triangular shape. These are used where the load force on the screw is only applied in one direction. They are as efficient as square threads in these applications, but are easier to manufacture.

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1.3 1.4 1.5 Fig.1.2, 1.3 Buttress thread Fig 1.4 Acme threads Fig 1.5 Square thread

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1.2

1)Design variable:Any system or component is defined by a set of quantities some of which are viewed as variable during the deign process. In general, certain quantities are usually fixed at outset . All other quantities are treated as variable in design process and are called design variable xi =1,2,3,....,n. X1 = pitch X2 = mean diameter Design variable :X1 X2 2) Objective function :Def :- the critarion with respect to which the design is optimized, when expressed as a function of design variable, is known as criterion or merit or objctive function.

In this we have selected one power screw jack. For optimization there are so many methods, we used graphical method approach & with MATLAB programming software.

to minimize the weight of power screw The screw length is 340 mm. considering the following constraints. i) The screw should be self locking assume coefficient of friction between screw & nut is 0.16. ii) The screw is safe in buckling. iii) Permissible stress should be less than or equal to yield strength/FOS. iv)The screw should be safe in shear failure. v) Tensile strength should be between 630-710 N/mm2

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Fig 1 Detail drawing of Power Screw Jack

Data :Load = P = 4 Tonn = 38.83 KN bcoz [Axial load / force = mass x accn = 4000 x accn n For acc : - we know that, V= u + at (law of motion ) At starting condition, initial velocity = u = 0 m/s Consider final velocity V = 35km/hr. V = at a = 9.72 m/s2 Force/Load=P=38.83 KN] Length = L = 340 mm Yield Strength = Syt = 240 N/mm2 Tensile strength = St = 154 N/mm2 Mathematical Formulation :-

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Reena M Arora* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 9, Issue No. 2, 171 - 177

Design variable :-

c)The permissible stress should be less than or equal to yield strength/FOS

X= { dm } = {x1} p x2 Objective Function = f(x) = To minimize weight I) Primary Design Equation (PDE):( Weight of screw is the criterion of optimization) Let, ρ = Mass Density = 7850 kg/m3 =7.8 x 10-6 N/ mm2 2 A = Area in mm L = Length in mm Weight of screw = mass x accn. due to gravity = density x volume =ρxAxL

g3 (x) = contraints ∂ ≤ Syt. / Fos σ ≤ Syt / fos It is observed that ρ, Syt, St are material parameter A is geometrical parameter P, L, fos are functional requirement. We observed that, there is one free variable (A) & one SDE i.e. nf = Ns It is a case of normal specification. P / A ≤ 360/2 38.83 x 10 3 ≤ 180 π /4 x dm2 dm ≤ 16.57 mm X1 ≤ 16.57 mm

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Objective Function = f(x) = To minimize weight =ρxAxL = ρ x (π/4) x dm2 x L F (x) = 0.00208287 X12 3)Design Constraints :-

d) Screw safe in shear failure. Assume shear stress is to be 0.57 Syt. = 0.57 x Syt. = 0.57 x 240 = 136.8 N/mm2 g4 = (x) = contraints

a) For Self locking

Now Frictional torque T = F. dm/2 x tan (۲ +Ø) Also, T = π x di 3 x FS 16 FS = 16 x P x dm/2 (Effort = P= Fx tan (۲ +Ø)) π di 3 = 8 x 38.83x103 x dm π di 3 We know that , do= dm + P/2 di= dm - P/2 di 3 = (dm - P/2) 3 di 3 = dm 3 [ 1 - 3P ] 2dm FS = 310.64x103 ≤ 136.8 π x dm 2 [ 1 - 3P ] 2dm = 310.64x103 ≤ 136.8 π X12 [ 1 - 3 X2 ] 2 X1 = 310.64x103x2 ≤ 136.8 π X1 [ 2x1- 3X2 ] g4(x) :X12-1.5 X1 X2 = 1445.61

α –helix angle θ – friction

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g 1( x) = constraints α<θ tan α < tan θ P / π dm < 0 .16 angle X2 / π dm <0 .16 X2 / X1 < 0 .502 X1 / X2 ≥ 1.99 g1 (x) :- X1/ X2 ≥ 1.99 X1= 1.99 X2

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Def :- The restriction that must be satisfied to produce an acceptable design are collectively called design constraints.

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X2 2 4 6 8 10 X1 3.98 7.96 11.94 15.9 19.9 16 18 20 22 24 31.84 35.82 39.8 43.78 47.76

12 23.88 26 51.74

14 27.86 28 55.72

b) Screw safe in buckling

g2 (x) = constraint Fn = n x π 2 x E x I L2 ( n = 1 both end hinge ) E = 2x1011 N/m2 38.83x103 = 1 x π 2 x 200x103 x (π/64) dm4 3402 3 38.83 x 10 =.8381 dm4 46.33 x 103 = dm4 dm ≥ 14.67 X1 ≥ 14.67

X2

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3

4

6

8

10

X1

39.55

40.33

41.63

42.78

44.49

46.25

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Graphical solution for power screw jack

disp('------------------------------------------------------------------'); disp(' '); P=input('Enter Axial Load (Tonn) : '); L=input('Enter Screw Length (mm): '); mu=input('Enter co-efficient of friction : '); ro=input('Enter the Mass Density (kg/m3) : '); %A=input('Enter the Area in (mm) : '); acc=9.72; ForceByLoad=P*acc; fprintf('\n Force/Load = P = %5.2f KN \n',ForceByLoad'); ro=ro/1000000000; fprintf('\n Mass Density : %5.2e N/mm2\n',ro); X2=[2 4 6 8 10 12 14 16 18 20 22 24 26 28 30]; mf=1/(mu*pi); for i=1:15 X1(i)=mf*X2(i); %% g1(x) end

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Graph 1 From above graph :x1=dm=14.67 mm F(x) = Minimum mass F (x) = 0.00208287 X12 Minimum Mass of screw = 0.448 kg Thus, It is necessary to select a material having maximum yield strength for maximum load carrying capacity & minimum mass is = 0.448 kg.

disp(' '); disp('------------------------------------------------------------------'); fprintf('X2 : %5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \n',X2); disp(' '); fprintf('X1 : %5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \t%5.2f \n',X1); disp('------------------------------------------------------------------'); X1=[2 4 6 8 10 12 14 16 18 20 22 24 26 28 30]; X1=X1/2; X2=X1/mf; X1=[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15]; figure; plot(X1,X2,'-or') grid on; title('Graph'); xlabel('X1--->'); ylabel('X2') Nr=((pi^3)*200000)/64; Dr=L^2;

3. optimal solution by METLAB of power screw jack

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By metlab various results of optimization obtained after graphical solution can be checked or verified . The program is prepared with the help of constraints, design variables. Here we get desired output because we can get many iterations & from these values it gives optimum value. It gives various graphs with iterations

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Program:-

%%%%%%% Optimization in Design of POWER SCREW %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% AUTHOR : Reena M Arora %%%%%%% Date : 24/7/2011 clc clear all disp('------------------------------------------------------------------'); disp('OPTIMIZATION FOR DESIGN OF POWER SCREW '); disp(' '); disp('REENA M ARORA');

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B]

D=Nr/Dr; dm=sqrt(sqrt(ForceByLoad*1000/D)); %%g2(x) a=(ForceByLoad*1000)/(pi/4); dm2=sqrt(a/180);

………………………………………………………………………………………………………………… OPTIMISATION FOR DESIGN OF POWER SCREW REENA M ARORA …………………………………………………………………………………………………………………

X1=dm; msgbox('Condition is fulfilled'); ObjFun=ro*(pi/4)*X1^2*L; fprintf('\n The minimum Wieght is : %f\n',ObjFun);

Enter Axis load (Tonn) : 4 Enter Screw Length (mm) : 360 Enter co-efficient of friction : 0.24 Enter the Mass Density (kg/m³) : 7850 Force / Load = p = 38.88 KN

Output of MATLAB program:-

Mass Density : 7.85e-006 N/mm² X2 :

2.00 4.00 6.00

………………………………………………………………………………………………………………

X1 :

3.98 7.96 11.94 15.92 19.89 23.87 27.85

OPTIMISATION FOR DESIGN OF POWER SCREW

16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00

REENA M ARORA

31.83 35.81 39.79 43.77 47.75 51.73 55.70 59.68

…………………………………………………………………………………………………………………

The minimum weight is : 0.506148

Enter Axis load (Tonn) : 4 Enter Screw Length (mm) : 340 Enter co-efficient of friction : 0.16

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Enter the Mass Density (kg/m³) : 7850 Force / Load = p = 38.88 KN

8.00 10.00 12.00 14.00

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Mass Density : 7.85e-006 N/mm² X2: 2.00 4.00 6.00 8.00 10.00 12.00

14.00

X1 :

27.85

3.98 7.96 11.94 15.92 19.89 23.87

16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00 31.83 35.81 39.79 43.77 47.75 51.73 55.70 59.68

The minimum weight is : 0.451472

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The optimized values obtained from MATLAB sofware are matching with the graphical method of optimization. Now we will see the graphs obtained from MATLAB software.

Fig 3.2 Graph for Length 360mm.

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C] ………………………………………………………………………………………………………………… OPTIMISATION FOR DESIGN OF POWER SCREW REENA M ARORA ………………………………………………………………………………………………………………… Enter Axis load (Tonn) : 4 Enter Screw Length (mm) : 400 Enter co-efficient of friction : 0.16 Enter the Mass Density (kg/m³) : 7850 Force / Load = p = 38.88 KN Mass Density : 7.85e-006 N/mm² X2 :

2.00 4.00 6.00 8.00 10.00 12.00 14.00

X1 :

2.65 5.31 7.96 10.61 13.26 15.92 18.57

16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00 21.22 23.87 26.53 29.18 31.83 34.48 37.14 39.79

Fig 3.1 Graph for length 340mm

The minimum weight is : 0.624875

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Reena M Arora* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 9, Issue No. 2, 171 - 177

4. Result & discusion:After going through various processes of optimization we come to the conclusion that the minimum weight of power screw jack under given set of constrains by graphical method of optimization is found to be 0.448. In MATLAB software result we found various result these are as follow.

In A] section if we enter

Fig 3.3 Graph for length 400mm.

Axis load (Tonn) : 4

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D]

Screw Length (mm) : 340

…………………………………………………………………………………………………………………

co-efficient of friction : 0.16

……….

It gives the minimum weight : 0.451472

REENA M ARORA

In B] section if we enter

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OPTIMISATION FOR DESIGN OF POWER SCREW ………………………………………………………………………………………………………………… ……….

Axis load (Tonn) : 4

Screw Length (mm) : 360

Enter Axis load (Tonn) : 4 Enter Screw Length (mm) : 500

co-efficient of friction : 0.24

Enter co-efficient of friction : 0.46

It gives the minimum weight : 0.506148

Enter the Mass Density (kg/m³) : 8256 Force / Load = p = 38.88 KN

In C] section if we enter

Mass Density : 8.26e-006 N/mm²

Axis load (Tonn) : 4

2.00 4.00 6.00 8.00 10.00 12.00 14.00

X1 :

1.38 2.77 4.15 5.54

Screw Length (mm) : 400

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X2 :

6.92

8.30

9.69

16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00 11.07 12.46 13.84 15.22 16.61 17.99 19.38 20.76

co-efficient of friction : 0.16 It gives the minimum weight : 0.624875 In D] section if we enter

The minimum weight is : 1.026864

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Axis load (Tonn) : 4 Screw Length (mm) : 500 co-efficient of friction : 0.46 It gives the minimum weight : 1.026864 In this results we have seen if we increase the value of screw length & co-efficient of friction the weight increases. After going through various processes of optimization we come to the conclusion that the minimum weight of power screw jack under given set of constrains by graphical method of optimization is found to be 0.448 , which is verified by MATLAB software.

Fig 3.4 Graph for length 500mm.

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5. Symbols X1 X2 P V u ∂ α θ Syt St A L

- design variable for pitch. - design variable for mean diameter - load - velocity - initial velocity - permissible stress - helix angle - friction angle - Yield Strength - Tensile strength - Area - Length

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6. Conclusion:The objective of design of power screw jack using ASME Codes is achieved through this work. The components of power screw jack are first designed by using ASME Codes.

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The optimization of power screw jack done by graphical method & also with the help of MATLAB software. We compared both the results, i.e. analytical and the results with software and our results are matching. Thus we have achieved our goal of optimization .

References:-

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There is unlimited scope of future work in design optimization of power screw jack, by increasing number of constrains by considering different geometrical and design aspect losses can be minimizes more efficiently.

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[1] J.E Shigley and C.R Mischke , Mechanical Engineering Design , McGraw Hill Publication [2] Engineerig Optimization (theory & practice ) by S.S. Rao , third edition. [3] S.L.S. Jacoby, J.S. Kowalik, and J.T.Pizzo, iterative methods for Nonlinear Optimization Problem, prentice-Hall, Englewood cliffs, N.J., 1972 [4] Matlab programming by David C. Kuncicky, Pearson Education, 2004 [5]Applied optimization with MATLAB programming Wiley Interscience Engineering Mechanical by P. Venkataraman Wiley-IEEE, 2002. [6] H. Rothbart , “ mechanical design and systems handbook” , new york:mcgraw-hill book co., 1964, pp.7–173

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