ES2D5 - A2 Planar Structures and Mechanisms Assessment Answer

Planar Structures

Four engineers, ψ, Γ, Λ, and Φ, of body

Weights

M ψ = 50+ N 3 N 4 = 50 + 10 = 60Kg 

M Γ= 50+ N 4 N 5 = 50+07=57 Kg 

M Λ= 50+ N 5 N 6 =129 Kg 

M Φ= 50+ N 6 N 7 = 50 +90 = 140 Kg 

Having lost their jobs following the crash of the automotive industry, found employment moving a pile of 20+ N 6 kg = 20 + 9 = 29 kg sacks of potato over an H = 5+ N 2/4 m = 5 + 0/4 = 5 m brick wall. 

t =1+N5 = 1 + 7 = 8 mm 

w = (10+ N 4)·2 cm 

w = (10+0)*2 = 20 cm 

a = thickness of plank 

Area of plank = 0.008a x 0.2 = 0.0016a m2 

Young’s Modulus, E=9 GPa 

UTS, σmax=40 MPa\ 

Design the Seesaw and Suggest the Sack-tossing Procedure.

Free Body Diagram

Applying equilibrium of force, 

F1 + F2 = F3 

M Γ + M ψ + M Φ = 113 + 50 +50 = 257 Kg 

Input (F1)= 257*9.81 = 2521.17 N 

Output (F2)= 29*9.81 = 284.49 N 

Therefore, F3 = 2521.17 + 284.49 = 2805.66 N 

>>Matlab_code Results 

N1 = 2 

N2 = 0 

N3 = 1 

N4 = 0 

N5 = 7 

N6 = 9 

N7 = 0 

N3N4 = 10 

M_psy = 60 

NN5 = 7 

M_gama = 57 

N5N6 = 79 

M_lemda = 129 

N6N7 = 90 

M_phy = 140 

t = 8 

w = 20 

a = 8 

Area = 0.0640 

E = 9.0000e+09 

Sigma_max = 40000000 

F1 = 2521.17 

F2 = 284.4900 

F3 = 2805.66 

Moment at F1 = 0 

F3*(L-x) - F2*L =0 

2521.17 (L-x) - 284.49 L = 0 

2521.17L – 2521.17x – 284.49 L = 0 

2236.68L – 2374.04x = 0 Equation (1) 

Or L= (2374.04/2236.68)*x= 1.0614 * x 

All simple machines produce a mechanical advantage. 

Input = L –x 

Output = x 

IMA = Output side/ Input side

IMA = x/ (L-x)=1.1362/0.1362= 8.3421 

F2 = (L-x)/x. F1 

284.49 = (L-x)/x. 2089.3 

0.13616 = (L-x)/x 

L=1.13616*x 

0.13616 = (8.3421-x) / x 

x = 8.3421/ 1.13616= 7.3424 m 

Putting into (1), 

L = 7.7933 m 

As per thumb rule of the beam, width of the beam can be stated as :- 

W = D/1.5 = 20 cm /1.5 = 13.33 cm 

Since the plank thickness is 8 mm 

So the complete thickness will come out to be in a multiple of 8 which is coming out to be 360mm or 36 cm. 

Hang Time and Initial Velocity 

Z = sack stuffing on which engineers are standing. 

Let assume L = 10m 

Vx = 10 m / T 

Position of Y = -1/2 * 9.81 m/s2 x T2 + Vy x T

0 = -1/2 * 9.81 m/s2 x T2 + Vxtan(30) x T 

0 = -1/2 * 9.81 m/s2 x T2 + 10mtan(30) x T 

-5.773 m / -(1/2 * 9.81)= T2 

T = 1.084 sec 

Vx = 9.21 m/s 

Vy = 5.321 m/s 

Vx = Z/T 

9.21 = Z / 1.084 

Z = 8.4968 m 

This is not safe. 

If they fall from 8.5m, their bones will break down.

Hence reconsidering the length of the beam to L = 5m. 

Taking θ = 85 degree 

T = 3.41 sec 

Vx = 1.466 m/sec 

Z = 2.326 m 

It is somewhat safe. 

h = L-x tan(15)

h = 7.79-7.34 tan(15)

h = 0.120m 

Appendix A

MATLAB code

% Data 

N1 = 2 

N2 = 0

N3 = 1 

N4 = 0 

N5 =7 

N6 = 9 

N7 = 0 

N3N4= N3*10+N4 

M_psy = 50+N3N4 % M_psy = 60 

N4N5= N4*10+ N5 

M_gama = 50+N4N5 % M_gama = 57 

N5N6= N5*10+ N6 

M_lemda = 50+N5N6 % M_lemda =129 

N6N7= N6*10+ N7 

M_phy = 50+N6N7 % M_phy = 140 

t =1+N5 % 1 + 7 = 8 mm 

w = (10+N4)*2 % in CM 

w = (10+0)*2 % w =20 cm 

a = 8 % thickness of plank in mm 

Area =0.008*a % = 0.0016 m*m Area of plank in m2 

E =9*10^9 % Young’s Modulus GPa 

Sigma_max= 40*10^6 % UTS, MPa 

F1 = 213*9.81 % 2089.53 N; F1 =2.0895e+03 

F2 = 29*9.81 % Output = 284.49 N 

F3 = F1+F2 % F3 = 2374.04 N 

L= 2374.04/2089.5 

IMA = 1.1362/0.1362 

x = 8.3421/ 1.13616 

L=1.13616*x 

References

Uicker, John J.; Pennock, G. R.; Shigley, J. E. (2011). Theory of machines and mechanisms. New York: Oxford University Press. ISBN 978-0-19-537123-9. 

Usher, A. P. (1929). A History of Mechanical Inventions. Harvard University Press (reprinted by Dover Publications 1988). p. 94. ISBN 978-0-486-14359-0. OCLC 514178. Retrieved 7 April 2013. 

John L Synge & Byron A Griffith (1949). Principles of Mechanics (2nd ed.). McGraw-Hill. 

Beer FP, Johnston ER, Mazurek DF, Cornell PJ, and Eisenberg, ER (2009). Vector Mechanics for Engineers: Statics and Dynamics (9th ed.). McGraw-Hill. p. 158. 

Lakshmana C. Rao; J. Lakshminarasimhan; Raju Sethuraman; Srinivasan M. Sivakumar (2004). Engineering Mechanics. PHI Learning Pvt. Ltd. p. 6. ISBN 81-203-2189-8. 

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