IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 1

Supplementary problem 5

A force P is applied as shown. Calculate the reaction force (x-direction) on the block, using the principle of virtual work ?A force P is applied as shown. Calculate the reaction force (x-direction) on the block, using the principle of virtual work ?

P

A

C

B

θθ

l l

Z

XY

Sliding block

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 2

Supplementary problem 5

1. In this machine, we know it is definitely fixed at A and will not lift up at B1. In this machine, we know it is definitely fixed at A and will not lift up at B

P

A

C

B

θθ

l l

Z

XY

Sliding block

2. We are interested in a virtual displacement at B and C2. We are interested in a virtual displacement at B and C

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 3

Supplementary problem 5

3. Assign reactions at A and B3. Assign reactions at A and B

P

A

C

B

θθ

l l

Z

XY

Ax Az

Bx

Bz

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 4

Supplementary problem 5

4. Create a displacement for point B and thereby for point C4. Create a displacement for point B and thereby for point C

P.δz + Bx.δx = 0

5. Use the principle of virtual work : ∑Fi.δi = 05. Use the principle of virtual work : ∑Fi.δi = 0

Pδz Cos0 + Bxδx Cos180 = 0

δz = δ (-l Cosθ) = l Sinθ δθ

δx = δ (2l Sinθ) = 2l Cosθ δθ

P (l Sinθ δθ) Cos0 + Bx (2l Cosθ) δθ Cos180 = 0

Bx = (P/2) Tanθ

δz

Bx

Bz

z = -l CosθB’

Ax Az

δx

θ δθ

x = 2l Sinθ

C’

X

Z

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 5

Supplementary problem 6

P

M

A

C

B

E

D

F

Find the couple M required to maintain the mechanism in equilibrium !Find the couple M required to maintain the mechanism in equilibrium !

AC = EC = CB = DB = DF = l

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 6

Supplementary problem 6

P

M

A

C

B

E

D

F

1. Assign reactions1. Assign reactions

Ax

Ex Ez

X

Z AC = EC = CB = DB = DF = l

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 7

Supplementary problem 6

P

M

A

C

B

E

D

F

1. Create a virtual displacement for point D1. Create a virtual displacement for point D

Ax

Ex Ez

X

Z AC = EC = CB = DB = DF = l

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 8

Supplementary problem 6

P

M

A

C

B

E

D

F

1. Create a virtual displacement for point D1. Create a virtual displacement for point D

Ax

Ex Ez

X

Z

θ

δθ

δxx = 3l Sinθδx = 3l Cosθ δθ

D’

A’B’

F’

C’

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 9

Supplementary problem 6

P

M

A

C

B

E

D

F

2. Apply the principle of virtual work2. Apply the principle of virtual work

Ax

Ex Ez

X

Z

D’

A’

θ

δθ

δxx = 3l Sinθδx = 3l Cos θ δθ

P(+δx) + M(-δθ) = 0

P(3l Cosθ δθ ) + M(-δθ) = 0

M = 3Pl Cosθ

B’

F’

C’

AC = EC = CB = DB = DF = l

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Problem 5

Micromechanics – WS 2011/2012/ Exercise 1 / sheet 10

Calculate the internal force variables and plot the trend curvesCalculate the internal force variables and plot the trend curves

F

A B

xz

y

a b

l=a+b

q0

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Problem 5

Micromechanics – WS 2011/2012/ Exercise 1 / sheet 11

1. Divide the beam into two sectors and calculate IFVs in each sector 1. Divide the beam into two sectors and calculate IFVs in each sector

1CQI 3222

1)( Cxq

bxQ oII

211)( CxCxM I 43

32

6)( CxC

b

xqxM o

II

b

Fabq

l

bCF

bq

l

baCF

bq

l

bC ooo .

6

..;

6

..

.;

6

.341

a bA B

F

qo

x1 x2

xy

z Sector IISector I

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Problem 5

Micromechanics – WS 2011/2012/ Exercise 1 / sheet 12

a+b

M

ax

Q

a a+b x

2. Plot IFVs along beam axis2. Plot IFVs along beam axis

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 13

Supplementary problem 7

P

F

A

C

B

E

D

F

Find the force F required to maintain the mechanism in equilibrium !Find the force F required to maintain the mechanism in equilibrium !

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Micromechanics – WS 2011/2012/ Exercise 3 / sheet 14

Supplementary problem 8

A force P is applied as shown. Find the angle ‘θ‘ for which the mechanism is in an equilibrium position ? Find the tension in the spring as well. The spring has a spring constant ‘k’. The unstretched length of the spring is ‘h’.

A force P is applied as shown. Find the angle ‘θ‘ for which the mechanism is in an equilibrium position ? Find the tension in the spring as well. The spring has a spring constant ‘k’. The unstretched length of the spring is ‘h’.

P

A

C

B

θθ

l l

Sliding block

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Problem 5

Micromechanics – WS 2011/2012/ Exercise 1 / sheet 15

F

A B

xz

y

2a 2a

3a

2a

q0

Find the reactions at A and B.Find the reactions at A and B.

IMTEKLehrstuhl Konstruktion von Mikrosystemen

Problem 6

Micromechanics – WS 2011/2012/ Exercise 1 / sheet 16

Calculate the internal force variables and plot the tend curvesCalculate the internal force variables and plot the tend curves

A

xz

y

la

q(x) = q0