_+14+m11+eq.+in+3d+no+conn

Module 11a Class Activity: Equilibrium in 3D

(without engineering connections)

Created by:

Paul S. Steif Department of Mechanical Engineering

Carnegie Mellon University, Pittsburgh, PA [email protected]

Anna DollárMechanical and Manufacturing Engineering Department

Miami University, Oxford, OH [email protected]

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Copies or derivatives of this work should acknowledge the original creators.

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Equilibrium in 3-D

SM|P = 0Vector equations:

SF = 0

Scalar equations:

SFx = 0SFy = 0SFz = 0

SM|P x = 0SM|P y = 0SM|P z = 0

These 6 scalar equations are both necessary and sufficient conditions for equilibrium in 3-D

These 2 vector equations are both necessary and sufficient conditions for equilibrium

Equilibrium in 3-D

Which equations are trivially satisfied, if all the forces act in the (x, y) plane?



There are only 3 scalar equations that are both necessary and sufficient conditions for equilibrium in 2-D:

Special case A: planar, 2-D systems of forces

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

SFx = 0SFy = 0

SM|P z = 0

Equilibrium in 3-D

Which equations are trivially satisfied?


SM|Ox = 0SM|Oy = 0SM|Oz = 0

There are only 3 scalar equations that are both necessary and sufficient conditions for equilibrium of 3-D concurrent forces:

Special case B: forces concurrent at a point

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi


SM|o ≡ 0

Equilibrium in 3-D




There are only 3 scalar equations that are both necessary and sufficient conditions for equilibrium of 3-D parallel forces:

Special case C: parallel forces

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

SFx = 0 SM|P y = 0SM|P z = 0

Equilibrium in 3-D




There are only 5 scalar equations that are both necessary and sufficient conditions for equilibrium of 3-D forces concurrent with a line:

Special case D: forces concurrent with a line

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi


SM|P y = 0SM|P z = 0

Equilibrium in 3-D

Equilibrium in 3-DExample 1:

Can the member be supported by one finger under some point of the body?

Yes

No AB

C5 N

7 N

Equilibrium in 3-D

Gr

Pi

Locate single force which is statically equivalent to the 7 N and 5 N forces

W=12 N

W must have the same magnitude and produce the same moments M|x and M|y as the weights of the two legs:

yc

xc

Center of Gravity in 3-D

5 cm

7 cm

xyz

5 N7 N

SM|By = 5(5/2) = 12 xcxc = 25/24 cm

SM|Bx = -7(7/2) = -12 ycyc = 49/24 cm

=

center of gravity

SFz = -7-5= -W

“L” center of gravity is located outside its boundary

W =12N

Bxyz

B

Equilibrium in 3-D

YesNo A

B

C

Consider supporting the member in the orientation shown with one finger (no friction) under each of the points A, B and C.

Gr

Pi

Can the body be supported this way?

What are the force magnitudes at A, B and C?

Case 1: Forces at A, B, and C

Equilibrium in 3-D

FBD: Which equation is trivially satisfied?

A50 mm

70 mm

xyz

B

C5 N

7 N

SM|Px = 0

SM|Py = 0

SM|Pz = 0

SFx = 0

SFy = 0

SFz = 0

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi


Equilibrium in 3-D

FBD: These equations are trivially satisfied:

SM|Pz 0

SFx 0SFy 0

A50 mm

70 mm

xyz

B

C5 N

7 N


These equations need to be satisfied:

SFz = 0

SM|Px = 0

SM|Py = 0

Equilibrium in 3-D

Which forces contribute to the moment summation SM|Ax = 0?

Yes NoGr PiA

B

C

5 N

7 N

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

(neglect thickness compared with long dimensions)

Imposition of equilibrium of moments in the x direction:

A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

In the moment summation SM|Ax = 0, the moment arm for C is:

70

50

120

[502+702]1/2

Gr

Pi

Bl

Ye


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

In the moment summation SM|Ax = 0, force C contributes a moment which is:

M|Ax > 0

M|Ax < 0

Gr

Pi


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

SM|Ax = C(70 mm) – 7N (70/2 mm) = 0

C = 7/2 N

Equilibrium in 3-D


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

Which forces contribute to the moment summation SM|By = 0?

Yes NoGr PiA

B

C

5 N

7 N

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Yes NoGr Pi

Equilibrium in 3-D

Imposition of equilibrium of moments in the y direction:

A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

In the moment summation SM|By = 0, the moment arm for A is:

70

50

120

[502+702]1/2

Gr

Pi

Bl

Ye


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

In the moment summation SM|By = 0, force A contributes a moment which is:

M|By > 0

M|By < 0

Gr

Pi


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

SM|By = -A(50 mm) + 5N (50/2 mm) = 0

A = 5/2 N


A50 mm

70 mm

xyz

B

C5 N

7 N


Equilibrium in 3-D

SFz = A + B + C – 7N – 5N = 0

B = 6 N

B > C > A

Imposition of equilibrium of forces in z direction:

A =5/2 N50 mm

70 mm

xyz

B

C =7/2 N5 N

7 N


Equilibrium in 3-D

xyz

AB

C

Consider supporting the member in the orientation shown by:• two fingers applying upward forces • a nut driver applying only a couple to the nut located near B

The member can be balanced by a couple of the right magnitude and direction acting at B and forces applied to the following pairs of points:

A and B:

A and C:

B and C:

Yes No

Yes No

Yes No

Gr Pi

Gr Pi

Gr Pi

Equilibrium in 3-D

nut driver &A and C:

nut driver &B and C:

Yes Gr

Yes Gr

C

C

B

A

Equilibrium in 3-D

AB

C

Consider supporting the member in the orientation shown by applying: a couple to the nut located near B + two forces with the fingers at B and C

The couple applied to the nut to maintain equilibrium is described by:

CMx > 0

CMx < 0

CMy > 0

CMy < 0

CMz > 0

xyz

Gr

Pi

Bl

Ye

Wh

Case 2: Forces at C, B and couple CM

Equilibrium in 3-D

xyz

CMB

C5 N

7 N

CMy < 0 Ye


Equilibrium in 3-D

Couple is CMy < 0.

What is the magnitude CM of the coupleapplied at the nut?

CM = (7.0)(35) N-mm

CM = (2.5)(25) N-mm

CM = (7.0)(25) N-mm

CM = (12) (25) N-mm

CM = (5.0)(25) N-mm

Gr

PiBl

Ye

Wh


50 mm

70 mm

xyz

CMB

C5 N

7 N


Equilibrium in 3-D

With fingers under B and C, what is the magnitude of the couple applied at the nut?

CM = (5)(25) N-mm

SM|By = 5(50/2) – CM = 0

Wh


50 mm

70 mm

xyz

CMB

C5 N

7 N


Equilibrium in 3-D

Given the whole member weighs 12 N, what are the forces exerted by the fingers?

B = 5.0 N, C = 7.0 N

B = 8.5 N, C = 3.5 N

B = 7.5 N, C = 4.5 NB = 3.5 N, C = 8.5 N

B = 9.5 N, C = 3.5 N

Gr

PiBl

YeWh

Imposition of equilibrium equations:

50 mm

70 mm

xyz

125 N mmB

C5 N

7 N


Equilibrium in 3-D

SM|Bx = 7(70/2)-C(70) = 0


C = 3.5 N

50 mm

70 mm

xyz

125 N mmB

C5 N

7 N


Equilibrium in 3-D

B = 8.5 N, C = 3.5 N

SFz = B + C – 12 = 0

Equilibrium in 3-D

Pi

Imposition of equilibrium of forces in the z direction:

50 mm

70 mm

xyz

125 N mmB

C =3.5 N5 N

7 N


Equilibrium in 3-D

AB

C


CMx > 0

CMx < 0

CMy > 0

CMy < 0

CMz > 0

xyz

Consider supporting the member in the orientation shown by applying: a couple to the nut located near B + two forces with the fingers at C and A

Gr

Pi

Bl

Ye

Wh

Case 3: Forces at C, A and couple CM

Equilibrium in 3-D


CMy > 0

CMy < 0

Consider supporting the member in the orientation shown by applying: a couple to the nut located near B + two forces with the fingers at C and A

Bl

Ye

Need to know the magnitude of force at Ato determine the sense!!!

50 mm

70 mm

xyz

CM

C5 N

7 N

A


Equilibrium in 3-D

Given the whole member weighs 12 N, what are the forces exerted by the fingers?

A = 5.0 N, C = 7 N

A = 8.5 N, C = 3.5 N

A = 7.5 N, C = 4.5 NA = 3.5 N, C = 8.5 N

A = 9.5 N, C = 3.5 N

Gr

Pi

BlYe

Wh

Imposition of equilibrium equations:

50 mm

70 mm

xyz

CM

C5 N

7 N

A


Equilibrium in 3-D

SM|Ax = 7(70/2)-C(70) = 0 C = 3.5 N


50 mm

70 mm

xyz

CM

C5 N

7 N

A


Equilibrium in 3-D

A = 8.5 N, C = 3.5 N

SFz = A + C – 12 = 0

Imposition of equilibrium of forces in the z direction:

50 mm

70 mm

xyz

CM

C5 N

7 N

A


Equilibrium in 3-D

50 mm

70 mm

xyz

CM

3.5 N5 N

7 N

8.5 NCM = 125 N-mm

CM = 425 N-mm

CM = 300 N-mm

CM = 87.5 N-mm

CM = 550 N-mm

What is the magnitude of the couple applied at the screw?

Gr

Pi

Bl

Ye

Wh



Equilibrium in 3-D

SM|By = +5(50/2) - 8.5(50) + CM = 0

CM = 300 N mm Bl



50 mm

70 mm

xyz

CM

3.5 N5 N

7 N

8.5 N

Can the member be supported by two fingers, each under a different point of the body?

Yes, in one way

No

Yes, in many ways

AB

C5 N

7 N

Equilibrium in 3-D

Gr

Pi

Ye

Select one support point D at a distance d from B along BC.

Can you always balance the member with a second support point E at a distance e from B along BC (no matter where D is between B and C)?

Yes

No

D

Eed A

B

C5 N

7 N

xyz

Equilibrium in 3-D

Gr

Pi

View member from top (gravity still acts along –z)One finger is placed at D near end of long member.

where E?

D

(xc,yc)

x

y

Equilibrium in 3-D

Select one support point D at a distance d from B along BC.

Can you always balance the member with a second support point E at a distance e from B along BC (no matter where D is between B and C)?

Where to place support E?

Support E should be along line connecting D and center of gravity. WHY?

yD

(xc,yc)

xE

Equilibrium in 3-D

• all three forces are perpendicular to plane.• moments about all axes must sum to zero. • take moments about axis P-P:

D

(xc,yc)

x

yP

PE

If E is not on line P-P, it produces an unbalanced moment

Equilibrium in 3-D

• let E be at end of short member. • again D must be on the line joining E and centroid. • if D is any lower, E would have to be off the end of member

D(xc,yc)

x

yP

E

D cannotbe in here

Equilibrium in 3-D

_+14+m11+eq.+in+3d+no+conn

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