Harvard University Division of Engineering and Applied Sciences Eng-Sci 240: Solid Mechanics...

Harvard UniversityDivision of Engineering and Applied Sciences

Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo

Failure Analysis of a Fin Design for

the Micro-Mechanical Fish

By Michael Petralia

Wood MicroRobotics LaboratoryFin Design by Kyla Grigg

December 12, 2006

Design of the fin



Goal Prototype Design

The final design will look like a fish fin, but to prototype the mechanical system, square elements are being used.

Design of the fin



Brass Plate

Carbon Fiber Tendon

Glass Fiber Laminas

Silicone Rubber Shape Memory Alloy

(SMA)

Forces acting on the fin



FSMA

FTendonFRubber

FSMA




FSMA

FTendonFRubber

FSMA

θmin

FSMA

θmin = 6.5o Assuming the rubber stretches at most 1 mm,

The SMA’s provide a maximum pull of,

FSMA = 1.47 N

The horizontal force from the SMA to the rubber will be,

Frubber = 2FSMAsin(θmin) = 0.333 N

Case 1: Maximum Deflection

Assuming this load is equally distributed over the right edge of the fin,

Frubber / A = 0.2641 N/mm2




FSMA

FTendonFRubber

FSMA

FSMA

The maximum angle is,

θmax = 22o

Fmax = Frubber = 1.10 N

Though this would mean the fin has not moved, let’s take this as our worst case scenario. The maximum horizontal force from the SMA will be,

Again, assuming this load is equally distributed over the right edge of the fin,

Frubber / A = 0.8740 N/mm2

Case 2: Maximum Force

θmax




FSMA

FTendonFRubber

FSMA

The force in the tendon is the force necessary to keep the fin in place during swimming. It will be the horizontal force from the SMAs for a given angle.

Because the deflection will be relatively small, it should be safe to assume the force in the tendon will act parallel to the surface of the fin.

Although it would mean there was no deflection of the fin, let’s use the maximum horizontal force provided by the SMA for analysis.

Force from the Tendon

Ftendon = 1.10 N

Properties of glass fiber



According to Chou* unidirectional lamina can be treated as a homogeneous, orthotropic continuum. Additionally, for circular cross-section fibers randomly distributed in the unidirectional lamina, the lamina can be considered transversely isotropic.

The result is that we only have five independent material constants:

E11, E22, v12, G12, v23

These values of these constants were provided by the manufacturer:

*Chou, Microstructural Design of Fiber Composites (1992)

+This value was not provided by the manufacturer, but it was necessary to assume a value in order to please ABAQUS.

E11 (GPa) E22 (GPa) v12 G12 (GPa) v23+

50 7 0.33 5 0.33

Orientation of the glass fiber



+ =

Horizontal Fibers Vertical Fibers Crossed Fibers

List of Analyses



The following situations were analyzed for the cases of maximum deflection and maximum forces. To look at the worst case scenario, the tendon force was included in each analysis.

One Layer: Horizontal fiber direction

Vertical fiber direction

Two Layers: Horizontal fiber direction

Vertical fiber direction

Crossed fiber direction

A Note on the Results



The maximum tensile and compressive stresses and strains in the 1 and 2 directions were compared to the maximum allowable values as provided by the manufacturer.

The orientation of the fibers was considered, and the graphs are based on the stresses and strains with respect to the fiber orientation, not with respect to the fin orientation.

Because of this, in the cross fiber analyses, the front and back square were considered separately.

Maximum value of σ1t

29 12 28 15 25 388

23 58 15 31 5

1400

0

200

400

600

800

1000

1200

1400

1600

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

(MP

a)

Maximum value of σ1c

29 14 41 19 31 4 36 16 22 36 5

1225

54

0

200

400

600

800

1000

1200

1400

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

(MP

a)

Maximum value of σ2t

47

2033

165

34

141

3730

155

31 35

0

20

40

60

80

100

120

140

160

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

(MP

a)

Maximum value of σ2c

61

3137

196

46

59

2935

176

44

140

0

20

40

60

80

100

120

140

160

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

(MP

a)

Maximum value of σ12

23

11

34

17

68

18

36

1813 15

63

0

10

20

30

40

50

60

70

80

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Max

Defl

ectio

n

2 Lay

ers C

rosse

d Max

For

ce

Ultimate

She

ar

(MP

a)

Maximum value of ε1t

0.00400.0017

0.0006 0.0003 0.0005 0.0005

0.0122

0.00110.0032

0.0003 0.0006 0.0006

0.0250

0.000

0.005

0.010

0.015

0.020

0.025

0.030

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers V

ertic

al M

ax F

orce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

Maximum value of ε1c

0.0042

0.00200.0008 0.0004 0.0006 0.0006

0.0052

0.00230.0011 0.0005 0.0007 0.0007

0.0220

0.000

0.005

0.010

0.015

0.020

0.025

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

Maximum value of ε2t

0.00090.0004

0.0047

0.0024

0.0007 0.0007

0.0026

0.0007

0.0043

0.0022

0.0006 0.0006

0.0050

0.000

0.001

0.002

0.003

0.004

0.005

0.006

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

Maximum value of ε2c

0.0012 0.0006

0.0054

0.00270.0009 0.0009 0.0012 0.0006

0.0051

0.00260.0009 0.0009

0.0200

0.000

0.005

0.010

0.015

0.020

0.025

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

2 Lay

ers C

rosse

d Fro

nt M

ax D

eflec

tion

2 Lay

ers C

rosse

d Bac

k Max

Defl

ectio

n

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Fro

nt M

ax F

orce

2 Lay

ers C

rosse

d Bac

k Max

For

ce

Maximum value of ε12

0.0045

0.0023

0.0068

0.0034

0.0136

0.0035

0.0071

0.00360.0026 0.0030

0.0130

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

1 Lay

er Vert

ical M

ax D

eflec

tion

2 Lay

ers V

ertic

al M

ax D

eflec

tion

1 Lay

er Hor

izonta

l Max

Defl

ectio

n

2 Lay

ers H

orizo

ntal M

ax D

eflec

tion

1 Lay

er Vert

ical M

ax F

orce

2 Lay

ers V

ertic

al M

ax F

orce

1 Lay

er Hor

izonta

l Max

For

ce

2 Lay

ers H

orizo

ntal M

ax F

orce

2 Lay

ers C

rosse

d Max

Defl

ectio

n

2 Lay

ers C

rosse

d Max

For

ce

Ultimate

She

ar

Vertical Fibers - Max Deflection



Deformation Scale Factor =100

σt,max = 58.5 MPa

σc,max = 65.3 MPa

Horizontal Fibers - Max Deflection



σt,max = 53.2 MPa

σc,max = 52.3 MPa




σt,max = 25.5 MPa

σc,max = 32.6 MPa

Horizontal Fibers - Max Deflection



σt,max = 26.7 MPa

σc,max = 26.1 MPa

Crossed Fibers - Max Deflection



Front Square:σt,max = 27.5 MPaσc,max = 33.1 MPa

Back Square:σt,max = 38.9 MPaσc,max = 48.2 MPa




σt,max = 25.5 MPa

σc,max = 32.6 MPa

Vertical Fibers - Max Force



Deformation Scale Factor =100

σt,max = 175.2 MPa

σc,max = 62.7 MPa

Horizontal Fibers - Max Forces



σt,max = 60.2 MPa

σc,max = 59.3 MPa




σt,max = 45.5 MPa

σc,max = 30.7 MPa

Horizontal Fibers - Max Force



σt,max = 28.6 MPa

σc,max = 28.4 MPa

Crossed Fibers - Max Force



Front Square:σt,max = 31.6 MPaσc,max = 37.7 MPa

Back Square:σt,max = 37.5 MPaσc,max = 44.8 MPa




σt,max = 45.5 MPa

σc,max = 30.7 MPa

Table of Maximum Principal Stresses and Strains



Number of Layers Fiber Direction Case

σt,max σc,max εt,max εc,max

1 Vertical 58.5 65.3 0.0044 0.0059

Horizontal 53.2 52.3 0.0064 0.0051

Vertical Max Deflection 25.5 32.6 0.0022 0.0063

2 Horizontal 26.9 26.1 0.0032 0.0067

Crossed - Front 27.5 33.1 0.0017 0.0027

Crossed - Back 38.9 48.2 0.0017 0.0027

1 Vertical 175.2 62.7 0.0133 0.0025

Horizontal 60.2 59.3 0.0063 0.0031

Vertical Max Force 45.5 30.7 0.0034 0.0033

2 Horizontal 28.6 28.4 0.0032 0.0018

Crossed - Front 31.6 37.7 0.0019 0.0016

Crossed - Back 37.5 44.8 0.0019 0.0016

Conclusions



• As was previously thought, it is necessary to employ two layers of glass-fiber.

• The crossed-fiber configuration provides lower strains in the the 2 direction, but the horizontal stresses will approach the limit of the material in the vertical-fiber square.

• Despite the higher strains, the 2 layer, horizontal-fiber configuration does not experience any stresses or strains greater than 50% of the maximum allowable value.

• It is recommended that the fin be constructed using two horizontal-fiber squares.

Questions?



Harvard University Division of Engineering and Applied Sciences Eng-Sci 240: Solid Mechanics...

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Transcript of Harvard University Division of Engineering and Applied Sciences Eng-Sci 240: Solid Mechanics...