presentation tecca

7/23/2019 Presentation Tecca

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On the Use ofGeoGebra for Path

Curvature Analysis inLinkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation

Kinematic Synthesis forInfinitesimallySeparated Positions

Mechanisms simulation

Three InfinitesimallySeparated Link Positions

Inflection Circle

Four InfinitesimallySeparated Link Positions

How to use the .ggbfiles

GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

On the Use of GeoGebra for Path Curvature Analysisin Linkage Design

23 January 2014

University of Rome Tor Vergata

Federico Tecca
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Summary:

Introduction,

Kinematic Synthesis for Infinitesimally Separated

Positions:Mechanism simulation,Three Infinitesimally Separated Link Positions,Four Infinitesimally Separated Link Positions,

How to use the .ggb files,

GeoGebra STEM 2013,

Conclusions.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Introduction
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Aim of the work:

Is to summarise graphical methods of MechanicalEngineering formulated in the past and subsequentlyabandoned because of their complexity in representation

and construction when available tools were not suitable.
http://find/http://goback/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Aim of the work:

Is to summarise graphical methods of MechanicalEngineering formulated in the past and subsequentlyabandoned because of their complexity in representation

and construction when available tools were not suitable.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

In particular for Kinematic Synthesis for Infinitesimally

Separated Positions:

We want to draw the notable locii for the Kinematic

Synthesis for Infinitesimally Separated Positions.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

In particular for Kinematic Synthesis for Infinitesimally

Separated Positions:

We want to draw the notable locii for the Kinematic

Synthesis for Infinitesimally Separated Positions.

Question:

We want to develop parametric worksheets. In these itspossible to change the input parameters and then theworksheet designs the mechanism and the notable lociifor another configuration.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

The features of the sofware must be:
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


Flexibility,
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


Flexibility,

Intuitivness,


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


Flexibility,

Intuitivness,

Parametricity.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


Flexibility,

Intuitivness,

Parametricity.

Choice
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


Flexibility,

Intuitivness,

Parametricity.

Choice Software GeoGebra , is the answer to theproblem.


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

GeoGebra STEM 2013 GeoGebra InternationalInstitute
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

GeoGebra STEM 2013 GeoGebra InternationalInstitute

Supervised by CCITE(Cambridge Center forInnovation in Technological Education)

.


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

A new section of the web sitehttp://www.geogebraitalia.org/ was createdfor these materials.

this is dedicated to the engineering applications onGeoGebra,http://engineering.geogebraitalia.org/
http://www.geogebraitalia.org/http://engineering.geogebraitalia.org/http://engineering.geogebraitalia.org/http://www.geogebraitalia.org/http://find/http://goback/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

A new section of the web sitehttp://www.geogebraitalia.org/ was createdfor these materials.

this is dedicated to the engineering applications onGeoGebra,http://engineering.geogebraitalia.org/

Part of the work was discussed for the bachelor thesisParametric Kinematic Synthesis of Linkages for

Prescribed Infinitesimally Separated Positions.
http://www.geogebraitalia.org/http://engineering.geogebraitalia.org/http://engineering.geogebraitalia.org/http://www.geogebraitalia.org/http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Kinematic Synthesis for InfinitesimallySeparated Positions


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


= 0.6

d= 0.5

l= 1.7

s= 0.9

A0

= 0.93
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


= 0.6

d= 0.5

l= 1.7

s= 0.9

A0

= 0.93

A

A : (s cos, s sin)
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


= 0.6

d= 0.5

l= 1.7

s= 0.9

A0

= 0.93

A

A : (s cos, s sin)

r = l

saxis = x tan

d


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


= 0.6

d= 0.5

l= 1.7

s= 0.9

A0

= 0.93

A

A : (s cos, s sin)

r = l

saxis = x tan

dB
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


= 0.6

d= 0.5

l= 1.7

s= 0.9

A0

= 0.93

A

A : (s cos, s sin)

r = l

saxis = x tan

dB
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Three Infinitesimally Separated LinkPositions
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

The second expression of the Euler-Savary formula is usedto design the Inflection CircleF :

P0M2 = M MM, (1)

Point M, on the Inflection CircleF:

MM=P0M

2

M , (2)
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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

The second expression of the Euler-Savary formula is usedto design the Inflection CircleF :

P0M2 = M MM, (1)

Point M, on the Inflection CircleF:

MM=P0M

2

M , (2)

The distances P0M e Mof the known points arecalculated and then the Inflection CirlceF is designed,computing MM from (2).
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle

A0

L1 = 5.2= 0.5

A

L2 = 5.5

B0

Lgl= 11.5

BL3 = 3.3L4 = 3 C

P15
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle

A0

L1 = 5.2= 0.5

A

L2 = 5.5

B0

Lgl= 11.5

BL3 = 3.3L4 = 3 C

P15
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle

A0

L1 = 5.2= 0.5

A

L2 = 5.5

B0

Lgl= 11.5

BL3 = 3.3L4 = 3 C

P15

r = P15 B

2

B0 B


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle

A0

L1 = 5.2= 0.5

A

L2 = 5.5

B0

Lgl= 11.5

BL3 = 3.3L4 = 3 C

P15

r = P15 B

2

B0 B

B1


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Inflection Circle

A0

L1 = 5.2= 0.5

A

L2 = 5.5

B0

Lgl= 11.5

BL3 = 3.3L4 = 3 C

P15

r = P15 B

2

B0 B

B1


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Four Infinitesimally Separated LinkPositions


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Cubic of Stationary Curvature

The design of the Cubic of Stationary CurvatureK ismade with geometric construction too. In particular theCartesian form of the Cubic of Stationary CurvatureK:

x2 +y2

Ny +

x2 +y2

Mx = 1 , (3)

this is separated in a system of two equations:

x

2

+y2 x= 0

x2 +y2 y= 0 (4)

With this method the intersection between the twoequations, two circles, correspond to a point on the Cubic

of Stationary CurvatureK.


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

N ed Mare the geometric invariants of the Cubic ofStationary Curvature. These are determined from twopoints,A e B, on the Cubic of Stationary Curvature.

From these points its possible to reach the equations:N= hAhBsin(BA)(cosAcosB(hAsinBhBsinA))M=N hAcosA((NcosAhA)sinA)

, (5)

Where A, Bare the angles between the segments P0Ae P0Bwith the normal vector

N, vector of the Canonical

reference, and hA, hB the distances from A and B to P0.
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0
http://find/


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)
http://find/


40/69



Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)

A(, )
http://find/


41/69



Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)

A(, )


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Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)

A(, )(0, )

(, 0)


43/69



Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)

A(, )(0, )

(, 0)

K


44/69



Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

P0

S(M, 0)

R(0,N)

A(, )(0, )

(, 0)

K


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Curvature Analysis in

Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

a

A(, )

S(M, 0)

R(0,N)

K

P0

(0, )

(, 0)


46/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions


N

T

a

A(, )

S(M, 0)

R(0,N)

K

P0

(0, )

(, 0)
http://find/


47/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Example applied to a mechanism

C

= 0.87rad

L1 = 4.5B02 = 1.3

= 2rad

AL3 = 7

BLgl= 7.8

P14
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Pivot Point Curvature


49/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Example applied to a mechanism

A0

L1 = 1.5 = 1.26

A

L2 = 1.44

L3 = 2.12L4 = 2

B0

B

P13
http://find/


50/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Its possible to compute the centre of curvature of thepoints on the Cubic of Stationary Curvature

Kwith a

geometric construction on GeoGebra using the secondexpression of the Euler-Savary formula:

Pivot Point Curvature
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

How to use the .ggb files


52/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Notable locii varying the anomaly angle .
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Notable locii varying the input parameters.


54/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

GeoGebra STEM 2013
http://find/


55/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

For the competition organized by the GeoGebraInternational Institute, GeoGebra STEM 2013, othermaterials were made.

In particular the materials are developed in:Fluid Dynamics,

Strength of Materials.

All the worksheets are collected in the web sitehttp://engineering.geogebraitalia.org/.
http://engineering.geogebraitalia.org/http://engineering.geogebraitalia.org/http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Boundary Layer
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

In a thin Boundary Layer near the solid boundary, shearforces must necessarily be significant because of the largeshearing deformations resulting from the fluids being atrest at the solid boundary.

This fact generates high gradients at the interfaceflatbed-fluid and the creation of the Boundary Layer.


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Its possible to demonstrate that the function thatdescribed the Boundary Layer, with some approximation:

=LU

= L

Re, (6)


59/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Its possible to demonstrate that the function thatdescribed the Boundary Layer, with some approximation:

=LU

= L

Re, (6)

where:

L is the length of the flatbed,

Reis the Reynolds number,

U is the speed of the fluid,is the cinematic viscosity of the fluid.


60/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions
http://find/


61/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Force in a Nozzle
http://find/


62/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

The starting point is the conservation of the momentum:

F=dQ

dt , (7)
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

The starting point is the conservation of the momentum:

F=dQ

dt , (7)

Through the Reynolds theorem and some othersimplification its possible to obtain the application formof the conservation of the momentum:

F=

Ni=1 (

uun

+pn

)iSi (8)
http://find/


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Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

u2 = 6.2

2 = 3.32rad

E

r2 = 1.2

A

1 = 0.7rad

r1 = 1.4

u1 = 3.8

Force
http://find/


65/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Force in a Nozzle.
http://find/


66/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Conclusions
http://find/


67/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

A library with some of the materials created is made.These GeoGebra worksheets are totally parametric and

they resolve some problems for Mechanical Engineering.
http://find/


68/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

A library with some of the materials created is made.These GeoGebra worksheets are totally parametric and

they resolve some problems for Mechanical Engineering.

The solutions to these problems are graphic andgeometric constructions.
http://find/


69/69



Linkage Design

Federico Tecca

Introduction

Aim of the work

Choosen Software

Work partecipation




Inflection Circle



GeoGebra STEM 2013

Boundary Layer

Force in a Nozzle

Conclusions

Thank you for the attention

presentation tecca

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