G.K. AnanthasureshandAll the current and past members of the M2D2 LaboratoryMechanical EngineeringIndian Institute of ScienceBangalore

Mobility, Analysis, and Synthesis

2

Ananthasuresh, IISc, Dec. 2013

Outline

• Mobility

• Analysis

• Synthesis (design)

Ananthasuresh, IISc, Dec. 2013

MOBILITY

Ananthasuresh, IISc, Dec. 2013

Degrees of freedom (dof)

• Minimum number of scalar inputs needed to define the complete configuration of the mechanism.

• Also equal to the minimum number of inputs needed for deterministic motion.

Ananthasuresh, IISc, Dec. 2013

How many dof do these have?

Ananthasuresh, IISc, Dec. 2013

Kutzbach-Grübler’s formula

1 2 3 4 56( 1) 5 4 3 2dof n f f f f f

number of rigid bodies (called links)n

joints that allow relative motionsif i

3D

1 23( 1) 2dof n f f

2D

Ananthasuresh, IISc, Dec. 2013

Bricard linkage: formula fails!

A closed-loop 6R linkage

1 2 3 4 56( 1) 5 4 3 2

6(6 1) ( 6) 05 (4 0 0 0

0

) (3 ) (2 ) (1 )

30 30

dof n f f f f f

Ananthasuresh, IISc, Dec. 2013

Kaleidocycle mechanism with elastic pairs

Six-tetrahedrons joined together with six flexural hinges

Ananthasuresh, IISc, Dec. 2013

dof formula extended to compliant mechanisms

2 2 3

1 1 1

3( 1) 3 3 3seg Kj Cj fix scj

j j j

dof n j n j n n j n

5 5 6

1 1 1

6( 1) 6 6 6seg Kj Cj fix scj

j j j

dof n j n j n n j n

3D

2D

number of (rigid or compliant)

number of kinematic pairs allowing relative dof

number of allowing relative

segment

dof

number of

number of

s

elastic pairs

fixed connecti ns

segm

o

seg

Kj

Cj

fix

scj

n

n j

n j

n

n

segment complianent ces with of j

Midha, Murphy, and Howell (1995)Ananthasuresh and Howell (1996)

Ananthasuresh, IISc, Dec. 2013

Let us count…

Ananthasuresh, IISc, Dec. 2013

dof of the rolamite elastic pair

Ananthasuresh, IISc, Dec. 2013

dof of a nail-clipper

1

2

34

1 2 13(4 1) 2 3 3

3 3 2 1 1 3 2 1 1

9 2 1 6 1 1

K K fix scdof n n n n

2 2 3

1 1 1

3( 1) 3 3 3seg Kj Cj fix scj

j j j

dof n j n j n n j n

So, we can make up homework problems!

And it is useful in practice.

Ananthasuresh, IISc, Dec. 2013

Maxwell’s rule—structures perspective

3 6v b dof sss

vertices

bars

number of

number of

nu states mber of of self-stress

v

b

sss

2 3v b dof sss 2D

3D

Modified by Calladine

Ananthasuresh, IISc, Dec. 2013

Maxwell’s rule applied to trusses and tensegrity structures

2 3

16 3 12 1 2 1

v b dof sss

2 3

10 3 7 0

v b dof sss

2 3

12 3 9 0

v b dof sss

Ananthasuresh, IISc, Dec. 2013

Compliance and Equilibrium matrices

2 2 1 1b v v b C u e 2 1 2 1v b b v H p f

T T

T T T

T

p e f u

p C u p H u

C H

Compliance matrix

Disp. at vertices

Bar elongations

Equilibriummatrix

Bar forces

Forces at vertices

Rank-deficiency of C indicated dof.Rank-deficiency of H indicates sss.

Null-matrix of C indicated instantaneous rigid-body modes.

Null-matrix of H indicates sss modes.

Ananthasuresh, IISc, Dec. 2013

dof & sss

2 3

16 3 12 1 2 1

v b dof sss

12 16 16 1 12 1 C u e 16 12 12 1 16 1 H p f

Null-space “modes” of C.

Rank deficiency = 2 (not counting rigid-body modes) 2 dof

Rank deficiency = 1 1 sss

Ananthasuresh, IISc, Dec. 2013

Can also use stiffness matrix.(finite element framework)

2 2 1 1b v v b C u e 2 1 2 1v b b v H p f

Compliance matrix

Disp. at vertices

Bar elongations

Equilibriummatrix

Bar forces

Forces at vertices

1 1b bb b p eD22 2 12 1v vv v u fK

1 2 2 1

2 2 2 1

2 2

2 1

2 1 2 1

b b v v

v b b v v v b b

v b b b

b

b

b v vv

b

b

D

D

p C u

H C

C uD

u

fH

H p

Ananthasuresh, IISc, Dec. 2013

Maxwell’s rule can be applied to trusses and linkages alike…

2 3

48 3 48 3

v b dof sss

13( 1) 2

3(16 1) 2 24 3

Kdof n n

dof

Fowler and Guest have developed nice theory for counting symmetry groups and resolve the conflicts arising due to special geometric conditions.

Ananthasuresh, IISc, Dec. 2013

ANALYSIS

Ananthasuresh, IISc, Dec. 2013

Kinematic pairs

Rigid bodies

Elastic segments

Elastic pairs

elastic chains

Modeling elastic chains

Kinematic analysis with springs

Continuum elastic deformation analysis

Ananthasuresh, IISc, Dec. 2013

Kinetoelastic analysis

• Empirical modeling of elastic pairs

• Elastica analysis of beam segments

• Pseudo Rigid-Body (PRB) modeling

• Spring-Lever (SL) and Spring-Mass-Lever (SML) modeling

• Compliant ellipsoid modeling

• Non-dimensional maps

• Finite element analysis

Ananthasuresh, IISc, Dec. 2013

Synthesis (design) methods

• Kinetoelastic synthesis using PRB modeling

• Topology and shape optimization

• Selection and re-design

• Instant-centre method

• Design using building blocks

• Pragmatic design with non-dimensional maps

• Intuitive design using a kit

Ananthasuresh, IISc, Dec. 2013

• Empirical modeling of elastic pairs

• Elastica analysis of beam segments

• Pseudo Rigid-Body (PRB) modeling

• Finite element analysis

• Spring-Lever (SL) and Spring-Mass-Lever (SML) modeling

• Compliant ellipsoid modeling

• Non-dimensional maps

• Kinetoelastic synthesis using PRB modeling

• Topology and shape optimization

• Selection and re-design

• Instant-centre method

• Design using building blocks

• Pragmatic design with non-dimensional maps

• Intuitive design using a kit

Models for analysis Synthesis methods

*

#

@

%

!

$

Ananthasuresh, IISc, Dec. 2013

PhD studentsA Narayana Reddy

Sourav Rakshit Sudarshan Hegde

Sangamesh Deepak RSambuddha KhanSantosh Bhargav

Sudhanshu SekharKiran AkellaBiplab Sarkar

T. J. Ramnath BabuNandhini Nehru

Shantanu ChakravarthyAnasuya DigheG. vasundhara

Sreenath BalakrishnanM. S. Suma

Administrative AssistantsT. S. DeepaT. Bharath K. Bindhu

Faculty CollaboratorsAshitava Goshal (ME)H.S.Jamdagni (CEDT)Navakanta Bhat (ECE)

Saraswathi Vishveshwara(MBU)

K.J.Vinoy (ECE)Saumitra Das (MRDG)

G. Mohan Rao (ISU)C. S. Jog (ME)

Utpal Nath (MCB)

Multi-disciplinary and Multi-scale Device and Design Lab

SponsorsDST, BRNS,

NPOL, ADA,

NPSM, DRDO, ISRO, SBMT,

NPMASS, and

industries.

Thank you!

Project staffG. Balaji, Shantanu

Chakravarthy, B. Manjunath, M. S. .

Deepika, B. K. Deepthi, P. Dinesh, B. Pradeep, A.

Singh, A. Alwan, M. Kulkarni, R. Ganesh, K. Girish, Siddharth Sanan,

K. Manjunath, Vinod Kumar, Vedanandan, A. Ravi Kumar, Manoj Raj,

Chetan Kumar, G. Ramu, Duely Rakshit, Vijay Prakash, G. Ramu, A. Sajeesh Kumar, M. S.

Suma, Meenakshi Sundaram, Vishwaman Malaviya, R. Genesh,

Krishna Pavan, B. Varun, Mukund Madhav Nath,

Puneet Singh, Shilpa, Ashwin Rao, Gaurah

Singh, Gautham Kumar, Gautham Baichapur,

Nikhil Jorapur, Avinash Kumar, Kunal Patil.

Masters studentsGirish KrishnanV. S. S. Srinivas

M. DineshShyam Madhavan

Harish VarmaJagdish SinghSaurabh Mitta

Saurav RakshitP. Sivanagendra

Manish AgrawalM. Rajesh

Deepak SahuG. Bhargav

P. Mallikarjuna RaoPadmanabh Limaye

Meenakshi SundaramPakeeruraju Podugu

Subhajit BenerjeeNirmit DaveGaurav Nair

Rakesh PathakNavaneet Krishna

P. SandeepMohit Mathur

Ananthasuresh, IISc, Dec. 2013

PhD studentsA Narayana Reddy

Sourav Rakshit Sudarshan Hegde

Sangamesh Deepak RSambuddha KhanSantosh Bhargav

Sudhanshu SekharKiran AkellaBiplab Sarkar

T. J. Ramnath BabuNandhini Nehru

Shantanu ChakravarthyAnasuya DigheG. vasundhara

Sreenath BalakrishnanM. S. Suma

Administrative AssistantsT. S. DeepaT. Bharath K. Bindhu

Faculty CollaboratorsAshitava Goshal (ME)H.S.Jamdagni (CEDT)Navakanta Bhat (ECE)

Saraswathi Vishveshwara(MBU)

K.J.Vinoy (ECE)Saumitra Das (MRDG)

G. Mohan Rao (ISU)C. S. Jog (ME)

Utpal Nath (MCB)

Multi-disciplinary and Multi-scale Device and Design Lab

SponsorsDST, BRNS,

NPOL, ADA,

NPSM, DRDO, ISRO, SBMT,

NPMASS, and

industries.

Thank you!

Project staffG. Balaji, Shantanu

Chakravarthy, B. Manjunath, M. S. .

Deepika, B. K. Deepthi, P. Dinesh, B. Pradeep, A.

Singh, A. Alwan, M. Kulkarni, R. Ganesh, K. Girish, Siddharth Sanan,

K. Manjunath, Vinod Kumar, Vedanandan, A. Ravi Kumar, Manoj Raj,

Chetan Kumar, G. Ramu, Duely Rakshit, Vijay Prakash, G. Ramu, A. Sajeesh Kumar, M. S.

Suma, Meenakshi Sundaram, Vishwaman Malaviya, R. Genesh,

Krishna Pavan, B. Varun, Mukund Madhav Nath,

Puneet Singh, Shilpa, Ashwin Rao, Gaurah

Singh, Gautham Kumar, Gautham Baichapur,

Nikhil Jorapur, Avinash Kumar, Kunal Patil.

Masters studentsGirish KrishnanV. S. S. Srinivas

M. DineshShyam Madhavan

Harish VarmaJagdish SinghSaurabh Mitta

Saurav RakshitP. Sivanagendra

Manish AgrawalM. Rajesh

Deepak SahuG. Bhargav

P. Mallikarjuna RaoPadmanabh Limaye

Meenakshi SundaramPakeeruraju Podugu

Subhajit BenerjeeNirmit DaveGaurav Nair

Rakesh PathakNavaneet Krishna

P. SandeepMohit Mathur