Prof. Dr. Carlos Eduardo Nigro Mazzilli

PEF-5737

Non-Linear Dynamics and Stability

• Newton’s laws

1st law (inertia): there are special observers (called inertial observers)

with respect to whom isolated material points (that is, those subjected to

null resultant force) are at rest or perform URM (uniform rectilinear

motion).

3rd law (action and reaction): to every action of a material point over

another one it corresponds a reaction of same intensity and direction,

although in opposite sense.

Equations of motion for systems of material points

2nd law (fundamental): the resultant force acting on a material point is

proportional to its acceleration with respect to an inertial observer

( ). The constant of proportionality is a material point property

called mass, m>0 amF

Physical space: afine Euclidian space of dimension 3.

Configuration space: afine Euclidian space of dimension 3N (if the 3N

coordinates are independent).

• N material points mi

• position of mi caracterized by 3

coordinates: xi1, xi

2, xi3

• a “point” in this space fully characterizes

the material point system in time t

(coordinates of material points obtained by

“projections”)

Example: a material point moving on a parabole

x3 = 0

x2 = (x1)2

c = 2 constraint

equations

Configuration space of dim 3N = 3

Configuration space of dim n = 3N-c

n = 1

• if there are c constraint equations relating the coordinates it is

possible to use a configuration space os dimension n = 3N-c (called

number of “degrees of freedom”)

• Generalized coordinates Q1(t), Q2(t), ... , Qn(t), where n = number of

degrees of freedom, are conveniently chosen scalar quantities that allow

for stablishing a bi-univocal relationship with the 3N coordinates of the

material point system.

tQQQxx

tQQQxx

tQQQxx

nNN

n

n

,,...,,

,,...,,

,,...,,

21

33

21

2

1

2

1

21

1

1

1

1

3N holonomic (reonomic) constraint

equations

• the functions are finite of class C1 tQQQx n

i ,,...,, 21

• Transformation matrix

• Non-null Jacobian

...

• Particular case of holonomic constraint: scleronomic constraint

03

2212

1

x

Qxx

Qx

a transformation “matrix” T (of

order n = 1) with det T 0

Let it be

n

ii QQQxx ,...,, 21

Example: a material point moving on a parabole

11

Q

x

Q

xT

1

J= det T = 1

t

mt+dt

constraint equation in t2

0,,, 1

321

1 txxxf

0,,, 2

321

2 txxxf

constraint equation in t1

real displacement

virtual displacement

ix

jQ

Virtual displacements – holonomic constraint

• Virtual displacements are kinematically admissible displacements in a

fixed time t, that is, satisfy the constraint equations for that time t.

• The class of real displacements do not necessarily coincide with that of

the virtual displacements for holonomic constraints.

• However, for scleronomic constraints, since the constraint equations are

independent of t, the class of real displacements coincides with that of

virtual displacements, that is, the real displacements are a particular case

of the virtual displacements.

• Ideal constraint reactions are orthogonal to the respective virtual

displacements. Hence, ideal constraint reactions do not give rise to virtual

work:

0. RFW vi

2

2

dt

RdmF

2nd Newton’s law

D’Alembert’s Principle

= 1 a N

02

2

dt

RdmF

= 1 a N

resultante força vnvia FFFF

active ideal

constraint

non-ideal constraint

inércia deforça 2

2

dt

RdmF I

The sum of the resultant

and the inertial forces is

the null vector

(“closing” of force

polygon, as in statics)

0.0.11

RRFFFFNN

Ivnvia

0.1

RFFFN

Ivna Generalized

D’Alembert’s Principle

resultant force

inertial force

vnae FFF

• Remark 1 Effective force

(it is not necessary to know the constraint forces to

formulate the equations of statics/dynamics)

equilibrium

• Remark 2 System of ideal constraints, only:

0.1

RFFN

Ia

• Remark 3 Principle of virtual displacements in statics is a particular

case:

0.1

RFN

a

2nd Newton’s law

Hamilton’s Principle

2

1

0

t

t

nc dtWVT

N

dt

Rd

dt

RdmT

1

.2

1cinética energia

D’Alembert’s Principle Hamilton’s Principle

N

RRmT1

.

dt

d notação

N

c RFV1

vasconservati forças das virtual trabalho.

N

ncnc RFW1

vasconservati não forças das virtual trabalho.

kinetic energy

notation

virtual work of conservative forces

virtual work of non-conservative forces

Hamilton’s Principle

Lagrange’s Equations

i

iii

NQ

V

Q

T

Q

T

dt

d

cinética energia,,...,,,,...,, 2121 tQQQQQQTT nn

Lagrange’s Equations

p o tencial energia,,...,, 21 tQQQVV n

N

i

nc

iQ

RFN

1

. vaconservati-não dageneraliza força

• Remark

n

i

Nncnc

ii WRFQN1 1

.

virtual work of non-

conservative forces

kinetic energy

potential energy

generalized non-conservative force

Example 1: One-degree-of-freedom linear oscillator

Formulation of equations of motion

QmQck Qtp

2a Newton’s law:

0 QmQck Qtp D’Alembert’s Principle:

QQQmQck Qtp 0 Generalized D’Alembert’s Principle:

tpk QQcQm

2

1

0

t

t

nc dtWVT

Hamilton’s Principle:

2

2

1QmT

2

2

1kQV

QQctpQNW n c

Substituting:

tpk QQcQm

QQmT

Qk QV

02

1

2

1

QdtQctpkQdtQQm

t

t

t

t

integrating by parts

02

1

2

1

QdttpkQQcQmQQm

t

t

t

t

0

021 tQtQ

Hence, QQdttpkQQcQm

t

t

02

1

NQ

V

Q

T

Q

T

dt

d

Lagrange’s equation:

2

2

1QmT

2

2

1kQV

QQctpQNW n c QctpN

tpk QQcQm

0;

Q

TQm

Q

T

kQQ

V

QmQ

T

dt

d

Qctpk QQm Substituting:

Example 2

AB e BC rigid bars

BC massless bar

Linear dynamics: horizontal displacements at B and C are neglected,

which is reasonable for small Q.

AB

C

x

m

m2

c1 c2 k2k1

a a a a a2a

A'

B'

C'

Q

)(),( ta

xptxp

a

QmdxQa

xmQmT

4

0

2*

22

22

1

42

1

3

2

2

1

ammm3

4

9

4com 2

*

a

QgmdxQa

xgmQkQkV

4

0

2

2

2

2

13

2

43

1

2

1

4

3

2

1

QpQkV *

0

2*

2

1

21

*

9

1

16

9com kkk gmamp

2

*

03

22ewith

with

Lagrange’s equation:

QNdxQa

xt

a

xpQQc

QQcW

a

nc

4

0

21444

tpQcN **

21*

16com c

cc taptp

3

16e *

NQ

V

Q

T

Q

T

dt

d

tppQkQcQm **

0

***

with and

Example 3: Simple pendulum

NQ

V

Q

T

Q

T

dt

d

Lagrange’s equation:

22

1QLmT

Qm gLV cos1 m

Q

mg

L - Q(1 cos )

L

Qm gLQm L sen2

linear)-não (análise0senou QL

gQ

0 0

linear) (análise0 QL

gQ

(non-linear)

(linear)

Example 4: Simple pendulum subjected to support excitation

2222222 sen2sencos2

1fQQfLQQLQQLmT

Q1Lfm gV c o s

jQLfiQLR

)cos(sen

jQQLfiQQLR

)sen(cos

QQfmLfmQmLT sen2

1

2

1 222

m

Q

mg

L

ft ()

j

i

R

NQ

V

Q

T

Q

T

dt

d

Lagrange’s equation: 0

QQfm LQfm LQm L co ssen2

QQfm L cos Qm g L sen

0sen)(2 Qfgm LQm L

linear)-não (análise0sen)(1

ou QfgL

Q

Q

T

dt

d

Q

T

Q

V

QQfm LQfm LQm L co ssen2

Substituting:

QQfm L cos Qm g L sen

linear) (análise0)(1

QfgL

Q

(non-linear)

(linear)

Example 5: Rigid bar with non-linear rotational spring and imperfection,

subjected to dynamical load p(t) and static pre-loading mg

Non-linear

“constitutive” law 1oimp e rfe içã

21 QQKQM

22

2

1QmLT

QmgL

QQKQmgLdKV

Q

coscos

42coscos1

42

2

0

imperfection

NQ

V

Q

T

Q

T

dt

d

Lagrange’s equation:

QQLtPQNW n c sen QLtPN sen

QLtPm gQQKQm L sen1)(22

QLtPm gQQKQm L sen1)(22