Dinamica Non Lineare di Strutture e Sistemi Meccanici

Prof. Carlos Eduardo Nigro Mazzilli

Universidade de São Paulo

Dinamica Non Lineare di Strutture e Sistemi

Meccanici

Lezione 1

• Dynamics based on classic mechanics, whose fundamental laws are

credited to Newton (1646-1727), ‘standing on giant’s shoulders”...

• Greeks: axiomatic reasoning disconnected from experimentation

Forces were necessarily caused by contact; what about field forces?

Aristotle (384BC-322BC): a force causes constant velocity?

Terrestrial mechanics vs celestial mechanics?

• Ptolemy (90-168): geocentric system vs Aristarco (310BC-230BC)

heliocentric system (three centuries before)

• ...Galileo (1564-1642): ‘e pur si muove’

• Romans?

Historical overview on dynamics

• Moslems: from VIII to XIV centuries (Alexandria, Iberic Peninsula)

Barakat (1080-1165) denied Aristotle: force causes velocity to change...

Newton’s second law?

Alhazen (965-1040): body moves perpetually unless force obliges it to

stop or change direction... Newton’s first law?

Avempace (1095-1138): to an action corresponds a reaction... Newton’s

third law?

• Kepler, Copernicus and Galileo: celestial mechanics

• Galileo: terrestrial mechanics (displacement of a falling body

proportional to the square of time)

• Newton: law of universal gravitation and much more...


Newton’s laws

First law (inertia): there are priviledged observers, called inertial observers, with

respect to whom isolated material points – that is, those subjected to null resultant

force – are at rest or in uniform rectilinear motion.

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in

directum, nisi quatenus a viribus impressis cogitur statum illum mutare


Second law (fundamental): the resultant force of a mass point is proportional to its

acceleration defined with respect to an inertial observer. The proportionality constant is

termed mass, which is positive and it is a property of the material point.

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et eri secundum

lineam rectam qua vis illa imprimitur

2

2

d RF m

dt

Newton’s laws


Third law (action and reaction): to every action of a material point upon another one

corresponds a reaction of same intensity and direction, yet in oposite sense.

Lex III: Actioni contrariam semper et aequalem esse reactionem: sive corporum

duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi

Newton’s laws


• Newton: differential and integral calculus

• Leibniz (1646-1716): independent development of differential

calculus & fundamentals of analytical dynamics

• D’Alembert (1717-1783): principle...

• Lagrange (1736-1813): Mécanique Analytique and variational

principles

• Hamilton (1805-1865): principle...

• ...


Physical space: affine Euclidian space of dimension 3

Configuration space: affine Euclidian space of dimension 3N (provided the 3N co-

ordinates of the N material points are independent)

• N material points mi

• position of mi given by cartesian co-

ordinates: xi1, xi

2, xi3

• a “point” in this space caracterizes completely

the configuration of the system of material

points in a given time t (co-ordinates of material

points obtained by “projections”)

Example: a material point moving along a parabolic curve

Original configuration space of dim 3N = 3

• If there are c constraint equations relating these co-ordinates, it is possible to

define another configuration space with dimension n = 3N-c , termed “number

of degrees of freedom” of the system

x3 = 0

x2 = (x1)2

c = 2 constraint

equations

Configuration space of dim n = 3N-c=1

• Generalised coordinates Q1(t), Q2(t), ... , Qn(t), n = number of degrees of freedom,

are scalars conveniently chosen, so that they uniquely define the original 3N

physical coordinates of the system

tQQQxx

tQQQxx

tQQQxx

nNN

n

n

,,...,,

,,...,,

,,...,,

21

33

21

2

1

2

1

21

1

1

1

1

3N holonomic constraint equations

• the functions are are finite of class C1 tQQQx n

i ,, . . . ,, 21

• Jacobian of the transformation is non-null

• Particular case of holonomic constraint: scleronomic constraint

03

2212

1

x

Qxx

Qx

a transformation “matrix” (of order n = 1)

with det T 0

Let it be

n

ii QQQxx , . . . ,, 21

Example: a material point moving along a parabolic curve

11

Q

x

Q

xT

1

J= det T = 1

constraint equation in t2

0,,, 1

321

1 txxxf

0,,, 2

321

2 txxxf

constraint equation in t1

real infinitesimal displacement

R

virtual displacementix jQ

Virtual displacements in holonomic constraints

t

mt+dt

Rd

• Virtual displacements are kinematically admissible at a fixed time t, that is, they

satisfy the constraint equations at that time t

0. RFW v i

• The class of real displacements doesn’t necessarily coincide with the class of real

displacements for holonomic constraints

• For scleronomic constraints, however, since the constraint equations are independent

of t, the class of real displacements coincides with the class of virtual displacements,

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

• Ideal (constraint) reactions are orthogonal to the virtual displacements at the points

they are applied. Hence, the virtual work of ideal reactions is null.

D’Alembert’s principle

the sum of the resultant force

and the inertial force is the null

vector

2

2

d RF m

dtNewton’s 2nd law = 1 a N

2

20

d R

F md t

= 1 a N

r e s u l t a n t f o r c e a v i v nF F F F

active

ideal constraint

non-ideal constraint

2

2in e rt ia fo rc e

I d RF m

d t

“closing” of the force

polygon, as in statics

1 1

. 0 . 0

N N

a v i v n IF F F F R R

1

. 0

N

a vn IF F F RGeneralised D’Alembert’s

principle

v nae FFF

• Remark 1 Effective force

(it is not necessary to know a priori the reactions to write down the

equations of equilibrium/motion)

equilibrium

• Remark 2 System with ideal constraints:

0.1

RFFN

Ia

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

0.1

RFN

a

Newton’s 2nd law

Hamilton’s principle

2

1

0 t

nc

t

T V W dt

1

1kinetic energy .

2

N dR dR

T mdt d t

D’Alembert’s principle Hamilton’s principle

v i r t u a l v a r i a t i o n o f k i n e t i c e n e r g y T

1

.

N

T m R R n o ta tio n d

x xd t

1

. v i r tu a l w o rk o f c o n s e rv a t iv e fo rc e s

N

cV F R

1

. v i r tu a l w o rk o f n o n -c o n s e rv a t iv e fo rc e s

N

n c n cW F R

v i r t u a l v a r i a t i o n o f p o t e n t i a l e n e r g y V

v i r t u a l w o r k o f n o n - c o n s e r v a t i v e f o r c e s n cW

Lagrange’s equation

i

i ii

d T T VN

dt Q QQ

1 2 1 2, , . . . , , , , . . . , , k i n e t i c e n e r g yn nT T Q Q Q Q Q Q t

1 2, , . . . , , p o t e n t i a l e n e r g ynV V Q Q Q t

1

g en e ra lized n o n -co n se rv a tiv e fo rce .N

n c

i

i

RN F

Q

• Remark

1 1

.n N

n c n c

i i

i

N Q F R W

virtual work of the non-

conservative forces

Hamilton’s principle Lagrange’s equation

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

Formulation of equations of motion

QmQck Qtp

Newton’s 2nd law:

0 QmQck Qtp

D’Alembert’s principle:

QQQmQck Qtp 0

Generalised D’Alembert’s principle:

tpk QQcQm

2

1

0

t

nc

t

T V W dt Hamilton’s principle:

n cW N Q p t c Q Q

Substituting...

m Q c Q k Q p t

2 2

1 1

0

t t

t t

m Q Q d t kQ p t cQ Q d t

integrating by parts

2

2

11

0

tt

tt

m Q Q m Q cQ kQ p t Q d t 1 2 0Q t Q t

2

1

0

t

t

m Q cQ kQ p t Q d t Q

21

2T mQ

21

2V kQ

T m Q Q

V k Q Q

d T T VN

dt Q QQ

Lagrange’s equation:

n cW N Q p t c Q Q N p t c Q

m Q c Q k Q p t m Q k Q p t c Q Substituting...

21

2T mQ

21

2V kQ

; 0T T

m QQQ

VkQ

Q

d Tm Q

dt Q

Example 2: sistem of rigid rods

AB and BC rigid rods

BC massless rod

Linear dynamics: horizontal displacements of B and C are negligible for small Q

AB

C

x

m

m2

c1 c2 k2k1

a a a a a2a

A'

B'

C'

Q

)(),( ta

xptxp

2 24

* 2

2

0

1 2 1 1

2 3 2 4 2

ax

T m Q m Q d x m Qa

*

2

4 4w ith

9 3m m m a

2 2 4

1 2 2

0

1 3 1 1 2

2 4 2 3 4 3

ax

V k Q k Q m g Q d x m g Qa

* 2 *

0

1

2V k Q p Q

*

1 2

9 1w ith

1 6 9k k k *

0 2

2an d 2

3p m a m g


4

1 2

04 4 4

a

n c Q Q x xW c c Q Q p t Q d x N Q

a a

* *N c Q p t

* 12w ith

1 6

cc c * 1 6

an d 3

p t p a t

d T T VN

dt Q QQ

* * * * *

0m Q c Q k Q p p t

Example 3: Simple pendulum

d T T VN

dt Q QQ


21

2T m L Q

1 c o sV m g L Q m

Q

mg

L - Q(1 cos )

L

2 s e nm L Q m g L Q

o r s e n 0 ( n o n -lin e a r a n a ly s is )g

Q QL

0 ( lin e a r a n a lys is )g

Q QL

Example 4: Simple pendulum subjected to support excitation

2 2 2 2 2 2 21c o s s e n 2 s e n

2T m L Q Q L Q Q L f Q Q f

1 c o sV m g f L Q

s e n ( c o s )R L Q i f L Q j

c o s ( s e n )R L Q Q i f L Q Q j

2 2 21 1s e n

2 2T m L Q m f m L f Q Q

m

Q

mg

L

f(t)

j

i

R

d T T VN

dt Q QQ


2 ( ) s i n 0m L Q m L g f Q

1o u ( ) s in 0 ( n o n -lin e a r a n a ly s is )Q g f Q

L

2 s in co sd T

m L Q m L f Q m L f Q Qd t Q

co sT

m L f Q QQ

sinV

m gL QQ

2 s i n c o s c o s s i nm L Q m L f Q m L f Q Q m L f Q Q m g L Q

1 ( ) 0 ( lin e a r a n a ly s is )Q g f Q

L

Example 5: Rigid rod with non-linear spring and geometric imperfection, subjected to

static and dynamic loading

Non-linear “constitutive” lawg e o m e t r i c i m p e r f e c t i o n 1

2

1M Q K Q Q

2 21

2T m L Q

2 4

2

0

1 cos cos cos cos2 4

Q Q QV K d m gL Q K m gL Q

d T T VN

dt Q QQ


s e nn cW N Q P t L Q Q s e nN P t L Q

22 ( ) 1 s i nm L Q K Q Q m g P t L Q

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