PEF-5737 Non Linear Dynamics and Stability
Transcript of PEF-5737 Non Linear Dynamics and Stability
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