Dynamical Modeling of Movement

UNIVERSITÁ POLITECNICA DELLE MARCHE

FACOLTÁ DI INGEGNERIA

Dynamical Modeling

of Movement

Notes by

Luca A. Pettinari

compiled with L TEX

A

regheliuk61@gmail.com

Contents

1 Introduction 1

1.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Free and applied vectors . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Mechanics principles 17

2.1 Dynamics of material point . . . . . . . . . . . . . . . . . . . . . 17

2.2 Newton’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Classical interactions . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Conservation of total energy . . . . . . . . . . . . . . . . . . . . . 24

3 Rigid body kinetics 27

3.1 Center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Koenig’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Kinetics of rotations . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Time derivatives of linear and angular momentum . . . . . . . . 41

4 Rigid multi-body dynamics 46

4.1 Dynamic and static principles . . . . . . . . . . . . . . . . . . . . 46

4.2 Joint modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Lagrange’s equation of motion 58

5.1 Coordinates and constraints . . . . . . . . . . . . . . . . . . . . . 59

5.2 Virtual power and generalized forces . . . . . . . . . . . . . . . . 62

5.3 Lagrange’s equations of motion . . . . . . . . . . . . . . . . . . . 65

i

Chapter 1

Introduction

Classical mechanics, and in particular, dynamics, is the discipline that stud-

ies the movement of material systems and researches how interaction beetween

them can cause or modify their movements. In order to study and analyze hu-

man movement, notions of dynamics of the material point, rigid bodies, and

articuleted rigid bodies are necessary. The branch of biomedical engineering

biomechanics.

that studies these facts is called In non-relativistic framework

8

·

c = 3 10

(velocities are much lesser than , and there is no need to apply quan-

tum mechanics), where we will place ourselves as observers of these phenomena,

the determination of the movements of a system and of the actions that cause

or oppose to these movements consist in establishing a system of equations by

applying four basic principles:

1. The law of mass conservation, which states that for any closed system to

all transfers of matter and energy, the mass of the system must remain

constant over time, as system mass cannot change quantity if it is not

added or removed.

2. The fundamental principles of dynamics, that is to say the conservation of

linear momentum in the hypothesis that mechanical system we approach

in this subject are closed systems.

3. The first law of thermodynamics, that is to say the conservation of the

total energy of an isolated system.

4. The second law of thermodynamics, that is to say the principle of evolution

of systems, which describe the degradation of energy in other form like

heat, friction or other, using the concept of entropy.

As a remark, it’s useful to recall these definitions of system in thermodynamics:

1

1.1. FUNDAMENTAL CONCEPTS 2

Definition. A closed system can exchange energy (as heat or work) but not

matter, with its surroundings. An isolated system cannot exchange any heat,

work, or matter with the surroundings, while an open system can exchange en-

ergy and matter.

Equation of motion can be established based on the fundamental laws of the

dynamics, which in turn are obtained by the physical principles below. This

approch generally leads to a system of equation where the number of equation

is less than the number of unknowns, so that it is always possible to seek for an

acceptable solution of the motion. Describing the motion of a material point, a

rigid body, or an articulated one means to know at each instant where is located

every particle of the system in the space, with respect to the chosen reference

frame.

1.1 Fundamental concepts

Concept of time is absolute, flow of time is the same for any observer. That

is false according to Eisten’s theory of relativity, but this assuption is suitable

for our observation, which are embedded in a "world" where phenomena involve

E

lower velocities by far than light speed. Let be the euclidean tridimensional

points.

space containig the primay objects of geometry: Between two not

distance,

overlaped points it can be defined an invariant of the space, as the

shortest path to reach one starting from the other:

E

∀ ∈ ⇒ kP −

O, P d = Ok

OP

E

∈

O

Given a point called origin, an operation of summation and multiplication

E

∈

P v,

for a scalar element, for any other point can be associet a quantity

V

vector,

called as an element of a vectorial space (a set with a particular

algebric structure): −

v = P O

Every vector sharing the same origin is a quantity related to the same "ob-

reference frame.

server", which in this contest take name of

E E

∈

O

Definition. Let be the euclidian tridimensional space, given a point

{e }

, e , e

and a tern of versors mutually orthogonal and with unitary norm.

1 2 3

The resulting set of these elements is called reference frame:

{O, }

R = ê , ê , ê

1 2 3

Reference frame is a point where an observer measures physical phenomena:

2

1.1. FUNDAMENTAL CONCEPTS 3

as it, we can associate to reference frame the velocity of its origin or the ori-

entation of its associated basis with respect to another reference frame (with

special properties) called absolute. To summarize, vectors are quantities defined

E: E

∈ ∈

O, P O

starting from two different points for any origin point it can

V,

be associated a vectorial space with its structures and of course its basis.

~ −

OP (P O).

Another notation is , equivalent to Note that versors, which are

unitary vectors, are noted as a bold style hatted letter, while vectors in general

are in bold style. With the idea of vectors we are able to define the concepts

of motion, speed, force, momentum and many others. Observe that different

quantities in different frame referencies can be expressed in the same basis of

V, whereas this is somehow a local algebric structure suitable for every point

of the euclidean space. Since we use numbers to express things and we cannot

R,

treat properly vectors as abstract objects, given a frame reference the same

coordinates

vectors can be expressed by means of in the associated basis of

R, being understood that each of the following representation define exaclty the

same object in the space: ⇐⇒ {v }

v , v , v

1 2 3

V

In particular, the link between the space and coordinates of one of its element

is: v = v ê + v ê + v ê

1 1 2 2 3 3 −

v v v v = P O

The scalar quantities , and expresses the vector in the given

1 2 3 V

R.

reference It is true that, while the representation as element of is universal

v

(and abstact in a certain manner), the same vector can be expressed in many

ways for any given reference frame. When it is important to express a vectorial

quantities by means of its coordinates, the notation we use is:

 

v 1

v = v

 

2

 

R v 3 ∈

i S

Remark (Einstein notation). Quantity containg an index (where for

≡

S

example expresses objects of vectorial spaces. Generally, upper index

N)

are used for vectors, lower for covectors and in tensors they are mixed to express

their covariant and countervariant part (tensors are applications that generalize

every object of vectorial spaces, like vectors, covectors, matrices, scalar products

and more). When an index reapets in the same term, there should be summation

3

1.2. FREE AND APPLIED VECTORS 4

for that term on the whole set where the repeated index vary. For example:

n

X

j j

→ ⇐⇒

i, j = 1, . . . , n a = B c a = B c

i j i j

i i

j=1 j

a b B is matrix

In this case and are covectors with the same dimension, while

i j i

×

n n

of elements.

Every reference frame has its associated basis, but the two concept are differ-

ent: since the first has the role of the observer, the measured quantities depends

on it, while the representation of these object (they are vectors) is not unique,

so that they can be expressed in any basis, usually the one related with the

frame. This is true if we think that interpretation of phenomena can depend

on the observer (as relativity shows) but they are actually bound by the same

thing, which properly is a law of the physic, and such as, equal for any observed

in any place on the space at any time. When it needs to change representation

(i.e. for easier calculus), vectors are converted in other representation with ex-

act laws, known as law of covariancy and countervariancy. In particular, vectors

transform as countervariant objects with this relation:

i i k

v = R u

k

B B

0)

In other terms, given two basis (briefly indicated with and (briefly

0 1

V

∈ v

1), v v , and

and

indicated with a vector is expressed respectively 1

0

the relation between the coordinates is: 10

v = R v

0 1

The previous one is a relation between coordinates and differs from any other

10

R change

relation we will write, which will be all vectorial relations; is called

of basis matrix, and it allows to write a vector in the old basis once known

its coordinates in the new one (that’s why vectors are so called countervariants

quantities). We will also see that this relation (and a couple of more) allows

us to express every vectorial relation in the basis we prefer, letting said that to

do calculus, quantities must be expressed in the same basis (so once chosen the

basis it must be the same for every term involved in a relation).

1.2 Free and applied vectors

One of the most important thing when we describe velocity, force, torque

application point.

of a particle or a body is the In particular, let us start

4

1.2. FREE AND APPLIED VECTORS 5

P

considering velocity of a point :

Figure 1.1: Definition of velocity

The previous picture shows as the velocity vector is applied in each istant of

P

time to ; from what we have learnt from the last paragraph is that, given an

O, P

origin a vector points , starting from the origin. In this case, component

R

of velocity are expressed with respect to (that is to say, with respect to the

R), P

associated basis of but the vector is applied on . This become an evidence

if we use definition of velocity with difference quotience:

− − −

P (t + h) O (P (t) O)

d −

(P (t) O) = lim =

v = dt h

h→0

− − −

P (t + h) O P (t) + O P (t + h) P (t)

= lim = lim

h h

h→0 h→0

v

Thereby we can see that the concept of is yielded by quantities in a certain

reference frame, though giving as a result a vector that do not share the same

v P

applied vector

origin. For this reason is called to the point , because it’s

free vectors:

nature is strictly related to it. On the other hand we have they are

not peculiar of the point of application, hence they are used to express a global

property of the euclidian space. For instace, in fluid the concept of velocity is

no more a characteristic of a point (or a set of point) but rather a characteristic

of a manifold of them, described by a vectorial field, which associates a velocity

for each point of the fluid in terms of precise laws. Angular velocity and torque

are free vectors, while linear velocity, acceleration, force are applied ones.

V,

∈

a, b

Definition. Given two vectors cross product associates to them a

third vector which direction is orthogonal to the plane they lies on and verse is

determined according to the right-hand rule. It’s magnitude is given by:

ka × kakkbk

bk = sin θ (1.1)

θ

where is the angle between the lying directions.

5

1.2. FREE AND APPLIED VECTORS 6

V W B =

Definition. Let it be and two vectorial spaces with respectively basis 1

B 0

{e }, ∈ {e }, ∈

i I = j J,

and then the unique vectorial space spanned by

i 2 j V W, B

0 ⊗

(b , b ) =

each pair of any versors is which basis is indicated as

i j V W,

0

{e ⊗ } ∈ × ∈ ∈

e (i, j) (I J). v w

as Given two vectors, and they

i j V W:

⊗ ∈ ⊗

T = (v w)

associate a unique element X 0

i j ⊗

T = v w e e

i j

ij

It can be shown that the following relation including tensorial product:

⊗ ·

(a b)c = (a c)b

Cross product allows to express severals concepts in dynamics (as tensor product

is mostly used in fluid mechanics and continuum mechanics). In particular,

B {ê }

= , ê , ê

given a basis (or in the same way a reference frame), being

1 2 3

(a , a , a ) (b , b , b ) a, b

and respectively the coordinates of expressed in the

1 2 3 1 2 3

same reference, the cross product is given by:

ê ê ê

1 2 3

×

a b = a a a

1 2 3

b b b

1 2 3

Another useful relation is the so called Gibb’s formula, which involves also dot

product. It states that:

× × · − · ⊗ − ⊗

a (b c) = b(a c) c(a b) = (a b b a)c (1.2)

Other sensible mention properties are:

• Bilinearity: × × ×

(ka) b = k(a b) = a (kb)

× × ×

(a + c) b = a b + c b

× × ×

a (b + c) = a b + a c

• × ×

a b = 0, a b a b = c, c

If then and are lineary dependent. If then is

orthogonal to the plane where the pair of them lie. It’s obvious that:

×

a a =0

6

1.2. FREE AND APPLIED VECTORS 7

• Anticommutativity: × −b ×

a b = a

• {ê },

, ê , ê

Given the standard basis it stands that:

1 2 3

×

ê ê = ê

1 2 3

×

ê ê = ê

2 3 1

×

ê ê = ê

3 1 2

• ×

a b

Any given product can be rearranged as multiplication of a suitable

×

3 3 S(a)

matrix, called axial matrix and indicated as with the second

b,

argument so that: ×

a b = S(a)b

{a }

, a , a a

If are the components of in a given basis, the axial matrix

1 2 3

a

of is an antisymmetric matrix defined by:

 

−a

0 a

3 2

−a

S(a) = a 0

 

3 1

 

−a a 0

2 1

Now we have sufficient notion to declare the Possoin theorem, which allows to

compute the time-derivative of a versor changing its orientation in time.

B {ê }

= , ê , ê

Theorem (Poisson’s rule). Given a basis of orthonormal vec-

1 2 3 V

∈

ω

tors, varying their orientation with time, it exists a vector such that:

d ×

ê (t) = ω ê (t) (1.3)

i i

dt

e

Proof. Existence. Being unitary normed vector for each time istant, it holds

i

that: 2

ke k ·

= e e = 1

i i i

Applying time derivative to both members and exploiting product rule:

d d

· ·

ê (t) ê (t) + ê (t) ê (t) = 0

i i i i

dt dt

d ·

2 ê (t) ê (t) = 0

i i

dt

This shows that: d ⊥

ê (t) ê (t)

i i

dt 7

1.2. FREE AND APPLIED VECTORS 8

Thus, it will exist a vector, according to the properties of cross product, such

that: d ×

ê (t) = ω ê (t) i = 1, 2, 3

i i

dt ω

Unicity. Let us show that (whose physical interpretation is angular velocity)

{1,

i = 2, 3},

is unique. Considering that scrolling the index ad absurdum let

B

i ω

us assume that for any given it is associated a different vector . Since

i

contains orthonormal versors, it can be written:

× 6

ê ê = 0 i = j

i j

Therefore, applying time derivative on both members of the previous we obtain:

d

d ·

· ê + ê ê = 0

ê j i j

i dt dt d

d − ·

· ê = ê ê

ê j i j

i dt dt

According to what has been shown below:

× · −ê · ×

ω ê ê = ω ê

j j i j i i

Exploiting anticommutative property and mixing dot and cross product prop-

erties, the last relation can be reduced like:

−(ê × · −ω × ⇔

ê ) ω = (ê ê ) ω = ω

i j j j i j i j

ω

It can be shown that the quantity that appear in the Poisson’s rule coincide

with the angular velocity of a frame reference. The Poisson rule allows us to

a

write a relation between components of any time-varying vector with respect

B {ê }

= , ê , ê

to fixed and mobile basis. Let us introduce the pair of them as 0 1 2 3

B 0 0 0

{ê }

0) = , ê , ê

(the fixed one, indicated as and (the mobile one, indicated

1 1 2 3

1), a

as then components of are different with respect to one or other:



  

0

a

a

1 1

0

a = a =

a a

   

2 2

1

0    

0

a a

3 3

Therefore it stands that: 0 0 0 0 0 0

a = a ê + a ê + a ê = a ê + a ê + a ê

1 1 2 2 3 3 1 1 2 2 3 3

8

1.3. KINEMATICS 9

B

Applying time derivative