Dynamical Modeling of Movement

Università Politecnica delle Marche

Facoltà di ingegneria

Dynamical modeling of movement

Notes by Luca A. Pettinari compiled with LaTeX

Aregheliuk61@gmail.com

Contents

• Introduction 1
• Fundamental concepts 2
• Free and applied vectors 4
• Kinematics 9
• Mechanics principles 17
• Dynamics of material point 17
• Newton’s law 18
• Classical interactions 21
• Conservation of total energy 24
• Rigid body kinetics 27
• Center of mass 28
• Koenig’s Theorems 30
• Kinetics of rotations 32
• Kinetic energy 39
• Time derivatives of linear and angular momentum 41
• Rigid multi-body dynamics 46
• Dynamic and static principles 46
• Joint modeling 50
• Lagrange’s equation of motion 58
• Coordinates and constraints 59
• Virtual power and generalized forces 62
• Lagrange’s equations of motion 65

Chapter 1

Introduction

Classical mechanics, and in particular, dynamics, is the discipline that studies the movement of material systems and researches how interaction between them can cause or modify their movements. In order to study and analyze human movement, notions of dynamics of the material point, rigid bodies, and articulated rigid bodies are necessary. The branch of biomedical engineering that studies these facts is called biomechanics. In non-relativistic framework (velocities are much lesser than 8·c = 3 10, and there is no need to apply quantum 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:

Fundamental concepts

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 energy 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 approach 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 Einstein’s theory of relativity, but this assumption is suitable for our observation, which are embedded in a "world" where phenomena involve lower velocities by far than light speed. Let be the euclidean tridimensional space containing the primary objects of geometry: points. Between two not overlapped points it can be defined an invariant of the space, as the distance, shortest path to reach one starting from the other:

E ∀ ∈ ⇒ kP − O, P d = OkOP E ∈ O

Given a point called origin, an operation of summation and multiplication for a scalar element, for any other point can be associated a quantity V, vector, called as an element of a vectorial space (a set with a particular algebraic structure): −v = P O

Every vector sharing the same origin is a quantity related to the same "observer", which in this contest take name of reference frame. 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:

R = {O, ê₁, ê₂, ê₃}

Reference frame is a point where an observer measures physical phenomena: as it, we can associate to reference frame the velocity of its origin or the orientation of its associated basis with respect to another reference frame (with special properties) called absolute. To summarize, vectors are quantities defined starting from two different points for any origin point it can be associated a vectorial space with its structures and of course its basis. −OP (P O).

Another notation is v, equivalent to −OP (P O). 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 algebraic structure suitable for every point of the euclidean space. Since we use numbers to express things and we cannot treat properly vectors as abstract objects, given a frame reference R, the same vectors can be expressed by means of coordinates in the associated basis of R, being understood that each of the following representation define exactly the same object in the space:

V ⇔ {v₁, v₂, v₃}

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

v = v₁ê₁ + v₂ê₂ + v₃ê₃

−v = P O

The scalar quantities v₁, v₂ and v₃ expresses the vector V in the given reference R. It is true that, while the representation as element of V is universal (and abstract 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_R = ⎡v₁ v₂ v₃⎤

Remark (Einstein notation). Quantity containing an index (where for example i ∈ S ≡ N) expresses objects of vectorial spaces. Generally, upper index 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 repeats in the same term, there should be summation for that term on the whole set where the repeated index vary. For example:

n
Σ a^j = B^j_icⁱ ⇔ a = Bc
j=1

In this case a and b are covectors with the same dimension, while B is matrix i×n of elements.

Every reference frame has its associated basis, but the two concept are different: 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 exact laws, known as law of covariancy and countervariancy. In particular, vectors transform as countervariant objects with this relation:

vⁱ = Rⁱ_ku^k

In other terms, given two basis (briefly indicated with B₀ and B₁), and indicated with a vector V ∈ v₁, v₀, the relation between the coordinates is:

v₀ = R₁₀v₁

The previous one is a relation between coordinates and differs from any other relation we will write, which will be all vectorial relations; R₁₀ is called change 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 of a particle or a body is the application point. In particular, let us start considering velocity of a point P:

Figure 1.1: Definition of velocity

The previous picture shows as the velocity vector is applied in each instant of time to P; from what we have learnt from the last paragraph is that, given an origin O, a vector points P, starting from the origin. In this case, component of velocity are expressed with respect to R (that is to say, with respect to the associated basis of R), but the vector is applied on P. This become an evidence if we use definition of velocity with difference quotient:

v = d −(P(t) O) / dt = lim_h→0 [−P(t + h) O − (P(t) O)] / h

= lim_h→0 [−P(t + h) O + O P(t)] / h = lim_h→0 [P(t + h) P(t)] / h

Thereby we can see that the concept of v is yielded by quantities in a certain reference frame, though giving as a result a vector that do not share the same origin. For this reason v is called applied vector to the point P, because it’s nature is strictly related to it. On the other hand we have free vectors: they are not peculiar of the point of application, hence they are used to express a global property of the euclidian space. For instance, 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.

Definition. Given two vectors a, b ∈ V, 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 × bk = kak kbk sin θ (1.1)

where θ is the angle between the lying directions.

Definition. Let it be V and W two vectorial spaces with respectively basis B₁ = {e_i}, i ∈ I and B₀ = {e_j}, j ∈ J, and then the unique vectorial space spanned by each pair of any versors is V ⊗ W, which basis is indicated as B₀ ⊗ B₁ = {e_i ⊗ e_j} ∈ (I × J). Given two vectors v ∈ V and w ∈ W, they associate a unique element T = (v ⊗ w) ∈ V ⊗ W:

T = vⁱw^je_i ⊗ e_j

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, given a basis B = {ê₁, ê₂, ê₃}, being (a₁, a₂, a₃) and (b₁, b₂, b₃) respectively the coordinates of a and b expressed in the same reference, the cross product is given by:

a × b = | ê₁ ê₂ ê₃ / a₁ a₂ a₃ / b₁ b₂ b₃ |

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 linearly dependent. If then c is orthogonal to the plane where the pair of them lie. It’s obvious that: a × a = 0
• Anticommutativity: a × b = −b × a
• Given the standard basis {ê₁, ê₂, ê₃}, it stands that: ê₁ × ê₂ = ê₃
• ê₂ × ê₃ = ê₁
• ê₃ × ê₁ = ê₂
• Any given product a × b can be rearranged as multiplication of a suitable 3 × 3 matrix, called axial matrix and indicated as S(a) with the second argument b, so that: a × b = S(a)b

If {a₁, a₂, a₃} are the components of a in a given basis, the axial matrix of a is an antisymmetric matrix defined by:

S(a) = ⎡0 −a₃ a₂ / a₃ 0 −a₁ / −a₂ a₁ 0⎤

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.

Theorem (Poisson’s rule). Given a basis B = {ê₁, ê₂, ê₃} of orthonormal vectors, varying their orientation with time, it exists a vector ω ∈ V such that:

d ê_i(t) / dt = ω × ê_i(t) (1.3)

Proof. Existence. Being unitary normed vector e_i for each time instant, it holds that:

ke_ik² = e_i · e_i = 1

Applying time derivative to both members and exploiting product rule:

d ê_i(t) / dt · ê_i(t) + ê_i(t) · d ê_i(t) / dt = 0

2 d ê_i(t) / dt · ê_i(t) = 0

This shows that:

d ê_i(t) / dt ⊥ ê_i(t)

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

d ê_i(t) / dt = ω × ê_i(t)   i = 1, 2, 3

Unicity. Let us show that ω (whose physical interpretation is angular velocity) is unique. Considering that scrolling the index i = {1, 2, 3}, ad absurdum let us assume that for any given B_i it is associated a different vector ω_i. Since B contains orthonormal versors, it can be written:

ê_i × ê_j ≠ 0   i = j

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

d ê_i / dt · ê_j + ê_i · d ê_j / dt = 0

ê_j · d ê_i / dt = −ê_i · d ê_j / dt

According to what has been shown below:

(ω_i × ê_i) · ê_j = −ê_i · (ω_j × ê_j)

Exploiting anticommutative property and mixing dot and cross product properties, the last relation can be reduced like:

−(ê_i × ê_j) · ω_i = (ê_j × ê_i) · ω_j ⇔ ω_i = ω_j

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 write a relation between components of any time-varying vector a with respect to fixed and mobile basis. Let us introduce the pair of them as B₀ = {ê₁⁰, ê₂⁰, ê₃⁰} (the fixed one, indicated as 0) and B₁ = {ê₁, ê₂, ê₃} (the mobile one, indicated as 1), as then components of a are different with respect to one or other:

a₀ = ⎡a₁⁰ a₂⁰ a₃⁰⎤    a₁ = ⎡a₁ a₂ a₃⎤

Therefore it stands that:

a = a₁⁰ê₁⁰ + a₂⁰ê₂⁰ + a₃⁰ê₃⁰ = a₁ê₁ + a₂ê₂ + a₃ê₃

1.3 Kinematics

Applying time derivative