Rigid Body Dynamics for Movement Analysis in Biomechanics

Chap. I – dynamics of a material point

1. Reference frame

To specify positions and time, each observer may choose a zero of the time scale, an origin in space and a set of 3 Cartesian co-reference ordinates axes: we refer to these collectively as a reference frame.

2. The mass

At any particle M (point representing a certain amount of material), a mass m is associated, which is a constant and positive scalar (expressed in kg, SI).

Experience shows that the larger the mass m, the more the cause of movement to be important for producing the same effect: accelerating. The mass reflects the inertia of a body (here a particle) to be set in motion for a given cause.

3. The linear momentum

With respect to a given reference frame R, for a particle M whose mass is m moving with a velocity V⃗, the linear momentum is defined by: p⃗_R = m . V⃗_R.

Contrary to the mass, the linear momentum depends on the reference frame R.

4. The angular momentum

With respect to the same reference frame R, and for the same particle M, the angular momentum in any point A is defined by: σ⃗_R(A) = AM⃗ x p⃗_R.

It corresponds to the moment, in A, of the linear momentum and depends on the reference frame R.

Newton’s first law or law of inertia

The experiment highlights that in some specific reference frames, the dynamic laws are always expressed identically, these reference frames will be called inertial frames.

The experiment also shows that between each other, two material systems always interact (e.g., electrostatic attraction or repulsion), and that these interactions decrease when the distance between the two systems increases.

We will say that a particle (or a system) is isolated if it undergoes no action from other systems: Then, a particle sufficiently far from other material systems would be considered as isolated.

There exist specific reference frames (at least one) in which the movement of an isolated particle is a uniform motion (or the point is at rest). These reference frames are called inertial frames.

In other words: in an inertial frame, the acceleration of an isolated particle is zero.

Consequently:

• A particle that is at rest will stay at rest unless an external force acts upon it.
• A particle that is in motion will not change its velocity unless an external force acts upon it: so, it will move with uniform velocity in a straight line.

Formally, an inertial frame may be defined to be one with respect to which any isolated body, far removed from all other matter, would move with uniform velocity.

This is of course an idealized definition. For all practical purposes, however, an inertial frame is one whose position and orientation are fixed relative to Earth.

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Newton’s second law or fundamental principle of dynamics

Let an inertial frame R and a particle M of mass m, studied from R. The first Newton’s law states that if this particle is isolated (without interaction), its linear momentum p⃗_R = m . V⃗_R is constant, as m and V⃗_R are constant with respect to R.

Then: d p⃗_R / dt |_R = 0.

The experiment shows, on the contrary, that if external actions act on this particle, its movement will be modified so that its linear momentum (i.e., its velocity) will change.

It will exist an acceleration (modification of the velocity norm, or direction, or both) which appears as a result of a vector quantity named force.

The second law states that the net force acting on a particle is equal to the rate of change of its linear momentum in an inertial frame R (that is, the derivative):

F⃗ = d p⃗_R / dt |_R.

As in classic mechanics, the mass m is constant, it comes: F⃗ = m . d V⃗_R / dt |_R = m . γ⃗_R.

Comments:

The force is a vector, so if N forces act on the same particle M, we have: F⃗ = F⃗₁ + ... + F⃗_N.

F⃗ is then the resultant of all forces acting on M, also called net force. The dimension of a force is imposed by its definition: [F] = m . [γ] = m . L . T⁻².

SI Unit: The Newton N = kg. m. s⁻².

The moment of force in a given point O, M⃗_F(O), is a vector, representing the turning effect of the force about the point O.

If the force F⃗ is exerted on a point A: M⃗_F(O) = OA⃗ x F⃗.

The moment is a vector perpendicular to OA⃗ and F⃗.

The magnitude of the moment is: ‖M⃗_F(O)‖ = F . d = F. OA sin().

Where d is the perpendicular distance from O to the line of action of the force, called moment arm. Note: Moving a force along its line of action does not change its moment.

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Reference point transformation rule: The vector M⃗_F(O) depends on the choice of the point O. The moment of the same force F⃗ in another point Q is given by:

M⃗_F(Q) = QA⃗ x F⃗ = (QO⃗ + OA⃗) x F⃗ = OA⃗ x F⃗ + QO⃗ x F⃗ = M⃗_F(O) + QO⃗ x F⃗.

Then, the reference point transformation rule gives, for any points O and Q:

M⃗_F(Q) = M⃗_F(O) + QO⃗ x F⃗.

Newton’s third law or action-reaction law (mutual actions law)

The third law states that if one particle M₁ exerts a force F⃗_1→2 on a second particle M₂, then simultaneously M₂ exerts a force F⃗_2→1 on M₁, and the two forces are equal and opposite:

F⃗_1→2 = - F⃗_2→1.

Classical forces

1. The gravitational force

Gravitation is a natural phenomenon by which all physical bodies attract each other. It gives weight to physical objects and causes them to fall toward one another.

Newton's law of universal gravitation postulates that the gravitational force of two bodies of mass is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Gravitational constant: G = 6,67 M⁻¹¹ S.I.10 em.

The force exerted by the Earth (of mass), on a particle of mass, situated at a distance r of the Earth’s centre is:

If the altitude of the particle is negligible with respect to the Earth radius, we have:

F⃗ = m g⃗.

With: ‖g⃗‖  9,81 m. s⁻².

g⃗ directed along the descending vertical (toward the Earth’s centre).

2. The friction force

The friction force is the force exerted by a surface as an object moves across it, which opposes the motion of the object. It results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together.

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3. The spring force

The spring force is the force exerted by a compressed or stretched spring upon any object that is attached to it. An object that compresses or stretches a spring is always acted upon by a force that restores the spring to its rest or equilibrium position. The magnitude of the force is directly proportional to the amount of stretching or shortening of the spring.

F⃗ = - K x⃗.

K: Stiffness > 0.

x: Lengthening / Shortening.

Chap. 2 – work, kinetic energy, kinetic moment and related theorems

I. Work and power

1. Notion of work

Let consider a particle M subjected to a force F⃗, moving with a sufficiently small displacement δM⃗ that remains constant during it. The elementary work δW is defined as: δW = F⃗ . δM⃗.

Work exists only if there is displacement, and the force is not ⏊ to displacement …

If the particle, under the action of the force F⃗ (which may vary during movement), describes a path from position A to position B, the work done by the force between A and B is: W_AB = ∫_A^B F⃗ . dM⃗.

If the displacement is broken up into a number of smaller displacements, over each of which the force can be assumed to be constant, the total work is then the sum of the works associated with each small displacement: in the infinitesimal limit, this becomes an integral.

W = ∑_i^N F⃗_i . δM⃗_i.

[W] = [F] . [L] = m . L . T⁻² . L = m . L² . T⁻².

SI Unit: The Joule J = kg. m². s⁻² = N . m.

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2. Notion of power

If dW is the work done by F⃗ during the time interval dt, the corresponding power is: P = dW / dt.

[P] = [W] / [T] = m . L² . T⁻³. SI Unit: The Watt W = kg. m². s⁻³ = N . m. s⁻¹.

Other expression: P = dW / dt = F⃗ . dM⃗ / dt = F⃗ . V⃗_R.

W = ∫_tA^tB P . dt = ∫_tA^tB F⃗ . V⃗_R . dt.

W and P depend on the reference frame R.

II. Kinetic energy and work-energy theorem

1. Kinetic energy

The kinetic energy of an object is the energy that it possesses due to its motion.

The kinetic energy of a particle M is proportional to its mass and to the square of its speed: T_R = 1/2 m V².

Where: V² = ‖V⃗_R‖². The kinetic energy depends then on the reference frame R.

2. Work – energy theorem

The work of the resultant force acting on a particle M between a position A, where its speed is V_A, and a position B, where its speed is V_B, equals the change in the particle’s kinetic energy between both positions.

W_AB = ∫_A^B F⃗ . dM⃗ = 1/2 m V_B² - 1/2 m V_A² = T_R(B) - T_R(A).

Demonstration:

dW = F⃗ . dM⃗ = F⃗ . V⃗_R . dt.

F⃗ = m . γ⃗_R.

dW = m . γ⃗_R . V⃗_R . dt.

γ = dV/dt.

γ . V = V . dV/dt = ½ . dV²/dt.

dW = m . (½ . dV²/dt) . dt = ½ . m . dV².

Using normal and tangential components to express acceleration and velocity vectors:

dW = m . (γ_T T⃗ + γ_N n⃗) . V T⃗ . dt.

dW = m . γ_T . V . dt.

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dW = m . dV/dt . V . dt = m.V.dV = m d(V²/2) = d(1/2 m V²).

W_AB = ∫_A^B F⃗ . dM⃗ = ∫_A^B dW = ∫_A^B d(1/2 m V²) = 1/2 m V_B² − 1/2 m V_A².

III. Conservative forces and potential energy

1. Conservative force

The condition for a force to be conservative is that it can be expressed as the gradient of a potential function: the potential energy E_p.

F⃗_C = - grad⃗ E_p.

This result follows from the gradient theorem, which states that the integral: