Energy Basilio Bona DAUIN - Politecnico di Torino October 2013 - - PowerPoint PPT Presentation

Energy

Basilio Bona

DAUIN - Politecnico di Torino

October 2013

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 1 / 54

Introduction

Analytical Approach The multibody system is considered as a system in which the dynamic equations derive from a unifying principle. This principle is based on the fact that, in order to describe the motion of a system, it is sufficient to consider some scalar quantities. These were in origin called vis viva and work function, nowadays are called kinetic energy and potential energy. Both are state functions, i.e., those functions that map the value of the state vector into a scalar function. The concept of state will be defined later; for the moment we simply consider that the state corresponds to the two vectors q(t) and ˙ q(t).

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 2 / 54

This general principle is the principle of least action. Let us consider the space Q of the generalized coordinates q ∈ Q, as sketched in Figure for a two-dimensional space Q. A particle starts its motion at time t1 in Q1 = q(t1) and ends it motion at time t2 reaching the state Q2 = q(t2). Assume that the motion keeps constant the sum E = C + P of the kinetic energy C and the potential energy P that the particle has at time t1.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 3 / 54

Q1 and Q2 are connected by a continuous path (trajectory) called true trajectory, and is unknown, since it is what we want to compute as the result of the dynamical equation analysis. If we choose at random a different trajectory, with the only condition that the two boundary point remain fixed (a perturbed trajectory), the chance to obtain exactly the true trajectory will be very small. What characterizes the true trajectory with respect to all possible other perturbed trajectories? Euler contributed to the solution of this problem, but Lagrange developed a complete theory, that was later extended by Hamilton. The true trajectory is the one that minimizes the integral of the vis-viva (i.e., twice the kinetic energy) of the entire motion between Q1 and Q2. This integral is called action and has a constant and well defined value for each perturbed trajectory having constant E (E depends only on the initial state).

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 4 / 54

The least action principle states that the nature “chooses”, among the infinite number of trajectories starting in q(t1) and ending in q(t2), the trajectory that minimizes the definite integral S = t2

t1

C∗(q(t), ˙ q(t))dt

• f a particular state function C∗(q(t), ˙

q(t)). It is necessary to compute the trajectory in the space Q that minimizes S. The integral between the initial time t1 and the final time t2 must obey to the boundary constraints the two time instants.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 5 / 54

The minimization of a functional is based on a particular mathematical technique, called calculus of variations. A functional is a mapping between a function and a real number; the function shall be considered as a whole, i.e., not a single particular value; in this sense a functional is often the integral of the function. The conditions that guarantee the minimization of S provide a set of differential equations that contain the first and second time derivatives of the qi(t); this set completely describes the system dynamics.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 6 / 54

The differential equations specify the evolution of a physical quantity as the result of infinitesimal increments of time or position; summing up this infinitesimal variations we obtain the physical variables at every instant, knowing only their initial value and possibly some initial derivative: we can say that the motion has a local representation. The action characterizes the motion dynamics requiring only the knowledge of the states at the initial and final times; every intermediate value of the variables can be determined by the minimization of the action, that is a global, rather than a local, measure. The Lagrange approach is based on the definition of two scalar quantities, namely the total kinetic co-energy and the total potential energy associated with the body. The reason for using the term co-energy instead of the term energy, will be clarified later.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 7 / 54

Lagrangian approach

The Lagrange method allows to define a set of Lagrange equations, that have some advantages with respect to the vector equations provided by the Newton-Euler approach. The approach provides n second-order scalar differential equations, directly expressed in the generalized coordinates ˙ qi(t) e qi(t). If holonomic constraint are present, the constraint force do not appear in the equations. The kinetic co-energies and the potential energies are independent of the reference frame used to represent the body motion. The kinetic co-energies and the potential energies are additive scalars: in a multi-body system the total energies/co-energies are the sum of each energy/co-energy component.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 8 / 54

Kinetic energy and co-energy for single point-mass

The mechanical kinetic energy associated to a point-mass m is defined as the work necessary to increase the linear or angular momentum from 0 to h, i.e., C(h) = h dW where the symbol · indicates the scalar product. The infinitesimal work associated to the mass is given by dW = f · dr where f is the resultant of the applied forces on the mass and dr is the infinitesimal displacement increment and f = dh dt

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 9 / 54

The resulting infinitesimal work is therefore dW = f · dr = dh dt · dr = dh dt · vdt = v · dh and we can write C(h) = h v · dh The kinetic energy is a scalar state function associated to the particle states v and h. Another state function associated to the point-mass, called mechanical kinetic co-energy, is defined as C∗(v) = v h · dv As shown in Figure, between the mechanical energy and the co-energy a relation exists C∗(v) = h · v − C(h)

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 10 / 54

This relation is an example of the Legendre transformation

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 11 / 54

In particular, if the mass particle is moving at a velocity significantly smaller that the speed of light c, i.e., it is not a relativistic mass, the relation is h = mv with m constant, and the two “energies” become C(h) = h 1 mh · dh = 1 2mh · h = 1 2m h2 C∗(v) = v mv · dv = 1 2mv · v = 1 2m v2 As one can see, in this case the kinetic energy and co-energy are the same since h2 = m2 v2 This does not happen for relativistic masses where m = m(v(t)).

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 12 / 54

In an extended body composed by N masses mi the kinetic co-energy is the sum of the kinetic co-energy of each mass C∗(v) = 1 2

N

• i=1

mivi · vi

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 13 / 54

We consider the velocity vi = ˙ ri with respect to R0. The velocities in R0 can be computed from the general relation ˙ r0(t) = ω0

01(t) × ρ0(t) + R0 1˙

r1(t) + ˙ d

1(t) = ω0 01(t) × ρ0(t) + ˙

d

1(t)

where the term R0

b˙

ri(t) is zero, since the point-masses are fixed with respect to the body-frame. Now we consider a purely translatory motion and then a purely rotational motion.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 14 / 54

Translational motion

If the motion is purely translational ˙ r0

i (t) = ˙

d

b(t) ≡ v0(t)

where v0 is the total linear velocity with respect to R0. All point-masses mi have the same velocity v0 C∗ = 1 2v0 · v0

N

• i=1

mi = 1 2mtot v0 · v0 = 1 2mtot v02 = 1 2vT

0 (mtotI)v0

where the mass mtot is the total body mass. The kinetic co-energy is equivalent to that of one particle with total mass mtot with the translational velocity v0. The total mass mtot can be ideally concentrated in the body center-of-mass C, whose position is rc rcmtot =

• i

rimi → rc =

• i

mi mtot ri

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 15 / 54

Since the velocity is equal for all points of the body, v0 is also the velocity

• f the center-of-mass C; if we use the symbol v0c ≡ v0 for this velocity,

we can write C∗ = 1 2mtot v0c · v0c = 1 2mtot v0c2 = 1 2 vT

0c(mtotI)v0c

that gives the usual rule: “the kinetic energy is half the product of the total mass for the total velocity squared.”

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 16 / 54

Rotational motion

If the motion is purely rotational, then ˙ r0

i (t) = ω0 01(t) × rb i

i.e., vi = ω0 × ri where ω0

01 ≡ ω0 is the total angular velocity and ri is the position of the

i-th mass in R0. Considering all masses C∗ = 1 2

N

• i=1

mi(ω0 × ri) · (ω0 × ri) since a · (b × c) = b · (c × a) and assuming a ≡ (ω0 × ri), b ≡ ω0, c ≡ ri we obtain C∗ = 1 2

N

• i=1

miω0 · ri × (ω0 × ri) = 1 2 ω0 · N

• i=1

miri × (ω0 × ri)

• Basilio Bona (DAUIN - Politecnico di Torino)

Energy October 2013 17 / 54

The previous relation is equivalent to C∗ = 1 2 ω0 · N

• i=1

hi

• = 1

2 ω0 · h0 where h0 is the total angular momentum with respect to O. Since h0 = Γ0ω0, we have C∗ = 1 2 ωT

0 Γ0ω0

Notice the similarity: C∗ = 1 2 vT

0cp0 = 1

2 vT

0c(mtotI)v0c

pure translation C∗ = 1 2 ωT

0 h0 = 1

2 ωT

0 Γ0ω0

pure rotation

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 18 / 54

Total kinetic co-energy

C∗ = 1 2

• i

mivi · vi = 1 2

• vT

c (mtotI)vc + ωT 0 Γcω0

• This is a well known result, that can be expressed in words as: “the total

kinetic co-energy of a body is the sum of the translational kinetic co-energy of the center of mass plus the rotational kinetic co-energy around the center of mass.” This relation is valid also for extended bodies.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 19 / 54

Generalized coordinates

Considering the generalized coordinates q(t) and velocities ˙ q(t) and the Jacobians vc = Jℓ(q)˙ q

• r

ω = Jω(q)˙ q we obtain C∗(˙ q, q) = 1 2

• ˙

qTJT

ℓ (mI)Jℓ ˙

q + ˙ qTJT

ωΓcJω ˙

q

• and

C∗(˙ q, q) = 1 2 ˙ qT JT

ℓ (mI)Jℓ + JT ωΓcJω

• ˙

q = 1 2 ˙ qTΓtot ˙ q where Γtot = JT

ℓ (mI)Jℓ + JT ωΓcJω

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 20 / 54

Potential energy

Potential energy is a form of energy that depends only on position; two types of position-related energies exist. One is due to the gravitational field, the other is the energy stored in the elastic parts of the body, that store energy under the effects of deformation. Since in our approach the considered bodied are rigid, the elastic parts are external to the bodies and are represented by springs that connect various parts of the mechanical system.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 21 / 54

A potential function P(r), is a scalar function that depends only on the position r = x y zT A force f is said to be conservative when it is the negative gradient of P(r) f(r) = −∇P(r) = − ∂P(r) ∂x ∂P(r) ∂y ∂P(r) ∂z T If a potential function exists, it is called potential energy of the system and it is unique apart from an additive constant. This implies that the effects on the body dynamics depend only from the potential energy variation, and not on its absolute value.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 22 / 54

Gravitational energy

An example of conservative force field is the gravitational field around the

• Earth. The potential field produces the so-called weight forces.

The potential energy due to a gravitational field and associated to a generic mass m is given by the following relation: Pg = −mg · rc0 where g is the local gravitational acceleration vector and rc0 is the body center-of-mass position vector with respect to a plane orthogonal to g, that provides the conventional zero value of potential energy (zero potential energy plane) as shown in Figure.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 23 / 54

Figure: The potential energy due to the gravitational field.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 24 / 54

Elastic energy

Another force field is related to potential energy, that due to elastic elements: these elements represent the abstract model of a proportional relation between displacement and force. If we assume a one-dimensional linear spring the relation between the applied force f and the linear elongation e from the rest position of the spring is f = kee If we assume a one-dimensional torsional or torsion spring we can write a relation between the applied torque τ and the resulting angular deformation δ from the rest position of the spring τ = k′

eδ

ke and k′

e are the so-called elastic constants of the springs.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 25 / 54

Figure: A one dimensional linear spring. The rest length of the spring is x0, and e is the extension/compression occurring when the force f is applied.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 26 / 54

The potential energy is the integral of the virtual work performed by the spring deformation P(e) = e f · de

• r

P(δ) = δ τ · dδ We can define also in this case the potential co-energies, that are P∗(f) = f e · df

• r

P∗(τ) = τ δ · dτ the relation between P∗(f) and P(e) is is given by the Legendre transformation P∗(f) = f · e − P(e)

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 27 / 54

A nonlinear spring: the relation between f(t) and e is nonlinear, but the relation f · e = P(e) + P∗(f) holds.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 28 / 54

When the elastic elements are linear with constant ke, the potential energy and co-energy are given by P(e) = 1 2eT(keI)e = 1 2ke e2 and P∗(f) = 1 2fT(keI)−1f = 1 2ke f2 When torsion springs with constant k′

e are considered, the potential energy

and co-energy are given by P(δ) = 1 2δT(k′

eI)δ = 1

2k′

e δ2

and P∗(τ) = 1 2τ T(k′

eI)−1τ =

1 2k′

e

τ2 In linear case, energies and co-energies are equal P(e) = P∗(f) and P(δ) = P∗(τ)

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 29 / 54

If the elastic constants are different along the three directions a more general relation is valid P(e) = 1 2eTKee; P∗(f) = 1 2fTK−1

e f

and P(δ) = 1 2δTK′

eδ;

P∗(τ) = 1 2τ T(K′

e)−1τ

where Ke = diag(ke1, ke2, ke3) e K′

e = diag(k′ e1, k′ e2, k′ e3) are the elastic

constant matrices along the three dimensional axes.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 30 / 54

Generalized forces in holonomic systems

The forces fi acting on the i-th mass can be classified according to three groups: N′′′ constraint forces fv

i due to constraint reactions.

N′′ conservative forces fc

i due to conservative fields.

N′ non conservative forces fnc

i .

The total force is the sum of these three types of forces f =

N′

• i=1

fnc

i

+

N′′

• i=1

fc

i + N′′′

• i=1

fv

i

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 31 / 54

Constraint forces

The virtual displacements δri are always tangent to the constraints, while the constraint forces fv

i are always orthogonal to the constraints; from this

assumption it follows that fv

i · δri = 0

Therefore the work done by the constraint forces is zero (the forces “do not work”) δWv =

N′′′

• i=1

fv

i · δri = 0.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 32 / 54

Conservative forces

The N′′ conservative forces do a work that results δWc =

N′′

• i=1

fc

i · δri = − N′′

• i=1

∇Pi · δri = −

N′′

• i=1

∇Pi ·  

n

• j=1

∂ri ∂qj δqj   =

n

• j=1

N′′

• i=1

−∇Pi · ∂ri ∂qj

• Fc

j

δqj This last expression highlights the so called generalized conservative forces Fc

j

The virtual work can be expressed as a function of the generalized coordinates qj: δWc =

N′′

• i=1

fc

i · δri = n

• j=1

Fc

j δqj = Fc · δq

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 33 / 54

Non conservative forces

The N′ non conservative forces fnc

i

do a work equal to δWnc =

N′

• i=1

fnc

i

· δri =

N′

• i=1

fnc

i

·  

n

• j=1

∂ri ∂qj δqj   =

n

• j=1

N′

• i=1

fnc

i

· ∂ri ∂qj

• Fnc

j

δqj This last expression highlights the so called generalized non conservative forces Fnc

j

and allows to transform the virtual work from a function of the positions r to a function of the generalized coordinates qj: δWnc =

N′′

• i=1

fnc

i

· δri =

n

• j=1

Fnc

j δqj = Fnc · δq

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 34 / 54

In conclusion, only two types of forces will do work in a system subject to holonomic constraints the j-th generalized conservative forces: Fc

j (q) = − N′′

• i=1

∇Pi · ∂ri ∂qj the j-th generalized non conservative forces: Fnc

j (q) = N′

• i=1

fnc

i

· ∂ri ∂qj The generalized force, being the result of a scalar product, will be itself a scalar quantity.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 35 / 54

In case of torques acting on the system, the generalized forces due to them will give origin to the following generalized torques T c

j = N′′

• i=1

−∇Pi · ∂αi ∂qj ; T nc

j

=

N′

• i=1

τ nc

i

· ∂αi ∂qj The symbol used to identify both the generalized forces and the generalized torques will be F.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 36 / 54

Lagrange equations with holonomic constraints

In a multi-body system subject to holonomic constraints, the formulation

• f the Lagrange equations may take different forms.

From the knowledge of the total co-energy of the system C∗(q, ˙ q) =

N

• k=1

C∗

k(q, ˙

q)

• ne derives the Lagrange equations: they are a set of n equations (one for

each generalized coordinates qi) defined as d dt ∂C∗ ∂ ˙ qi

• − ∂C∗

∂qi = Fi i = 1, . . . , n where Fi = Fc

i + Fnc i

is the i-th generalized force, with a positive sign if applied by the environment to the body, or a negative sign if applied by the body to the environment.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 37 / 54

d dt ∂C∗ ∂ ˙ qi

• − ∂C∗

∂qi + N′

• k=1

∇Pk · ∂rk ∂qi

• = Fnc

i

The term inside the square bracket is equal to ∂P ∂qi ; therefore d dt ∂C∗ ∂ ˙ qi

• −

∂C∗ ∂qi − ∂P ∂qi

• = Fnc

i

P does not depend on ˙ qi, so ∂P ∂ ˙ qi = 0 and we have the new form of the lagrange equation d dt ∂C∗ ∂ ˙ qi − ∂P ∂ ˙ qi

• −

∂C∗ ∂qi − ∂P ∂qi

• = Fnc

i

i = 1, . . . , n

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 38 / 54

Lagrange state function

The Lagrange function L is defined as the difference between the total kinetic co-energy C∗ and the total potential energy P of he system L(q, ˙ q) = C∗(q, ˙ q) − P(q) We can write n differential equations d dt ∂L(q, ˙ q) ∂ ˙ qi

• − ∂L(q, ˙

q) ∂qi = Fnc

i (q)

i = 1, . . . , n each one relative to the i-th generalized coordinate The term ∂L ∂ ˙ qi is the generalized momentum and is usually indicated by the symbol µi; the vector of generalized momenta is indicated by µ(q(t)).

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 39 / 54

Dissipative and friction forces

The friction phenomena involve energy dissipated by the body as heat; they are due to complex interaction between solid/solid or solid/fluid surfaces; tribology is the science that studies the friction forces. If we keep the friction or other dissipative forces f fric

i

separate from the

• ther non conservative forces, the Lagrange equation becomes:

d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = Fi − f fric

i

We can approximately describe the friction force f fric

i

as a nonlinear function of the relative velocity v between the two contact surfaces of the involved bodies. We can model the total friction force as in Figure and write f fric

total = f fric stiction + f fric coulomb + f fric viscous

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 40 / 54

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 41 / 54

While stiction and Coulomb friction must be explicitly introduced as non-conservative forces, it is a common assumption to express the viscous dissipative phenomenon as the derivative of a dissipation function, also called Rayleigh function, given by: Di(˙ q) = 1 2 ˙ qT(βiI)˙ q = 1 2 βi ˙ q2 where the coefficient βi is the viscous friction coefficient, and ˙ q is the norm of the relative velocity between the moving body and the surface responsible of the viscous friction effect. This quadratic expression ins NOT a dissipation “energy”, but only a conventional way to introduce it in the Lagrange equation, as follows d dt ∂L ∂ ˙ qi

• − ∂L

∂qi + ∂Di ∂ ˙ qi = Fi i = 1, . . . , n Now the term Fi includes only the non conservative forces

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 42 / 54

Lagrangian systems are holonomic systems where the forces are solely due to generalized potential functions P(q, ˙ q), Hamiltonian systems are those where the kinetic co-energy and the potential energy explicitly depend on time C∗ = C∗(q, ˙ q, t) and P = P(q, ˙ q, t) ... or, if you prefer the Wikipedia definition ... A Lagrangian system is a pair (Y , L) of a smooth fiber bundle Y → X and a Lagrangian density L which yields the Euler–Lagrange differential

• perator acting on sections of Y → X.

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 43 / 54

Characterization of the Lagrange equations

The Lagrange approach generates n differential equations d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = Fi i = 1, . . . , n

• r

d dt (µi(˙ qi)) − ∂L ∂qi = Fi i = 1, . . . , n Collecting the n equations in one vector equation d dt ∂L ∂ ˙ q

• − ∂L

∂q + ∂D ∂ ˙ q = F If the equations are linear (or if we consider small perturbations around some equilibrium point), we will have a general formulatio expressed as a second order differential vector equation A1¨ q(t) + A2 ˙ q(t) + A3q(t) = F

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 44 / 54

The linear equation can be rewritten as M¨ q(t) + (D + G)˙ q(t) + (K + H)q(t) = F where M = MT is the mass or inertia matrix positive definite, symmetric D = DT is the viscous damping matrix symmetric G = −GT is the gyroscopic matrix skew-symmetric K = KT is the stiffness (elasticity) matrix symmetric H = −HT is the circulatory matrix (constrained damping) skew-symmetric

Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 45 / 54