IV. RESULTADOS
4.4 MATRIZ DE CONSISTENCIA
Forces
We assume that an applied force Faacts on the particle. In addition, we assume that the friction is of the Coulomb type. Consequently, if the particle is moving relative to the helix,
F= Fa+ λ1˜a2+ λ2˜a3+ Ff, where
Ff = −µdλ1˜a2+ λ2˜a3 ˙θ˜a ˙θ˜a11.
On the other hand, if the particle is not moving relative to the helix, i.e., θ is constant, then
F= Fa+ λ1˜a2+ λ2˜a3+ λ3˜a1.
The friction force in this case is subject to the static friction criterion:
Ff ≤µs||N|| ,
Balance of Linear Momentum and Lagrange’s Equations
For an unconstrained particle moving inE3, we have the three Lagrange’s equations:
d
For the present coordinate system θ, r, η, these equations read d
Equations of Motion for the Particle on the Helix
We obtain the equations of motion for the particle on the helix from the preceding equations by substituting for the resultant force and imposing the constraints. With
some algebra, for the case in which the particle is moving relative to the helix, we find three equations:
d dt
m(1+ α2)R2˙θ
= Fa· ˜a1− µdλ1˜a2+ λ2˜a3||˜a1|| ˙θ
| ˙θ|, d
dt
mα2Rθ ˙θ
− m(1 + α2)R ˙θ2= Fa· ˜a2+ λ1− µdλ1˜a2+ λ2˜a3 ˙θ˜a1· ˜a2
˙θ˜a1 , d
dt
mαR ˙θ
= Fa· ˜a3+ λ2− µdλ1˜a2+ λ2a3 ˙θ˜a1· ˜a3
˙θ˜a1 , where
˜a1= Reθ+ αRE3, ˜a2= er+ αθE3, ˜a3 = E3.
These three equations provide a differential equation for the unconstrained motion of the particle and two equations for the unknowns λ1and λ2.
For the case in which the motion of the particle is specified (i.e., the particle is not moving relative to the helix), we find, from F= ma, three equations for the three unknowns:
λ3 = −Fa· ˜a1, λ2= −Fa· ˜a2, λ1= −Fa· ˜a3.
It remains to use the static friction criterion, but this is left as an easy exercise.
The Particle on a Smooth Helix In this case,
F= Fa+ λ1˜a2+ λ2˜a3,
and because Fc· ˜a1= 0, Lagrange’s equations of motion decouple:
d dt
m(1+ α2)R2˙θ
= Fa· ˜a1, d
dt
mα2Rθ ˙θ
− m(1 + α2)R ˙θ2= Fa· ˜a2+ λ1, d
dt
mαR ˙θ
= Fa· ˜a3+ λ2. Consequently, the desired differential equation is
m(1+ α2)R2θ¨= Fa· (Reθ+ αRE3), and the constraint force is
Fc= λ1˜a2+ λ2˜a3
=
mα2R ¨θθ− mR ˙θ2− Fa· ˜a2
˜a2+
mαR ¨θ− Fa· ˜a3
˜a3.
Once θ as a function of time has been calculated from the ordinary differential equa-tion, then Fcas a function of time can be determined.
3.9 Summary 91
To illustrate the previous equations, consider the case in which the applied force is gravitational, Fa= −mgE3. Then, from the preceding equations,
m(1+ α2)R2¨θ= −mgαR. (3.18)
Subject to the initial conditions θ(t0) = θ0and ˙θ(t0) = ω0, this equation has the solu-tion
θ(t)= θ0+ ω0(t− t0)− gα
2R(1+ α2)(t− t0)2. Using this result, we find that the constraint force is
Fc= where θ(t) is as previously given.
Some Observations
Suppose one is interested in determining only the differential equation governing the unconstrained motion of the particle moving on a smooth helix. In other words, the constraint forces are of no concern. One can obtain this differential equation by imposing the constraints on the expression for T:
T˜ =m
A quick calculation shows that the resulting differential equation is identical to that obtained previously [(3.18)].
Clearly, Lagrange’s equations calculated with Approach II (i.e., with ˜T) have their advantages, but they cannot accommodate dynamic friction forces. It is, how-ever, the standard approach to Lagrange’s equations in the literature and textbooks.
You should note that ∂ ˜∂˙rT = ∂ ˜∂˙ηT = ∂ ˜∂rT = ∂ ˜∂ηT = 0. Consequently, we cannot recover the other two Lagrange’s equations once we have imposed the constraints.
3.9 Summary
In this chapter, several forms of Lagrange’s equations of motion for a particle were presented. The most fundamental of these forms is [see (3.1)]
d
In one of the following exercises, we establish two other forms of these equations by expanding the partial derivatives with respect to the coordinates and their velocities.
These two forms are a covariant form (3.22) and a contravariant form (3.23). If we
decompose the forces acting on the particle into conservative and nonconservative forces, then we can transform (3.1) to (3.2):
d dt
∂L
∂q˙i
− ∂L
∂qi = Fncon· ai.
Now suppose that an integrable constraint is imposed on the particle, that this con-straint can be written as q3− f (t) = 0, and that the constraint force associated with this constraint is Fc= λa3. In this case, Lagrange’s equations of motion can be used to readily provide a set of differential equations for the generalized coordinates q1 and q2:
d dt
∂ ˜L
∂q˙α
− ∂ ˜L
∂qα = Fncon· ˜aα, α= 1, 2. (3.19) These equations feature the constrained Lagrangian ˜L that we obtain from L by imposing the integrable constraint q3= f (t), and, most important, do not feature λ.
That is, equations of motion (3.19) are reactionless. This case, in which all the con-straints are integrable and the constraint forces are prescribed by use of Lagrange’s prescription, is an example of a mechanical system subject to “ideal constraints.”
We also discussed the situation in which nonintegrable constraints were im-posed on the system and outlined how the equations of motion could be obtained in these circumstances. The imposition of nonintegrable constraints will not affect the number of generalized coordinates, the configuration manifold, or the kinematical line-element.
The summary just presented will be identical for systems of particles, rigid bod-ies, and systems of both particles and rigid bodies. The only major differences are that the calculation of the kinetic energy becomes significantly more complicated for these systems and that the right-hand sides of Lagrange’s equations feature sev-eral forces and moments. Despite these differences, the decoupling of the equations of motion into a set of reactionless equations governing the generalized coordinates will hold if Lagrange’s prescription for the constraint forces is used. This is one of the most remarkable features of Lagrange’s equations for systems subject to inte-grable constraints.
EXERCISES
3.1. Recall from Exercise 1.5 in Chapter 1 that, for a parabolic coordinate system {u, v, θ},
a1 = ∂r
∂u = ver+ uE3, a2 = ∂r
∂v = uer− vE3, a3 = ∂r
∂θ = uveθ, and
a1 = 1
u2+ v2a1, a2= 1
u2+ v2a2, a3= 1 uveθ.
Exercises 3.1–3.2 93
(a) Consider a particle of mass m that is acted on by a force F and is free to move inE3. Show that the equations of motion of the particle are
d dt
m(u2+ v2) ˙u
− m( ˙u2+ ˙v2)u− mv2u ˙θ2= F · a1, d
dt
m(u2+ v2) ˙v
− m( ˙u2+ ˙v2)v− mu2v ˙θ2= F · a2, d
dt
mu2v2˙θ
= F · a3.
(b) Next, we are interested in a particle that is moving on the parabolic surface of revolution:
c2= −z + z2+ r2,
where c is a constant. A vertical gravitational force−mgE3acts on the par-ticle. Using the results of (a), derive the equations governing the uncon-strained motion of the particle and show that the normal force acting on the particle is
N= −
m ˙u2c+ mu2c ˙θ2+ mgc a2.
Show that the two second-order differential equations governing the generalized coordinates can be written as a single second-order differential equation:
m(u2+ c2)
¨
u+ m˙u2u− h2
mu3c2 = −mgu, (3.20) where h is a constant. (This constant is none other than HO· E3, which is an integral of motion). Noting that the units of u and c are meters1/2, what is a dimensionless form of equations of motion (3.20)?
(c) Show that the solutions of (3.20) conserve the energy E=m
2(u2+ c2) ˙u2+ h2
2mu2c2 +mg
2 (u2− c2).
How does one arrive at this expression for E?
3.2. For many mechanical systems a canonical form of Lagrange’s equations can be established that is suited to numerical integrations. Here, we establish one such form [see (3.23)].∗ This problem is adapted from the texts of McConnell [139] and Synge and Schild [208]. We take this opportunity to note that (3.23) can be found in an early paper by Ricci and Levi-Civita [178].
We start by recalling the covariant component forms of Lagrange’s equations of motion for a particle that is in motion under the influence of a resultant external
∗ As will become evident from the developments of later chapters, a related form can be established for any mechanical system that features scleronomic integrable constraints and constraint forces and moments that are prescribed by use of Lagrange’s prescription.
force F =3
Here, we wish to show that Lagrange’s equations can be written in two other equivalent forms. The first one is the covariant form:
m
where a Christoffel symbol of the first kind is defined by [si, k]= ∂as
∂qi · ak.
It is important to note that [si, k]= [is, k]. The second form of Lagrange’s equations is known as the contravariant form:
m ¨qk+ m
where a Christoffel symbol of the second kind is defined by
kij= ∂ai
∂qj · ak.
Notice that kij= kji. This form of Lagrange’s equations is used in numerical simu-lations of mechanical systems.†
(a) Show that the Christoffel symbols have the representations [si, k]= 1
† Most numerical integration packages assume that the differential equations to be integrated are of the form ˙x= f(x, t). By defining the set of variables (states) x1= q1, . . . ,x3= q3,x4= ˙q1, . . . ,x6=
˙
q3, the contravariant form of Lagrange’s equations can be easily placed in the form ˙x= f(x, t).
Exercises 3.2–3.3 95
(b) Starting from Lagrange’s equations, d
derive the following representation for the covariant component form∗:
m
(c) Starting from Lagrange’s equations in the form
m
derive the following representation for the contravariant component form†:
m ¨qk+ m
(d) For which coordinate system do the Christoffel symbols vanish?
3.3. Recall that for spherical polar coordinates,{R, φ, θ}, the covariant basis vectors are
a1 = eR, a2= Reφ, a3= R sin(φ)eθ, and the contravariant basis vectors are
a1= eR, a2= 1
Reφ, a3= 1 R sin(φ)eθ.
Furthermore, the linear momentum and kinetic energy of a particle of mass m are G= m ˙Ra1+ m ˙φa2+ m ˙θa3, T= m
2
R˙2+ R2φ˙2+ R2sin2(φ) ˙θ2
. (a) For a particle of mass m that is in motion in E3 under the influence of a
resultant force F, establish the three covariant components of Lagrange’s equations of motion. In your solution, avoid explicitly calculating the 27 Christoffel symbols of the first kind.
(b) For a particle of mass m that is in motion inE3under the influence of a re-sultant force F, establish the three contravariant components of Lagrange’s equations of motion. In your solution, avoid explicitly calculating the 27 Christoffel symbols of the second kind.
∗ Hint : Expand the partial derivatives of T using the representation (3.21). Then, take the appropriate time derivative and reorganize the resulting equation by using the aforementioned symmetries. You may need to relabel certain indices to obtain the desired results.
† Hint : Multiply the covariant form by askand sum over k. After some rearranging and relabeling of the indices, you should get the final desired result. Notice that the covariant component and contravariant component forms of these equations can be viewed as linear combinations of each other.
3.4. Consider a particle that is in motion on a rough surface. A curvilinear coordi-nate system q1,q2,q3is chosen such that the surface can be described by the equa-tion
q3= d(t), where d(t) is a known function of time t.
(a) Suppose that the particle is moving on the rough surface.
(i) Argue that vrel= ˙q1˜a1+ ˙q2˜a2.
(ii) Give a prescription for the constraint force acting on the particle.
(b) Suppose that the particle is stationary on the rough surface. In this case, two equivalent prescriptions for the constraint force are
Fc= N + Ff =
3 i=1
λiai, where the tildes are dropped for convenience.
(i) Show that
where N, and Ff1and Ff2uniquely define the normal force N and friction force Ff, respectively, and aik= ai· akwith i, k= 1, 2, 3.
(ii) For which coordinate systems do F1f = λ1, Ff2= λ2, and N= λ3? Give an example to illustrate your answer.
(c) Suppose that a spring force and a gravitational force also act on the particle.
Prove that the total energy of the particle is not conserved, even when the friction force is static.
3.5. Consider a particle of mass m that is in motion on a helicoid. In terms of cylin-drical polar coordinates r, θ, z, the equation of the right helicoid is
z= αθ,
where α is a constant. A gravitational force−mgE3acts on the particle.
(a) Consider the following curvilinear coordinate system forE3: q1= θ, q2= r, q3= ν = z − αθ.
Exercises 3.5–3.6 97
(b) Consider a particle moving on the smooth helicoid:
(i) What is the constraint on the motion of the particle, and what is a pre-scription for the constraint force Fcenforcing this constraint?
(ii) Show that the equations governing the unconstrained motion of the particle are
d dt
m
r2+ α2
˙θ
= −mgα, d
dt(m ˙r)− mr ˙θ2= 0. (3.24) (iii) Prove that the angular momentum HO· E3is not conserved.
(c) Suppose the nonintegrable constraint r ˙θ+ h(t) = 0
is imposed on the particle. Establish a second-order differential equation for r(t), a differential equation for θ(t), and an equation for the constraint force enforcing the nonintegrable constraint. Indicate how you would solve these equations to determine the motion of the particle and the constraint forces acting on it.
3.6. Consider a particle of mass m that is free to move on the smooth inner surface of a hemisphere of radius R0(cf. Figure 3.6). The particle is under the influence of a gravitational force−mgE3.
m O
E1
E2 E3
r g
Figure 3.6. Schematic of a particle of mass m moving on the inside of a hemisphere of radius R0.
(a) Using a spherical polar coordinate system, what is the constraint on the motion of the particle? Give a prescription for the constraint force acting on the particle.
(b) Using Lagrange’s equations, establish the equations of motion for the par-ticle and an expression for the constraint force.
(c) Prove that the total energy E and the angular momentum HO· E3 of the particle are conserved.
(d) Show that the normal force acting on the particle can be expressed as a function of the position of the particle and its initial energy E0:
N=
−2E0
R0 + 2mg sin (φ) + mg cos (φ)
eR.
(e) Numerically integrate the equations of motion of the particle and show that there are instances for which it will always remain on the surface of the hemisphere.
3.7. As shown in Figure 3.7, consider a bead of mass m that is free to move on a smooth semicircular wire of radius R0. The wire has a constant angular velocity
0E3 and whirls about the configuration shown in the figure. The particle is also
m O
E1
E2
E3
r g
(t)
(t)
Figure 3.7. Schematic of a particle of mass m moving on a semicircular path that is being whirled about the vertical at a speed (t)= 0.
Exercises 3.7–3.8 99
under the influence of a gravitational force−mgE3. This is a classical problem that is discussed in several textbooks (see, for example, [78]).
(a) Using a spherical polar coordinate system, what are the two constraints on the motion of the particle? Give a prescription for the constraint force Fc
acting on the particle.
(b) Using Lagrange’s equations, establish the equation of motion for the particle:
φ¨=
20cos (φ)+ g R0
sin (φ) . (3.25)
After nondimensionalizing (3.25), numerically integrate the resulting dif-ferential equation and construct its phase portrait for values of 0.5, 1.0, 1.5 of the parameter g
R020.
(c) Recall that an equilibrium point x= x0of the differential equation ¨x= f (x) is such that ˙x= 0 and f (x0)= 0. Show that (3.25) has three equilibria:
φ0 = 0, φ0= π, φ0= cos−1
− g
R020
.
Give physical interpretations for these equilibria and show that the third one is possible if, and only if, 20is sufficiently large. How do these results correlate to your phase portraits?
(d) Starting from the work–energy theorem ˙T= F · v, prove that the total energy of the particle is not conserved:
E˙ = NθR00sin (φ) ,
where Nθis the eθcomponent of the normal force acting on the particle.
3.8. Consider a particle of mass m moving inE3. If coordinate system (1.6) is used to describe its kinematics, then establish expressions for the velocity vector v and the kinetic energy T of the particle.
(a) Suppose a particle is constrained to move on a rough parabolic surface de-scribed by the equation
x− y2= −4.
Give a prescription for the constraint force Fc acting on the particle, and establish the equations of motion for the particle.
(b) As illustrated in Figure 3.8, suppose a particle is constrained to move on the smooth parabola
x− y2= −4, z= 0.
Give a prescription for the constraint force Fc acting on the particle, and establish the equations of motion for the particle.
y m
6 x 2
−4
−4
q2= −4
Figure 3.8. Schematic of a particle moving on a parabola in the x− y plane.