1.Notes Introduction 1

8/7/2019 1.Notes Introduction 1

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Introduction

2.1 Multibody mechanical systems–examples

• road vehicles – cars, motorcycles, trucks

• satellites

• robots

• air vehicles – aeroplanes, helicopters, UAVs

• mems

• submersible vehicles

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for non conservative systems as well. The Hamiltonian formulation of the principle of least action asserts that the actual motion realised in nature is that particular motion forwhich this action assumes its smallest value.

2.3 Comparison between treatments

The vectorial and variational theories of mechanics are two different mathematical descriptionsof the same physical phenomena. In Newton’s theory everything is based on two fundamentalvectors: “momentum” and “force”. The variational theory bases everything on two scalar

quantities: “kinetic energy” and “work function”.

In the case of particles whose motion is not restricted by given constraints, i.e free particles,the two treatments lead to equivalent results. For systems with constraints, the unknown natureof the interaction forces makes it is necessary to introduce additional postulates when usingvectorial methods. Newton thought that his third law of motion, “action equals reaction”, wouldtake care of all dynamical problems. This, however, is not true in general but it holds for the

dynamics of rigid bodies. The analytical form of description is simpler and more economicalwhen it comes to constrained systems. The kinematical conditions that exist between theparticles of a moving system do not exist a priori but they are maintained by strong forces.The analytical treatment does not require the knowledge of these forces, but can take the givenkinematical conditions for granted. We can develop the dynamical equations of a rigid bodywithout knowing what forces produce the rigidity of the body.

Vectorial mechanics construct a separate acting force for each moving particle; analyticalmechanics consider one single function: the work function (or potential energy). This onefunction contains in most cases all the necessary information concerning forces.

In the analytical method, the entire set of equations of motion can be developed from one

unified principle: the principle of least action. Such a minimum principle is independent of anyspecial reference frame, the equations of analytical mechanics hold for any set of coordinates.This permits one to adjust the coordinates employed to the specific nature of each problem.

The problems that are well suited to the vectorial treatment are those which can be handledwith a rectangular frame of reference. Many elementary problems are solvable with this methodbut for more complicated problems the variational treatment is superior.

2.4 Simple examples

2.4.1 Horizontal spring-mass system

x

k

m

Figure 2.1: Horizontal spring mass system.

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↓The spring force is

F = −kx, k > 0.

Newton’s Second Law givesmx + kx = 0.

If we define the natural frequency ωn as

ωn =

k

m,

we get the equation of a linear harmonic oscillator

x + ω2nx = 0.

This is a second order linear differential equation with a general solution

x(t) = A cos(ωnt) + B sin(ωnt).

↑

2.4.2 Vertical spring-mass system

x

x0k

m

m

equilibriumposition

Figure 2.2: Vertical spring mass system.

↓There is no dissipation and therefore the total energy is conserved:

T + V = constant.

The potential energy V is given by

V = 12

k(x + x0)2 − mg(x + x0

2).

Modelling and Control of Multibody Mechanical Systems7



At equilibrium−kx0 + mg = 0,

and therefore by substitution

V =1

2kx2.

The kinetic energy is given by

T = 12

mx2.

By conservation of energy

d

dt(T + V ) =

d

dt(

1

2mx2 +

1

2kx2) = 0,

which givesmxx + kxx = 0.

The governing equation is thusmx + kx = 0.

↑2.4.3 Spring-damper-mass system

x

k

c

m

Figure 2.3: Spring-damper-mass system.

↓The spring force is given by

F s = −kx, k > 0,

and the damper force is obtained from

F d = −cx, c > 0.

Use of Newton’s Second Law (F total = mx) gives

F s + F d = mx,

ormx + cx + kx = 0.

If we define the natural frequency ωn =

k/m and the damping ratio ζ = c/(2mωn) theequation of motion becomes

x + 2ζωnx + ω2nx = 0.

It is straightforward to show that the general solution of this equation is

x(t) = A1e(−ζ +√

ζ 2−1)ωnt + A2e(−ζ −√

ζ 2−1)ωnt,

where A1 and A2 are integration constants and can be found when specific initial conditions aregiven. ↑

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2.4.4 Forced spring-damper-mass system

x

k

cm

F ext

Figure 2.4: Forced spring-damper-mass system.

↓The spring force is given by

F s = −kx, k > 0,

the damper force is given by

F d = −cx, c > 0,and the external input that forces the system is F ext. Use of Newton’s Second Law (F total =mx) gives

F s + F d + F ext = mx,

ormx + cx + kx = F ext.

If we define the natural frequency ωn =

k/m and the damping ratio ζ = c/(2mωn) theequation of motion becomes

x + 2ζωnx + ω2nx =

F ext

m.

The general solution of this equation is the solution of the unforced system (previous example)plus the particular solution with the forcing term included.

↑

2.4.5 Double spring-damper-mass system

x1

k1

x2

k2

c2

m1 m2

F ext

Figure 2.5: Forced double spring-damper-mass system.

↓The total force on mass m1 is given by

F 1 = −kx1 − k2(x1 − x2) − c2( x1 − x2) + F ext,






where

A =

0 0 1 00 0 0 1

− k1m1

− k2m1

k2m1

− c2m1

c2m1

k2m2

− k2m2

c2m2

− c2m2

,

and

B =

001m1

0

.

These equations are linear and in order to solve them we can make use of well developedtechniques.

↑


1.Notes Introduction 1

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