harmonic oscillator

harmonic oscillator

1

Introduction

Harmonic oscillator

is the simplest model but one of the most important vibrating system. It is often encountered in engineering systems and commonly produced by the unbalance in rotating machinery, isolation, earthquakes, bridges, building, control and atomatization devices, just for naming a few examples. Although pure

harmonic

oscillation

is less likely to occur than general periodic oscillations by the action of arbitrary types of excitation, understanding the

behavior

of a system undergoing harmonic oscillation is essential in order to comprehend how the system will respond to more general types of excitation. Such harmonic oscillations may be in the form of free vibrations or by the action of an exciter force applied at some point of the system, as we shall see later. Indeed, harmonic oscillator is a single degree of freedom system (it is like a vibration module) which is tipically constituted by an ideal spring in

parallel

with an ideal (but not

conservative

) viscous damper, both attached to a mass in one end and the other one

grounded

.

In this entry we will make an

elementary

but nonconventional discussion about the fundamental

concepts

involved in a single vibrating system. Unlike the conventional method, which is usually discussed in inherent Literature about this topic, where, by one side, the solution of the corresponding

differential equation

of motion is expressed in terms of circular

trigonometric functions

, having to deal with complex arbitrary

constants

(obviously depending on the

initial conditions

) and, on the other hand, studying separately the three cases of damping. Alternatively we shall use

hyperbolic functions

and their properties to obtain the solution of the mentioned differential equation. This way we gain a little in brevity and hilarity in the exposition, as we may study all the three cases of damping simultaneously, avoiding at the same time calculating complex constants.

Besides preliminars, we will separate the discussion in three paragraphs namely: free damped vibrations, forced damped vibrations and mechanical resonance.

2

Preliminars

The purpose of this

section

is concept definifitions.

Response:

x

=

x

⁢

(

t

)

, the

general solution

of the linear differential equation involved in the motion of harmonic oscillator. We will assume

x

>

0

downward, like the sense of gravitatory field.

Ideal spring:

an unidimensional model, weightless, linear, elastic and that obeys the Hooke’s law

F

s

=

k

⁢

x

, where

k

is an elastic constant so-called

stiffness

,

F

s

is the restitutive spring force and

x

an arbitrary spring elongation.

Ideal damper:

an unidimensional model, weightless, where the damping force is proportional to the velocity of oscillator, i.e.

F

d

=

c

⁢

x

˙

, being

c

the

damping constant

.

Natural angular frequency:

ω

n

=

k

/

m

(rad/sec), a specific property of the system;

m

is the mass of oscillator.

Damping factor:

ζ

=

c

/

2

⁢

k

⁢

m

≡

c

/

2

⁢

m

⁢

ω

n

(physical dimensionless).

Critical damping:

c

c

=

2

⁢

k

⁢

m

≡

2

⁢

m

⁢

ω

n

. Thus

ζ

=

c

/

c

c

, and it is also called

damping ratio

.

Static equilibrium

configuration

:

a static position at

t

=

0

-

, reached because the action of gravitatory field over the mass of oscillator, i.e. the weight

m

⁢

g

(

g

is the gravity acceleration), thus deflecting the spring a quantity

δ

, from its natural length, so-called

spring static deflection

. As a result, from Newton’s equation

∑

F

x

=

0

, the spring force

k

⁢

δ

(damper does not work!) due to its static deflection is balanced, at every instant, by the weight of oscillator and therefore both forces do not appear in the dynamical equation.

3

Free Damped Vibrations

With the help of an oscillator’s free body diagram and by applying the Newton’s second law

∑

F

x

=

m

⁢

x

¨

, we obtain the differential equation of motion

m

⁢

x

¨

+

c

⁢

x

˙

+

k

⁢

x

=

F

⁢

(

t

)

,

(1)

being

F

⁢

(

t

)

the so-called

exciter force

. It is convenient to introduce

ω

n

and

ζ

. So that (1) becomes

x

¨

+

2

⁢

ζ

⁢

ω

n

⁢

x

˙

+

ω

n

2

⁢

x

=

f

⁢

(

t

)

,

(2)

where

f

⁢

(

t

)

=

F

⁢

(

t

)

/

m

. But in this section we are interested in free vibrations. So that we set

f

⁢

(

t

)

≡

0

and thus we get the

associate

homogeneous differential equation

x

¨

+

2

⁢

ζ

⁢

ω

n

⁢

x

˙

+

ω

n

2

⁢

x

=

0

.

(3)

In this case, about free oscillations,

x

means an arbitrary displacement from the static equilibrium position. Let us suppose initial conditions

x

⁢

(

0

)

=

x

0

,

x

˙

⁢

(

0

)

=

x

˙

0

. These conditions may be interpreted as follows: from the static equilibrium position we give an initial displacement

x

0

and an initial velocity

x

˙

0

to the mass

m

and then we let it free at instant

t

=

0

. Accordingly, the

characteristic equation

of (3) is

s

2

+

2

⁢

ζ

⁢

ω

n

⁢

s

+

ω

n

2

=

0

,

(4)

whose roots are given by

s

1

=

-

ζ

⁢

ω

n

+

ω

n

⁢

ζ

2

-

1

≡

-

ζ

⁢

ω

n

+

ω

d

,

(5)

s

2

=

-

ζ

⁢

ω

n

-

ω

n

⁢

ζ

2

-

1

≡

-

ζ

⁢

ω

n

-

ω

d

,

(6)

where we are defining

ω

d

:=

ω

n

⁢

ζ

2

-

1

which is called

frecuency of damped oscillation

.

In the

nullspace

of the

linear operator

associate to (3), we find the general solution

x

⁢

(

t

)

=

A

⁢

e

s

1

⁢

t

+

B

⁢

e

s

2

⁢

t

⟹

x

˙

⁢

(

t

)

=

A

⁢

s

1

⁢

e

s

1

⁢

t

+

B

⁢

s

2

⁢

e

s

2

⁢

t

.

(7)

By substituting the initial conditions in (7), we obtain the

symmetric

linear system of equations

x

0

=

A

+

B

,

x

˙

0

=

A

⁢

s

1

+

B

⁢

s

2

.

(8)

From (5), (6) and solving (8) for

A

and

B

, leads to

A

=

x

˙

0

+

(

ζ

⁢

ω

n

+

ω

d

)

⁢

x

0

2

⁢

ω

d

,

B

=

-

x

˙

0

+

(

ζ

⁢

ω

n

-

ω

d

)

⁢

x

0

2

⁢

ω

d

,

(9)

which substituted into the solution (7) and after an elementary

algebraic

manipulation, leads to

x

⁢

(

t

)

=

e

-

ζ

⁢

ω

n

⁢

t

ω

d

⁢

{

(

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

)

⁢

e

ω

d

⁢

t

-

e

-

ω

d

⁢

t

2

+

ω

d

⁢

x

0

⁢

e

ω

d

⁢

t

+

e

-

ω

d

⁢

t

2

}

,

(10)

that, in terms of hyperbolic functions, it is expressed as

x

⁢

(

t

)

=

e

-

ζ

⁢

ω

n

⁢

t

ω

d

⁢

{

(

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

)

⁢

sinh

⁡

ω

d

⁢

t

+

ω

d

⁢

x

0

⁢

cosh

⁡

ω

d

⁢

t

}

.

(11)

It is now easily seen that the general solution of the

homogeneous equation

(3) satifies

lim

t

→

∞

⁡

x

⁢

(

t

)

=

0

, since we are dealing with a transitory solution which agree with the fact that an exciter force

f

⁢

(

t

)

is not present.

Damping factor lets a classification about the kind of damping in concordance with the values that

ζ

may take. Thus,

{

1

.

ζ

>

1

⟹

overdamped case

,

2

.

ζ

=

1

⟹

critical damping case

,

3.  0

<

ζ

<

1

⟹

underdamped case

.

Let us proceed to study all the three cases.

1.

ζ

>

1

.

Overdamped case

. The response is

nonperiodic

and is given directly by (11), i.e.

x

⁢

(

t

)

=

e

-

ζ

⁢

ω

n

⁢

t

ω

d

⁢

{

(

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

)

⁢

sinh

⁡

ω

d

⁢

t

+

ω

d

⁢

x

0

⁢

cosh

⁡

ω

d

⁢

t

}

.

(12)

Roots

s

1

>

s

2

∈

ℝ

. Hence

e

s

1

⁢

t

,

e

s

1

⁢

t

are

linearly independent

and form a basis in the above mentioned nullspace.

2.

ζ

=

1

.

Critical damping case

. The response is also

nonperiodic

. In

addition

,

s

1

=

s

2

<

0

, so that

e

s

1

⁢

t

,

e

s

2

⁢

t

are linearly dependent and therefore they do not form a

complete

basis in the space of solutions of the differential equation (3). Nevertheless, we may find out a second solution linearly independent because in this case

ω

d

≡

0

, and recalling (11) we note that (by applying

L’Hospital

),

lim

ω

d

→

0

⁡

sinh

⁡

ω

d

⁢

t

ω

d

=

t

.

Thus, the general solution for critical damping may be expressed as

x

⁢

(

t

)

≡

x

c

⁢

(

t

)

=

e

-

ω

n

⁢

t

⁢

{

x

0

+

(

x

˙

0

+

ω

n

⁢

x

0

)

⁢

t

}

.

(13)

3.

0

<

ζ

<

1

.

Underdamped case

. Since in this case

ω

d

∈

ℂ

, we redefine it as

ω

d

=

i

⁢

ω

n

⁢

1

-

ζ

2

:=

i

⁢

ω

d

′

,

i

=

-

1

.
Recalling the hyperbolic equations

sinh

⁡

ω

d

⁢

t

=

sinh

⁡

i

⁢

ω

d

′

⁢

t

=

i

⁢

sin

⁡

ω

d

′

⁢

t

,

cosh

⁡

ω

d

⁢

t

=

cosh

⁡

i

⁢

ω

d

′

⁢

t

=

cos

⁡

ω

d

′

⁢

t

which substituted into (11) gives the general solution on this case. That is,

x

⁢

(

t

)

=

e

-

ζ

⁢

ω

n

⁢

t

⁢

{

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

ω

d

′

⁢

sin

⁡

ω

d

′

⁢

t

+

x

0

⁢

cos

⁡

ω

d

′

⁢

t

}

.

(14)

This solution is clearly harmonics. Alternatively, we may introduce definitions about the

amplitude

X

of the response and the

phase angle

ϕ

. They are,

X

:=

x

0

2

+

(

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

ω

d

′

)

2

,

tan

ϕ

:=

ω

d

′

⁢

x

0

x

˙

0

+

ζ

⁢

ω

n

⁢

x

0

⋅

(15)

With these definitions placed into (14), it becomes

x

⁢

(

t

)

=

X

⁢

e

-

ζ

⁢

ω

n

⁢

t

⁢

sin

⁡

(

ω

d

′

⁢

t

+

ϕ

)

.

(16)

4

Forced Damped Vibrations

In this case an exciter force is present. Many useful and important applications, a few mentioned above in the

Introduction

, agree with an excellent degree of accuracy if we assume an

intensive

periodic exciter force (exciter force by mass unit) on the form

f

⁢

(

t

)

=

f

0

⁢

cos

⁡

ω

⁢

t

1

1

Naturally, an exciter force on the form

F

⁢

(

t

)

=

F

0

⁢

sin

⁡

ω

⁢

t

is also valid, but the

conclusions

are the same.

, being

f

0

:=

F

0

/

m

the

amplitude

of the intensive exciter force, and

ω

the

frecuency of the exciter force

. In such case, the differential equation of motion (2) takes the form

x

¨

+

2

⁢

ζ

⁢

ω

n

⁢

x

˙

+

ω

n

2

⁢

x

=

f

0

⁢

cos

⁡

ω

⁢

t

.

(17)

It is well-known, from the linear differential equations theory, that the general solution of (17) may be expressed as the

superposition

of an homegeneous or transitory response and a particular or

permanent

response, i.e.

x

⁢

(

t

)

=

x

h

⁢

(

t

)

+

x

p

⁢

(

t

)

.

(18)

The general

homogeneous

solution

x

h

already have been stablished, so that in this section we focus our attention in find out a particular solution

x

p

. We search that solution on the form

x

p

⁢

(

t

)

=

C

⁢

cos

⁡

ω

⁢

t

+

D

⁢

sin

⁡

ω

⁢

t

.

(19)

Its first and second time derivatives are

x

˙

p

⁢

(

t

)

=

-

C

⁢

ω

⁢

sin

⁡

ω

⁢

t

+

D

⁢

ω

⁢

cos

⁡

ω

⁢

t

⟹

x

¨

p

⁢

(

t

)

=

-

C

⁢

ω

2

⁢

cos

⁡

ω

⁢

t

-

D

⁢

ω

2

⁢

sin

⁡

ω

⁢

t

.

(20)

Next we substitute (19) and (20) into (17), setting

x

⁢

(

t

)

=

x

p

⁢

(

t

)

and by collecting terms in

cos

⁡

ω

⁢

t

and

sin

⁡

ω

⁢

t

, we have

{

(

ω

n

2

-

ω

2

)

⁢

C

+

(

2

⁢

ζ

⁢

ω

n

⁢

ω

)

⁢

D

}

⁢

cos

⁡

ω

⁢

t

+

{

(

ω

n

2

-

ω

2

)

⁢

D

-

(

2

⁢

ζ

⁢

ω

n

⁢

ω

)

⁢

C

}

⁢

sin

⁡

ω

⁢

t

=

f

0

⁢

cos

⁡

ω

⁢

t

.

We are dealing with a

subspace

of

dimension

2

, where the set of

functions

{

cos

⁡

ω

⁢

t

,

sin

⁡

ω

⁢

t

}

⊂

𝒞

⁢

[

0

,

∞

)

form an

orthogonal basis

in the spaces of solutions of (17).
In other words,

cos

⁡

ω

⁢

t

and

sin

⁡

ω

⁢

t

are there linearly independent. Therefore,

(

ω

n

2

-

ω

2

)

⁢

D

-

(

2

⁢

ζ

⁢

ω

n

⁢

ω

)

⁢

C

≡

0

,

and

(

ω

n

2

-

ω

2

)

⁢

C

+

(

2

⁢

ζ

⁢

ω

n

⁢

ω

)

⁢

D

≡

f

0

.

By solving for

C

and

D

,

C

=

(

ω

n

2

-

ω

2

)

⁢

f

0

(

ω

n

2

-

ω

2

)

2

+

4

⁢

ζ

2

⁢

ω

n

2

⁢

ω

2

,

D

=

2

⁢

ζ

⁢

ω

n

⁢

ω

⁢

f

0

(

ω

n

2

-

ω

2

)

2

+

4

⁢

ζ

2

⁢

ω

n

2

⁢

ω

2

⋅

These one are introduced into (19) and we obtain a particular solution of (17), i.e. the

permanent response

of harmonic oscillator’s forced vibrations. That is,

x

p

⁢

(

t

)

=

(

ω

n

2

-

ω

2

)

⁢

f

0

(

ω

n

2

-

ω

2

)

2

+

4

⁢

ζ

2

⁢

ω

n

2

⁢

ω

2

⁢

cos

⁡

ω

⁢

t

+

2

⁢

ζ

⁢

ω

n

⁢

ω

⁢

f

0

(

ω

n

2

-

ω

2

)

2

+

4

⁢

ζ

2

⁢

ω

n

2

⁢

ω

2

⁢

sin

⁡

ω

⁢

t

.

(21)

One also may define the

frecuency ratio

r

=

ω

/

ω

n

and continue to discuss an important set of physical considerations, but that is not the sight of this entry

2

2

Indeed the author mostly wanted to indicate the alternative

elementary method

corresponding to free vibrations but, for completeness reasons, the latter two sections have been added.

.

5

Mechanical Resonance

In absence of damping, i.e.

ζ

=

0

, (21) reduces to

x

p

⁢

(

t

)

=

f

0

ω

n

2

-

ω

2

⁢

cos

⁡

ω

⁢

t

.

(22)

One realize (theoretically) that as the frequency of exciter force coincides with the natural frequency of the system, i.e.

ω

=

ω

n

(or equivalently

r

=

1

), then

x

p

=

∞

. This is what is defined as

mechanical resonance

phenomena. In the practice, however, low values of

ζ

(

ζ

≪

1

) make feasible high values of the permanent response and thus, this fact, is also called

resonance

. Such phenomena is undesirable in any system, unlike the magnetic resonance that provides beneficial applications in several fields of science.

6

References

P. Fernández,

Un método elemental alternativo para el estudio de las vibraciones libres amortiguadas en un oscilador armónico

, (ComunicaciÃÂ³n interna), Departamento de Mecánica, Facultad de Ingeniería, Universidad Central de Venezuela (UCV), Caracas, 1997.

Title

harmonic oscillator

Canonical name

HarmonicOscillator

Date of creation

2013-03-22 17:22:46

Last modified on

2013-03-22 17:22:46

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

10

Author

perucho (2192)

Entry type

Application

Classification

msc 34C05

Classification

msc 34A30

Classification

msc 34-01

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