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 equationMathworldPlanetmath of motion is expressed in terms of circular trigonometric functionsDlmfMathworldPlanetmath, having to deal with complex arbitrary constants (obviously depending on the initial conditionsMathworldPlanetmath) and, on the other hand, studying separately the three cases of damping. Alternatively we shall use hyperbolic functionsDlmfMathworldPlanetmath 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 Fs=kx, where k is an elastic constant so-called stiffness, Fs 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. Fd=cx˙, being c the damping constant.
Natural angular frequency: ωn=k/m (rad/sec), a specific property of the system; m is the mass of oscillator.
Damping factor: ζ=c/2kmc/2mωn (physical dimensionless).
Critical damping: cc=2km2mωn. Thus ζ=c/cc, and it is also called damping ratio.
Static equilibrium configurationPlanetmathPlanetmath: a static position at t=0-, reached because the action of gravitatory field over the mass of oscillator, i.e. the weight mg (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 Fx=0, the spring force kδ (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 Fx=mx¨, we obtain the differential equation of motion

mx¨+cx˙+kx=F(t), (1)

being F(t) the so-called exciter force. It is convenient to introduce ωn and ζ. So that (1) becomes

x¨+2ζωnx˙+ωn2x=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 associateMathworldPlanetmath homogeneous differential equation

x¨+2ζωnx˙+ωn2x=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)=x0,x˙(0)=x˙0. These conditions may be interpreted as follows: from the static equilibrium position we give an initial displacement x0 and an initial velocity x˙0 to the mass m and then we let it free at instant t=0. Accordingly, the characteristic equationMathworldPlanetmathPlanetmathPlanetmath of (3) is

s2+2ζωns+ωn2=0, (4)

whose roots are given by

s1=-ζωn+ωnζ2-1-ζωn+ωd, (5)
s2=-ζωn-ωnζ2-1-ζωn-ωd, (6)

where we are defining ωd:=ωnζ2-1 which is called frecuency of damped oscillation.
In the nullspaceMathworldPlanetmath of the linear operator associate to (3), we find the general solution

x(t)=Aes1t+Bes2tx˙(t)=As1es1t+Bs2es2t. (7)

By substituting the initial conditions in (7), we obtain the symmetricPlanetmathPlanetmathPlanetmathPlanetmath linear system of equations

x0=A+B,x˙0=As1+Bs2. (8)

From (5), (6) and solving (8) for A and B, leads to

A=x˙0+(ζωn+ωd)x02ωd,B=-x˙0+(ζωn-ωd)x02ωd, (9)

which substituted into the solution (7) and after an elementary algebraicMathworldPlanetmath manipulation, leads to

x(t)=e-ζωntωd{(x˙0+ζωnx0)eωdt-e-ωdt2+ωdx0eωdt+e-ωdt2}, (10)

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

x(t)=e-ζωntωd{(x˙0+ζωnx0)sinhωdt+ωdx0coshωdt}. (11)

It is now easily seen that the general solution of the homogeneous equation (3) satifies limtx(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>1overdamped case, 2.ζ=1critical damping case, 3.  0<ζ<1underdamped case.

Let us proceed to study all the three cases.

  1. 1.

    ζ>1. Overdamped case. The response is nonperiodic and is given directly by (11), i.e.

    x(t)=e-ζωntωd{(x˙0+ζωnx0)sinhωdt+ωdx0coshωdt}. (12)

    Roots s1>s2. Hence es1t,es1t are linearly independentMathworldPlanetmath and form a basis in the above mentioned nullspace.

  2. 2.

    ζ=1. Critical damping case. The response is also nonperiodic. In additionPlanetmathPlanetmath, s1=s2<0, so that es1t,es2t are linearly dependent and therefore they do not form a completePlanetmathPlanetmathPlanetmathPlanetmath 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 ωd0, and recalling (11) we note that (by applying L’Hospital),


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

    x(t)xc(t)=e-ωnt{x0+(x˙0+ωnx0)t}. (13)
  3. 3.

    0<ζ<1. Underdamped case. Since in this case ωd, we redefine it as ωd=iωn1-ζ2:=iωd,i=-1. Recalling the hyperbolic equations sinhωdt=sinhiωdt=isinωdt, coshωdt=coshiωdt=cosωdt which substituted into (11) gives the general solution on this case. That is,

    x(t)=e-ζωnt{x˙0+ζωnx0ωdsinωdt+x0cosωdt}. (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:=x02+(x˙0+ζωnx0ωd)2,tanϕ:=ωdx0x˙0+ζωnx0 (15)

    With these definitions placed into (14), it becomes

    x(t)=Xe-ζωntsin(ωdt+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)=f0cosωt11Naturally, an exciter force on the form F(t)=F0sinωt is also valid, but the conclusionsMathworldPlanetmath are the same., being f0:=F0/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¨+2ζωnx˙+ωn2x=f0cosωt. (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 permanentMathworldPlanetmath response, i.e.

x(t)=xh(t)+xp(t). (18)

The general homogeneousPlanetmathPlanetmathPlanetmathPlanetmath solution xh already have been stablished, so that in this section we focus our attention in find out a particular solution xp. We search that solution on the form

xp(t)=Ccosωt+Dsinωt. (19)

Its first and second time derivatives are

x˙p(t)=-Cωsinωt+Dωcosωtx¨p(t)=-Cω2cosωt-Dω2sinωt. (20)

Next we substitute (19) and (20) into (17), setting x(t)=xp(t) and by collecting terms in cosωt and sinωt, we have


We are dealing with a subspacePlanetmathPlanetmath of dimensionPlanetmathPlanetmath 2, where the set of functionsMathworldPlanetmath {cosωt,sinωt}𝒞[0,) form an orthogonal basis in the spaces of solutions of (17). In other words, cosωt and sinωt are there linearly independent. Therefore,


By solving for C and D,


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,

xp(t)=(ωn2-ω2)f0(ωn2-ω2)2+4ζ2ωn2ω2cosωt+2ζωnωf0(ωn2-ω2)2+4ζ2ωn2ω2sinωt. (21)

One also may define the frecuency ratio r=ω/ωn and continue to discuss an important set of physical considerations, but that is not the sight of this entry22Indeed 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

xp(t)=f0ωn2-ω2cosωt. (22)

One realize (theoretically) that as the frequency of exciter force coincides with the natural frequency of the system, i.e. ω=ωn (or equivalently r=1), then xp=. This is what is defined as mechanical resonance phenomena. In the practice, however, low values of ζ (ζ1) 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, (Comunicación 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