Aerospace Structures & Computational Mechanics

Lecture NotesVersion 1.6

AE21

35-I I

-Vib

ratio

ns

Forced Damped Vibrations (Harmonic)

6 Forced damped vibrations (harmonic) 17

6 Forced damped vibrations (harmonic)

Let’s look at the harmonic forced vibrations of an underdamped system, this has the followingequation of motion:

x+ 2ζωnx+ ω2nx = f cos (ωxt)

6-1 Solution method

To solve for the displacement, xp = Re(Apeiωxt

)is used, because: eiωt = cos (ωt) + i sin (ωt),

so cos (ωt) = Re(eiωt

). Entering this into the EOM gives:

Re((−ω2

x + 2ζωniωx + ω2n

)Apeiωxt

)= Re

(f eiωxt

)−→ Ap = Re

(f

ω2n − ω2

x + 2ζωniωx

)The denominator can be rewritten as a complex number u = v + iw, with:

v = ω2n − ω2

x

w = 2ζωnωxNow, denoting u as the complex conjugate of u:

Ap = Re(f

u

)= Re

(fu

uu

)= Re

(fu

v2 + w2

)Also:

u = v − iw = |u| (cos(θ)− i sin(θ)) = |u| (cos(−θ) + i sin(−θ)) = |u| e−iθ

with θ = arctan(wv

). Hence:

Ap = Re(f

√v2 + w2e−iθ

v2 + w2

)= Re

(f e−iθ√v2 + w2

)= Re

f e−iθ√(ω2n − ω2

x)2 + (2ζωnωx)2

So the particular solution becomes:

xp = Re(Apeiωxt

)= Re

f ei(ωxt−θ)√(ω2n − ω2

x)2 + (2ζωnωx)2

= f cos (ωxt− θ)√(ω2n − ω2

x)2 + (2ζωnωx)2

6-2 Initial conditions

x = Ahe−ζωnt sin (ωdt+ ϕ) +Ap cos (ωxt− θ)

x = Ah(−ζω−ζωnt

n sin (ωdt+ ϕ) + e−ζωntωd cos (ωdt+ ϕ))−Apωx sin (ωxt− θ)

So when x(0) = 0 and x(0) = 0, then:{Ah sin (ϕ) +Ap cos (θ) = 0Ah (−ζωn sin (ϕ) + ωd cos (ϕ)) +Apωx sin (θ)

from which ϕ and Ah can be derived as a function of the known Ap and θ.

AE2135-II - Vibrations Lecture Notes