1. Proceedings of the ASME 2015 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2015
August 2-5, 2015, Boston, MA, United States
DETC2015-47715
DRAFT: TRANSIENT FREQUENCY RESPONSE BEHAVIOR AS AN INDICATOR OF
STABILITY IN PARAMETRICALLY EXCITED SYSTEMS
Johnathan Losek
Mechanical Engineering Student
York College of Pennsylvania
York, PA 17403
jlosek@ycp.edu
Tristan M. Ericson∗
Assistant Professor
Department of Physical Sciences
York College of Pennsylvania
York, PA 17403
tericson@ycp.edu
ABSTRACT
Parametrically excited systems are often analyzed for sta-
bility without considering the response in the time and frequency
domains. This work, however, looks more closely at the tran-
sient response, particularly in the frequency domain, as the re-
sponse of a parametrically excited time-variant system settles to
equilibrium (stable) or approaches an unbounded response (un-
stable). There are a number of peaks in the frequency domain,
other than the primary resonance near the natural frequency,
that shift along the frequency axis through a sweep. A number
of unique features of this behavior are considered in light of the
stability properties of the system. Unstable regions occur when
one of the secondary peaks merges with the primary resonance.
The primary resonance is not at a constant frequency; it moves
asymptotically near instability regions. The research is driven by
numerical analyses, but experimental and analytical validation
techniques will be discussed in the presentation. This research
shows that multiple response peaks are important in paramet-
ric response. Their proximity to the primary resonance indicates
how close the operating frequency of the system is to an insta-
bility region. This form of analysis can be used in practical ap-
plications to predict how close a stable system is to an unstable
response if a transient signal can be measured.
∗Address all correspondence to this author.
INTRODUCTION
Mechanical system vibration depends on the parameters of
the system and the source of excitation. Parametric excitation is
a type of stimulus that commonly occurs in gear systems [1–3].
Gears are important components in many commercial and mili-
tary applications, but they are vulnerable to vibration issues be-
cause tooth contact fluctuates during operation. This periodic
fluctuation results in a time varying tooth stiffness. Large am-
plitude vibration occurs at particular parametrically excited fre-
quencies leading to instability. Current analytical models provide
solutions for frequencies at which parametric instability occurs in
geared systems [4] and general dyanmic systems [5,6], but work
considering transient response behavior deeply is limited.
An analytical model is developed for a single-degree-of-
freedom system with sinusoidal time varying stiffness. Numeri-
cal result are obtained in the frequency and time domains. Fre-
quency response plots show unbounded response behavior oc-
curring at twice the natural frequency, the natural frequency, and
lower frequencies as predicted by current models. Unbounded
regions away from the natural and twice the natural frequency
are detected as the amplitude of stiffness fluctuation is increased.
The domain of an unbounded response region increases linearly
with increasing amplitude of stiffness fluctuation. Analysis of the
frequency spectrum reveals multiple response peaks. Unbounded
response occurs as sub-peaks coalesce with the primary output
frequency.
1 Copyright c 2015 by ASME

2. ANALYTICAL MODEL
An analytical model of a single-degree-of-freedom, mass-
spring-damper system with time varying stiffness
k(t) = k +αksin(ωwt) (1)
is used in this analysis where ωe is the parametric excitation fre-
quency. The amplitude of the fluctuating portion of the stiffness
is defined as a percentage of the mean value k by the nondomen-
sional parameter 0 ≤ α ≤ 0. The equation of motion for the para-
metrically excited system with no external excitation is
m
d2x
dt2
+c
dx
dt
+kx+αksin(ωet)x = 0. (2)
Dimensionless parameters are obtained for convenience. Let
x(t) = ε (τ(t))xc and τ(t) = t
tc
, where ε is the nondimensional
displacement, τc is the nondimensional time, and xc and Tc are
constants. Thus, the derivatives of x are
dx
dt
=
dε
dτ
dτ
dt
xc =
dε
dτ
xc
tc
(3)
and
d2x
dt
=
d2ε
dτ2
d2τ
dt2
xc =
d2ε
dτ2
xc
tc
. (4)
Substituting these dimensionless variables into the equation
of motion, we have
d2ε
dτ2
+2ζωntc
dε
dτ
+ω2
nt2
c ε +αω2
nt2
c sin(ωτtc) = 0 (5)
where ζ is the damping ratio and ωn is the natural frequency.
Letting ω2
nt2
c = 1, or tc = 1
ωn
, the nondimansional equation
¨ε(τ)+2ζ ˙ε(τ)+ε(τ) = −α sin
ω
ωn
τ ε(τ) (6)
is used in the numerical analysis.
NUMERICAL RESULTS
A numerical solver is used to obtain the transient response
of the parametrically excited time-variant system represented in
equation 6 due to an initial displacement of εo = 1. The solver
gives nondimensionalized displacement ε in the time domian.
The simulation is run through a frequency sweep from ωe
ωn
= 0.5
to ωe
ωn
= 3.0. An FFT gives the frequency spectrum for each so-
lution set. Even though the transient signals die out (or grow
unbounded within instability regions), their frequency content
shows some interesting behavior. First, there is typically a pri-
mary response peak near the natural frequency, but it is not at the
natural frequency, and it is not constant. Figure 1 shows the spec-
trum zoomed in near the primary response peak for two different
parametric excitation frequencies. The primary peak can appear
below or above the system natural frequency. In subsequent anal-
ysis, we see that the primary response peak dips to lower fre-
quencies before instability regions (during a speed sweep) and
appears at higher frequencies after the sweep emerges from the
instability region.
0
0.01
0.02
0.03
0.04
0.05
0.6 0.8 1 1.2 1.4
ε
ω
nω
ω
nω =1.03eω
nω =0.87e
FIGURE 1: Frequency spectrum of the stable transient response
zoomed in near the primary response peak with ζ = 0.01, α =
0.80 and (solid black) ωe
ωn
= 0.87 and (dashed red) ωe
ωn
= 1.03
Continuing with frequency domain analyis of the transient
response, there are a number of peaks other than the primary re-
sponse peak. As system damping ζ is decreased, the number of
observable peaks increases. With zero damping, there may be
infinite number of them. Figure 2 shows the transient response
when the excitation frequency ωe = 1.60ωn. At this point, the
frequency sweep is past the instability region around ωn and it
is approaching the largest instability region near 2ωn typical of
parametric systems. The figure shows that there are several fre-
quency peaks besides the primary peak ωp, which is found near
the natural frequency ωn. The arrows indicate the direction that
2 Copyright c 2015 by ASME

3. these peaks are moving with the increasing speed sweep. The
system is currently stable, but it is approacing the instability re-
gion around 2ωn. As the parametric excitation frequency is in-
creased, the peaks will move in the directions indicated. The
instability region is entered with three simultaneous events:
1. The lowest peak (below ω
ωn
= 0.5) reaches zero.
2. The peak just below ωp collides with ωp.
3. The peak just below ω
ωn
= 1.5 catches up with the next peak
above ω
ωn
= 1.5 (because the former is moving faster than
the latter).
The combined peaks move together, to higher frequencies,
throughout the instability regions. The system becomes stable
again when the peaks separate. Thus, the proximity of peaks to
one another, and particularly a secondary peak to the primary
peak, indicats closeness to an instability region. This is true
for all instability regions observed numerically (around 2ωn, ωn,
2
3 ωn, and 1
2 ωn).
0 0.5 1 1.5 2 2.5 3
0
0.02
0.04
0.06
0.08
0.10
0.12
ε
ω
nω
pω
FIGURE 2: Frequency spectrum of the stable transient response
with ζ = 0.01, α = 0.80, and ωe
ωn
= 1.60. The arrows indicate the
direction that the peaks are moving as the parametric excitation
frequency is increased.
Previously, Figure 1 showed that the primary response peak
ωp is not stationary along the frequency axis for all parametric
excitation frequencies ωe. We have noted that it is decreasing
in frequency in Figure 2 as the system approaches an instability
during a speed sweep. Figure 3 shows the frequency of the pri-
mary response peak
ωp
ωn
as a function of the parametric excitation
frequency ωe
ωn
obtained throuh a frequency sweep for three dif-
ferent fluctuation stiffness amplitudes, α = 0.5, 0.65, and 0.80.
The unstable regions are highlighded for the middle value of α.
These unstable regions are characterized by a steady linear in-
crease in the frequency of the primary respose peak ωp. They
are preceeded by a sharp drop in primary response frequency
and followed by a high primary response peak that settles back
down quickly. This asymptotic behavior is noted at 2ωp, ωp, and
3
2 ωp
1. Thus, instability regions occur when a secondary peak ap-
proaches the primary peak ωp and when the primary peaky drops
rapidly on increaseing parametric excitation frequency (or rises
rapidly on decreasing excitation frequency).
0.5 1.0 1.5 2.0 2.5 3.0
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Parametric excitation frequency (ω /ω )e n
Frequencyofprimaryresponsepeak(ω/ω)np
FIGURE 3: Location of the primary response peak ωp as a func-
tion of the parameteric excitation frequency ωe with ζ = 0.01
and (dotted red) α = 0.50, (solid black) α = 0.65, and (dash
blue) α = 0.80. Unstable regions for α = 0.65 are highlighted
gray.
Figure 4 illustrates more thoroughly the frequency domain
behavior at parametric excitation speeds near and approaching
an instability region. As speed increases through regions of sta-
bility (top two plots), the primary response peak shifts to lower
frequencies. At the same time, a unique secondary peak below
the primary peak moves into higher frequencies. The secondary
peak eventually converges with the primary peak (middle plot).
This convergence corresponds to an instability region character-
ized by unbounded response in the time domain. During instabil-
ity, both peaks increase in frequency. As ωe
ωn
moves past the in-
stability region, the secondary peak continues to shift into higher
1Further instability regions around 2
n ωn (n is an integer) [7] are noted with
decreased damping.
3 Copyright c 2015 by ASME

4. frequencies while the primary peak shifts back down to lower
frequencies. The primary peak will eventually converge with
another secondary peak. This sequence of events repeats until
ωe
ωn
> 2.
CONCLUSIONS
Parmateric stability is a primary concern in paramtrically ex-
cited systems. Though analytical means are capable of deter-
mining system stability, this analaysis shows that the frequency
spectrum of a transient response shows indicators of closeness
to instability regions. The frequency domain shows a primary
response peak near the natural frequency. This frequency peak
is not stationary along the frequency axis. It moves toward
low/high values when the excitation frequency approaches an
instability region with increasing/decreasing frequency, respec-
tively. In addition, there are secondary peaks that approach the
primary peak and coalesce with it as the system enters an in-
stability region. After the instability, the primary resonance has
jumped up to a higher frequency value. This behavior indicates
that the spectum of a system can be used to predict whether a
system is close to an instability region. A system is nearing in-
stability if the primary response peak moves away from the nat-
ural frequency and a secondary peak approaches it. Continued
work will seek to validate these results analytically and experi-
mentally. We expect to discuss this verification at the time of the
presentation.
REFERENCES
[1] Velex, P., and Flamand, L., 1996. “Dynamic response of
planetary trains to mesh parametric excitations”. Journal of
Mechanical Design, 118(1), Mar., pp. 7–14.
[2] Vaishya, M., and Singh, R., 2001. “Sliding friction induced
non-linearity and parametric effects in gear dynamics”. Jour-
nal of Sound and Vibration, 248(4), May, pp. 671–694.
[3] Lin, J., and Parker, R. G., 2002. “Planetary gear parametric
instability caused by mesh stiffness variation”. Journal of
Sound and Vibration, 249(1), Jan., pp. 129–145.
[4] Liu, G., and Parker, R., 2012. “Nonlinear, parametrically
excited dynamics of two-stage spur gear trains with mesh
stiffness fluctuation”. Journal of Mechanical Engineering
Science, 226(C8), pp. 1939–1957.
[5] Han, Q., Wang, J., and Li, Q., 2010. “Frequency response
characteristics of parametrically excited system”. Journal of
Vibration and Acoustics, 132(4), Aug.
[6] Neal, H. L., and Nayfeh, A. H., 1990. “Response of a single-
degree-of-freedom system to a non-stationary principal para-
metric excitation”. International Journal of Non-Linear Me-
chanics, 25(2-3), pp. 275–284.
[7] Nayfeh, A. H., and Mook, D. T., 1995. Nonlinear Oscilla-
tions. Wiley-VCH.
0
0.02
0.04
0.06
ε
0 0.5 1.0 1.5 2.0 2.5 3.0
ω
n
0
0.02
0.04
0.06
ε
0
0.5
1.0
1.5
2.0
2.5 x 105
ε
0
0.1
0.2
0.3
ε
0
0.04
0.08
0.12
0.16
ε
ω
Stable
ω
nω = 1.05e
Stable
ω
nω = 1.03e
Stable
ω
nω = 0.875e
Stable
ω
nω = 0.865e
Stable
ω
nω = 0.855e
pω
pω
pω
pω
pω
FIGURE 4: Frequency spectrum of the transient response of
the parametrically excited single-degree-of-freedom system with
α = 0.8 and ζ = 0.01 for parametric excitation frequencies
ωe
ωn
= 0.86, ωe
ωn
= 0.87, ωe
ωn
= 0.88, ωe
ωn
= 1.03, ωe
ωn
= 1.05.
4 Copyright c 2015 by ASME