# Effects of Variations in Nonlinear Damping Coefficients on the Parametric Vibration of a Cantilever Beam with a Lumped Mass

Authors: Demian G. Silva; Paulo S. Varoto
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Uncertainties in damping estimates can significantly affect the dynamic response
of a given flexible structure. A common practice in linear structural
dynamics is to consider a linear viscous damping model as the major energy
dissipation mechanism. However, it is well known that different forms of
energy dissipation can affect the structure's dynamic response. The major
goal of this paper is to address the effects of the turbulent frictional damping
force, also known as drag force on the dynamic behavior of a typical flexible
structure composed of a slender cantilever beam carrying a lumped-mass
on the tip. First, the system's analytical equation is obtained and solved by
employing a perturbation technique. The solution process considers variations
of the drag force coefficient and its effects on the system's response.
Then, experimental results are presented to demonstrate the effects of the
nonlinear quadratic damping due to the turbulent frictional force on the system's
dynamic response. In particular, the effects of the quadratic damping
on the frequency-response and amplitude-response curves are investigated.
Numerically simulated as well as experimental results indicate that variations
on the drag force coefficient significantly alter the dynamics of the
structure under investigation.


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## Body

## 1. Introduction

Characterization and quantification of uncertainties
have been a topic of major importance in the context of structural dynamics.
Generally speaking, the term uncertainty can be associated to variations of the
system's physical parameters due to inaccuracies present either in the system's
model or experimental data. In a broad view, the sources of uncertainties can
be grouped into two main categories, namelystatistical and nonstatistical [1], where the former
is associated to fluctuations in the system's parameter mostly due to
variations in material and/or geometry, and the later reflects inaccuracies
present in the system's model caused by adoption of inappropriate assumptions
or variations in numerical errors, for instance. More recently, a new
terminology has been used to this classification by employing the words aleatory or aleatoric and epistemic [2, 3] to refer to these two groups of uncertainties,
respectively. Several statistical and fuzzy theory-based procedures have been
recently proposed (see [4–8]) to characterize and quantify uncertainties in complex
structural systems. Similarly, uncertainties have also been subject of
investigation in wave propagations and vibroacoustics (see [9–11]) as well as aerospace structures (see [3, 12, 13]). In the field of nonlinear
structural dynamics, a reduced number of works have  been reported. Nichols et al.
[14] has developed a
procedure for the detection of quadratic nonlinearities while Adhikari
[2] has discussed
uncertainties in damping models.Parametrically excited cantilever beams have been
extensively investigated in the last two decades, specially in the case of the
principal parametric resonance [15–18]. Although most of these investigations have dealt
with various aspects of the parametric resonance phenomenon, the majority of
analysis was done by neglecting the effects of fluid medium. In this work, we
perform an experimental and theoretical investigation on the effects of the
viscous quadratic damping on the dynamic response of a cantilever beam with tip
mass to a principal parametric resonance [19, 20]. To investigate the quadratic damping effect, the
structure shown in Figure 1 was built. It is composed of a slender stainless
steel ASTM A240 beam, with dimensions of 100mm in length, 20mm in width, and 1mm in thickness. The lumped mass is composed of
carbon steel ASTM A36, with dimensions of 10mm in length, 40mm in width, and 20mm in height. The opposite beam's end is clamped
to a rigid base built from carbon ASTM A36 steel.Physical system under investigation: (a)
frontal view, (b) lateral view [19, 21, 23].
(a)(b)Figure2 depicts the results of an experiment that was carried out by using the system
shown in Figure 1. The experiment consisted of driving the structure into a
principal parametric resonance condition through an input base sinusoidal
signal. The structure's vibration and interaction with the surrounding fluid
medium could be observed through the smoke-wire arrangement [19] as shown in the sequence of
pictures of Figure 2. This experiment qualitatively indicates that the drag
force plays an important role on the dynamics of the structure under
investigation. Thus, this paper is concerned in studying theoretically and
experimentally the effects of variations in the quadratic damping coefficients
on the lateral vibration of a cantilever beam undergoing a principal parametric
excitation.Qualitative
effects of quadratic damping on the lateral vibration of the system [19].
(a)(b)(c)(d)(e)(f)


## 2. Mathematical Modeling

In a previous
work by the authors [21], a general mathematical model for the structure shown
in Figure 1 was developed by using the model shown in Figure 3. According to this
model, the OXYZ orthogonal coordinates system is fixed at the
base of the beam at its unstressed position and directed such that the X axis is taken as the centerline of the beam.
The origin O of the coordinate system may be subject to a
dynamic displacement UB(t) in the X direction that represents the external driving
signal.Figure 3
Physical model of the structure
under investigation, combined acoustic, and base excitations [19].


### 2.1. Energy Relationships

The beam is
modeled as a continuum solid with displacement field described byui(i=1,2,3) and uio(i=1,2,3).
The kinetic energy of the lumped-mass
system can be described by (2.1)T=12∫Vρu˙iu˙idV+12mou˙iou˙io in which the dot
denotes time derivative, ρ and V are, respectively, the material density
and volume of the beam, and mo is the value of the lumped mass. In order to
simplify (2.1), the contribution of the distributed mass of the beam will be
ignored as well as the rotatory energy of the lumped mass. Hence, the kinetic
energy is simplified to (2.2)T=12mo[(u˙1o)2+(u˙3o)2].The task to findT consists in performing several steps. First,
the time derivative of the displacement field must be computed which results in
expressions for (u˙1)2 and (u˙3)2. Second, the terms in the right-hand side of
the expressions of (u˙1)2 and (u˙3)2 are described as functions of the w(x,t) and its spatial derivatives. Third, a spatial
reduction is necessary so that the deflection on the center of the lumped mass
can be obtained. This can be done by using an expression of the
form (2.3)w(x,t)=ϕ(x)wo(t) in which ϕ(x) represents the first linear natural mode of
the structure and wo(t) represents the modal coordinate associated with this natural
mode. As a final result, the expressions for (u˙1o)2 and (u˙3o)2 are found and truncated to result in
nonlinearities of third order as
follows: (2.4)(u˙1o)2=(A1)2wo2w˙o2−2A1wow˙oU˙B+83A2wo3w˙oU˙B+(U˙B)2,(u˙3o)2=w˙o2.Substituting the
expressions of(u˙1o)2 and (u˙3o)2 described above into (2.2), the kinetic energy
is then given as (2.5)T=12mow˙o2+12mo[(A1)2wo2w˙o2−2A1wow˙oU˙B+83A2wo3w˙oU˙B+(U˙B)2], in which A1 and A2 are geometrical constants given
as (2.6)A1=∫0L(∂ϕ∂x)2dx,A2=∫0L(∂ϕ∂x)4dx, and the first
linear mode shape function ϕ(x) is given as (2.7)ϕ(x)=1−cos(πx2L).Once the final expression for the kinetic energy is
known, the next step towards the derivation of equation of motion is obtaining
the system's strain or potential energy which may be written as function of the
stress and strain in theX direction as (2.8)U=12∭VσxxεxxdV. By using the
assumption that the material follows the constitutive Hooke law, and ignoring
Poisson's effects, (2.8) reduces to (2.9)U=12∭VEεxx2dV=12∫0L∫AEεxx2dAdx.However,uc is very small when compared to ur.
Therefore, to simplify the analysis, the
contributions of uc and of the gravitational field to the strain
energy are ignored, thus giving (2.10)U=12∫0L∫AE[z2(w′′)2−z2(w′′)2(w')2]dAdx. By writing w(x,t) as a function of ϕ(x) and wo(t) (see (2.3)), the final expression for the strain
energy truncated cubic terms in the system's equation of motion is given
by (2.11)U=12EIyB1wo2−32EIyB2wo4+⋯, where Iy is the area moment of inertial  about the Y
axis, and the geometrical constants B1 and B2 are given as (2.12)B1=∫0L(ϕ′′)2dx,B2=∫0L(ϕ′′)2(ϕ′)2dx.The last step before deriving the system's equation of
motion consists in obtaining the expression for the nonconservative forces
acting on the system. Herein, it will be considered the action of two
nonconservative forces. The first is the structural damping force which is
modeled in terms of the generalized coordinates asc1w˙o.
The second is the aerodynamic drag damping force acting on the system (when in
motion) and is proportional to the squared of the generalized velocity c2w˙o|w˙o|.
Both damping forces act in the negative direction of the virtual transversal
displacement δwo.
Therefore, the nonconservative virtual work δWnc which is done on the system is given
by (2.13)δWnc=(−c1w˙o−c2w˙o|w˙o|)δwo.Since the nonconservative virtual work is defined as a
function of the nonconservative generalized forceQnc as δWnc=Qncδwo,
the generalized force Qnc is obtained as (2.14)Qnc=δWncδwo=−c1w˙o−c2w˙o|w˙o|.


### 2.2. Equation of Motion

In the earlier section, the expressions for the
kinetic energyT,
strain energy U, and nonconservative generalized force Qnc were obtained. From these results, it is
possible to derive the system's equation of motion by using the well-known
Lagrange equation [22] which, in turn, for the
system under investigation is written as (2.15)ddt(∂T∂w˙o)−∂T∂wo+∂U∂w˙o=Qnc.Through the computation of
each term of Lagrange's equation and substitution of the result
(2.15), the following result is obtained: (2.16)(1+A12wo2¯)w¨o+(c1mo+c2mo|w˙o|¯)w˙o+(A12w˙o2¯−A1U¨B¯+EIyB1mo)wo+[43A2U¨B¯¯−(6EIyB2mo)¯¯]wo3=F(t)mo.Equation (2.16) represents an ordinary inhomogeneous
nonlinear time-dependent differential equation. In addition, this equation
holds both the axial contraction and the curvature nonlinear effects. If both
the underlined and double underlined terms are ignored, this equation reduces
to a classical linear damped forced model. On the other hand, if only the
double underlined terms are ignored, this equation reduces to the same equation
obtained in [23] plus
a forced term. Still, if the double underlined terms plus the nonlinear damping
were ignored, this equation reduces to the same model obtained in [24] plus a forced term.Since the present work is focused on the dynamic
response of a structure under parametric sinusoidal excitation, it is
considered that this excitation can be written as(2.17)U¨B(t)=−Qλ2cos(λt+φ)=−Qocos(λt+φ) in which Qo is the magnitude of the input base
acceleration,  λ is  the parametric excitation frequency, and φ is a phase shift. Then, (2.16) can be rewritten
in the dimensional final form as (2.18)(1+A12wo2)w¨o+(c1mo+c2mo|w˙o|)w˙o+(A12w˙o2+A1Qocos(λt+φ)+EIyB1mo)wo+[−43A2Qocos(λt+φ)−6EIyB2mo]wo3=0.From the numerical viewpoint, it is interesting to
work with the differential equation in dimensionless form. Therefore, by
setting new dimensionless variableswo∗=A1wo and t∗=t/Tn in which Tn is the period of free vibration, the system's
equation of motion is given by (2.19)(1+wo∗2)w¨o∗+H1w˙o∗+H2w˙o∗|w˙o∗|+[1+w˙o∗2+H3cos(Θt∗+φ)]wo∗−[H5+H4cos(Θt∗+φ)]wo∗3=0 in which (2.20)H1=c1Tnmo,H2=c2moA1,H3=QoA1Tn2,H4=4A2QoTn23A12,H5=6B2EIyTn2moA12,Θ=λTn,Ψ=ΩTn. Further, we will discuss the
effects of variations of the dimensionless quadratic damping coefficient H2 on the response of the parametrically driven
cantilever beam.


## 2.1. Energy Relationships

The beam is
modeled as a continuum solid with displacement field described byui(i=1,2,3) and uio(i=1,2,3).
The kinetic energy of the lumped-mass
system can be described by (2.1)T=12∫Vρu˙iu˙idV+12mou˙iou˙io in which the dot
denotes time derivative, ρ and V are, respectively, the material density
and volume of the beam, and mo is the value of the lumped mass. In order to
simplify (2.1), the contribution of the distributed mass of the beam will be
ignored as well as the rotatory energy of the lumped mass. Hence, the kinetic
energy is simplified to (2.2)T=12mo[(u˙1o)2+(u˙3o)2].The task to findT consists in performing several steps. First,
the time derivative of the displacement field must be computed which results in
expressions for (u˙1)2 and (u˙3)2. Second, the terms in the right-hand side of
the expressions of (u˙1)2 and (u˙3)2 are described as functions of the w(x,t) and its spatial derivatives. Third, a spatial
reduction is necessary so that the deflection on the center of the lumped mass
can be obtained. This can be done by using an expression of the
form (2.3)w(x,t)=ϕ(x)wo(t) in which ϕ(x) represents the first linear natural mode of
the structure and wo(t) represents the modal coordinate associated with this natural
mode. As a final result, the expressions for (u˙1o)2 and (u˙3o)2 are found and truncated to result in
nonlinearities of third order as
follows: (2.4)(u˙1o)2=(A1)2wo2w˙o2−2A1wow˙oU˙B+83A2wo3w˙oU˙B+(U˙B)2,(u˙3o)2=w˙o2.Substituting the
expressions of(u˙1o)2 and (u˙3o)2 described above into (2.2), the kinetic energy
is then given as (2.5)T=12mow˙o2+12mo[(A1)2wo2w˙o2−2A1wow˙oU˙B+83A2wo3w˙oU˙B+(U˙B)2], in which A1 and A2 are geometrical constants given
as (2.6)A1=∫0L(∂ϕ∂x)2dx,A2=∫0L(∂ϕ∂x)4dx, and the first
linear mode shape function ϕ(x) is given as (2.7)ϕ(x)=1−cos(πx2L).Once the final expression for the kinetic energy is
known, the next step towards the derivation of equation of motion is obtaining
the system's strain or potential energy which may be written as function of the
stress and strain in theX direction as (2.8)U=12∭VσxxεxxdV. By using the
assumption that the material follows the constitutive Hooke law, and ignoring
Poisson's effects, (2.8) reduces to (2.9)U=12∭VEεxx2dV=12∫0L∫AEεxx2dAdx.However,uc is very small when compared to ur.
Therefore, to simplify the analysis, the
contributions of uc and of the gravitational field to the strain
energy are ignored, thus giving (2.10)U=12∫0L∫AE[z2(w′′)2−z2(w′′)2(w')2]dAdx. By writing w(x,t) as a function of ϕ(x) and wo(t) (see (2.3)), the final expression for the strain
energy truncated cubic terms in the system's equation of motion is given
by (2.11)U=12EIyB1wo2−32EIyB2wo4+⋯, where Iy is the area moment of inertial  about the Y
axis, and the geometrical constants B1 and B2 are given as (2.12)B1=∫0L(ϕ′′)2dx,B2=∫0L(ϕ′′)2(ϕ′)2dx.The last step before deriving the system's equation of
motion consists in obtaining the expression for the nonconservative forces
acting on the system. Herein, it will be considered the action of two
nonconservative forces. The first is the structural damping force which is
modeled in terms of the generalized coordinates asc1w˙o.
The second is the aerodynamic drag damping force acting on the system (when in
motion) and is proportional to the squared of the generalized velocity c2w˙o|w˙o|.
Both damping forces act in the negative direction of the virtual transversal
displacement δwo.
Therefore, the nonconservative virtual work δWnc which is done on the system is given
by (2.13)δWnc=(−c1w˙o−c2w˙o|w˙o|)δwo.Since the nonconservative virtual work is defined as a
function of the nonconservative generalized forceQnc as δWnc=Qncδwo,
the generalized force Qnc is obtained as (2.14)Qnc=δWncδwo=−c1w˙o−c2w˙o|w˙o|.


## 2.2. Equation of Motion

In the earlier section, the expressions for the
kinetic energyT,
strain energy U, and nonconservative generalized force Qnc were obtained. From these results, it is
possible to derive the system's equation of motion by using the well-known
Lagrange equation [22] which, in turn, for the
system under investigation is written as (2.15)ddt(∂T∂w˙o)−∂T∂wo+∂U∂w˙o=Qnc.Through the computation of
each term of Lagrange's equation and substitution of the result
(2.15), the following result is obtained: (2.16)(1+A12wo2¯)w¨o+(c1mo+c2mo|w˙o|¯)w˙o+(A12w˙o2¯−A1U¨B¯+EIyB1mo)wo+[43A2U¨B¯¯−(6EIyB2mo)¯¯]wo3=F(t)mo.Equation (2.16) represents an ordinary inhomogeneous
nonlinear time-dependent differential equation. In addition, this equation
holds both the axial contraction and the curvature nonlinear effects. If both
the underlined and double underlined terms are ignored, this equation reduces
to a classical linear damped forced model. On the other hand, if only the
double underlined terms are ignored, this equation reduces to the same equation
obtained in [23] plus
a forced term. Still, if the double underlined terms plus the nonlinear damping
were ignored, this equation reduces to the same model obtained in [24] plus a forced term.Since the present work is focused on the dynamic
response of a structure under parametric sinusoidal excitation, it is
considered that this excitation can be written as(2.17)U¨B(t)=−Qλ2cos(λt+φ)=−Qocos(λt+φ) in which Qo is the magnitude of the input base
acceleration,  λ is  the parametric excitation frequency, and φ is a phase shift. Then, (2.16) can be rewritten
in the dimensional final form as (2.18)(1+A12wo2)w¨o+(c1mo+c2mo|w˙o|)w˙o+(A12w˙o2+A1Qocos(λt+φ)+EIyB1mo)wo+[−43A2Qocos(λt+φ)−6EIyB2mo]wo3=0.From the numerical viewpoint, it is interesting to
work with the differential equation in dimensionless form. Therefore, by
setting new dimensionless variableswo∗=A1wo and t∗=t/Tn in which Tn is the period of free vibration, the system's
equation of motion is given by (2.19)(1+wo∗2)w¨o∗+H1w˙o∗+H2w˙o∗|w˙o∗|+[1+w˙o∗2+H3cos(Θt∗+φ)]wo∗−[H5+H4cos(Θt∗+φ)]wo∗3=0 in which (2.20)H1=c1Tnmo,H2=c2moA1,H3=QoA1Tn2,H4=4A2QoTn23A12,H5=6B2EIyTn2moA12,Θ=λTn,Ψ=ΩTn. Further, we will discuss the
effects of variations of the dimensionless quadratic damping coefficient H2 on the response of the parametrically driven
cantilever beam.


## 3. Perturbation Analysis and Numerical Simulations

In order to
address the effects of the quadratic damping (H2) on the structure's response, a solution of
the (2.19) is required. Such a solution is here developed by employing the method
of multiple scales (MME) [19]. For that purpose, it is more convenient to rewrite
(2.19) in a slight different way by considering zero-order (ε0) and first-order (ε1) terms as
follows: (3.1)ε0w¨o∗+ε0(2π)2wo∗=−ε1wo∗2w¨o∗−ε1w˙o∗2wo∗−ε1H1w˙o∗−ε1H2w˙o∗|w˙o∗|−ε1H3cos(Θt∗+φ)wo∗+ε1H4cos(Θt∗+φ)wo∗3+ε1H5wo∗3 in which the dimensionless coefficients ε1H1,ε1H2,…,ε1H5 are defined according to (2.20). To apply the
MME technique, first we express wo∗ as (3.2)wo∗(ε;t∗)=w0(T0,T1)+εw1(T0,T1), where T0=t is the fast time scale associated with changes
occurring at the frequency Θ and T1=ε1t is a slow time scale associated with the
modulations in amplitude and phase.As it is known, the principal parametric resonance
occurs when the parametric excitationΘ assumes a value that is equal to twice the
undamped natural frequency ωn.
Therefore, the normalized undamped natural frequency ω¯n=2π can be written as (3.3)(2π)2=(12Θ)2−εσ in which εσ is a tuning parameter that flags the proximity
of the principal parametric resonance.By carrying out the standard details of the method of
multiple scales, the first approximation to the solution of (3.1) is obtained
as (3.4)wo∗=a(T1)cos[2πT0+β(T1)]+O(ε) in which a(T1) and β(T1) are given by (3.5)a'=(−14ΘεH4a3+12ΘεH3a)sin(2β)−12εH1a−23πεH2Θa2,aβ'=(−12ΘεH4a3+12ΘεH3a)cos(2β)+ε8Θ(−6H5−Θ2)a3−εσaΘ and (3.5) are known as modulation equations. As stated in
[19], steady state
motions correspond to fixed points (constant
solutions) of these modulation equations. Mathematically, this condition is
reached when ai'=βi'=0.
Hence, in steady-state condition the modulation equations are rewritten
as (3.6)(−14ΘεH4a¯3+12ΘεH3a¯)sin(2β)−12εH1a¯−23πεH2Θa¯2=0,(−12ΘεH4a¯3+12ΘεH3a¯)cos(2β)+18Θ(−6εH5−Θ2)a¯3−εσa¯Θ=0, where a¯ describes the steady-state vibration
amplitude. Trivial solutions of this system of equations are immediately
apparent and correspond to the case where a¯=0.
Nontrivial solutions (a¯≠0) are obtained by solving (3.6) in
terms of the amplitude a¯ and phase angle β.
The solution process for these quantities is laborious and can be found in
detail in the work by da Silva [19]. The final expression for a¯ is given as (3.7)a¯=±−2E4E2±2E4E22−4E4E02E4, where the Ei are given as (3.8)E4=−9π2εH4εH3Θ2εσ−54π2εH4εH3εH5εσ+9π2εH42εσ2−1174π2εH42εH32−32Θ4εH4εH3εH22+916π2εH32Θ4+814π2εH32εH52+274π2εH32εH5Θ2,E2=27π2εH4εH33+54π2εH32εH5εσ+16Θ4εH32εH22+9π2εH32Θ2εσ−36π2εH4εH3εσ2,E0=36π2εH32εσ2−9π2εH34, and the phase angle β is written as (3.9)β=12tan−1(G1G2) with coefficients G1 and G2 given as (3.10)G1=8Θ(4εH2ΘεH4a¯3−4εH2ΘεH3a¯),G2=3π(6εH5εH4a¯4+εH4Θ2a¯4+8εH4εσa¯2)+3π(−12εH3εH5a¯2−2εH3Θ2a¯2−16εH3εσ). Hence, the solution for the modulation equations is given as a function
of the parametric excitation frequency Θ,
the frequency tuning parameter εσ,
and the dimensionless coefficients Hi.
Particularly, we are interested in investigating the effects of variations of
the dimensionless quadratic damping coefficient H2 on the amplitude of the response. For that
purpose, a series of simulations were performed by varying this parameter on
the above equations and computing the resulting response. Figure 4 shows the
computed response a¯ from (3.7) for different values of εH2.
It is seen that variations of this parameter do not alter the critical points C and E.
Additionally the nonsymmetric shape of the amplitude-frequency curve tends to
decrease as εH2 increases. It is also noticed that two
characteristics of the response are strongly influenced by variations on the
quadratic damping coefficient, namely, the amplitude of the response and
stability of the nontrivial ramification CG. The maximum value of the amplitude of
response a¯max represents an important information of the
system under investigation. In this sense, it is equally important to assess
the influence of variations of the nonlinear damping coefficient on a¯max.
Figure 5 shows the behavior of this parameter when variations on εH2 are introduced. The results were obtained for
three different values of the amplitude of the parametric excitation,
represented by the dimensionless coefficient εH3.
Figure 5(a) shows that for small values of  εH2(0<εH2<0.1),a¯max is relatively insensitive to variations on εH2.
The largest impact of εH2 on a¯max occurs in the 0.2<εH2<0.5 range, becoming less sensitive as εH2 approaches the end of the range. Figure 5(b)
shows essentially the same trend where the values of the relative reduction of
the amplitude are depicted. As previously pointed out in Figure 4, the
stability of the nontrivial ramification CG is strongly affected by the nonlinear
quadratic damping since it involves the definition of the bifurcation shown in
point G.
This bifurcation is responsible for the jump phenomenon when the values of the
tuning parameter εσ are varied in the ascending order. From the
numerical solution of (3.7) we can identify critical values of εH2 in the response-frequency curve that will make
the bifurcation disappear. These critical values can be found from the
following expression: (3.11)εH2crit=−0.012εH32+0.14εH3+0.14 and they are represented in Figure 6. Hence, from this equation it is
possible to estimate values for the εH2 for a given known excitation condition (εH3) that would make the jump phenomenon to completely
disappear from the system's response. Figure 7 shows the effects of variations of
the εH2 in the system's response when the excitation
amplitude is varied. It can be seen that the critical point C is not affected by the different values of εH2.
On the other hand, the vibration amplitude and the stability of the nontrivial
ramification CF are strongly affected when varying the values
of the quadratic damping coefficient. It is also interesting to observe in
Figure 7 the magnitude of the response at D and F,
here referred to as a¯D and a¯F,
respectively. The value of a¯D indicates the minimum value of the vibration
amplitude as soon as the parametric resonance occurs, or, if the critical point C is exceeded. The value of a¯F reflects the minimum value of the amplitude at
the moment that the principal parametric resonance condition ceases. The
variation of a¯D and a¯F with respect to εH2 is shown in Figure 8.Figure 4
Typical response-frequency curve showing the
effects of varying the quadratic dimensionless termH2 and εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH3=2.1352, εH4=3.4674, εH5=62.7218 [19].Influence of
quadratic damping variation on the nontrivial response solutiona¯:
(a) variation of the maximum value of a¯ with the nonlinear damping and (b) percent
reduction of the maximum value of a¯ for vacuum operation. Curves obtained for (—) εH1=0.0, εH3=2.62,εH4=4.25,εH5=62.72;
(⋯) εH1=0.0,εH3=2.13,εH4=3.46,εH5=62.72;
(– –) εH1=0.0,εH3=1.57,εH4=2.55,εH5=62.72 [19].
(a)(b)Figure 6
Critical values ofεH2 as a function of εH3 [19].Figure 7
Typical response-amplitude curve
demonstrating the effects of varying the quadratic nonlinear damping term (εH2) for εσ=−0.73.
Also εH1=0.0 (nontrivial), εH1=0.016 (trivial), and εH5=62.7218 [19].Figure 8
Typical plots showing the effects of the
nonlinear quadratic damping on pointsD and F.
Results obtained with εσ=−0.73,εH1=0.0 (nontrivial), εH1=0.016 (trivial), and εH5=62.7218 [19].


### 3.1. Response-Nonlinear Damping Curves

In the previous
analysis, a series of numerically simulated results have shown in detail how
the nonlinear quadratic damping affects the response of the cantilever beam
under parametric excitation. In this section, we continue to explore these
effects from the numerical standpoint by defining the response-nonlinear
damping curves. This curve is obtained from a specific vibration condition
imposed to the structure under test by the excitation mechanism, and Figure9
defines three distinct operating regions (marked as I, II, and III) that differ
essentially in terms of the excitation frequency imposed to the system as well
as resulting vibration amplitude. Two operation  points P1 and P2 are chosen in regions I and II, respectively,
with corresponding amplitudes given by a¯P1 and a¯P2. Figure 10 shows how variations on εH2 affect the amplitude a¯P1.
Two ramifications form this curve, one stable (solid line) and one unstable
(dashed line). In case εH2 is decreased to the value εH2=0, the structure still remains vibrating but with
an amplitude approximately 4% larger. Similarly, if εH2 increases beyond point P1, the vibration amplitude decreases on the
stable nontrivial ramification until point C is reached. At this point, a¯P1 is reduced by 19%.
If εH2 exceeds point C,
the nontrivial stable solution looses stability through a bifurcation and the
vibration is extinct. When the structure vibrates according to
point P2, the magnitude of the response is given by a¯P2. Figure 11 shows the effects on the system's response a¯P2 produced by varying the values of εH2.
In this case, a single stable ramification is observed, and independently on
how εH2 is varied the response continues being
nontrivial and stable. On the other hand, there is a strong reduction in the
value of a¯P2.Figure 9
Typical response-frequency curve showing two
specific operation points, designated asP1 and P2.
Results obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH2=0.1,εH3=2.1352,εH4=3.4674,εH5=62.7218 [19].Figure 10
Typical
response-quadratic damping curve showing the effects of variations ofεH2 on the magnitude of the response. Results
obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH3=2.1352,εH4=3.4674,εH5=62.7218, and εσ=−2 [19].Figure 11
Typical response-quadratic damping curve
showing the effects of variations ofεH2 on the magnitude of the response. Results
obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH3=2.1352, εH4=3.4674, εH5=62.7218, and εσ=0 [19].


## 3.1. Response-Nonlinear Damping Curves

In the previous
analysis, a series of numerically simulated results have shown in detail how
the nonlinear quadratic damping affects the response of the cantilever beam
under parametric excitation. In this section, we continue to explore these
effects from the numerical standpoint by defining the response-nonlinear
damping curves. This curve is obtained from a specific vibration condition
imposed to the structure under test by the excitation mechanism, and Figure9
defines three distinct operating regions (marked as I, II, and III) that differ
essentially in terms of the excitation frequency imposed to the system as well
as resulting vibration amplitude. Two operation  points P1 and P2 are chosen in regions I and II, respectively,
with corresponding amplitudes given by a¯P1 and a¯P2. Figure 10 shows how variations on εH2 affect the amplitude a¯P1.
Two ramifications form this curve, one stable (solid line) and one unstable
(dashed line). In case εH2 is decreased to the value εH2=0, the structure still remains vibrating but with
an amplitude approximately 4% larger. Similarly, if εH2 increases beyond point P1, the vibration amplitude decreases on the
stable nontrivial ramification until point C is reached. At this point, a¯P1 is reduced by 19%.
If εH2 exceeds point C,
the nontrivial stable solution looses stability through a bifurcation and the
vibration is extinct. When the structure vibrates according to
point P2, the magnitude of the response is given by a¯P2. Figure 11 shows the effects on the system's response a¯P2 produced by varying the values of εH2.
In this case, a single stable ramification is observed, and independently on
how εH2 is varied the response continues being
nontrivial and stable. On the other hand, there is a strong reduction in the
value of a¯P2.Figure 9
Typical response-frequency curve showing two
specific operation points, designated asP1 and P2.
Results obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH2=0.1,εH3=2.1352,εH4=3.4674,εH5=62.7218 [19].Figure 10
Typical
response-quadratic damping curve showing the effects of variations ofεH2 on the magnitude of the response. Results
obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH3=2.1352,εH4=3.4674,εH5=62.7218, and εσ=−2 [19].Figure 11
Typical response-quadratic damping curve
showing the effects of variations ofεH2 on the magnitude of the response. Results
obtained with εH1=0.0 (nontrivial), εH1=0.016 (trivial), εH3=2.1352, εH4=3.4674, εH5=62.7218, and εσ=0 [19].


## 4. Experimental Analysis

This section
describes an experimental analysis that was performed on the structure shown in
Figure1. Initially some basic properties of the cantilever beam-mass system
such as first bending damped natural frequency and modal damping ratio were
obtained by standard modal testing procedures. In this case, the step
relaxation method was employed to excite the system in order to get the driving
point frequency response function (FRF) at the beam's end point [25]. The resulting values found
for the first bending damped natural frequency and viscous modal damping ratio
were ωd=18.066Hz and ζm=0.1272%,
respectively. These results were used to correlate the experimental results
with the analytical prediction as well as in the planning of nonlinear tests. A
detailed explanation of this procedure can be found in [19, 23].Once these basic linear characteristics were found,
the system of Figure1 was subjected to a base driven test according to the
experimental setup shown in Figure 12. The beam carrying the lumped mass at one
end is first attached at the opposite end to a steel block in order to properly
simulate the fixed end boundary condition. This assembly is then mounted on the
vibrating table of a B&K type 4810 electrodynamic vibration exciter that
will drive the system in the vertical direction. The excitation signal is
provided by the HP Agilent E1432A data acquisition board that is controlled by
the MTS I-Deas 10 modal testing software. The sinusoidal input signal was first
sent to a B&K power amplifier type 2707 and further sent to a B&K type
4810 electrodynamic vibration exciter. The beam's transverse output signal was
monitoring using an HP Agilent oscilloscope type 54621D. To minimize the
rocking and translation, the shaker was clamped securely to the floor in the
testing room. The sensing mechanism employed is three
piezoelectric accelerometers, two for monitoring the base's motion and one for
the beam-mass-system's motion. The base's linear translation motion was
measured using an accelerometer B&K model 4371 (9.84pC/g), and possible rocking motion about Y axis
was measured using a Kistler angular accelerometer model 8836M01 (34μV/rad/s2). The beam-mass-system's translation motion
was measured using an accelerometer B&K model 4374 (1.06pC/g) and mass of 0.64g.Figure 12
Experimental layout employed to obtain the experimental
frequency-response and amplitude-response curves [19–21].In order to perform a coherent comparative analysis
with the theoretical results, four experimental tests were conducted. In two of
these tests, the amplitude of the base excitation acceleration was maintained
constant at39.24m/s2 and the sinusoidal excitation signal was
slowly varied upward and downward in the frequency range of interest. The
remaining tests were performed by keeping the excitation frequency constant at 36.132Hz,
and then increasing the amplitude of the input excitation signal in the power
amplifier. It should be noticed that the excitation frequency of 36.132Hz corresponds to the so-called principal
parametric excitation frequency since it is approximately twice the system's
first bending natural frequency thus satisfying the 2 : 1 relationship that is required to drive such a
system into a principal parametric resonance condition [26].The process of obtaining the experimental
frequency-response curve consists of varying the excitation frequency while
keeping the magnitude of the input base acceleration constant. The input base
acceleration was maintained constant during the tests. A similar process was
used to obtain the amplitude-response curves, however the frequency of
excitation was kept constant and the amplitude was changed in small increments.Figure13 shows the experimental and theoretical
frequency-response curves for the first bending mode. The theoretical curve was
obtained in the absence of quadratic damping (i.e., εH2=0.00), while the experimental was obtained in
atmospheric conditions.Figure 13
Experimental and theoretical response-frequency curves
of the first flexural mode whenεH2=0.00 and for εH1=0.0160 (trivial), εH1=0.0160 (nontrivial), εH3=2.1352;εH4=3.4674;εH5=62.7218.
Experimental results for parametric excitation frequency at 36.132Hz and 39.24m/s2.
(—) stable
solution; (⋯) unstable solution [20].The theoretical results in the upward direction show
that by increasing the excitation frequency starting at pointA,
the trivial solution loses stability at point C,
which corresponds to the critical value εσ=−εH3/2,
through a subcritical pitchfork bifurcation and jumps up to point D.
This point belongs to the nontrivial stable branch GDE.
From point D,
the steady-state amplitude of parametric response a˜ decreases as the excitation frequency is
increased, until point E is reached. From this point, the nontrivial
solution loses stability through a supercritical pitchfork bifurcation and the
trivial solution is reached again.On the other hand, theoretical results in the downward
direction show that decreasing the frequency of excitation form pointF,
the trivial solution looses stability at point E,
corresponding to the critical value εσ=εH3/2,
through a supercritical pitchfork bifurcation, and the nontrivial stable branch GDE is reached. From the point E,
the response amplitude a˜ increases as the frequency is decreased. The
solution loses stability though a turning point at point G and the response amplitude jumps down to point B where only the trivial solution exists
thereafter.When the theoretical behavior described above is
confronted with the experimental results, it can be claimed that a good
qualitative match between theory and experiment exists, mainly in the vicinity
of the bifurcation frequencies. However, quantitatively there is an enormous
difference between the amplitudes of theoretical and experimental
results. The maximum theoretical value for the response amplitude
is abouta˜≈0.225,
while the maximum experimental value is about a˜≈0.1,
that is, about 125% smaller. This difference suggests the
existence of important dissipative forces acting on the structure.From this last result, it can be seen that by
introducing the nonlinear quadratic damping a strong reduction of response amplitude
has been achieved. Particularly, by usingεH2=0.333,
an excellent agreement between the experimental and theoretical results was
obtained. However, a slight discrepancy was observed in terms of the
bifurcation points mainly in the upward sweep. Consequently, there is no
exact agreement around point C.
An analogous result was obtained for bifurcation point E.Figure14 shows the experimental and theoretical
amplitude-response curves also for the first bending mode. Since an exact
resonance condition (εσ=0) is difficult to be achieved, the theoretical
curve was obtained for εσ<0.
In addition, the result shown was obtained in the absence of quadratic damping
(i.e., εH2=0.00) while the experimental result was obtained
in atmospheric conditions.Figure 14
Experimental
and theoretical amplitude-response curves of the first flexural mode whenεH2=0.00 and for εH1=0.0160 (trivial), εH1=0.0160 (nontrivial), εH3=2.2236;εH4=3.6109;εH5=62.7218 and εσ=−0.725.
Experimental results for parametric excitation frequency at 36.132Hz.
(—) stable solution; (⋯) unstable solution [20].The theoretical results in the upward direction shows
that by increasing the excitation amplitude from pointA,
the trivial solution loses stability at point C,
corresponding to the critical value εH3=4εσ2+εH12Θ2, through a subcritical pitchfork bifurcation and jumps up to point D.
This point belongs to the nontrivial stable branch FDE.
A further increase in the excitation amplitude leads to higher response
amplitudes tracing the branch DE.On the other hand, the theoretical results in the
downward direction shows that by decreasing the excitation amplitude from pointE,
the amplitude response continually decreases until point F is reached. Then, the nontrivial solution
loses stability through a turning point bifurcation, leading to a jump down to
point B whereby only the trivial solution exists
thereafter.When the theoretical behavior described above is
confronted with the experimental results, a very good qualitative as well as
quantitative disagreement between them exists. These two characteristics also
suggest the existence of important dissipative force acting on the structure.
In order to prove this point, an additional theoretical amplitude-response
curve was obtained, but now, including the nonlinear quadratic damping effects.
The results are shown in Figure15.Figure 15
Experimental and theoretical amplitude-response curves
of the first flexural mode whenεH2=0.227 and for εH1=0.0160 (trivial), εH1=0.00 (nontrivial), εH3=2.2236,εH4=3.6109,εH5=62.7218, and εσ=−0.725.
Experimental results for parametric excitation frequency at 36.132Hz.
(—) stable solution; (⋯) unstable solution [20].In this last figure, it can be claimed that by
introducing the nonlinear quadratic damping a much better match between theory
and experiment was obtained, specially whenεH2=0.227 is used. However, some discrepancies related
to response amplitude, as well as with the bifurcation point F, were observed. A close view in region between 30m/s2 and 50m/s2 revealed 20% difference in the amplitude response. On the
other hand, this discrepancy tends to decrease with the increase of the
excitation amplitude. Also, there is a discrepancy related with the bifurcation
point F.
In the theoretical prediction, it occurs about 35m/s2,
whereas in the experimental results it is shown at 30m/s2 showing a 17% difference.


## 5. Concluding Remarks

This article
addressed numerically and experimentally the effects of viscous fluid medium on
the dynamic response of a cantilever beam carrying a lumped mass. Numerically
simulated results showed the effects of variations induced in the nonlinear
damping on the acceleration response of the test structure when it undergoes a
principal parametric resonance condition. Experimental assessment on the
effects of quadratic damping due to frictional turbulent force on the
structure's dynamic response has been obtained. Generally speaking, good
agreement between experimental and numerically simulated results was achieved
in terms of frequency and amplitude response curves. It was observed that the
quadratic damping due to frictional turbulent force plays an important role in
the response of parametrically excited cantilever beam carrying a lumped mass.
The inclusion of the quadratic damping significantly improves the theoretical
predictions, and it should be included in the mathematical models when the
problem involves the principal parametric response. Although the results shown
in this paper were obtained for the first bending mode, similar conclusions may
be obtained for higher natural frequencies.



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*Source: 185351-2008-08-04.xml*