Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

57, 1, pp. 263-272, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.263

INFLUENCE OF THE DAMPING EFFECT ON THE DYNAMIC

RESPONSE OF A PLATE

Łukasz Borkowski

Lodz University of Technology, Department of Strength of Materials, Łódź, Poland

e-mail: lukasz.borkowski@p.lodz.pl

The subject of the research is analysis of the influence of the damping effect on the dynamic
response of a plate. During the tests, the areas of dynamic stability and instability for the
plate with and without damping are compared. Besides, exact analysis of the nature of
the solution by applying criteria such as phase portraits, Poincarémaps, FFT analysis, the
largest Lyapunov exponents are carried out and found.

Key words: damping effect, dynamic stability, dynamic response, phase portraits, Poincaré
maps

1. Introduction

Thebeginnings of studies concerning the dynamic stability of plates canbe found inpublications
from the middle of the twentieth century. The first publication regarding dynamic stability of
plateswaspresentedbyZizicas (1952). In thatpaper, theoretical solutions for the joint supported
plate with a time-dependent load were reported. Subsequent years of research led to creation of
dynamic stability criteria which were divided into: geometric (Cooley and Tukey, 1965), energy
(Raftoyiannis and Kounadis, 2000) and failure ones (Petry and Fahlbusch, 2000).

One of the major criterion was a Budiansky-Hutchinson criterion (Hutchinson and Budian-
sky, 1966) which concerned rods and cylindrical shells with an axial load. They analyzed the
load in the formof a pulse of a finite and infinite duration. They proved that the loss of stability
of dynamically loaded constructions occurs when small load increments cause a rapid increase
of deflection. Budianskywas one of the authors of a similar criterion regarding cylindrical shells
with a transverse load – the Budiansky-Roth criterion (Budiansky and Roth, 1962). This cri-
terion was willingly used in research of other scientists who were involved in the similar topics
(Shariyat, 2007; Kubiak, 2007; Zhang et al., 2004).

Another important criterion is the Petry-Fahlbusch criterion (Petry and Fahlbusch, 2000).
The researchers said that the analysis of the stress state should determine the dynamic critical
load for the constructionwith a stable post-critical equilibriumpath. Based on such an analysis,
it is possible to determine the load for which destruction of the structure takes place. According
to Petry-Fahlbusch’s theory, if the condition – the reduced stress is smaller or equal to the
boundary stress – is fulfilled at any time and at any point of the studied structure, then a
dynamic response of the construction under the pulse load is dynamically stable.

The next important criterion is the Volmir criterion (Volmir, 1972). He analyzed pulses of a
finite duration: a rectangular pulse and an exponentially decreasing pulse, pulses of an infinite
duration and a linearly increasing load. He studied pulses that caused both compression and
shear. Using the Bubnov-Galerkin (Michlin and Smolnicki, 1970) and Runge-Kutta (Collatz,
2012; Fortuna et al., 2005) methods, he said that the loss of stability of pulse loaded plates
occurs when the maximum deflection of the plates is equal to their thickness or half thickness.



264 Ł. Borkowski

Ari-Gur and Simonetta (1977) proposed four criteria for the loss of stability. They described
the critical load depending on the following parameters: themeasured deflection in themiddle of
lengthandwidthof theplate and the intensity of load forplates fixedat all edges and loadedwith
a pulse of half-wave shaped (a pulse of finite duration). The first concerns the value of deflection
and the intensity of the load pulse – if a slight increase in the load pulse intensity causes a
significant increase in the value of deflection then dynamic buckling takes place. According to
the second criterion – if a slight increase in the amplitude of the load pulse causes a decrease
in the value of deflection then dynamic buckling happens. The next two criteria are failure
criteria which are based on the response analysis of the loaded edge of a plate. According to the
third criterion – if a small increase in the force pulse amplitude causes a sudden increase in the
shortening value of the loaded edge of the plate then dynamic buckling occurs. According to the
fourth criterion – if a small increase in the pulse intensity of displacement of the loaded edge
causes a change in the reaction sign on the plate edge then dynamic buckling takes place.

The behavior of rod systems was analyzed by the finite element method by Kleiber et al.
(1987). They formulated a quasi-bifurcation criterion of dynamic stability for a construction
under a jump loaded (Heaviside pulse) by using properties of a tangent stiffness matrix in the
point of bifurcation. According to this criterion, the structure loses stability and the deflection
begins to grow boundlessly when the determinant of the tangent stiffness matrix is equal to
zero and the absolute value of the smallest eigenvalue is greater than the absolute value of the
nearest maximum reached by the smallest eigenvalue.

All the above criteria are widely used in the research of many scientists who deal with the
analysis of dynamic stability (Bolotin 1972; Hsu andForman, 1975; Kołakowski, 2007; Kołakow-
ski andKubiak, 2007; Kowal-Michalska, 2010; Kubiak et al., 2010;Mania andKowal-Michalska,
2007; Moorthy et al., 1990; Wu and Shih, 2006).

However, the analysis of plate structures with application of dynamic criteria such as phase
portraits, Poincaré maps, FFT analysis, the largest Lyapunov exponents is less used (Alijani
et al., 2011a,b; Gilat and Aboudi, 2000; Touati and Cederbaum, 1995; Wang et al., 2010; Yeh
and Lai, 2002; Yuda and Zhiqiang, 2011). Therefore, this paper presents the influence of the
damping effect on the dynamic response of the plate using the above tools.

2. Studied plate

A square isotropic plate with dimensions b = l = 100mm, h = 1mm and material constants
E = 200GPa, ν = 0.3 is analyzed (Fig. 1). The analyzed plate is simply supported on the all
edges. The plate is loaded with a dynamic compressive load. The dynamic load means the load
that has been introduced suddenly and lasts for an infinitely long time.

Fig. 1. Studied plate



Influence of the damping effect on the dynamic response of a plate 265

2.1. The plate without damping

According to the researchbyVolmir (1972), the above plate canbedescribedby the following
equation

ζ̈ +ω20

(

1−
σ∗
x

σ∗
cr

)

ζ +ηζ3 =0 (2.1)

After transformations, the test plate without damping can be described using the equation

ζ̈ +Ω20(1−kcosθt)ζ +ηζ
3 =0 (2.2)

where

k =
σ∗
t
/σ∗
cr

1−σ∗0/σ
∗

cr

Ω20 = ω
2
0

(

1−
σ∗0
σ∗
cr

)

and ζ is the deflection of the plate, ω0 – natural frequency, σ
∗

cr
– critical stress, σ∗0 – medium

stress,σ∗
t
– stress amplitude,η –parameterwhosevalue isdependenton theboundaryconditions.

Transform now equation (2.2) into a dimensionless form

ẍ+a(1−kcosψτ)x+ bx3 =0 (2.3)

where

a =1−
σ∗0
σ∗
cr

b =
η

ω20
ẍ =

1

ω20
ζ̈ x = ζ x3 = ζ3 ψ =

θ

ω0

and τ is the dimensionless time. For the studied plate supported on all edges, the values of
parameters are: ω0 =3014.3rad/s, η =0.23rad/s

2, σ∗
cr
=72.3MPa. For the purpose of further

numerical analysis, equation (2.3) is replaced by two first-order differential equations

ẋ1 = x2 ẋ2 =−a(1−kcosψτ)x1− bx
3
1 (2.4)

2.2. The plate with damping

Introducing damping into equation (2.2) and transforming into a dimensionless form, one
obtains

ẍ+cẋ+a(1−kcosψτ)x+ bx3 =0 (2.5)

where: c = 2h/ω0 – the dimensionless damping ratio, h = 0.02 (Kołakowski and Teter, 2013),
the other parameters are the same as for the plate without damping.
Writing equation (2.5) in the form of two first-order differential equations, we get

ẋ1 = x2 ẋ2 =−cx2−a(1−kcosψτ)x1− bx
3
1 (2.6)

All studies are carried out for the following initial conditions: x1 =0.01, x2 =0.

3. Numerical analysis of the plate

Figure 2 shows the areas of dynamic stability and instability (circled areas) for the plate with-
out (a) and with (b) the damping effect after earlier presentation of the full compliance of the
results presented byVolmir (1972) and the results obtainedwith the dynamic tools for the plate
without the damping effect (Borkowski, 2017).



266 Ł. Borkowski

Fig. 2. Graphs of dynamic stability and instability areas for the plate without (a) and with (b) the
damping effect

Both graphs in k − ψ/2Ω coordinates (ψ = θ/ω0,Ω = Ω0/ω0) by changing values of the
parameters σ0 and σt have been made. Calculations of the parameters k and ψ/2Ω changing
every 0.01 were executed. Figure 2 has been obtained by using the criteria of phase portraits,
Poincaré maps and FFT analysis.

Analyzing both charts, it can be concluded that there are larger areas of dynamic instability
for the plate without damping as against the plate with damping. In addition, small dynamic
stability areaswithin the dynamic instability range in both cases are observed (Figs. 3a and 3b).

Fig. 3. Detailed graphs of dynamic stability and instability areas for the plate without (a) and with (b)
the damping effect

For the plate without damping, the dynamic stability area is represented by a quasi-periodic
solution. In the range of dynamic instability, both quasi-periodic as well as chaotic solutions can
be specified.



Influence of the damping effect on the dynamic response of a plate 267

For the plate with damping, in the dynamic stability area the trajectory is heading to the
critical point. In the range of dynamic instability, the periodic solutions as well as the series of
period-doubling bifurcations, which lead to a chaotic response, are obtained.

Therefore, for thepurposeof amoredetailed analysis andpresentation of the above solutions,
the criterion of the largest Lyapunov exponents has been used.

Figure 4 shows the areas of the chaotic solution (gray areas) for the plate without (a) and
with (b) damping. The dashed lines indicates the boundary for the dynamic stability/instability
areas which corresponds to the circled part in Fig. 2. Comparing the two graphs, it can be
clearly stated that the introduction of damping to the analyzed plate results in obtainingmuch
smaller areas of dynamic instability with a chaotic solution. Figure 5 presents amagnification of
Figs. 4a and 4b. Gray dots correspond to specific values and gray lines to ranges for the chaotic
solution.

Fig. 4. Graphs of areas representing the chaotic solution for the plate without (a) and with (b) the
damping effect

Fig. 5. The detailed graphs of areas representing the chaotic solution for the plate without (a) and
with (b) the damping effect



268 Ł. Borkowski

In order to present the solutions more clearly, three points from Fig. 2 for individual ran-
ges have been selected. These points represent solutions from the area of dynamic stability –
(k = 0.50, ψ/2Ω = 0.30), from the area of dynamic instability with a periodic/quasi-periodic
solution – (k = 0.25, ψ/2Ω = 1.00) and from the area of dynamic instability with a chaotic
solution – (k =1.50, ψ/2Ω =0.30).
Analyzing the obtained results and using the criteria of phase portraits as well as Poincaré

maps, it can be concluded that the loss of dynamic stability is associatedwith a sudden increase
in the displacement x1 and velocity x2 (Figs. 6d, 6f, 6g, 6i) when compared to the dynamic
stability areas (Figs. 6a, 6c). According to the research presented in (Bazant andCedolin, 2010),
the loss of stability is related to displacement of the phase trajectory into infinity. This is the
case when the analysis time corresponds to the period of natural vibration of a construction.
In order to use dynamic tools such as phase portraits or Poincaré maps, the presented research
concerns the analysis duration many times greater than the period of natural vibration. For a
long duration, the phase trajectory does not move into infinity. It achieves some limit values of
the displacement x1 and velocity x2, the value of which depends on the parameter k. However,
applying the criterion of phase portraits and analyzing the plate for both short and long analysis
duration, the same results are obtained.

Fig. 6. The plate without damping – phase portraits (a), (d), (g), FFT analysis (b), (e), (h) and
Poincarémaps (c), (f), (i) for the areas of dynamic stability (a), (b), (c), dynamic instability –
quasi-periodic solution (d), (e), (f) and dynamic instability – chaotic solution (g), (h), (i)

Applying FFT analysis, it can be concluded that it is possible to precisely determine domi-
nant frequencies in the stability range (Fig. 6b).Also in the instability rangewithaquasi-periodic
solution, dominant frequencies can be specified (Fig. 6e). In both cases, the appearance of two



Influence of the damping effect on the dynamic response of a plate 269

disproportionate to each other frequencies can be observed. The so-called two-dimensional torus
(2D torus) is created. In both Fig. 6b and Fig. 6e, the largest Lyapunov exponents are appro-
ximately equal to zero (λ1 =0.000002, λ2 =−0.000002 – for the point k =0.50, ψ/2Ω =0.30;
λ1 =0.000004, λ2 =−0.000004 – for the point k =0.25, ψ/2Ω =1.00). It should be noted that
the two zero Lyapunov exponents for the stability area are the result of the absence of damping
in the system (2.3). As a consequence, there is no attractor (attractors) to which the trajectory
would converge.
In the instability range with a chaotic solution (Fig. 6h), the frequency spectrum is con-

tinuous. It is not possible to specify the dominant frequencies. The amplitude of the tested
signal increases significantly, which is expressed in decibels. The value of the largest Lyapunov
exponent is positive (λ1 =0.043531, λ2 =−0.043531).
Similarly to the plate without damping, the loss of stability for the plate with damping is

associated with a sudden increase in the displacement x1 and velocity x2 (Figs. 7d, 7f, 7g, 7i)
when compared to the dynamic stability areas (Fig. 7a). In the stability area – as a result of
the introduced damping – the trajectory goes to the critical point (Fig. 7a). The Lyapunov
exponents are negative (λ1 =−0.019993, λ2 =−0.020007) and there is no solution in the FFT
graphs (Fig. 7b) as well as Poincaré maps (Fig. 7c).

Fig. 7. The plate with damping – phase portraits (a), (d), (g), FFT analysis (b), (e), (h) and Poincaré
maps (c), (f), (i) for the areas of dynamic stability (a), (b), (c), dynamic instability – periodic

solution (d), (e), (f) and dynamic instability – chaotic solution (g), (h), (i)

In the areas of dynamic instability, a periodic solution has been obtained (Figs. 7d, 7e, 7f).
Together with the series of period-doubling bifurcations, it leads to a chaotic solution (Figs. 7g,
7h, 7i). Figures 7d, 7e, 7f show a solution with a period equal to 2. The FFT analysis (Fig. 7e)



270 Ł. Borkowski

enablesprecise representationof thedominant frequencies.TheLyapunovexponents arenegative
(λ1 =−0.020000, λ2 =−0.020000).

Similarly to the plate without the damping effect, in the areas of dynamic instability with
a chaotic solution, the frequency spectrum is continuous, and it is impossible to distinguish the
dominant frequencies (Fig. 7h). The amplitude of the signal also increases. The value of the
largest Lyapunov exponent is positive (λ1 =0.043397, λ2 =−0.043397).

4. Summary

The subject of the research is to present the influence of the damping effect on the dynamic
response for an isotropic plate. The areas of dynamic stability and instability for the plate with
and without damping are compared. Additionally, using the criteria such as phase portraits,
Poincaré maps, FFT analysis, the largest Lyapunov exponents, the nature of the solution of the
analyzed plate has been presented.

After the tests, it can be concluded that the impact of damping causes changes in the
instability areas of the studied structure. In addition, the introduction of damping to the system
results in a significant difference in the occurrence of areas in which the solution is chaotic.

For the plate without damping, a quasi-periodic solution in the dynamic stability areas
has been observed. The occurrence of two disproportionate to each other frequencies as well
as formation of a 2D torus have been proved. Whereas, both the quasi-periodic as well as the
chaotic solution in the instability range have been specified.

For the plate with the damping effect in the area of dynamic stability, the phase trajectory
is going to the critical point. In the range of dynamic instability, the periodic solutions as well
as the series of period-doubling bifurcations, which lead to the chaotic response, have been
obtained.

In both analyzed cases (for the platewithout andwith damping), the loss of dynamic stabili-
ty is associated with a significant increase in the displacement x1 and velocity x2 in comparison
to the dynamic stability areas – the criteria of phase portraits and Poincaré maps. Using the
FFT analysis, the loss of dynamic stability results in inability to precisely specify the dominant
frequencies in the spectral signal (what is possible in the areas of dynamic stability), and a
significant increase in their amplitude is found. Implementing the criterion of the largest Lyapu-
nov exponents, it is possible to clearly present significant differences between the areas with a
chaotic solution for plates without and with the damping effect.

Conflict of interest

The author declares that there is no conflict of interest concerning publication of this article.

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Manuscript received July 30, 2018; accepted for print October 23, 2018