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Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 18
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular
Plate
Ekpruke, E.O*, Ossia, C.V and Big-Alabo, A
Applied Mechanics & Design (AMD) Research Group, Department of Mechanical Engineering, University of Port Harcourt,
Port Harcourt, Nigeria. *Corresponding author’s email: [email protected]
Abstract
The existing studies on flexible plate response to spherical impact have focussed on an initially-stationary
plate i.e. a plate that is at rest prior to impact. This study applied theoretical models to analyze the
response of an initially-vibrating rectangular plate to the impact of a rigid spherical impactor. The
classical plate theory was used to model the small-amplitude vibrations for the initially-vibrating plate.
Thereafter, the pre-impact vibration response was integrated into the infinite plate impact model to
simulate the effect of pre-impact vibrations on the impact response. The results showed that the impact
response of the initially-vibrating plate is considerably different from the corresponding impact response
of an initially-stationary plate. Parametric analysis conducted on different pertinent impact parameters
showed that the plate thickness, impactor mass, impactor velocity and size of the impactor all have
significant influence on the impact response of the initially-vibrating plate. This paper represents a
paradigm shift in the study of rigid-flexible impact and has potential for more realistic impact analysis.
Keywords: Pre-impact vibrations, Low-velocity impact, Hertz contact law, Rigid-flexible impact,
Spherical impact
Received: 6th April, 2020 Accepted: 23
rd April, 2020
1. Introduction
A major reason for studying impact-related
problems is attributable to the effect of impact-
generated forces, which often leads to structural
damage. Damage in metallic plates can result in
deformity and wear in the region of contact
(Johnson, 1985), and for composite plates, it may
involve fiber cracking and delamination, which
often reduces structural stiffness (Zukas, 1982).
Another reason for impact studies lies in the
necessity for impact damage mitigation through
design of more responsive structural materials and
forms. Knowledge of structural response to impact
loading can aid design of structures that can better
resist impact damage. Some mitigation strategies
against structural damage arising from impact
loading have been suggested (Yigit and
Christoforus, 2000; Saravanos and Christoforou,
2002; Khalili et al., 2007; Angioni et al., 2011;
Grunenfelder et al., 2014; Big-Alabo, 2015) but
these strategies are still being developed because of
lack of understanding and accurate prediction of
the complex dynamics of rigid-flexible impact
response.
Rigid-flexible impact can be investigated
experimentally, theoretically and using finite
element analysis (Abrate, 1998). Theoretical
investigations are usually conducted using one of
the following modeling approaches (Abrate, 1998):
(i) spring-mass model (ii) energy balance model
(iii) infinite plate model and (iv) the complete
model. The spring-mass and energy balance
modeling approaches are normally used for large-
mass impact, where the mass of the rigid impactor
is at least twice the mass of the plate (Olsson
2000). For instance, investigations on the response
of low-velocity large-mass impact were conducted
by Khalili et al. (2007) using a two degree-of-
freedom (DOF) spring-mass model, while the
energy balance approach was applied by Foo et al.
(2011) to predict the low-velocity impact response
of sandwich composite plates. On the other hand,
the infinite plate modeling approach is applicable
to analysis of small-mass impacts where the
boundary conditions of the plate have no effect on
Uniport Journal of Engineering and Scientific Research (UJESR)
Vol. 5, Special Issue, 2020, Page 18-27 ISSN: 2616-1192
© Faculty of Engineering, University of Port Harcourt, Nigeria.
(www.ujesr.org)
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 19
the impact response. This model was first proposed
by Zener (1941) and has been applied to investigate
the small-mass impact response of composite plates
subjected to low-velocity spherical impact (Olsson,
1992; Zheng and Binienda, 2007). Finally, the
complete modeling approach is applicable when
the boundary conditions of the plate influence its
impact response. This happens when the impact
duration is significantly longer than the time it
takes for the vibration waves to hit and rebound
from the boundary. According to Olsson (2000),
the complete modeling approach should be used for
an intermediate-mass impact where the impactor-
plate mass ratio is between one-fifth and two. The
complete modeling approach takes into account the
displacement of the impactor, the vibrations of the
plate and the local contact mechanics of the plate-
impactor interaction, and it is the most
comprehensive theoretical modeling approach
(Big-Alabo, 2015). However, its implementation is
computationally expensive (Nosier et al., 1994;
Big-Alabo, 2015) and therefore some
simplifications are usually introduced to reduce the
computational effort e.g. use of a linearized contact
model (Christoforou and Swanson, 1991). The
present study is limited to low-velocity small-mass
impact. Hence, the infinite plate modeling
approach was applied.
Many studies have been conducted on the
impact response of plates to spherical impact. An
early study involving the impact of a steel sphere
on a steel plate was conducted by Karas (1939) and
still remains a benchmark reference in the literature
on rigid-flexible impact. Goldsmith (1960)
reviewed the Karas problem while Abrate (1998)
and Big-Alabo et al. (2015a) used Karas’ solution
to verify their solutions for sphere-plate impact. In
the last three decades, studies on sphere-plate
impact have concentrated on composites plates
(Christoforou and Swanson, 1991; Christoforou
and Yigit, 1995&1998; Abrate, 1998,2001&2011;
Chai and Zhu, 2011; Park, 2017) because
composites have a high strength to weight ratio and
permit a wide variety of material properties
including active properties for structural response
control (Saravanos and Christoforou, 2002; Khalili
et al., 2007). In addition, studies on the impact
response of functionally graded plates have been
reported (Shariyat and Farzan, 2013; Shariyat and
Jafari, 2013).
A common feature of all previous studies on
sphere-plate impact, irrespective of type of plate
material and geometry considered, is that the plate
is assumed to be initially at rest prior to impact.
Experience shows that many plate-like structures
are already experiencing vibration prior to impact.
For instance, the body of a vehicle in motion
experiences vibration due to excitations from its
engine and can be impacted by a rigid object. Also,
hailstone impact or bird strike on flying aircraft can
occur while the aircraft body is already
experiencing vibration. Therefore, it is necessary to
study the response of initially-vibrating plates to
rigid body impact as this represents a more realistic
scenario for some plate-like structures that are
exposed to accidental rigid body impact.
This paper presents a theoretical analysis of the
impact response of an initially-vibrating plate
subjected to low-velocity impact of a rigid
spherical mass. Since the plate is vibrating prior to
impact, it can be said that it is experiencing ‘pre-
impact’ vibrations. The goal of this study is to
investigate how pre-impact vibrations can influence
the response of a plate to spherical impact, in terms
of the indentation produced and the impact force
generated. The present study assumed a small-mass
impact and the infinite plate modeling approach
was applied together with the Hertz contact law to
formulate the governing nonlinear impact model.
The effect of pre-impact vibration was accounted
for in the expression for the indentation rate and
was confirmed by comparing the impact response
of the initially-vibrating plate with the response of
an initially-stationary plate. Finally, parametric
studies were conducted to determine the influence
of critical impact parameters on the impact
response of the initially-vibrating plate.
2. Materials and methods
2.1 Model formulation for vibration analysis of
thin rectangular plate
The vibration of a thin rectangular plate subject
to a spatially distributed external excitation (see
Fig. 1) can be derived from the well-known
classical plate theory (CPT).
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 20
Fig. 1: (a) Pre-impact vibration of a thin rectangular plate subject to uniformly distributed load (b) Onset
of contact for thin rectangular plate impacted by a rigid spherical surface
According to the CPT, the vibration model for
small displacements of a thin rectangular isotropic
plate is given as (Szilard, 2004):
(
)
( ) ( )
where ( )⁄ . Equation (1) is a
partial differential equation and can be solved for
different boundary conditions by using the double
series expansion method. Therefore, the
displacement and load can be expressed as (Big-
Alabo et al., 2015a):
( ) ∑ ∑ ( )
( )
( ) ∑ ∑ ( )
( )
where and are shape functions that depend
on the boundary conditions, ,
and is the load coefficient. For uniformly
loaded rectangular plate ( ).
Equation ( ) can be reduced to an ODE by
applying Galerkin’s variational approach as follows
(Szilard, 2004):
∫ ∫ (
(
)
( ))
( ) For a simply supported plate, with boundary
conditions: ( ( ) and
( )) and without loss of generality,
Equation ( ) reduces to Equation (5).
( ) ( ) ( ) ( )
where is the plate’s modal displacement,
√ ,
(
) and
Under resonance condition (i.e.
) and for zero initial conditions, is
given by (Chopra, 2012):
( )
* (
√ ) + ( )
where √ and √ . Therefore,
the displacement of the plate is given by:
∑ ∑
* (
√ )
+ (
) (
) ( )
In Equation (7), only the odd modes are applicable
since the even modes vanish due to the sine
functions. Therefore, and the
number of modes is calculated as ( )( ) .
2.2 Formulation of impact model incorporating
pre-impact vibrations
For low-velocity small-mass impact, the impact
event is not influenced by the boundary conditions
or external excitations since the impact is complete
before the vibration waves rebound from the
boundary of the plate. Therefore, the infinite plate
impact model can be used to predict the impact
response for such small-mass impact events.
According to Olsson (1992), the differential
equation that describes the indentation generated
by the small-mass impact is given by:
√
( )
The impact force, ( ), is determined using
an appropriate quasi-static contact law, while the
indentation, , is equal to the relative displacement
between the plate and impactor.
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 21
For an elastic impact event, the impact force can
be estimated using Hertz contact law as shown:
⁄ ( )
where ( )
⁄ , [( )
( ) ]
, ( )
and
.
For elastoplastic impact events, more accurate
contact laws are required and are available (e.g.
Brake, 2012; Big-Alabo et al., 2015b; Stronge,
2000). The elastoplastic contact laws are more
complicated and are computationally complex. In
the present study, the Hertz contact model was
applied to investigate the effect of pre-impact
vibrations. Moreover, this simplifies the
computational complexity while allowing one to
conduct the required investigation. Substituting
Equation (9) in (8) results in Equation (10).
√
⁄
⁄ ( )
Considering the initial conditions to Equation (10),
there is no relative displacement between the plate
and impactor at the onset of impact i.e. ( ) ,
while the initial relative velocity is ( ) . Hence, applying Equation (7) gives the initial
relative velocity as:
( ) ∑ ∑
* (
√ )
(
√ )
+ ( )
Equation ( ) incorporates the effect of pre-impact
vibrations in the initial relative displacement. This
is the main distinguishing factor between the
present analysis and previously published studies
where the plate is initially at rest and ( ) .
3. Results and discussion
3.1 Plate and impactor properties
A typical aluminum rectangular plate and a steel
spherical impactor were used for all simulations.
The material and geometric properties of the plate
and impactor are given in Table 1. Other inputs
used in the simulations are: , and . To ensure
that the initial impact velocity does not produce
elastoplastic response, Stronge’s velocity limit for
elastic impact was applied in selecting the initial
velocity of the impactor. This velocity limit is the
initial velocity required to cause yield and is given
as (Stronge, 2000):
(
) ( )
Based on the input values in Table 1,
. Therefore, the initial speed of the
impactor ( ) was selected so that the initial impact
speed is less than .
Table 1: Properties of the Aluminum – Steel
impact system
Geometric and material properties
Plate (Aluminum): ;
; ; ;
; ;
Impactor (Steel): ; ; ; ;
3.2 Verification of accuracy of numerical
solution
Equations (10) and (11) were solved
numerically using the default settings of the
NDSolve function in Mathematica computational
software. The NDSolve function has been applied
in previous studies on rigid-flexible impact (Big-
Alabo et al., 2015c&2017) and vibration analysis
(Big-Alabo and Ossia, 2019&2020). However, to
verify the accuracy of the NDSolve function for the
present model, the results obtained from the present
model were compared with those provided in the
literature (Zener, 1941).
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 22
Fig. 2: Dependence of COR on inelastic parameter
for impact of a spherical steel ball on a glass plate.
Zener (1941) was the first to derive Equation (10)
and he presented numerical and experimental
results for an infinite glass plate impacted by a
spherical steel ball. The properties of the steel ball
are given in Table 1 while the properties of the
glass are (Structural Glass, 2016): ,
, and . During
impact, the glass plate absorbs impact energy and
experiences local indentation and vibrations. The
coefficient of restitution (COR) is a measure of the
degree of the initial impact energy absorbed by the
plate that is not restored back to the impactor.
Zener (1941) obtained numerical COR solutions by
solving Equation (10), and the numerical COR
results were found to compare well with
experimental COR results. Hence, the accuracy of
the NDSolve function was tested by comparing its
COR estimates with the experimental results
reported in Zener (1941). Figure 2 depicts this
comparison and the good agreement between the
experiments and NDSolve results confirms the
accuracy of the latter. A further verification was
conducted by comparing COR estimates of the
NDSolve function with numerical COR results
reported by Zener (1941) as shown in Table 2.
Both numerical results are close thus verifying the
accuracy of the NDSolve solution.
Table 2: Numerical solutions for COR of a spherical steel ball impacting a glass plate
Inelastic parameter, 0.0 0.5 1 1.5
COR Zener (1941) 1.0 0.44 0.18 0.067
This study 1.0 0.436896 0.184123 0.066404
Fig. 3: Modal convergence test (left) and exploded view (right)
Fig. 4: Transient displacement (left) and steady-state displacement (right) based on nine modes
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
CO
R
λ
NDSolve solution
Experiment (Zener, 1941)
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 23
3.3 Modal convergence analysis
In Section 3.2, the accuracy of the NDSolve
function was tested for the case of an initially-
stationary plate in which the initial relative
indentation is zero. The model for the impact of an
initially-vibrating plate shows that the initial
relative indentation depends on the vibration modes
(see Equation (11)). Although the model requires
an infinite number of modes, it is only practical to
use a finite number of modes. The higher the
number of modes used in the analysis the more the
computational complexity. It is then necessary to
determine the minimum number of modes that can
give a sufficient accuracy in order to limit the
computational complexity. Figure 3 shows the
transient mid-point displacement of the pre-impact
vibrations for different number of modes when the
damping ratio is 0.05. The Figure reveals that
modal convergence was achieved with nine modes
(i.e. ) since the plots for four and nine
modes match and are indistinguishable. Therefore,
nine modes approximation was used for all
analyses discussed subsequently. Figure 4 shows
the transient and steady-state mid-point
displacement profile obtained using nine modes.
The steady-state response shows that the
displacement is within small-amplitude vibration
since . This is important because it
implies that the simulation is within the bounds of
application of the classical plate theory which was
used to model the pre-impact vibration response
(see Section 2.1).
3.4 Effect of pre-impact vibration on the impact
response Figures 5 – 7 are plots showing the effect of
pre-impact vibration on the indentation, indentation
rate (or relative velocity) and impact force
respectively. For the small-amplitude pre-impact
vibrations, a significant difference between the
impact response of the initially-stationary and
initially-vibrating plates was observed. In Figures 5
– 7, the initial velocity of the plate was opposite to
that of the impactor thus leading to a higher
maximum impact force, a higher maximum
indentation and a shorter impact duration compared
to the initially-stationary plate.
Fig. 5: Effect of pre-impact vibration on the
indentation response. The legend in this figure
applies to Figures 6 to 8.
Fig. 6: Effect of pre-impact vibration on the rate of
indentation
Fig. 7: Effect of pre-impact vibration on the impact force
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.5 1 1.5 2
δ [μ
m]
t [ms]
Initially stationary plate
Initially vibrating plate
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5
Re
lati
ve
ve
loci
ty [
mm
/s]
t [ms]
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
F [
N]
t [ms]
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 24
Fig. 8: Dependence of the of an initially-vibrating plate on initial impact time, . Impact initiated
during transient (left) and steady-state (right) phases of the pre-impact vibrations
Figure 8 shows the maximum impact force
( ) obtained when the impact was initiated at
different pre-impact times over a vibration cycle.
This investigation was based on vibration cycles in
the transient and steady-state phases of the pre-
impact vibration. The same trend is observed in the
profile of except that the steady-state
vibration cycle produces a higher or lower
magnitude of as the case may be. This is
expected since the plate velocity in the steady-state
phase is higher than that of the transient phase.
Figure 8 shows that when the initial velocities of
the plate and impactor are in the same direction,
reduces compared to the initially-stationary
plate and vice versa. Furthermore, changes
over the vibration cycle because the velocity of the
plate is constantly changing with time during the
pre-impact vibration. Hence, the time during the
pre-impact vibration at which impact commence
dictates the impact response experienced by the
initially-vibrating plate. The implication is that
small-amplitude pre-impact vibrations can alter the
impact response of a plate significantly and may
form a practical basis for adaptive impact damage
mitigation.
3.5 Parametric analysis
A parametric analysis was conducted to study
the consequence of varying critical impact
parameters, such as impactor mass, impactor size,
initial velocity of impactor and thickness of plate,
on the impact response of an initially-vibrating
plate. The results of the parametric study are shown
in Figures 9 – 12. The effect of the impactor mass
on for impact commencing at different times
within a vibration cycle of the steady-state pre-
impact vibration is shown in Figure 9. In practical,
the impactor mass can be changed independent of
its radius by using a cylindrical impactor with blunt
spherical end. It was observed that increasing the
impactor mass leads to a lower when the
velocities of the plate and impactor are in the same
direction and a higher when both velocities
are in opposite directions. However, the effect on
the higher is more pronounced than the effect
on the lower .
Fig. 9: Effect of impactor mass on
Fig. 10: Effect of impactor size on
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
20.1 20.15 20.2 20.25 20.3
Fm
ax [
N]
t0 [s]
mi = 1gmi = 2gmi = 3gmi = 4g
0
0.5
1
1.5
2
2.5
3
3.5
20.1 20.15 20.2 20.25 20.3
Fm
ax [
N]
t0 [s]
Ri = 6.35mmRi = 11.35mmRi = 16.35mm
Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate
Uniport Journal of Engineering & Scientific Research Vol. 5, Special Issue, 2020 Page 25
Fig. 11: Effect of initial velocity of impactor on
Fig. 12: Effect of plate thickness on
The observed effect that increasing the impactor
mass produced on was the same qualitative
effect observed when the impactor size (see Fig.
10) and initial velocity of the impactor (see Fig. 11)
were independently increased. However, the
observed trend in as the plate thickness
changes is that it undergoes more oscillations with
increase in plate thickness (see Fig. 12). This
happens because a thicker plate is stiffer and
exhibits a higher natural frequency, thereby
resulting in the oscillations observed in .
Hence, it can be concluded that each of the critical
impact parameters considered have significant
influence on the impact response of an initially-
vibrating plate and can be used as design
parameters for new generation impact mitigation
materials.
4. Conclusions
In this study, a clear departure was made from
the traditional rigid-flexible impact studies of
initially-stationary plates to investigate the impact
response of an initially-vibrating plate. This study,
though theoretical, has practical implications for
more realistic impact modeling since many plate-
like structures are already undergoing vibration
prior to being impacted by rigid bodies e.g. the
body of a moving automobile impact by a rigid
body. Also, the present study has potential for
adaptive impact mitigation as it shows that induced
vibrations prior to impact can significantly
influence the impact response of thin plates.
In conducting the present study, the vibration
model of a thin rectangular plate was combined
with the infinite plate impact model for small mass
impact. The resulting models incorporated the pre-
impact vibration effect into the initial relative
velocity of the impact model. The NDSolve
function in Mathematica was used to solve the
models for the impact response of the initially-
vibrating plate after its accuracy was verified by
comparing with results from literature. The analysis
showed that the small-amplitude pre-impact
vibrations of a thin plate have significant influence
on its impact response and the impact response is
strongly dependent on the pre-impact velocity at
the time of impact. Parametric investigations on the
effect of impactor mass, impactor size, initial
velocity of impactor and plate thickness all showed
a strong correlation between these parameters and
impact response of the initially-vibrating plate. The
impact parameters investigated in the parametric
analysis define a toolset that can be used to design
more responsive materials for impact damage
mitigation.
Funding/Acknowledgement
The author(s) received no financial support for the
research, authorship, and/or publication of this
article. The first author wishes to acknowledge the
encouragement of his friend, Dr. Okulonye Ofejiro,
who passed on sadly while this research was being
conducted.
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Nomenclature
Symbol Description
Length of plate in - direction
Length of plate in - direction
Damping coefficient per unit area
Effective bending stiffness of plate
Effective contact modulus
Elastic modulus of impactor
Elastic modulus of plate
Impact force
Thickness of plate
Contact stiffness
Modal number in and direction respectively
Mass of impactor
Mass of plate
Effective radii
Radius of impactor
Radius of plate
Poisson ratio of impactor
Poisson ratio of plate
( , , ) Spatially distributed excitation
Uniformly distributed pressure
( ) Transverse displacement of plate
( ) Displacement of impactor
0 Natural frequency of plate
Indentation
Indentation rate
Density of plate
Mass per unit area of plate
Yield strength of plate
Damping ratio