Uniport Journal of Engineering and Scientific Research (UJESR)

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)

http://www.ujesr.org/

Effect of Pre-Impact Vibrations on the Low-Velocity Impact Response of a Rectangular Plate


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).



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.



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).


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)



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]



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



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

Documents

Uniport Journal of Engineering and Scientific Research (UJESR)