Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 1, pp. 169-178, Warsaw 2018
DOI: 10.15632/jtam-pl.56.1.169

DETERMINATION OF THE INTERNAL RESISTANCE OF

A HAMMER DRILL CHISEL

Nikolay Nikolov

Technical University – Sofia, Strength of Materials Department, Sofia, Bulgaria

e-mail: nyky@tu-sofia.bg

Petko Sinapov

Technical University – Sofia, Department of Mechanics, Sofia, Bulgaria

e-mail: p sinapov@tu-sofia.bg

The present work examines damped oscillations of a chisel, represented as a distributed
parameters system. The system is discretized with the finite element method. Rayleigh’s
law is used for the modelling of the resistance. The internal resistance of the mechanical
system has been determined by an experiment, then enshrined in the numerical model.
Comparison and analysis of the results have been made.

Keywords: free damped oscillations, FEM, Rayleigh dampingmatrix, damping coefficients

1. Introduction

Precise determination of the internal resistance is a crucial point in the research on the di-
stributed parameter dynamics of mechanical systems. Without knowing its value, it would be
impossible to study numerically different processes, such as free damped or forced oscillations,
nonlinear oscillations, shock occurrences and others. The Rayleigh law is most commonly used
to describe the internal resistance

C=αM+βK (1.1)

where C is the Rayleigh damping matrix, α is a coefficient taking into account the influence
of the mass distribution on the resistance, β is a coefficient accounting for the influence of
the system elasticity on the resistance, while M and K are, respectively, mass and stiffness
matrices. According to many researchers (Stelzmann et al., 2008), the coefficients α and β are
to be determined for each specific structure. It turns out that by varying these two coefficients,
certain oscillation frequencies could be supressed,while others prevail. A number of studies have
shown (Man andCorman, 1995) that theα-damping affects low frequencies, and the β-damping
affects high frequencies, as shown in Fig. 1. A great advantage when using the Rayleigh law is
that after orthogonal transformation the system ofn differential equations, describingmotion of
the system, can be represented by n independent equations (Chowdhury and Dasgupta, 2003),
i.e. it can break the connection between the equations.
Different methods could be used for determining the α and β (Craig and Kurdila, 2006;

Alipour and Zareian, 2008; Adhikari, 2001). If ω∗1 and ω
∗

2 are the first two frequencies of the
system damped oscillations, then for themost commonmethod it is assumed that at both ends
of the frequency range [ω∗1;ω

∗

2] the damping ratio ξ has equal values (Fig. 1).When amechanical
system is dominated by free oscillations with frequencies within thementioned range, the values
of α and β are determined by the equations

α= ξ
2ω∗1ω

∗

2

ω∗1 +ω
∗

2

β= ξ
2

ω∗1 +ω
∗

2

(1.2)



170 N. Nikolov, P. Sinapov

Fig. 1. Rayleigh damping

The damping ratio ξ depends on thematerial and the studied structure. Information about the
value of ξ for steel can be found in the bibliography. For example, the following intervals from
0.02 to 0,08; from 0.02 to 0.03 and from 0.0008 to 0.0025 are given respectively in (Chowdhury
andDasgupta, 2003; Chopra, 1995; Dresig andHolzweissig, 2006). In (Mevada andPatel, 2016)
ξ was experimentally determined and the obtained value was 0.0069. It was found by Stevenson
(1980) that ξ depended on the dynamic values of the stresses and strains, and the value for steel
was set to 0.01. In (Zare et al., 2011) ξ has the value of 0.03 and in (Sangeetha et al., 2014)
ξ has the value of 0.2. It appears that for each structure the resistance is strictly individual and
it often depends even on the operation mode.

The purpose of the present work is to model damping oscillations of the mechanical system
once the internal resistance has been experimentally determined.

2. Research object

A schematic drawing of a hammer drill is shown on Fig. 2. It is well known that this type
of machine use a set of working tools with different shape, size and weight. As parts of the
mechanical system, theseworking tools also affect the force of the impact.Therefore, theworking
tools, just like theoperatingmodes, are subject tooptimization inorder tomaximize themachine
efficiency. The present study is focused only on the chisel.

Fig. 2. Schematic drawing of a hammer drill

A suitable dynamic model has to be developed in order to give a numerical representation
of the law for amending the force of impact. All parameters involved should be set down with
their real values, preferably determined by an experiment.

After being created, the model should be validated with experimental data from the actual
working process of the machine. The present study is the first step to the creation of such a
model. It is aimed at determining the coefficients α and β of the working tool of the hammer
drill.



Determination of the internal resistance of a hammer drill chisel 171

3. Dynamic model

Drillingmachineswhose tool performs complexmotion including rotation are examined in (Khu-
lief et al., 2005; Yigit et al., 1998; Zare et al., 2011). In our case, the chisel moves only in the
axial direction and it is assumed that the oscillations along this axis will be themost significant.
Bending and twisting vibrations due to additional factors at this stage are neglected. Figure 3
shows the studiedmechanical systemwhich consists of a chisel and support. The latter is a pair
of metal jaws clenching the chisel.

Fig. 3. Mechanical system: (a) sections and dimensions of the chisel, (b) fixing of the chisel

Figure 4 shows the dynamicmodel of themechanical system. For the purpose of themodel-
ling, the system is presented as a rod, fixed at its lower end, oscillating in the axial direction.
The movement of each section of the rod with constant cross-section is described with the

following partial differential equation

E

ρ

∂2ui(x,t)

∂x2
−

∂2ui(x,t)

∂t2
=0 (3.1)

where E is the elasticity modulus, ρ – density of the material, ui(x,t) – coordinate, describing
oscillations of the i-cross section of the rod. Since the cross section of the chisel is variable, it is
divided into four sections, as shown in Fig. 3a (the dimensions are given in mm):

• upper part of the shank, with diameter 10mm and cut out channels,

• lower part of the shank, cylindrical, with diameter 10mm,

• ram – hexagonal, with diameter of the inscribed circle 15mm,

• tip with a relatively complex shape and cross section, decreasing from the ram to the end.

The finite element method has been used for solving equation (3.1). The rod is represented by
15 finite elements, shown in Fig. 4. The tip is discretized with 3 finite elements and all the other
areas – with 4 elements. The cross section of the upper part of the shank is reduced by 20%
compared to its lower part, due to fixing channels. The lengths of the ram elements are chosen



172 N. Nikolov, P. Sinapov

Fig. 4. Dynamic model with finite elements: (a) arrangement of the finite elements,
(b) single finite element

in such a way as to obtain a node at the place of bonding of the strain gauges, as shown in
Fig. 3b. The elements of the tip are 10mm long and thus its lower part (No. 15 in Fig. 4a)
reaches the support. The cross section area of each element at the top is set according to the
actual geometry.
All finite elements are of type shown in Fig. 4b (Reddy, 1984; Rao, 2004; Stelzmann et al.,

2008). Each node has one degree of freedom, i.e. the finite element has two local degrees of

freedom (u
(i)
1 andu

(i)
2 ). Themass and stiffnessmatrices for the finite element used are as follows

M
(e) =

ρAili
6

[
2 1
1 2

]
K
(e) =

EAi
li

[
1 −1
−1 1

]
(3.2)

It is more convenient to work with global coordinates for the formation of global matrices. The
following connections between local and global coordinates should be taken into account during
the transition to such global matrices

u
(1)
1 =u1 u

(1)
2 =u

(2)
1 =u2 etc.

whereu(i) (i=1,2, . . . ,15) are the global coordinates (Reddy, 1984). Thus, equations (3.1) turn
into a system of 15 ordinary differential equations of the second order, which take into account
the boundary conditions.
The differential equation describingmotion of the mechanical system is

Mq̈+Cq̇+Kq=0 (3.3)

where M is the mass matrix (15× 15), K is the stiffness matrix (15× 15), C = αM+βK is
the Rayleigh damping matrix, q = [u1,u2, . . . ,u15]

T is the coordinate vector, 0 is zero matrix
(15×1).
The natural frequencies of the system are calculated inMatlab by solving equation (3.3) and

without taking into account the internal resistance, i.e.

Mq̈+Kq=0 (3.4)



Determination of the internal resistance of a hammer drill chisel 173

Equation (3.4) is brought into a normal appearance (Genov et al., 2007)

Ẋ= ÃX (3.5)

by introducing the following vector

X= [q, q̇]T (3.6)

and thematrix à is

Ã=

[
O15×15 E15×15
−MK −MO15×15

]
(3.7)

In equation (3.7), O15×15 is the zero matrix,E15×15 is the identity matrix.
The natural frequencies of the system are obtained by determining the eigenvalues of the

matrix à byQ-R algorithm – orthogonal projections (Watkins, 2004).
The first four values of the natural frequencies of the system thus obtained are given in

Table 1.

Table 1.Natural frequencies

Natural fi ωi
frequencies [Hz] [rad/s]

1 5640 35419

2 16920 106257

3 26810 168366

4 39530 248248

4. Influence of the internal resistance

It has been assumed that the occurrence of the first two natural frequencies at free oscillations in
the longitudinal direction is physically possible in the researchmodel. The interval in Fig. 1 has
been chosen to be from 35419 to 106257rad/s (according to Table 1). According to equations
(1.1), different values of the coefficients α and β have been obtained for different values of the
damping ratio ξ, as given in Table 2.

Table 2.Values of α and β depending on ξ

ξ α β

0.001 53.1288 1.41 ·10−8

0.010 531.288 1.41 ·10−7

0.100 5312.88 1.41 ·10−6

Differential equations (3.3) describing motion have been solved numerically by Matlab-
-Simulink, using the scheme in Fig. 5. For that purpose, the dimensions from Fig. 2 as well
as the following values of the physical constants: E = 2.1 · 1011Pa and ρ = 7850kg/m3 have
been taken.
The numerical solution of the problem using the values from Table 2, returned the results

shown in Fig. 6. The figure shows the relative deformation variation in time ε=(u11−u10)/l10.
Figure 6 shows that for the three values of ξ, the movement differs considerably (the time of
damping is different), i.e. for successfulmodelling, it is of particular importance to set the exact
value of ξ.



174 N. Nikolov, P. Sinapov

Fig. 5. Scheme for solving the differential equations inMatlab-Simulink

Fig. 6. Damped oscillations for different values of ξ

5. Experimental determination of the chisel internal resistance by the method of

damping oscillations

One of the future goals of this project will be tomeasure the impact force of themachine under
real operating conditions. Therefore, a decision has been taken to make tensometrical (strain)
measurements. Free damped oscillations of the chisel arising from shock impact, have been thus
registered in the course of the present study.

Figure 7 shows a scheme of the experimental installation. Strain gauges have been glued at
two opposite points in the middle of the ram (Fig. 3b). Two equal half bridge T-rosettes have
beenused, eachwith two serially connected strain gauges, oneof them in the longitudinal and the
other – in the transverse direction to the chisel axis. The rosettes are connected in a full bridge
circuit. The oscillation frequencies of the studied object are high, which makes the standard
tensometric equipment inapplicable for strain registration. It has been, therefore, decided to use
a high-frequency universal system for data acquisition (DAQ system) –National InstrumentsNI
USB-6211 (250kS/s, 16 bit). Excitation voltage is applied to the bridge circuit with a suitable
12V Li-ion rechargeable battery. The incoming information from the DAQ system is processed
and recorded using software package LabVIEW R○. Calibration with reference weights has been
made before the experiment. The chisel has been consecutively loaded with increasing weights
of pure pressure in order to check the linearity and establish the conversion factors stress/strain.

In Fig. 7, the T-rosettes are not in their actual positions.

Figure 8 shows the recordings of damped oscillations after the impact on the studied system.
Three recordings have beenmadewith random forces of the impact, designated as FDO1, FDO2



Determination of the internal resistance of a hammer drill chisel 175

Fig. 7. Experimental installation scheme for registering strain upon impact

Fig. 8. Measurement results and frequency analysis

and FDO3 (FDO – free damping oscillations). On the left, the relative strain ε is shown as a
function of time and on the right – frequency analysis determining the damped oscillations
frequency. Fast Fourier Transform (FFT) algorithm in Matlab is used for frequency analysis.
The values obtained for the first and second frequency of the damped oscillations are given in
Table 3. The results from the processing of all three recordings are similar.

Table 3. Frequencies of the damped oscillations

Recording
f∗
e1 f

∗

e2

[Hz] [Hz]

FDO1 5682 15882

FDO2 5495 16483

FDO3 5618 16854

Let us consider the first recording fromFig. 8 – FDO1. Damped oscillations have been obse-
rved with frequencies (f∗

e1 =5682Hz and f
∗

e2 =15882Hz) close to the first two natural frequen-
cies of themechanical system (Table 1). It is assumed that the specific resistance coefficient ξ of
the damped oscillations is the same for the first and second frequency of the damped oscillations
found from the experiment (ω∗

e1 = 2πf
∗

e1 = 35683rad/s and ω
∗

e2 = 2πf
∗

e2 = 99739rad/s). Then
its value can be determined on the basis of the lower frequency ω∗

e1.



176 N. Nikolov, P. Sinapov

The differential equation, describing the damped oscillations with frequency ω∗
e1, is

ü+2ξωe1u̇+ω
2
e1u=0 (5.1)

The relationship between ξ and ωe1 is given by the equation

ωe1 =
ω∗
e1√
1− ξ2

(5.2)

whereω∗
e1 is the first frequency of the damped oscillations, determined byFig. 8. Figure 9 shows

four consecutive oscillations from the record, given in Fig. 8.

Fig. 9. Determining the resistance of damped oscillations

From Fig. 9 it can be determined that the ratio of the amplitudes a1 and a7 is 3.2. The
following relation is valid for the amplitudes i and i+2k (i and k being positive integers)

ai
ai+2k

=exp(kξωe1τ
∗) (5.3)

Equation (5.3) can be resolved in relation to ξ, assuming that: a1/a7 =3.2, i=1, k=3, ω
∗

e1 is
expressed by (5.2), and the period τ∗ =2π/ω∗

e1. The result is ξ =0.06. Replacing ξ with 0.06,
ω∗
e1 with 35683rad/s and ω

∗

e2 with 99739rad/s in (1.1), we obtain the Rayleigh law coefficients

α=3153.7 β=8.86 ·10−7 (5.4)

Setting thevalues fromequations (5.4) in numericalmodel (3.3) andapplying theQ-R algorithm
– orthogonal projections (Watkins, 2004), we obtain the damped oscillation frequencies shown
in Table 4.

Table 4. Frequencies of the damped oscillations

Natural f∗
i

ω∗
i

frequencies [Hz] [rad/s]

1 5616 35270

2 16837 105737

3 26630 167238

4 39135 245769

The solution to differential equation (2.3) with these coefficients is shown in Fig. 10.



Determination of the internal resistance of a hammer drill chisel 177

Fig. 10. Simulated and experimentally determined damped oscillations

6. Summary

Free damped oscillations of the chisel represented as a distributed parameters system have
been studied in the present work. The dynamic model of the chisel was discretized with FEM
and the problem was solved numerically with Matlab. The values of the coefficients α and β
(5.4), forming the Rayleigh dampingmatrix, have been experimentally determined in the given
frequency range, after which they have been set in the numerical model. A comparison of the
modelled oscillations with a real record has beenmade – see Fig. 10. The resulting close match
shows that the proposed numerical model is acceptable and that the constants set in it have
been correctly determined.

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Manuscript received May 29, 2017; accepted for print August 30, 2017