Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 715-726, Warsaw 2008

VIBRATIONAL ENERGY TRANSFER VIA MODULATED

IMPACTS FOR PERCUSSIVE DRILLING

Marian Wiercigroch

University of Aberdeen, Centre for Applied Dynamics Research, Aberdeen, Scotland, UK

e-mail: m.wiercigroch@abdn.a.uk

Anton M. Krivtsov

St.-Petersburg State Technical University, Department of Theoretical Mechanics, St.-Petersburg,

Russia

Jerzy Wojewoda

Technical University of Lodz, Division of Dynamics, Łódź, Poland

In this paper, a new method of vibrational energy transfer from high-
frequency low-amplitude to low-frequencyhigh-amplitudemechanical vi-
brations is proposed and investigated, for the purpose of percussive dril-
ling. The mechanism uses an impact loading comprised of two frequen-
cies formingabeatwhich excites a structure tuned to thebeat-frequency.
This concept has beenmodelled and experimentally verified. It is postu-
lated that the efficiency of this technique is strongly dependent on the
strength of nonlinearity in the system.The vibrational energy transfer is
significantly reduced for superharmonic responses of the system. Prac-
tical applications of this mechanism can been seen as new generation
percussive drilling tools.

Key words: vibration, energy transfer, impacts, experimental studies

1. Introduction

This work was inspired by the project funded by The Centre for Marine and
Petroleum Technology entitled ’Enhanced Resonance Drilling’ (Wiercigroch,
1999), which was intended to ascertain the extent to which an introduction of
high frequency loading can improve drilling rates and reduce drilling forces, in
particular, the weight-on-bit (WOB). This was accomplished by carrying out
experimental studies in the laboratory on a carefully chosen selection of the



716 M. Wiercigroch et al.

rocks most likely to be encountered while drilling in the North Sea (Wierci-
groch et al., 2005).

The studies performed so far have confirmed that the percussive drilling
has a very clear advantage of achieving much higher drilling rates (up to
10 times) (Krivtsov and Wiercigroch, 1999; Wiercigroch et al., 1999, 2000)
whilst compared to the traditional shearing type drilling. It has to be noted
here that these results were obtained for a small-scale rig under conditions
which differ from the typical downhole ones. These high drilling rates are
achieved for the resonance conditions, which vary with respect to different
rocks.

It is clear that the presented idea of transferring high-frequency low-
amplitude vibrational energy to low-frequency high-amplitudewill havemuch
wider and generic applications. In fact, there are already some initial studies
(Anderson et al., 1996; Tabaddor and Nayfeh, 1997) addressing the issue of
high and low frequency modal interactions. The reported energy transferred
is modest due to weak nonlinearities in the system transferring energy.

The aim of this paper is to prove a novel concept of transferring vibra-
tional energy via impact (percussive) loading, which can be applied to the
percussive downhole drilling. The basic idea originates from the experimental
and theoretical studies on impact oscillators (Natsiavas, 1998, 1990; Peterka
and Vacik, 1992; Pavlovskaia and Wiercigroch, 2003a,b, 2004; Pavlovskaia et
al., 2001, 2003;Wiercigroch and Sin, 1998), which are strongly nonlinear. The
main nonlinearity in impact oscillators comes from the discontinuity in the
governing equation at themoment of impact. It is postulated here that strong
nonlinearity creates necessary conditions for efficient energy transfer.

This paper is organised as follows. Section 2 presents the results of experi-
mental studies for a beam system. Section 3 contains amathematical descrip-
tion and analysis of the experimental rig. In Section 4, an averaging procedure
was applied to obtain an approximate analytical model. The analytical solu-
tion is used to compute amplitude-frequency curves for a single, harmonic and
beat excitation. In Section 5, we draw some conclusions.

2. Experimental study

The experimental studypresented here is to verify the proposed concept of the
vibrational energy transfer. Transforming a beatwise excitation into a single
frequency response was achieved by using a simple cantilever beam system.
Thebeamhasanendmass to control thefirstnatural frequencyand to amplify



Vibrational energy transfer via modulated impacts... 717

the lateral displacement whilst operating in the resonant mode. It is excited
from a vibration exciter by an extension rod located close to the support of
the beam. This arrangement guarantees generation of hard impacts.
The excitation from the shaker is in the form of impacts shaped as a

beat forcing, x(t), as depicted in Fig.1. Two harmonic signals of the same
amplitude and having a frequency difference equal to the beat frequency are
summed up together to obtain a beat excitation as shown in the upper panel
of Fig.2. Then the beat signal is amplified and sent to the vibration exciter,
which impacts the beamwith a beatwise pattern.

Fig. 1. Experimental beam system

As mentioned earlier, the first natural frequency of the beam system is
chosen to match the beat-frequency. The gap between the beam at motion is
only a small fraction (a few percent) of the end mass motion amplitude. To
monitor the lateral vibration of thebeam y(t), a pair of strain gauges arranged
in a half bridge was used. The time history recorded from the strain gauges is
given in the lower part of Fig.2. As can be seen, the beam system responds
well to the high-frequency beat excitation.
The amplitude-frequency curve (AFC) have been obtained experimentally

by varying the excitation frequency and recording the amplitude of the dy-
namic response. Figure 3 compares the amplitude of the system responses for
the single frequency and beat excitation impactsmodes. Apart from themain
resonance, the 1/2 and 1/3 superharmonics are clearly visible for both types
of excitation. It is worth noting that when exciting with beat impacts, a large
amount of vibrational energy is transferred. By comparing the amplitudes for
the single and beat-frequency impact excitations at the main resonance, ap-
proximately seventy percent of the single frequency response is obtained for



718 M. Wiercigroch et al.

Fig. 2. Beat-wise excitation (upper plot) and the endmass response (lower plot)

the beat loading. The transfer is significantly reduced for the superharmonic
responses. To obtain a better insight of the system dynamics, a robustmathe-
matical modelling and analysis is needed, and this is undertaken in the next
Section.

Fig. 3. Experimental amplitude-frequency plot

3. Modelling and analysis

Consider a conceptual model presented in Fig.4, where x is the displacement
of the mass m, which is expected to vibrate with a low frequency and high
amplitude; y is thedisplacement of the exciter vibratingwith ahigh frequency
and lowamplitude,where z is thedisplacement of themassless plate P, which



Vibrational energy transfer via modulated impacts... 719

is being impacted by the exciter head. The oscillations are then transferred
to the mass m. The plate is connected to the support by a short spring c1
and to the mass with a long spring c2, where c1 ≫ c2. The difference in the
length of the springs allows one to obtain a high amplitude vibration of the
mass while the plane is vibrating with a low amplitude. The displacements y
and z are measured from the same position determined by the initial length
of the spring c2. The displacement x is measured from the right tip of the
spring c2 at the initial state of the springs.

Fig. 4. A physical model for beat-frequency vibrational energy transfer

Let us denote the equivalent stiffness of the springs c1 and c2 as c, which
for a parallel spring connection is

c=
c1c2
c1+ c2

(3.1)

When the plate P is not in contact with the exciter, then its displacement
can be calculated as

z=
c

c1
x=αx α≪ 1 (3.2)

where α is a small parameter due to the mentioned above differences in the
spring stiffnesses. Then the equation of motion for the mass m can take the
following piecewise linear form

{

y¬αx : mẍ+ cx=0

y­αx : mẍ+ c2(x−y)= 0
(3.3)

Hence, there are two equations of motion; the first one is applicable when
the exciter is not in contact with the plate (y < αx), the second one is for
the contact case (y ­ αx). It can be easily shown that when y = z0, both



720 M. Wiercigroch et al.

equations are identical. Expressing c2 in terms of c and α, one can rewrite
the second equation as

mẍ+ cx=
c

1−α
(y−αx) (3.4)

which allows one to rewrite system (3.3) as a single equation

ẍ+k2x=
k2

1−α
P(y−αx) (3.5)

where k =
√

c/m is the natural frequency of the system; P(ζ) is a switch
function defined as

P(ζ)=

{

0 for ζ ¬ 0

ζ for ζ ­ 0
(3.6)

4. Approximation and averaging

Assume that the exciter is oscillating harmonically according to the formula

y= y(t)=A(t)sin(Ωt)−s Ω≫ k (4.1)

where Ω represents the high (excitation) frequency, s – gap between the
exciter head and the plate, A(t) – amplitude of thehigh frequency oscillations,
carrying low frequency oscillations. For beat oscillations A(t)=A0 sin(ωt/2),
the beat excitation is depicted in Fig.5, where A0 is the amplitude, ω is the
low frequency. Here, the influence of the oscillating system on the exciter is
neglected. Let us average Eq. (3.5) over a period τ of the high frequency
oscillations, τ =2π/Ω

〈ẍ〉+k2〈x〉=
k2

1−α

〈

P(y−αx)
〉

(4.2)

where

〈ζ〉=
1

τ

τ
∫

0

ζ dt

The displacement of the mass can be represented as a sum of two compo-
nents

x=X+ ξ X = 〈x〉 (4.3)



Vibrational energy transfer via modulated impacts... 721

Fig. 5. Beat excitation

where X is the low frequency high amplitude component, and ξ is the high
frequency low amplitude component.
For α≪ 1 and ξ≪X, the αξ term can be neglected, which leads to

〈

P(y−αX−αξ)
〉

≈
〈

P(y−αX)
〉

(4.4)

Now let us evaluate expression (4.4)

〈

P(y−αX)
〉

=
|A(t)|

τ

τ
∫

0

P(±sin(Ωt)− q) dt q=
s+αX

|A(t)|
(4.5)

where ”±” is the sign of A(t). It here is assumed that the low frequency
function A(t) does not change over the short period of time τ. The above
integral can be represented as

〈

P(y−αX)
〉

= |A(t)|I(q) (4.6)

where

I(q)=
1

2π

2π
∫

0

P(sinθ−q) dθ θ=Ωt

The sign ”±” can be omitted here because it does not have any influence on
the value of the integral. The integral I(q) is calculated as



















q¬−1 ⇒ I(q)=−q

−1¬ q¬ 1 ⇒ I(q)=
1

π

(

√

1−q2− qarccosq
)

1¬ q ⇒ I(q)= 0

(4.7)

and shown in Fig.6.



722 M. Wiercigroch et al.

Fig. 6. Function I(q)

Finally, by substituting (4.3), (4.4) and (4.6) to (4.2), one can obtain an
equation for the slow component X of the mass motion in the form

Ẍ+k2X = k2G(X,t) (4.8)

where k2G(X,t) is the excitation force defined by

G(X,t) =
|A(t)|

1−α
I
(s+αX

|A(t)|

)

(4.9)

The function I(q) is shown in Fig.6 and determined from (4.7). The func-
tion G(X,t) can be understood as an equivalent forcing of a simple oscillator
depicted in Fig.7, whose its dynamics is governed by (4.8).

Fig. 7. Model corresponding to the averaged equation

Consider now averaged equation (4.8) for the simplest case: α→ 0, s=0.
Here G(X,t) is directly proportional to |A(t)|

G(X,t) = |A(t)|I(0) =
1

π
|A(t)| (4.10)

Let us assume A(t) = A0 sin(ωt/2), where A0 is a constant, ω is the
low excitation frequency. This corresponds to the following beat excitation:



Vibrational energy transfer via modulated impacts... 723

y(t) =A0 sin(ωt/2)sin(Ωt), where Ω is the high frequency (see Fig.5). Equ-
ation of motion (4.8) takes the following form in this case

Ẍ+2βẊ+k2X = k2
A0
π

∣

∣

∣sin
ωt

2

∣

∣

∣ (4.11)

To represent the systemmore adequately, a viscous damping is added to (4.8),
where the damping coefficient is a small parameter, β ≪ k. The computed
amplitude-frequency characteristic (AFC) is depicted in Fig.8, from which it
can be deduced that the main resonance occurs for ω= k. This is due to the
fact that if one takes the absolute value of sin(ωt/2), then themain frequency
increases twice. In addition, the AFC in Fig.8 has clear superharmonic reso-
nances, corresponding to 1

2
, 1
3
, 1
4
, 1
5
, . . . of themain resonance frequency. These

resonances are the result of the non-harmonic form of the excitation force.

Fig. 8. Amplitude frequency characteristic

To show better the nature of the superharmonic resonances, let us expand
the right part of equation (4.11) into Fourier series

∣

∣

∣sin
ωt

2

∣

∣

∣=
2

π
−
4

π

∞
∑

n=1

cosnωt

4n2−1
(4.12)

For a small dissipation, the resonance amplitudes are proportional to the
expansion coefficients, whichmeans that the subharmonic resonance amplitu-
des are decreasing with 1/n2.
To assess the efficiency of the beat frequency excitation mechanism, a

comparison with the low frequency excitation y = A0 sinωt is made. In this
case, equation of motion (3.5) takes the form

Ẍ+2βẊ+k2X = k2A0P(sinωt) (4.13)



724 M. Wiercigroch et al.

where X ≡ x due to absence of the high frequency component. By analysing
Fig.9, one can easily deduce that the beat mechanism can effectively ’pump’
approximately a quarter of the energy transferred via the single frequency
impact excitation, which is approximately twelve times more than previously
reported.

Fig. 9. Comparison ofAFC for single frequency excitation (1) and beat excitation (2)

The expansion of the right-hand sides of equations (4.11) and (4.13) into
the Fourier series retaining only twomain terms gives

∣

∣

∣sin
ωt

2

∣

∣

∣≈
2

π
−
4

3π
cosωt P(sinωt)≈

1

π
+
1

2
sinωt (4.14)

where the coefficients with cosωt and sinωt are responsible for the main
resonance. Thus the ratio between the main resonance amplitudes in these
two cases is equal to 3π/8≈ 3.70.

5. Closing remarks

In this paper, the proposed vibrational energy transfer mechanism from high-
frequency low-amplitude to low-frequency high-amplitude mechanical vibra-
tions has been investigated experimentally and theoretically. The mechanism
uses an impact loading comprised of two frequencies forming a beat, which
excites a structure tuned to the beat-frequency. The best efficiency is ob-
tained for the main resonance, for which a large amount of energy can be
transferred. The transfer is significantly reduced for the superharmonics. It is
also postulated throughout the paper that the efficiency of this technique is
strongly dependent on the strength of nonlinearity in the system.



Vibrational energy transfer via modulated impacts... 725

Although all precautions have been taken to ensure the same parameters
and conditions as for the theoretical model, there is no guarantee that the
coefficient of damping and the gap were the same. The obtained results con-
firmed that the proposed impact beat mechanism can be effective in energy
transfer.

The authors would like to kindly acknowledge financial support from the
Royal Society of London and from theCentre forMarine andPetroleumTech-
nology.

References

1. Anderson T.J., Nayfeh A.H., Balachandran B., 1996, Coupling betwe-
en high-frequency modes and a low-frequency mode: Theory and experiment,
Nonlinear Dynamics, 11, 1, 17-36

2. Krivtsov A.M., Wiercigroch M., 1999, Dry friction model of percussive
drilling,Meccanica, 34, 6, 425-434

3. Natsiavas S., 1998, Stability of piecewise linear oscillators with viscous and
dry friction damping, Journal of Sound and Vibration, 217, 3, 507-522

4. Natsiavas S., 1990, Stability and bifurcation-analysis for oscillators withmo-
tion limiting constraints, Journal of Sound and Vibration, 141, 1, 97-102

5. PavlovskaiaE.E.,WiercigrochM., 2003a,Modelling of vibro-impact sys-
tem driven by beat frequency, International Journal of Mechanical Sciences,
45, 4, 623-641

6. PavlovskaiaE.E.,WiercigrochM., 2003b,Periodic solutions finder for an
impact systemwith a drift, Journal of Sound and Vibration, 267, 4, 893-911

7. Pavlovskaia E.E., Wiercigroch M., 2004, Analytical drift reconstruction
for an impact systemoperating inperiodic and chaotic regimes,Chaos, Solitons
and Fractals, 19, 1, 151-161

8. Pavlovskaia E.E., Wiercigroch M., Grebogi C., 2001, Modelling of an
impact oscillator with a drift,Physical Review E, 64, 056224

9. Pavlovskaia E.E., Wiercigroch M., Woo K.-C., Rodger A.A., 2003,
Modelling of a vibro-impact groundmoling systemby an impact oscillatorwith
a frictional slider,Meccanica, 38, 1, 85-97

10. Peterka F., Vacik J., 1992, Transition to chaoticmotion inmechanical sys-
tems with impacts, Journal of Sound and Vibration, 154, 1, 95-115



726 M. Wiercigroch et al.

11. Tabaddor M., Nayfeh A.H., 1997, An experimental investigation of mul-
timode responses in a cantilever beam, Journal of Vibration and Acoustics-
Transactions of the ASME, 119, 4, 532-538

12. Wiercigroch M., 1999, Enhanced Resonance Drilling Report from CMPT
Research, Grant No. PMF008/CMPTPathfinder

13. Wiercigroch M., Krivtsov A.M., Wojewoda J., 2000, Dynamics of Ul-
trasonic, In: Applied Nonlinear Dynamics and Chaos of Mechanical Systems
with Discontinuities, edit. Wiercigroch M. and Kraker B., World Scientific,
403-444

14. WiercigrochM.,NeilsonR.D.,PlayerM.A., 1999,Material removal rate
prediction for ultrasonic drilling of hard materials using an impact oscillator
approach,Physics Letters A, 259, 2, 91-96

15. Wiercigroch M., Sin V.T.W., 1998, Experimental study of based excited
symmetrically piecewise linear oscillator, Journal of Applied Mechanics, 65,
657-663

16. Wiercigroch M., Wojewoda J., Krivtsov A., 2005, Dynamics of percus-
sive drilling of hard rocks, Journal of Sound and Vibration, 280, 3/5, 739-757

Transfer energii drgań poprzez modulowanie uderzeń w wierceniu

udarowym

Streszczenie

W niniejszej pracy zaproponowano i pokazano wyniki nowego sposobu transferu
energii z obszarudrgańowysokichczęstościach imałychamplitudachwobszarniskich
częstości i dużych amplitud. Badania przeprowadzone były pod kątem zastosowań
w wierceniu udarowym. Mechanizm działania wykorzystuje obciążenie uderzeniami
modulowane kombinacją dwu częstości tworzących obwiednię w kształcie dudnienia
o strukturze dostrojonej do częstości uderzeń. Pokazano opracowanymodel i wyniki
jego weryfikacji eksperymentalnej.Wydaje się, że wydajność tej techniki jest mocno
zależna od siły nieliniowościw układzie. Transfer energii drgań jestmocno zmniejszo-
ny dla superharmonicznychodpowiedzi układu.Możliwe jest praktyczne zastosowanie
tego mechanizmuw nowej generacji wierteł udarowych.

Manuscript received November 8, 2007; accepted for print December 5, 2007