High power piezoelectric axial
shockwave generation
Hp. Schad, HILTI AG, Schaan FL
L. Pickelmann, Piezomechanik GmbH, München

Content

1.

Introduction

3

2.

The elastic collision

4

2.1

Empirical approach

4

2.2

Collision mechanics

5

3.

Shockwave generation by piezo-stacks

6

3.1

The piezo-stack: an active bar

6

3.2

Symmetrical shock generator

7

3.3

Single end piezo-shock-generator

8

4.

Characterizing piezo-stack based shockwave generators

9

5.

Electrical pulse excitation

10

6.

Experiment studies on piezo-shock-generators

12

6.1

Experimental set up

13

6.2

Symmetrical piezo-shock-generator

14

6.3

Single output shock generation

15

7.

Discussion

16

8.

Outlook

17

9.

Shock wave theory in bars and rods

18

9.1

Impact propagation in the ideal rod

18

9.2

Stress impulse generation by a ramming rod

20

9.3

Piezo-actuator as active rod

222

Stand derIntroduction
Piezo-Multilayer-Technik

Mechanical impacting is involved in a lot of technical processes, e.g. for demolition of concrete bychiselling, for structure borne sound analysis,for impact-echo analysis of extended structures,for characterization of material properties at highstrain rates or indentation hardness tests.
For a precise understanding of this paper aredefinition of the used "shock-related" terms isneeded:When two rigid bodies (bars or rods) are colliding,then for a short time they are interacting by
inducing a rapid change of mechanical stress within
their materials. It is assumed, that this interaction ispurely elastic and no plastic deformation occurs.
This elastic stress propagates into the bodies with
their particular speed of sound and this phenomenon is called "stress impulse". In English
literature often the inconsistent expression "shockwave" is used to describe this situation.
When it is done here, "shockwave" and "stress
impulse" are synonyms.
The above ideal situation of shock propagation has
to be distinguished from the technically usedeffects, when an impact is to be created by the
indentation of a tool's cutting edge (e.g. from achisel or drill) into the surface of a rigid counterpart.
This is generally a non-linear interaction, where therelated constitutive law depends self-evidently on
the properties of the target material.
The extreme situations are plasticity deformations
without material removal on one side andchipping/crushing of highly brittle material likeceramics on the other side.
A detailed understanding of shock wave generationis the key for increasing the efficiency of impact
processes.
For generating an impact, in most cases, a hardmass body is accelerated during a starting phase(e.g. a hammer head), colliding then with a hardcounterpart (e.g. chisel). By this interaction of µs
duration, impulse and energy are transferred
between the bodies. The details of this process aredefined by the acousto-elastic properties of the
impact partners.

01

The starting and contact phases of such a classicalimpact scenario are hard to be reproduced exactly.
Common setups of precision impact measurement
do not allow high repetition rates.
Further the special feature of a µs-precise timing of
the impact event cannot be done by classicalsetups.
These restrictions can be overcome by piezomechanical technology providing an adaptive
impact generation principle, where the impact
parameters and the timing can be controlled by theelectrical driving conditions.
This paper describes the shockwave generation by
axially active piezo-stacks featuring following
essentials.
No pre-impact acceleration phase:● The impact partners are brought into mechanicalcontact before the impact generation.
● The complete system is at rest before igniting theimpact process.
●

Shockwave parameters like energy content, pulse
duration, acceleration rates are preset by theelectrical pulse parameters.

●

Shocks can be generated with high repetition
rates and high reproducibility (1kHz within aburst).

●µs precise time of the shock event●

Impact generation at non accessible sites●

Miniaturization (e.g. for shock sensor calibration)●

Impact generation under exotic driving conditions(e.g. cryogenic temperatures)●High acceleration rates (up to 500.000 m/s² )●High forces (up to several 10 kN
depending on the shock generator design).

The cooperation of HILTI AG and PIEZOMECHANIKGmbH aimed for the investigation of high power
levels by using big-sized high load piezo-stacks.
3

2

The elastic collision
2.1 Empirical approach

A physical shockwave is defined as the generationof local deformation/disturbance within an elastic
medium (e.g. a steel bar) by a rapid process (e.g.
collision with a hammer head). The resulting stress
distribution is then propagating through the medium
with the velocity of sound (approx. 5 km/s in steel).
Physical impulse and energy are carried by thisshock wave.
Related to the stress pulse propagation is a displacement of the material by the compression act. The
velocity of this displacement is the so called"particle velocity". This velocity is smaller by several
orders of magnitude compared to the speed ofsound (range of several m/s).
Particle velocity and involved mass displacement
are defining the related physical impulse.
The details of shock propagation depend strongly
on the nature and shape of the collision partners.
The response of interacting spheres differs from thecollision of bars (due to the excitation of internal
degrees of freedom).
Collision experiments (e.g. for material characterization with high strain rates) are often carried out
by using metal bars, because this situation can behandled straight forward by the mechanical theoryof thin bars (see chapter 9).
The dynamic deformation/strain of such metal bars
is detected by strain gages, the particle velocity by
LDAs (LaserDopplerAnemometer) monitoring the
bar's surface.
Shockwave propagation becomes more complex
when passing either into another kind of material orif the cross section of the bar is changing. The
pulse is split into reflected and transmitted parts.
The quantitative analysis of this situation allows thecharacterization of the high dynamic elastic
properties of material probes under very high strain
rates as carried out by split Hopkinson bar
experiments (Figure 2).

Strain gauge

Test-specimen
LDA-spot

Straingauge

LDA-spot
Shockwave

Figure 1: Schematic of a shock wave propagation within asolid bar, detected by strain gage and LDA measurements.

4

Straingauge

LDA-spot
Transmitted
shock-wave

Figure 2: Hopkinson bar experiment. The exciting and transmitted shockwaves are measured by strain gages and LDA.

2.2 Collision mechanics

In practice, the following basic shockwave situations
are usually discussed
A Free moving bar end: no collision partner.
A propagating shockwave will be completelyreflected back into the bar.
Bar's end is moving with maximum particle
velocity.
The acceleration rate of bar's end is doubled by
the reversed motion.
B Bar's end blocked and cannot move:
At bar's end the particle velocity is zero mechanical stress/compression is maximal.

C Acoustical matching:
Describes the shockwave transfer at an interface
e.g. between two metal bars.
The shockwave passes the matched interface
without losses in energy and impulse.
For optimizing the acoustical match, thematerials and cross sections of the interacting
bars are essential.
D Acoustical mismatch:complementary situation to c.
The incoming mechanical pulse is partiallyreflected and transmitted at the contact point.
Energy is transferred incompletely (see figure 2).
The shockwave reflection at a seismic mass isthe extreme case of an acoustical mismatch:
The complete shockwave energy is reflected(see figure 5).

5Shock wave generation
by piezo-stacks

3

3.1 The piezo-stack: an active bar

A piezo-stack can be described as a solid bar of
PZT-ceramics. When this PZT bar is electricallycharged sufficiently fast, the internal stress jumps
instantaneously to a high level:
The initial pressure is the blocking pressure, causing
now an accelerated expansion of the PZT-stack:a shock is created. By coupling the PZT rod toanother solid body, the shock impulse can betransferred and a shock wave is propagating.
In this terminology a PZT-stack is an "active bar",

6

generating inherently mechanical shock pulses byelectrical pulse excitation.
Because the active bar is at rest, two pulses are
generated propagating in opposite direction tobalance the total impulse to zero.
This kind of pulse generation is called to be "superelastic", because the kinetic energy of the systemafter the shock generation is higher than before theshock event.
This situation is similar to the use of an explosive.

3.2 Symmetrical shock generator

The above mentioned basic symmetry of the shockgeneration within a resting active bar can be used
for a symmetrical arrangement with two opposing
shock outputs of identical impulse content.
This allows elegant designs for calibration
experiments.

PZTImpulse stack

LDA
spot

LDA
spot
steel rod

Straingauge

HVElectric
current
powerswitch puls

Straingauge

TriggersignalFigure 3: Symmetrical piezo-shock-generator with two-sided
shock propagation. The shock parameters are detected bystrain gages and LDAs.

Figure 4 b: High load piezo-stack with 35 mm diameter, withshock proof wiring.

Design of a symmetrical piezo-shock-generator
The shockwave is generated within the piezo-bar
and transferred to both ram elements (steel, titanium, brass etc.). Acoustical matching for maximum
energy extraction is done by adoption of the crosssections of PZT-stack and the metal rams.

High power
PZT stackInternalprestress

Figure 4 c: Symmetrical shock generator: shock output via
the rams (left and right).
Ram 2
Steel
Titanium

Ram 1
Steel
Titanium

CasingElectricalwiringFigure 4 a: Schematic of a symmetrical piezo-shockgenerator.
7

Stand3.3
Singleder Technik
end piezo-shock-generator

The above mentioned symmetrical generator setup
can be modified towards a single ram element witha nearly doubled energy and impulse output. This isdone by applying a bigger mass for supporting the
PZT-stack at one side (seismic mass). The impulse
will be reflected there and directed towards theother output. A double pulse is created bycontaining nearly double the pulse energy (byelongating the total pulse output duration). The
compensating impulse is transferred to the seismic
mass as recoil (similar to a rifle).

High power
PZT-stack
Seismic body

Internalprestress

Ram Steeltitanium brass

Casing

PZT-stackImpuls reflection

Design of a single end piezo-shock-generator

Electricalwiring

LDA-spot
Steel rod

Figure 6 a: Schematic of a single output piezo-shockgenerator.

Seismic mass SM
HVcurrentElectric
power switch puls

Straingauge

HVP

TriggersignalFigure 5: Schematic of a single end piezo-shock-generated
system by application of a seismic mass. The interactionof the travelling pulses results in an elongated double pulse
output.

Figure 6 b: Single ouput piezo-shock-generator: brass part:seismic countermass.

8

Characterizing piezostack based
shockwave generators
The piezo-mechanical performance parameters of apiezo-shock-generator can be derived straightforward from the characterization of piezo-stackactuators.
●●

Free ram stroke:
Depends on PZT-stacks length, the applied electrical field strength and type of the used PZT
ceramics. Strokes are ranging from µm to hundreds of µm.
An efficient energy transfer must be completed
within this stroke range limit.
Piezo-shock-generators need therefore hard collision partners.
Internal starting force:
Generated force level within the PZT-bar uponcomplete electrical charge transfer.
When the electrical rise-time is significantly shorter than the shock generator's mechanical risetime, the initial force corresponds to an actuator's
blocking force. It depends mainly on stack's
cross section, applied electrical field strength andtype of used PZT material.
This starting force accelerates then the massesof the active PZT rod.

4●

Pulse-width:= shock propagation time through the active PZTrod.
It depends therefore on speed of sound in the
PZT stack and stack's length.
By pulse reflection at a seismic base, the resulting pulse length is doubled.

●

Particle velocity:
It corresponds to the maximum achievable shift
speed of a piezo-actuator.
Order of magnitude: several m/s.

●

Mechanical energy content:
It depends on volume or mass of the PZT-stack,the applied electrical field strength and the properties of the PZT ceramics.

●

Impulse content P of the mechanical shock:P = moved mass within the shock front multipliedby the related particle velocity.
Order of magnitude: kgm/s.

●

Piezo-ceramics:
High dielectric, high strain PZT materials provide
a much higher shock energy density than "lowcapacitance" actuator PZT materials (up to afactor 2).

9Electrical pulse excitation

The electrical equivalent circuitry describes a piezoceramic component mainly as a capacitor. By
applying a huge charge impulse to this PZTcapacitor (= generator stack), a mechanical shockwave will be produced. Our experiment aimed forthe characterization of large volume, high dielectric
stack actuators for a maximum mecha-nical pulse
energy output.
A typical arrangement was:
Piezo-generator capacitance: order of magnitude
10 µF
Impedance of charging circuitry during pulse
ignition: 1 ohm
Equivalent RC time constant: 10 µs
This time constant is significantly shorter than thetransition time of the mechanical pulse.
Applied peak voltage: up to +800 V
Peak current: up to 800 A
It has to be taken into account, that by high field
excitation, the effective capacitance is remarkably
higher than the above stated small signal value.
This effective capacitance must be explicitly derived
from the electrical charging parameters.
For shock generation, short-term a huge power
level is needed in the order of magnitude 0.5 Megawatts.
It was produced by the following circuitry (seeFigure 7).

105

The HighVoltagePulser HVP consists of a condensator bank of parallel and seriel high voltage capacitors with a total capacitance of typically 500 µRF>> piezo-shock-generator's capacitance.
This storing capacitor is charged up to a level up to+800 V by a standard power supply (not shown inschematic).
By an IGBT power transistor switch (2), the piezoshock-generator is coupled to this electrical charge
reservoir. Due to the very low resistance of thiscircuitry, a fast charging pulse of the piezo-element
occurs.
In terms of maximizing the shock wave energy, two
operating modes have been compared, differing inthe start conditions for the electrical pulse:
It is well-known from common piezo-actuators, thata much higher mechanical energy output isachieved by the so-called semi-bipolar operation
instead of the unipolar mode. "Unipolar mode"means, the electrical pulse is applied to adischarged piezoelement (pre-pulse voltage level 0,
voltage step 0V/Umax V).
A dramatic increase of the mechanical power outputof the shock generator is achieved by a pre-pulse
conditioning of the piezo-shock-generator by connecting it to a negative voltage power supply withlevels down to -200 V. (switch position 1). To one
part, this is simple caused by the wider voltage step
applied to the actuator (voltage step -200V/+Umax).

5. Electrical pulse excitation

Schematic of semi-bipolar pulse generation (HVP)
800

800
negative prepoledvoltage on
voltage off
stand bay

700
600

700
600

300
200

Stress (V)Stress (V)Switch position
1: negat. precharge
2: pulse charging: shock !
3: discharge

400

400
300
200

1 2 3

100

1 23

0

100

-100

0

-200
-10

-100
-200
-8

40 Hz

500

500

-6

-4

-2

0
2
Time (ms)

4

6

8

-5

0

5

10
15
Time (ms)

20

25

30

35

10

Switch47<=1-200 V

800 V
500 µF

1
2
3
47

Piezo-stack
10 µF

Figure 7: Schematic of bipolar switching of a power piezo-actuator by a HighVoltagePulser (HVP). The unipolar mode uses
0V instead of the -200V pre-pulse conditioning.

A further contribution to the power balance is doneby the nonlinear poling response by applying anelectrical counter field to a high dielectric PZTceramic (here by applying the -200V pre-pulse
setting).

For resetting the system after the shock wave
generation, the piezo-shock-generator is simply
discharged by a simple resistor (here 47) via theswitch position 3. Burst operation with 100 Hz on
the described big size PZT-elements has beensuccessfully carried out.

11

Experimental studies
on piezo-shock-generators

The cooperation between HILTI/Schaan/Liechtenstein and PIEZOMECHANIK/Munich/Germany
was focusing on high power mechanical shock
generation by using high voltage high load piezostacks.
HILTI is the leading manufacturer of highest quality
tools for demolition, chiselling based on the electro-

126pneumatic hammer principle. The high efficiency ofthese tools is based on the very exact matching of
all involved parts for the shock transfer from thehammer head down to the tip of the chisel.
Piezo-shock-generators with their high power
levels, high repetition rates and reproducible shock
parameters are used for R&D activities to shorten
test periods on these components significantly.

05
6.1 Experimental set up

High voltage high load piezo-stacks with an active
diameter 35 were used as the electro-mechanicalimpulse converter.
Emphasis has been put onto a shock proof designof the stack itself and the applied high current electrodes.
At the very beginning of this project, standard "lowdielectric, low capacitance" PZT-actuator ceramics
has been ruled out because of its reduced "shock
power" efficiency. Consequently all shock generator
stacks were based on the highly dielectric highpower PZT-material HP from PIEZOMECHANIK.

For an optimum acoustical matching, the piezostacks where combined with steel bars of 18 mm
diameter.
The shock pulses propagating within the steel bars
have been characterized by Laser-Doppler-Anemometry (LDA) determining the particle velocity atbar's surface, when the shock wave passes by.
From this particle velocity, the related strain
variation has been derived. Together with the wellknow elastic properties of steel, the energy andimpulse content of the shock front can beevaluated.

13

6.2 Symmetrical piezo-shock-generator

Comparison of the shock profiles within the twosteel bars derived from the above mentioned
symmetrical arrangement:

45
40
35

Stress (MPa)30
Supper rod

25

Slower rod

20
15
10
5
0
-5

0

10

20

30

40

50
60
Time (µs)

70

80

90

100

Figure 9: Shock profiles of a 50 mm active length symmetrical pulse generator, electrical pulse level 500 V.

Figure 8: Symmetrical pulse generator vertical arrangement.
Both sides are coupled to steel bars for two sided impulse
extraction. Notice the slits in casing and mounting flanges
steel for getting optical access for the LDA-measurements.

14

Timing is triggered by the transistor signal. The
delay of the onset of the mechanical pulse is due tothe distance from shock generator to the LDAmeasuring spot on the steel bar.
Take notice of the nearly identical shock profiles.
The slight offset between the two pulses is due toslight differences of the distances of the LDA-spots
from the pulse generator.

6.3 Single output shock generation

120
100
80
Stress (MPa)The asymmetric shock generator with seismic baseuses the same piezo-stacks like the symmetricaldesign. Figure 10 shows a "head down" arrangement of the shock generator: the contact between the piezo-shock wave generator and the steel
bar is thereby preloaded with the weight force ofapprox. 100 Newtons.

60
40
20
0
-20
0

20

40

60

80

100

120

Time (µs)

Figure 11: Mechanical stress/time profile time for a burst ofshocks. Active stack length: 120 mm. Shockwave energy:
3.25 J. Take notice of the remarkable coincidence of themechanical pulses.

Figure 11 shows the typical single output pulse
profile exhibiting a double peak structure of thestress response. This is due to superposition of the
primary impulse and the impulse, reflected at theseismic mass. This elongates the total propagation
time and leads to longer lasting output pulse.
The energy content is increased nearly to a factor of
two compared to a single pulse of a symmetricarrangement.

Figure 10: Test set up using a single output piezo-shockgenerator with seismic mass for impulse reflection.

Piezo-stacks with a length up to 200 mm have beentested with a mechanical shock energy content of
more than 4 J (energy measured in the steel bar ! ).
Physical impulse contents up to 2 kgm/s have beenverified.
For a "free end" bar experiment acceleration rates
up to 500.000 m/s² can be expected at the bar's
tip.

15

Discussion

Our experiments verified the specific properties ofaxially shock wave generation by using piezo-stackactuators.
A The results fit well into the bar theory of shockpropagation as the interaction of axially activatedelements.

7
C No pre-pulse "external mass acceleration" phaseis needed for generating a shock. The wholesystem is at rest and the shock generator isalready in contact with the mechanical counterpart immediately before to shock release.
D µs-precise timing of the shock-event byelectrical means.

B High levels of extractable mechanical energy inthe range of 7 J/kg piezo-ceramics are achievaE Variation of shock parameters like amplitude andble, provided a mechanically high efficient material
pulse width by electrical means.
is used (like PIEZOMECHANIK's HP-PZT) and thesemi-bipolar operating mode is applied. Then the
F Excellent reproducibility of shock profiles.
power balance exceeds that of standard actuator
PZT by a factor of two.
G High repetition rates (in bursts).

16

Stand derOutlook
Piezo-Multilayer-Technik

The normalized mechanical parameters of a piezostack like strain, blocking pressures, energy density
are (nearly) independent of the actual dimensions of
the piezo-element.
Therefore piezo-shock-generators can be successfully designed over a wide range of dimensions,
ranging from "big block" structures down tominiature shockers with similar electro-mechanicalconversion efficiencies.
Shock testing as a kind of DIRAC-pulse application
with its very fast rise time, high acceleration level,
excellent reproducibility and elevated repetition
rates allow mechanical testing/calibration ofcomponents like acceleration/crash sensors. New
approaches for quality inspection, mode analysis,
structure borne sound evaluation are under08

investigation as an alternative to harmonic highfrequency shaking with frequencies > 10 kHz.
A complete new feature in shock generation is theµs precise timing capability, what allows synchronizing of mechanical shock events with other fastphysical processes.
Fixed "Phased Arrays" of piezo-stack based shock
generators will allow shock front generation withvariable propagation characteristics.
Piezo-stacks are robust devices, which allow thepermanent integration into extended mechanicalstructures like buildings etc. for long term structure
health monitoring.
In an inverted operation mode, the described piezosystem can be used as shock absorber.

17

Shock wave theory in bars
and rods (Impact in rods)

9

9.1 Impact propagation in the ideal rod

The ideal rod is characterized by a small diameterto length ratio. An elastic signal is propagated withthe related speed of sound without changing theform of the signal according to the ansatz ofd'Alembert to solve the wave equation (s. Timoshenko). To be exact this stress impulse should not
be described as wave or shock impulse, but asthese designations are already very common inliterature we used it in the first part of this paper.
Here in the theoreticaI section we shall use stress
impulse instead of shock wave.
For the axial propagation of elastic stress impulses
the theory of thin rods (Graff, Johnson) can beapplied. A stress signal launched from the left in agiven homogenous rod with cross section a will be
propagated to the right with the speed of sound(Figure 12) c given byc = 앀(E /ρ)cSynchronously with the stress impulse sigma (t) thesignal of the particle velocity v(t) propagates along
the rod (Figure 12) where the fundamental relation
holds
σ(t) = -I v(t)A consequence of the 2. law of Newton. I represents the acoustical impedance given by
I=ρc
The law of Hooke correlates stress σ and strain εaccording to
σ = Eεwhere E is Young's modulus.
A fundamental property of a stress impulse is theequality of elastic and kinetic energy during propagation, except at the ends of the rod and as longas cross section and/or impedance do not change.
A harmonic wave behaves different, it periodicallyexchanges elastic and kinetic energy completely
with the frequency of the wave.
In general two stress impulses propagate independently in both directions. The resulting stress is thesum and given as a function of position and time
σ (x,t) = σ right (x-c t,t) + σ left (x+c t,t).
For steel c is about 5200 m/s.
The corresponding particle velocity as a function of
position and time readsv(x,t) = 1/ I (σ right (x,t) - σ left (x,t))c

Figure 12: Stress impulse (top) and related particle velocity(bottom).

18

with the designationstressσparticle velocityv
impulse momentum P
energyW
impedanceI9.1 Impact propagation in the ideal rod

The impedance of a steel rod is about 4 107 kg/(m2s)and for a stress impulse of 200 MPa the particle
velocity is about 5 m/s. The impulse energy as
mentioned above is to one half elastic which is givenby the integral

case kinetic and in the second case completely
elastic.
For the change of the rod cross section areas froma1 to a2.
Simple transmission - τ and reflection - r rules arevalid for the stress:
τ = 2 a1 / (a1 + a2)
r = (a2 - a1) / (a1 + a2)

or the kinetic part
Equivalent rules are given, if the impedances I1, I2changeτ = 2 I2 / (I1 + I2)
r = (I2 - I1) / (I1 + I2)
The techniques to measure these two energy contributions are different. For the elastic part strain gages or if both, areas and impedances change
are used, for the kinetic energies Laser Doppler
τ = 2 a1I2 / (a1I1+ a2I2)
Anemometry (LDA) is applied.
r = (a2I2- a1I1) / (a1I1+ a2I2)
The propagated impulse momentum PImpulse is
The reflection will be zero if the relation I2/I1 = a1/a2
determined by the integral
holds.
For the particle velocities similar rules are valid, notgiven here (Johnson).
The impulse reaching the end of the rod will bereflected. If the end is acoustically free (soft) the
stress amplitude changes sign, the particle velocity
not. If the end is acoustically hard (fixed), it is vica
versa. At the end of the rod all energy is in the first

The limit of the simple rod theory is reached if thelateral dimension of the rod has to be taken into
account leading to additional inertia effects. The
stress state is no longer constant along the crosssection (no plane situation).

199.2 Stress impulse generation by a ramming rod

Usually a stress impulse can be generated by the
impact of two cylindrical rods. An example are theimpacts of flying piston, ram and shaft of the Hilti
EP-tool (Electropneumatic Percussion). Here now
the fundamental difference between the real
hammer and the piezo-hammer is based: a hammer
is ramming - the piezo-actuator is impacting.

Dynamically the Hertzian contact between rod 1
und rod 2 with the effective mass meff
1/meff = 1/m1 + 1/m2)
and the dynamic force
fcontact(w(t)) = kHertzw(t)3/2

If the surfaces of contact are spherical the contact
theory of Hertz (Johnson) can be applied. The
contact of ideal plane surfaces has to be treatedacoustically or as the numerical limit of very large
contact radii.
Definitions:
The indices 1 and 2 are related to the rod 1 androd 2, respectively.
E is the Young's modulus, ν is Poisson's ratio. For
the effective modulus Eeff:
Eeff = (1 - ν12 ) / E1 + (1 - ν22 ) / E2
holds. For the effective contact radius reff theequation hold:1/reff = 1/r1 + 1/r2
With Eeff und reff the stiffness constant of the
Hertzian contact kHertz is written

lasts during the contact time Tcontact
Tcontact = 2.9432 (15/16)2/5 (1/Eeff)2/5 (meff)2/5 (1/reff)1/5 (1/v)1/5
with v beeing the relative velocity of the two rods
before impact.
Before impact, the rod (e.g. the flying piston) is
flying freely and can be represented in terms of twostress impulses one to the right σ right and a second
to the left σ left both with the same amount σ 0
σ 0= Iv0/2
but with opposite sign. For the particle velocity the
relation holds
1/I(σ rechts - σ links) = v0
During the contact energy and impulse momentum
are exchanged.

kHertz = 4/3 Eeff reff1/2
The force of contact is given byfcontact (w) = kHertzw3/2
where w is the interpenetration of the two surfaces.

20

For a dynamic Hertzian contact the force of contact
is in good approximation given by an Gaussian
signal form where impulse maximum and impulse
width depend on the given parameters. In theexample of figure 13 the maximum is assumed to
50 kN and the width to 50 µs.

9.2 Stress impulse generation by a ramming rod

Stress impulse

Motive force
200

50

180
160

40

Impulse (MPa)

Force (kN)

140
30

20

10

120
100
80
60
40
20

0

0
0

10

20
30
Time (µs)

40

50

0

10

20
30
Time (µs)

4050Figure 13: Force of contact as a function of time (left) and stress impulse as a function of time (right).

The maximum force of 50 kN (5t) in the rod of
18 mm diameter corresponds to a stress of about
200 MPa (the tension stress of Steel St52 is about
520 MPa).

The Gaussian signal form can be represented ingood approximation by

Force and stress are related byand the energy can then be analytically calculatedto
W = 4/3 T/π a / I σ 02

a is the cross section of the rod. The change of
impulse momentum Δp:

It is worthwhile to mention that impact and a single
harmonic wave do not correspond. The Gaussian
like stress impulse needs a Gaussian Fourier
distribution of frequencies in time and space.
A stress impulse running for- and backward within
the rod has only the period in common with thefundamental resonance vibration.

is the integral over the force as a function of time(second law of Newton). Δp = 1.3 kgm/s.

σ = σ 0 sin(πt/T)3/2.

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9.3 Piezo-actuator as active rod

There are two piezo-effects the direct and theinverse (both 1880 invented by J. und P. Curie atcristalls of quartz).
The direct piezo-effect will not be treated here. The
inverse piezo-effect is described by the state
equation in scalar form
ε = cEσ + d33 E
ε = strain
E = electrical fieldcE = elastic constant of compliance at constantelectrical field (e. g. E = 0).
σ = stress
d33 is the axial piezo-electrical constantFor σ = 0 the strain is direct proportional to theelectrical field. If the strain is blocked (ε = 0), thenσ = -d33/cE E
representing a compression. This relation is thebasis of the stress impulse generation using piezoactuators. As the piezo-actuator is a cylindrical rod
we call this special type of rod active.

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Connecting all piezo-disks of the stack at an instantof time with an electrical voltage (see chapter 5) the
piezo-rod (stack) will react with an instant strain.
The form of an active rod can be represented by
two stress impulses σ right und σ left with equal amountσ 0 and equal sign but opposite propagation
directions. I.e. the particle velocities are also equalin amount but opposite in propagation direction.
They are compensating at the beginning, contrary
to the free flying rod where the stresses arecompensating.
For the stress energy follows
W = ½ Vpiezo Epiezo d332 (U0/d)2
VpiezoEpiezoU0/ddvolumen of the piezo-stack
Young's modulus of the stackelectrical field strength given by the ratio ofvoltage U0 and dthickness of an individual piezo-disk

In reality the stress impulse is not ideal square. Due
to finite rise time of the electrical voltage signal
convolution effects are rounding the ideal squarestress impulse form.

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