generation of building and ground vibrations

endurance test

impact echo method

material testing

Piezo compositesBeyond positioning

Piezo composites for vibration excitation and pulse generation

Contents

1 Introduction 2

1.1 Characteristics of piezo-ceramics . . . . . . . . . . . . . . . . . . . 2

1.2 Structure of the actuators . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Actuator designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Stack configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Heat management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Load capacity and force limit . . . . . . . . . . . . . . . . . . . . . . 8

2 Beyond positioning 8

2.1 Force generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Blocking force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Dynamic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Piezo-shakers PiSha 12

3.1 Functional principle of piezo-electric shakers . . . . . . . . . . . . 12

3.2 Important characteristics of piezo electric shakers. . . . . . . . . . 13

3.3 Comparison of PiSha to electromagnetic shakers . . . . . . . . . 13

3.4 Benefits of piezo-electrical shakers . . . . . . . . . . . . . . . . . . 14

3.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Piezo-electrical shock-wave generators PIA 15

4.1 The application objective of piezo-electric shock generators . . . 17

4.2 The physical shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Piezo composite stacks as shock-wave generators . . . . . . . . . 19

4.4 The piezo stack as shock-wave generator . . . . . . . . . . . . . . 20

4.5 Electrical driving of piezo shock-wave generators . . . . . . . . . 23

4.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.7 Theory of the bar shock . . . . . . . . . . . . . . . . . . . . . . . . . 25

Nomenclature 32

i

1 Introduction

For common piezo-actuators no quartz or single crystals are used, but

special oxidic ceramics based on the PZT-System (Pb lead, Zr zirconate,

Ti titanate). This compound class shows much better piezo-electrical and

piezo-mechanical efficiency than quartz.

1.1 Characteristics of piezo-ceramics

The PZT-formulation can be varied with regard to stoichiometry and doping

resulting in a broad spectrum of material properties optimized for different ap-

plication profiles. Not all desirable properties can be put into one compound.

Because of the diverse requirements it is always a compromise when selecting

a suitable material for a certain application. Developing new piezo-materials is

a steadily ongoing process in the ceramic industry.

PZT is the most widely used smart material for solid-state actuation. Alternative

materials with enhanced strain capability are under study, but all these”inno-

vative“materials have severe drawbacks regarding common driving conditions

in practice.

Response characteristics

With piezo-composites from piezosystem jena response times of a few micro

seconds are viable. Thereby accelerations over 10'000 g are reachable. See

chapter 4 Piezo-electrical shock-wave generators PIA.

High force generation

Piezo-composites from piezosystem jena are able to produce forces of several

kilonewtons. Please find detailed information in chapter 2.1 Force generation.

Influence of magnetic fields

Piezo-composites are not influenced by magnetic fields and do not influence

magnetic fields respectively. Furthermore piezo-composites can run in mag-

netic field free environments because they don't producemagnetic fields. That's

because the piezo electric effect is based on electric fields.

2

Wear

The elongation of piezo-composites is based on a solid state effect as a result

of the deformation of the crystalline unit cell. Therefore piezo-composites do

not have parts that are movable relatively to each other, resulting in a wear-free

operation. Because of that piezo-composites are well suited for clean-room ap-

plications.

Temperature behavior

Capacity increase The dielectric coefficient of piezo-ceramics depends on the

temperature. The capacitance of common ceramic materials raises with the in-

crease of temperature. An increase of 60°C raises the capacitance of 20%. This

is important for the selection of the proper amplifier.

Thermal expansion High voltage actuators from piezosystem jena have a very

low thermal expansion coefficient. This is based on the composite-structure of

the stack. The ceramic has a negative coefficient, the metal electrodes have a

positive coefficient this leads to a partially compensation of the thermal expan-

sion.

Temperature dependency of the piezo-electrical effect The temperature depen-

dency of the expansion is based on the curie-temperature of the selected ce-

ramic material. A high curie-temperature leads to a slightly higher expansion

whereas a lower curie-temperature causes a slightly lesser expansion. In both

cases the changing is around 5% for a temperature changing of around 80°C.

piezosystem jena would be happy to advise you about the

temperature behavior of piezo-composites and will find

the optimal solution for your applications.

3

1.2 Structure of the actuators

High voltage actuators for voltage ranges up to 1000 V are made from com-

pletely sintered and finished individual PZT discs or plates prior to stacking.

The inserted layer electrodes are made from separate thin metal foils. There-

fore high voltage actuators are not monolithic, but a composite material.

Fig. 1: Piezo composite stack. 1. front discs without electrical contact, 2. completely sintered piezo ceramic discs,3. opposite poled electrodes.

4

1.3 Actuator designs

Cylindrical shapes are most common for piezo composite actuators. They can

be classified into solid and hollow cylinders. The length of the actuator depends

on the number of discs and electrodes used. They are manufactured in discrete

joining technology. The method is the same for both types. The aperture of

hollow cylinders can be used for active or passive cooling systems in dynamic

applications. See therefore chapter 1.5 Heat management.

Fig. 2: Piezo composite of different designs, diameters and lengths.

1.4 Stack configuration

Most customers relate the functionality of a piezo stack only to the used piezo

ceramic material. However, it is a common experience that all aspects of the

stack-design, manufacturing technique and finish of the whole component are

essentials regarding performance and reliability.

5

Following points require attention:

• joining techniques

• electrode design

• internal insulation techniques

• surface insulation techniques

All of the structural components of a stack are subject to strongly varying me-

chanical and electrical load conditions during dynamic cycling. This leads to very

different levels of reliability when considering different actuator concepts for a

distinct application. Piezo stacks from different sources will differ not only in the

used ceramic, but also in different manufacturing techniques. These differences

are not necessarily expressed in the short term performance data. Careful eval-

uation of the proposed piezo stacks must be undertaken when an application

requires high reliability. piezosystem jena stack actuators range from general-

purpose elements for low and medium dynamic applications up to specially

designed elements for very high mechanical dynamics. See chapter 4 Piezo-

electrical shock-wave generators PIA and chapter 3 Piezo-shakers PiSha.

Further upgrades of the stack actuator are the use of metal casings with

internal pre-stress mechanisms and other options like internal heat man-

agement, position sensing etc..

Fig. 3: Schematic of integration of a piezo stack into a preloaded casing.

6

1.5 Heat management

During dynamical operation of piezo-ceramics, electrical energy is dissipated

into heat due to”internal friction“ of the piezo ceramic structure. Approxi-

mately 5 to 20% of the electrical energy input is converted into heat. The loss

mechanisms are rather complex and are not represented in a realistic way in

the standard material data sheet for piezo-ceramics. The heat generation de-

pends to some extent on the mechanical operating conditions (stroke-force).

Blocked actuators show rather low losses, because the”internal structure“ of

the ceramics cannot move, therefore no”structural friction“ is present within

the ceramics.

The dynamic charging power of a piezo-actuator is partially dissipated into heat.

The actuator temperature reflects the equilibrium between heating by power

dissipation and cooling by the heat transfer to the environment. Too high tem-

peratures interfere with actuator performance and reliability. piezosystem jena

offers the option”thermostable“ to handle heat generation by dynamic opera-

tion. Heat is effectively removed from the ceramic stack and transferred towards

the casing. The”thermostable“ casing is made from thermally high conductive

metal (like copper, brass, aluminum etc.) providing effective heat sinking con-

tact to the environment. The benefits of the”thermostable concept are:

• high heat dissipation from the piezo ceramic

• homogenization of the temperature distribution

• avoidance of hot spots

Just the simple mounting of a”thermostable“ actuator to the operated me-

chanics provides a remarkable cooling effect. For very high thermal loads, all

common methods for enhanced cooling can be applied to the actuator casing

like forced air cooling or liquid cooling (Figure 4).

7

clamping

Fig. 4: Left: Thermal image of a clamped, dynamically cycled piezo composite actuator in common environment.The cooling because of the clamping is clear to see. Right: Various types of actuators with thermo-stableheat management. Air fin corpus can be added.

1.6 Load capacity and force limit

The maximum load force shown in the data sheet is the steady load. Within this

preload range, the full motion and force generation capability of an actuator

can be used. For higher preloads Fc, the PZT-ceramic performance decreases

as a consequence of de-poling effects (reversible!). Mechanical damage does

not occur (Damage threshold up to 10 times larger than the de-poling limit).

Only for very large aspect ratios length/diameter > 15, bending and buckling

of the stack becomes a concern.

2 Beyond positioning

Piezo-composites are increasingly used in applications with high forces,

accelerations, shock-waves, vibrations of high frequencies or impacts. In

these cases the precise positioning is playing a subordinate role.

2.1 Force generation

Piezo composite actuators generate forces only if they are built in a sys-

tem with a stiffness > 0, i.e. if they are limited in their elongation. This

circumstance is shown in the work-distance-diagram (Figure 5). Under a

constant load the elongation will be the same as without any load.

8

Fig. 5: Work-distance-diagram for piezo-composites. The spring characteristic shows the maximum elongationand force generation for a piezo composite actuator, which is working against a spring with this stiffness.It illustrates, that it is impossible to reach the maximum elongation and the maximum force generation atonce.

If an actuator with stiffness cT is working against an alterable external force, e.g.

a spring with stiffness cF , it will generate a force Feff according to :

Feff = cT ·∆l0 ·(1− cT

cT + cF

)

An external, stroke-dependent force (e.g. spring force) reduces the stroke in

comparison to ∆l0. On this note the pre-load represents an external force. The

pre-load of piezo-composite-actuators can be applied for each specific appli-

cation. The resulting reduction of the stroke will be considered. Pre-loaded

actuators can take-up tensile forces, therefore they are well suited for dynamic

applications. To avoid breakdowns the tensile forces always have to be smaller

than the pre-load.

9

2.2 Blocking force

A rigidly clamped piezo-actuator (stiffness→ ∞, elongation= 0) generates

its maximum force Fmax, the blocking force (Figure 5). In real nature, all

materials show a limited Young's modulus that does not allow infinitely high

stiffness by a passive clamping. But using a closed-loop active arrangement,

the blocking situation can easily be verified. Please note, that the piezo-actuator

must not be operated above its maximum voltage. With cF → ∞ the equation

for the maximum force is reduced to:

Fmax = cT ·∆l0

Blocking conditions are necessary for the generation of blocking forces and for

material testing, e.g. Hopkinson-bar. Please find further information in chap-

ter 4 Piezo-electrical shock-wave generators PIA. Please note that the piezo-

actuator exhibits no elongation under blocking conditions.

Example

A PSt 1000/35/80 in closed loop at 600 V reaches a position that should be sta-

bilized. The maximum load-increases, the piezo-actuator is able to compensate

by a Voltage rising to 1000 V, is according to the 400 V blocking force of the

actuator. This blocking force amounts 14 kN.

2.3 Dynamic Operation

Piezo stacks can be operated with very high dynamics with acceleration

levels of thousands of g. The expression”dynamic operation“ is used for op-

eration modes where the actuator and the whole piezo-mechanical system are

facing additional reactions compared to a nearly static driving situation.

Dynamic operation adds effects like higher acceleration rates and forces super-

imposed on to the static force balances. The increased electrical power con-

sumption results in self-heating of the actuator stack.

Application examples for high dynamic actuator excitation are:

• scanning,

• motion control,

10

• vibration generation,

• pulsed operation,

• fuel injection,

• shock generators,

• shakers and so on

Piezo-actuators are mostly operated non-resonantly and can cover a wide fre-

quency range from DC up to > 10 kHz. With special designed PiSha piezo-

shakers it is possible to work around or above the resonant frequency. This is in

contrast to resonating systems like ultrasonic generators. Those are operating

continuously on a single high frequency > 20 kHz with large amplitude.

11

3 Piezo-shakers for vibration generation PiSha

Piezo-electrical vibration generators from piezosystem jena can be driven both

at frequency's up to 80% of the resonant frequency and, with special adaption's,

around the resonant frequency and above. Therefore a prestress of the shaker

is indispensable. It should be noted that the amplitude underlies the magnifi-

cation factor if driven resonantly. Above the resonant frequency the amplitude

decreases strongly. A bigger part of the yielded energy is converted into”inner

friction“ . Due to this process a lot of heat is generated so that an active heat

management is necessary. Please see chapter 1.5 Heat management for further

information.

piezosystem jena is your specialist for high dynamic applications.

Fig. 6: Area of operation of piezo electric shakers.

3.1 Functional principle of piezo-electric shakers

Piezo electric shakers are based on special stack type actuators that transform

the electric input signal into an elongation. The amplitude is set by the oper-

ating voltage. The velocity, and therefore the maximum frequency, depends on

the charge current.

The internal layout of the shaker ensures that the shaker withstands the high

accelerations, forces and pressure even at high long term stress.

12

3.2 Important characteristics of piezo electric shakers.

The following characteristics depend on the shaker setup

as well as the electrical control. Shakers can be optimized

at a multitude of parameters. piezosystem jena will find

the best solution for your application.

Frequency domain: up to 100 kHz, depends on the shaker setup

Amplitude: µm up to several 100 µm depends on the op-

erating frequency and the shaker setup

Force modulation: up to several 10 kN, depends on shaker setup,

operating frequency and installation condi-

tions

Acceleration: ≤ 100′000 m · s−2(10′000 g), small actuators

with small amplitudes (extreme accelerations

over 10′000 g are possible with piezo electric

shock-wave generators)

Compact design: the dimensions of the piezo structures reach

down to a few millimeters

3.3 Comparison of PiSha to electromagnetic shakers

Relating to the dimension piezo electric shakers possess a higher stiffness and

a higher force potential at shorter motion than electromagnetic shakers. There-

fore piezo electric shakers are well suited for applications with high frequencies

and high forces. Another benefit is the miniaturization. The piezo technique

makes high power density possible. See figure 7 for a schematic representation

of the operating area.

13

Fig. 7: Comparison of the operating area of electromagnetic and piezo-electrical shakers.

3.4 Benefits of piezo-electrical shakers

In difference to ultrasonic transducers piezo-electrical shakers work over a wide

frequency range. Piezo-electrical shakers are designed for highest dynamics

and well suited for long term operation. They are characterized by:

• high stiffness

• high forces and pressures

• large frequency range

• miniaturized shakers down to a few millimeters

• setups with seismic mass up to 200 kg (440 lbs)

Therefore they are well suited for different applications like modal analyses

of mechanical structures,material characterization, fatigue testing, defect

investigation etc.

14

Fig. 8: Left: Comparison of the size of piezo micro shakers with thimble. Right: Schematic representation of thegeneration of ground vibrations via geo-shaker. Overall height ≈ 0.3 m.

3.5 Outlook

Due to a large number of variation possibilities concerning to the shaker setup

and the control piezo-composites are a good alternative to common shakers.

Piezo-electrical shakers are available in countless setup variations and scalable

lengths of few millimeters up to several 10 cm.

Other than ultrasonic generators piezo-shakers are able to cover a large fre-

quency domain so it is not necessary to use a couple of shakers for one ap-

plication. Piezo-electrical shakers display their full power at high frequencies

around the resonant frequency and above.

4 Piezo-electrical shock-wave generators PIA

Using piezo shock-wave generators it is possible to produce precisely trig-

gered mechanical impulses. The parameters of these pulses, e.g. accelera-

tion and pulse-width are adjustable. Before the shock the impact partners

remain in contact, therefore high repeatability's are possible.

Mechanical shocks occur in many technical tasks in everyday life, e.g.:

• demolition works at concrete and other building materials with chip-

ping hammers,

• impact based measurement technologies like solid-borne sound in-

vestigations,

15

• impact-echo-technique for geological and structurally investigations,

• at determination of material properties,

• at high strain rates,

• at indentation hardness tests.

The quantitative physical processes of the shock process are often considered

insufficiently. A deeper understanding and a possible control of the shock pro-

cess opens a large optimizing potential.

In most cases, a shock is generated by accelerating a mass in the start-up phase

and striking the shock partner. The momentum and energy are transferred to

the shock partner in the short contact phase (impulse). The temporal process of

the force shock is a consequence of the elastical conditions of the used bodies.

Because of the uncertainties of the start-up and the contact phase reproducible

shocks are only realizable with a huge effort and very complex experimental

setups. The shock repetition rates are limited. Especially, a timed triggering of

the shock incidence for measurement purposes in the micro-second-range is

not possible.

These restrictions can be overcome by the piezo technique as an adaptive me-

chanical shock generator principle.

In the following section, the shock generation is described by axial acting piezo

stacks. The main characteristics are:

Special basic situation:

• shock partners are in contact before the shock

• the shock system is in the state of rest, no starting phase

Adaptive shock generation:

• adjustable shock parameters like energy, acceleration, pulse width

• high reproducibility of the shock parameters

• variable shock repetition rates up to 1 kHz (Burst)

16

• micro-second precise triggering of the shock

Wide range for specific applications:

• shock generation at inaccessible places

• miniaturized shock generators/micro shock generation (e.g. for the cali-

bration of acceleration sensors)

• exotic operating conditions (shock generation in the low temperature range)

• highest accelerations (500'000 m · s−2 realizable)

• forces up to a few 10 kN

4.1 The application objective of piezo-electric shock gener-

ators

The piezo shock generators were designed to expose test objects, structures,

materials etc. to high accelerations or fast force modulations/shocks for gener-

ating statements about material characteristics, structure behavior, lifetime and

function under extreme dynamical operating conditions. Under this aspect, the

piezo shock generators aim to similar applications as piezo-shakers. But please

note the physical difference between shocks and oscillation.

There is another difference between them in their operations. Shakers work in a

continuous sine operation, the reached accelerations depends on the frequency.

This has a consequence for the expenditure of the electrical driving system with

increasing requirements for acceleration generation because of the higher con-

tinuous power consumption. From this it follows that shakers underlie high

thermal and mechanical strain.

However, shock generators work with approximate square-wave pulses. Their

acceleration potential does not depend on the repetition rate. Therefore, single

shocks with high energies can be generated. The single-pulse-power and the

repetition rate concerning the shock generators are completely decoupled. A

piezo shock generator can be operated with highest pulse power and acceler-

ation. The repetition rate is only limited by the power of the driving electronics

17

and the self heating problem.

An adaption to the different applications concerning energy content of the

shocks, rise times etc. can be made by the used piezo shock-generator and

the electrical control. Piezo shock generators are available in large and minia-

turized versions.

4.2 The physical shock

A physical shock occurs when a fast process generates elastic stress conditions

(pressure gradient) in a medium e.g. steel bar. This pressure gradient prop-

agates through the medium with the specific sonic velocity (≈ 5 km · s−1 for

steel).

The particle velocity is the speed of the particle while it oscillates around its ori-

gin position. It represents the material displacement in the medium as a result

of the pressure change. The particle velocity is considerably smaller than the

sonic velocity but it carries the momentum of the shock. Kinetics and propa-

gation depends on the shape of both shock partners. Therefore the results are

very different between spheres and bars as shock partners. Bars are established

in metrology for material testing, because of the easier type based on the bar

theory.

The compression, effected by the shock, is measured with strain gauges. A laser-

Doppler anemometer is used to detect the particle velocity on the bars surface.

The shock-wave splits into reflected and transmitted parts at inhomegeneities

in the material and changes of the cross section. This behavior is used at the

Hopkinson-bar (Figure 9) to determinematerial characteristics subjected to high

strain-rates.

18

Fig. 9: Hopkinson-Bar for material testing. By strain gauge and laser-Doppler anemometer the triggered and thereflected shock can be compared.

4.3 Characterization of the piezo composite stacks as shock-

wave generators

Piezo-electrical shock-wave generators can be characterized as the following.

• Displacement:

Depends on the length of the stack, the field strength and the ceramic

material. Depending on the design > 100 µm.

• Force generation at the beginning of the pulse (= blocking force):

Depends on the actuators diameter, the field strength and the ceramic

material. Up to several 10 kN.

• Pulse width:

Depends on the transit time of the mechanic pulse in the actuator, the

length of the actuator and the sonic speed. µs to a few 10 µs. The pulse

width increases by the reflection at a seismic mass.

• Particle velocity within the ceramic material:

Maximum: several m · s−1 (similar to the maximum displacement velocity

of a piezo-actuator).

• Mechanical energy content of a pulse:

Depends on the mass of the actuator, the field strength and the selected

ceramic material. Up to several Joule.

19

• Impact momentum:

Product of the moved mass in the shock front and the particle velocity.

Depends on the mass of the actuator and the ceramic material. Up to

1 kg ·m · s−1.

• Ceramic material:

For high effective piezo-electrical shock-wave generators a specialized

high dielectric ceramic material is used. The energy density of the shock

is twice as high as for common materials.

4.4 The piezo stack as shock-wave generator

The active bar

If the piezo is charged rapidly enough, the mechanical pressure in the ceramic

material instantly jumps to a high value. The blocking pressure is established

and the piezo bar starts an accelerated expansion. Thereby the actuator causes

a propagating pressure front in the coupled material.

A piezo-actuator represents an active bar, which produces mechanical shocks

when driven by power pulses.

Because of the momentum conservation two opposite pulses run to each end of

the stack. After the shock the energy in the whole system is higher than before

(super-elastic collision).

Symmetric piezo shock-wave generator

The mentioned opposite pulse can be used for symmetric arrangements for

simultaneous measurement and calibration.

20

Fig. 10: Schematic representation of a symmetric piezo shock-wave generator.

Single-sided piezo shock-wave generator

If the actuator is equipped with a seismic mass the backwards running pulse

will be reflected at this mass. Therefore the pulse in forward direction becomes

a double pulse (Figure 11). The delay of the reflected pulse results from the

elapsed time in the stack and therefore from the sonic velocity. This leads to

increased pulse duration. In the best case the energy of the pulse is nearly

doubled. Similar to the recoil of a gun the momentum is taken by the seismic

mass.

21

Fig. 11: Typical pulse shape of a shock wave generator with seismic mass.

Fig. 12: Schematic representation of a single-sided shock generator with a seismic mass. The double pulse isbuild by the overlay of reflected and forward pulse.

22

4.5 Electrical driving of piezo shock-wave generators

Piezo shock-wave generators are similar to capacitors. By a high current their

capacitance will be charged. Because of the piezo electric coupling a mechan-

ical shock is generated. The basic aim is the generation of high shock energy,

therefore a high volume of the actuator and a special, high dielectric ceramic

material is necessary. Therefore the energy content of the shock only depends

on the electrical driving conditions. Piezo-electrical shock-wave generators typ-

ical have a capacitance of ≈ 10 µF (low-level signal).

A capacitance of 10 µF under a resistance of 1 Ω has a charging time constant of

RC = 10 µs. Thereby the electrical rise time is shorter than the pulse width given

by the actuators dimensions. To charge a actuator to a voltage of U0 = 800 V a

maximum current of U0/R = 800 A is needed. The capacitor takes the charge

Q = C · U0 = 8 · 10−3 Coulomb and the energy 1/2 · C · U20 = 3.6 J. These

are approximate values because the large-signal behavior must be considered.

In particular in the so called semi bipolar operation the energy density can be

twice as high.

The electrical current is generated by a high power transistor switch as shown

in figure 13. A capacitor bank is charged to the selected voltage (typical up to

+800 V). At unipolar operation the piezo stack is discharged before the shock. At

semi bipolar operation the actuator is charged with negative voltage to -200 V

by a separate power supply. In this case the voltage pulse amounts 1000 V. Be-

cause of nonlinearities of the ceramic material the pulse power is 50% higher

instead of 25%. After the shock the actuator will be discharged by a larger re-

sistor (typ. 50 Ω, RC = 0.5 ms). After that the actuator is charged negative for

the next pulse.

With the sufficient power of the amplifier pulse repetitions to 100 Hz are possi-

ble. Because of the limited cooling of the actuators high repetition rates should

be done in burst-mode.

23

Fig. 13: Semi bipolar control by high voltage pulser. The switch with tree settings represents a combination ofhigh power transistors, which are driven by a logic device. 1. Negative charging to -200 V, 2. Shock likecharging to +800 V, 3. Discharge to 0 V.

4.6 Outlook

The parameters of piezo-electrical ceramic stacks (blocking pressure, energy

density and so on) are independent of the actuators dimensions. Therefore ef-

ficient piezo-electrical shock-wave generators are scalable in size, configuration

and range of performance in a wide range. They reach from miniature sizes

trough to high volume shock generators. Based on the high reproducibility

and potential high repetition rates, piezo shock generators are particularly use-

able for realistic dynamic load tests of components under high accelerations.

Thereby they beat common shaker arrangements.

Piezo electric shock-wave generators more and more replace the shaker mea-

surement technology in the field of modal analysis. They are ideally suited for

systematic generation of acceleration profiles with high dynamics. Therefore

they are the first class for the calibration of shock and impact sensors. The pre-

cise triggering in microsecond range allows the arrangement of multiple shock-

wave generators in phased arrays, to generate variable shock fronts. It is pos-

sible to synchronize the shock with other fast physical processes. Piezo shock

generators are fit for permanent integration (e.g. to cast into materials) at inac-

cessible areas.

By inverting the operational state piezo-actuators are able to compensate/damp

24

shocks and vibrations.

4.7 Theory of the bar shock

Impact propagation in the ideal bar

The ideal bar is thin compared to its length. According to the mathematical

approach of d'Alembert for the solution of the wave equation, a shock-wave

propagates through the ideal bar to both sides without dispersion. A short

term change in compression is referred to as a shock or shock-wave to distin-

guish it from harmonic waves.

The bar theory can be applied for the propagation of shock-waves in thin bars.

If a shock is triggered at one end of a homogenous bar with the cross section

A this shock will be propagate unchanged with sonic velocity in longitudinal di-

rection. The sonic velocity c equals:

c =

√E

ρ

Synchronously to the elastic pulse a velocity pulse of the particles (particle ve-

locity) propagates through the bar at sonic velocity. The particle velocity is ac-

cording to:

σ(t) = −I · v(t)

This is nothing other than to use the principle of linear momentum at the bar

with the so called acoustic impedance I.

I = ρ · c

The algebraic sign determines the direction of the shock. Hooke's law

σ = E · ε

combines the mechanical stress σ with the elongation ε . It is fundamental that

potential and kinetic energy coexist equally and don’t exchange each other

except at the end of the bar and during changes in cross section and/or the

25

impedance. A harmonic wave is characterized by the sinusoidal, harmonic en-

ergy exchange between kinetic and potential energy, this is an important dif-

ference with the shock impulse.

In general two shock impulses spread to the right and to the left independently

from each other. The mechanical stress at location x at the time t results in:

σ(x,t) = σr(x− ct,t) + σl(x+ ct,t)

shock propagation in the bar with the sonic velocity c. The bar is thus an ideal

conductor of the impact sound.

For the corresponding particle velocity

v(x,t) =1

I· (σr(x,t)− σl(x,t))

is true. The impedance of the bar for steel is approximately 4 · 107 kg ·m−2 · s−1.

The principle of linear momentum may also be interpreted as the first inte-

gral of the equation of motion. At σ = 200 MPa the particle velocity reaches

≈ 5 m · s−2. The impact energy is transported one half elastic and one half

kinetical. The impact energy is calculated with

W =A

2 · I·∫ T

0

σ2(t)dt

from the time profile of the stress or also the time profile from particle velocity.

A is the cross-section of the bar. In this case the two measurement techniques

for detecting the impact energy by strain gauges and laser Doppler velocity

measurement LDA are different.

W =A · I2

·∫ T

0

v2(t)dt

On the one hand the impact of the bar is generated by an impulse, on the

other hand it is transmitted through the bar and gets effective at the other end

depending on the boundary conditions. For the transported momentum p it is

26

valid:

p = A ·∫ Timpulse

0

σ(t)dt

If the end is acoustically non-reverberant (sound-absorbent) the tension does

change the sign, the particle velocity does not. Is the end acoustically hard (ide-

ally clamped), the particle velocity does change its sign the tension does not.

While in an infinitely long bar elastic and kinetic energy are equal, the energy

at the bar end is completely elastic (hard end) or completely kinetic (free end).

The integration limits must be considered in the calculation.

If the cross-section of the bar changes from A1 to A2, then simple laws of trans-

mission (τ ) and reflection (r) for the tension are valid (and analog for the particle

velocity).

τ = 2 · A1

A1 +A2

r =A2 −A1

A1 +A2

If the characteristic impedance changes at the interface from impedance I1 to

I2, it is true:

τ = 2 · I2I1 + I2

r =I2 − I1I1 + I2

For simultaneous change of the cross-section and impedance, the equations

are:

τ = 2 · A1 · I2A1 · I2 +A2 · I2

r =A2 · I2 −A1 · I1A1 · I1 +A2 · I2

In general, stress pulses spread independently in both longitudinal directions

of the bar. Furthermore, they can interfere at a defined location resulting in a

superposition of the single pulses. This also holds for the particle velocity in the

same way.

27

The limits of the bar theory are reached, when the transverse propagation can't

be neglected and no planar stress state is existent.

Impulse generation by a ramming bar

An impulse can be generated by impacting two cylindrical, bar-shaped bodies.

This impulse is the same as in mining equipment by air piston, ram and tool

shank. The contact surfaces are relatively slightly spherical, so that we can take

the”Hertzian contact theory“ as a basis. The collision of two cylinders with ideal

planar surfaces can either be viewed as a border case of the Hertzian contact

with very large radii or must be solved with an acoustic approach.

The index 1 refers to bar 1, 2 to bar 2. The effective Young's modulus Eeff is

given by:

Eeff =1− ν21E1

+1− ν22E2

The effective radius of the contact surfaces with the radii R1 and R2:

1

Reff=

1

R1+

1

R2

With Eeff and Reff a force constant k is calculated by Hertz:

kHertz =4

3· Eeff ·R

12

eff

With this, the contact force is a function of the penetration of the contact sur-

faces w in the static case:

Fk(w) = kHertz · w32

In the dynamic Hertzian contact between body 1 and 2 applies to the effective

mass meff:1

meff=

1

m1+

1

m2

and the dynamic force:

Fk(w(t)) = kHertz · w32 (t)

28

For the duration of the contact Tcontact applies:

Tcontact = 2.9432 ·(15

16

) 25

· 1

E25

eff

·m25

eff · 1

R15

eff

· 1

v15

The free flying bar can be described with two virtual tension pulses σl and σr

(same magnitude but opposite algebraic sign)

σ0 =I · v02

The resulting particle velocity is:

v0 =1

I · (σr − σl)

During the shock of the bar (hammer head or air piston) energy andmomentum

are transmitted to the bar with sonic speed.

For illustration (figure 14), we use calculated time distributions for force F(t)

(ideal: according Gauß) and tension σ(t). The pulse duration T = Timpulse in

this case is assumed to be 50 µs long.

Fig. 14: Left: Impulse. Right: Impulse tension.

The maximum force of 50 kN (5 tons) corresponds to a bar with 18 mm diam-

eter of a maximum tension of 200 MPa (tensile strength of S355J2 1.0577 is

≈520 MPa). In general force and tension correlate:

σ(t) =F (t)

A

29

Newton’s second law links the temporal integral of the force with the change of

the momentum ∆p to the impulse:

∆p =

∫ T

0

F (t)dt

This momentum can be related to a mass according to:

∆p = m ·∆v

The change in momentum of 1.3 kg ·m · s−1 corresponds to a change of the

particle velocity.

A good approximation of a Gaussian pulse is given by:

σ = σ0 · sin(π · tT

) 32

Thus, the energy can be calculated analytically:

W =4 · T ·A · σ2

0

3 · π · I

The momentum p cannot be calculated analytically.

It should be noted that the shock and a single harmonic oscillation do not

match. You need a temporal and spatial Fourier distribution.

A forth and back going shock pulse only shares the frequency with a resonant

oscillation, but not the mechanical stress distribution in the bar.

The piezo-electric actuator as a dormant bar

We distinguish the direct from the inverse piezo-electric effect (discovered in

1880 by the Curie brothers at quartz crystals). In the direct piezo-electrical ef-

fect a polarization is produced by an external force. The direct piezo-electric

effect applies in sensor applications.

The inverse piezo-electric effect is characterized by a mechanical strain ε gen-

30

erated by an electric field Eel. Represented by the equation of state:

ε = cE · σ + d33 · Eel

cE is the elastic constant at a constant electric field strength (e.g. Eel = 0). d33 is

the piezo-electric charge constant in the longitudinal (axial) direction of expan-

sion. If no mechanical stress is generated, i.e. reign free boundary conditions,

the expansion of the electric field strength is proportional to the elongation. If

the elongation is blocked, the result is a mechanical stress.

σ = −d33 · Eel

cE

This stress is negative (compressive stress). This equation is the basis for piezo-

electric shock generation.

Since the shock generation in the bar is made directly the bar is called active

bar.

If a stack of parallel contacted piezo discs is instantly driven by an electrical volt-

age, the active bar gives two shock impulses pr and pl. These shock impulses

have the same absolute value p0 and the same sign, but traveling into different

directions. For the impact energy it holds:

Wo =1

2· Vpiezo · Epiezo · d233 · E2

el

Vpiezo represents the volume of the actuator, Epiezo the Young's modulus of the

actuator.

In reality, the input signal, due to its limited rising time, must be convolved with

the mechanical voltage impulse. The otherwise ideal square pulse is rounded.

31

Nomenclature

Symbol Declaration Unit

A Cross section m2

c Sonic velocity m · s−1

cF Spring stiffness N · cm−1

cT Actuator stiffness N · µm−1

cE Elastic constant at constant electric field m2 ·N−1

d Thickness of a single ceramic disc µm

d33 piezo-electric strain coefficient for the longi-

tudinal effect

m · V−1

E Young's modulus N ·m−2

Eel Electric field V ·m−1

ε Elongation -

F Force N

I Acoustical field impedance kg ·m−2 · s−1

l0 Elongation without external force µm

m Mass kg

ν Poisson ratio -

p Momentum N · s

R Radius m

r Reflection coefficient -

ρ Density g · cm−3

σ Mechanical stress N ·m−2

T Pulse duration s

τ Transmission factor -

U Electrical voltage V

V Volume m3

v Particle velocity m · s−1

W Energy J

w Bend m

32