High power piezoelectric axial shockwave generation

High power piezoelectric axialshockwave generationHp. Schad, HILTI AG, Schaan FLL. Pickelmann, Piezomechanik GmbH, München

Content1. Introduction 32. The elastic collision 42.1 Empirical approach 42.2 Collision mechanics 53. Shockwave generation by piezo-stacks 63.1 The piezo-stack: an active bar 63.2 Symmetrical shock generator 73.3 Single end piezo-shock-generator 84. Characterizing piezo-stack based shockwave generators 95. Electrical pulse excitation 106. Experiment studies on piezo-shock-generators 126.1 Experimental set up 136.2 Symmetrical piezo-shock-generator 146.3 Single output shock generation 157. Discussion 168. Outlook 179. Shock wave theory in bars and rods 189.1 Impact propagation in the ideal rod 189.2 Stress impulse generation by a ramming rod 209.3 Piezo-actuator as active rod 222

Stand der01Piezo-Multilayer-Technik IntroductionMechanical impacting is involved in a lot of technicalprocesses, 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 byinducing a rapid change of mechanical stress withintheir materials. It is assumed, that this interaction ispurely elastic and no plastic deformation occurs.This elastic stress propagates into the bodies withtheir particular speed of sound and this phenomenonis called "stress impulse". In Englishliterature often the inconsistent expression "shockwave" is used to describe this situation.When it is done here, "shockwave" and "stressimpulse" are synonyms.The above ideal situation of shock propagation hasto be distinguished from the technically usedeffects, when an impact is to be created by theindentation 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 onthe properties of the target material.The extreme situations are plasticity deformationswithout 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 impactprocesses.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 µsduration, impulse and energy are transferredbetween the bodies. The details of this process aredefined by the acousto-elastic properties of theimpact partners.The starting and contact phases of such a classicalimpact scenario are hard to be reproduced exactly.Common setups of precision impact measurementdo not allow high repetition rates.Further the special feature of a µs-precise timing ofthe impact event cannot be done by classicalsetups.These restrictions can be overcome by piezomechanicaltechnology providing an adaptiveimpact generation principle, where the impactparameters and the timing can be controlled by theelectrical driving conditions.This paper describes the shockwave generation byaxially active piezo-stacks featuring followingessentials.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, pulseduration, acceleration rates are preset by theelectrical pulse parameters.Shocks can be generated with high repetitionrates and high reproducibility (1kHz within aburst).µs precise time of the shock eventImpact generation at non accessible sitesMiniaturization (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 kNdepending on the shock generator design).The cooperation of HILTI AG and PIEZOMECHANIKGmbH aimed for the investigation of high powerlevels by using big-sized high load piezo-stacks.3

The elastic collision 22.1 Empirical approachA physical shockwave is defined as the generationof local deformation/disturbance within an elasticmedium (e.g. a steel bar) by a rapid process (e.g.collision with a hammer head). The resulting stressdistribution is then propagating through the mediumwith 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 displacementof the material by the compression act. Thevelocity of this displacement is the so called"particle velocity". This velocity is smaller by severalorders of magnitude compared to the speed ofsound (range of several m/s).Particle velocity and involved mass displacementare defining the related physical impulse.The details of shock propagation depend stronglyon the nature and shape of the collision partners.The response of interacting spheres differs from thecollision of bars (due to the excitation of internaldegrees of freedom).Collision experiments (e.g. for material characterizationwith high strain rates) are often carried outby 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 barsis detected by strain gages, the particle velocity byLDAs (LaserDopplerAnemometer) monitoring thebar's surface.Shockwave propagation becomes more complexwhen passing either into another kind of material orif the cross section of the bar is changing. Thepulse is split into reflected and transmitted parts.The quantitative analysis of this situation allows thecharacterization of the high dynamic elasticproperties of material probes under very high strainrates as carried out by split Hopkinson barexperiments (Figure 2).Strain gaugeTest-specimenLDA-spotShockwaveLDA-spotStraingaugeStraingaugeLDA-spotTransmittedshock-waveFigure 1: Schematic of a shock wave propagation within asolid bar, detected by strain gage and LDA measurements.Figure 2: Hopkinson bar experiment. The exciting and transmittedshockwaves are measured by strain gages and LDA.4

2.2 Collision mechanicsIn practice, the following basic shockwave situationsare usually discussedA Free moving bar end: no collision partner.A propagating shockwave will be completelyreflected back into the bar.Bar's end is moving with maximum particlevelocity.The acceleration rate of bar's end is doubled bythe reversed motion.B Bar's end blocked and cannot move:At bar's end the particle velocity is zero mechanicalstress/compression is maximal.C Acoustical matching:Describes the shockwave transfer at an interfacee.g. between two metal bars.The shockwave passes the matched interfacewithout losses in energy and impulse.For optimizing the acoustical match, thematerials and cross sections of the interactingbars 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 generationby piezo-stacks33.1 The piezo-stack: an active barA piezo-stack can be described as a solid bar ofPZT-ceramics. When this PZT bar is electricallycharged sufficiently fast, the internal stress jumpsinstantaneously to a high level:The initial pressure is the blocking pressure, causingnow 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",generating inherently mechanical shock pulses byelectrical pulse excitation.Because the active bar is at rest, two pulses aregenerated 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.6

1.1 3.2 Piezostapel Symmetrical mit shock “on-stack generator Isolierung” (osi)The above mentioned basic symmetry of the shockgenerationwithin a resting active bar can be usedfor a symmetrical arrangement with two opposingshock outputs of identical impulse content.This allows elegant designs for calibrationexperiments.LDAspotImpulseLDAspotsteel rodStraingaugeElectricpowerswitchFigure 4a: Schematic of a symmetrical piezo-shockgenerator.PZTstackHVcurrentpulsStraingaugeTriggersignalFigure 3: Symmetrical piezo-shock-generator with two-sidedshock propagation. The shock parameters are detected bystrain gages and LDAs.Figure 4b: High load piezo-stack with 35 mm diameter, withshock proof wiring.Design of a symmetrical piezo-shock-generatorThe shockwave is generated within the piezo-barand transferred to both ram elements (steel, titanium,brass etc.). Acoustical matching for maximumenergy extraction is done by adoption of the crosssections of PZT-stack and the metal rams.High powerPZT stackInternalprestressFigure 4c: Symmetrical shock generator: shock output viathe rams (left and right).Ram 2SteelTitaniumRam 1SteelTitaniumCasingElectricalwiring7

Stand 3.3 Single der Technik end piezo-shock-generatorThe above mentioned symmetrical generator setupcan be modified towards a single ram element witha nearly doubled energy and impulse output. This isdone by applying a bigger mass for supporting thePZT-stack at one side (seismic mass). The impulsewill 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). Thecompensating impulse is transferred to the seismicmass as recoil (similar to a rifle).Design of a single end piezo-shock-generatorSeismic bodyHigh powerPZT-stackInternalprestressRam Steeltitanium brassPZT-stackCasingElectricalwiringImpuls reflectionLDA-spotSteel rodFigure 6a: Schematic of a single output piezo-shockgenerator.Seismic mass SMElectricpower switchHVcurrentpulsStraingaugeHVPTriggersignalFigure 5: Schematic of a single end piezo-shock-generatedsystem by application of a seismic mass. The interactionof the travelling pulses results in an elongated double pulseoutput.Figure 6b: Single ouput piezo-shock-generator: brass part:seismic countermass.8

Characterizing piezostackbasedshockwave generators4The 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 electricalfield strength and type of the used PZTceramics. Strokes are ranging from µm to hundredsof µm.An efficient energy transfer must be completedwithin this stroke range limit.Piezo-shock-generators need therefore hard collisionpartners.Internal starting force:Generated force level within the PZT-bar uponcomplete electrical charge transfer.When the electrical rise-time is significantly shorterthan the shock generator's mechanical risetime,the initial force corresponds to an actuator'sblocking force. It depends mainly on stack'scross section, applied electrical field strength andtype of used PZT material.This starting force accelerates then the massesof the active PZT rod.●●●●●Pulse-width:= shock propagation time through the active PZTrod.It depends therefore on speed of sound in thePZT stack and stack's length.By pulse reflection at a seismic base, the resultingpulse length is doubled.Particle velocity:It corresponds to the maximum achievable shiftspeed 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 propertiesof 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 providea much higher shock energy density than "lowcapacitance" actuator PZT materials (up to afactor 2).9

Electrical pulse excitation 5The electrical equivalent circuitry describes a piezoceramiccomponent mainly as a capacitor. Byapplying a huge charge impulse to this PZTcapacitor(= generator stack), a mechanical shockwavewill be produced. Our experiment aimed forthe characterization of large volume, high dielectricstack actuators for a maximum mecha-nical pulseenergy output.A typical arrangement was:Piezo-generator capacitance: order of magnitude10 µFImpedance of charging circuitry during pulseignition: 1 ohmEquivalent RC time constant: 10 µsThis time constant is significantly shorter than thetransition time of the mechanical pulse.Applied peak voltage: up to +800 VPeak current: up to 800 AIt has to be taken into account, that by high fieldexcitation, the effective capacitance is remarkablyhigher than the above stated small signal value.This effective capacitance must be explicitly derivedfrom the electrical charging parameters.For shock generation, short-term a huge powerlevel is needed in the order of magnitude 0.5 Megawatts.It was produced by the following circuitry (seeFigure 7).The HighVoltagePulser HVP consists of a condensatorbank of parallel and seriel high voltage capacitorswith 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-generatoris coupled to this electrical chargereservoir. Due to the very low resistance of thiscircuitry, a fast charging pulse of the piezo-elementoccurs.In terms of maximizing the shock wave energy, twooperating 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 operationinstead of the unipolar mode. "Unipolar mode"means, the electrical pulse is applied to adischarged piezoelement (pre-pulse voltage level 0,voltage step 0V/U max V).A dramatic increase of the mechanical power outputof the shock generator is achieved by a pre-pulseconditioning of the piezo-shock-generator by connectingit to a negative voltage power supply withlevels down to -200 V. (switch position 1). To onepart, this is simple caused by the wider voltage stepapplied to the actuator (voltage step -200V/+U max ).10

5. Electrical pulse excitationSchematic of semi-bipolar pulse generation (HVP)800800Stress (V)700600500400300200negative prepoledvoltage onvoltage offstand baySwitch position1: negat. precharge2: pulse charging: shock!3: dischargeStress (V)70060050040030020010040 Hz1 23 12310000-100-5 0 5 10 15 20 25 30 352 4 6 8 10-100-200 -10-200-8 -6 -4 -2 0Time (ms)Time (ms)47Switch

Experimental studies6on piezo-shock-generatorsThe cooperation between HILTI/Schaan/Liechtensteinand PIEZOMECHANIK/Munich/Germanywas focusing on high power mechanical shockgeneration by using high voltage high load piezostacks.HILTI is the leading manufacturer of highest qualitytools for demolition, chiselling based on the electropneumatichammer principle. The high efficiency ofthese tools is based on the very exact matching ofall involved parts for the shock transfer from thehammer head down to the tip of the chisel.Piezo-shock-generators with their high powerlevels, high repetition rates and reproducible shockparameters are used for R&D activities to shortentest periods on these components significantly.12

056.1 Experimental set upHigh voltage high load piezo-stacks with an activediameter 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 ceramicshas been ruled out because of its reduced "shockpower" efficiency. Consequently all shock generatorstacks were based on the highly dielectric highpower PZT-material HP from PIEZOMECHANIK.For an optimum acoustical matching, the piezostackswhere combined with steel bars of 18 mmdiameter.The shock pulses propagating within the steel barshave 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 strainvariation has been derived. Together with the wellknowelastic properties of steel, the energy andimpulse content of the shock front can beevaluated.13

6.2 Symmetrical piezo-shock-generatorComparison of the shock profiles within the twosteel bars derived from the above mentionedsymmetrical arrangement:454035Stress (MPa)30252015Supper rodSlower rod1050-50 10 20 30 40 50 60 70 80 90 100Time (µs)Figure 9: Shock profiles of a 50 mm active length symmetricalpulse generator, electrical pulse level 500 V.Figure 8: Symmetrical pulse generator vertical arrangement.Both sides are coupled to steel bars for two sided impulseextraction. Notice the slits in casing and mounting flangessteel for getting optical access for the LDA-measurements.Timing is triggered by the transistor signal. Thedelay of the onset of the mechanical pulse is due tothe distance from shock generator to the LDAmeasuringspot 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-spotsfrom the pulse generator.14

6.3 Single output shock generationThe asymmetric shock generator with seismic baseuses the same piezo-stacks like the symmetricaldesign. Figure 10 shows a "head down" arrangementof the shock generator: the contact betweenthe piezo-shock wave generator and the steelbar is thereby preloaded with the weight force ofapprox. 100 Newtons.Stress (MPa)120100806040200-200 20 40 60 80 100 120Time (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 pulseprofile exhibiting a double peak structure of thestress response. This is due to superposition of theprimary impulse and the impulse, reflected at theseismic mass. This elongates the total propagationtime and leads to longer lasting output pulse.The energy content is increased nearly to a factor oftwo compared to a single pulse of a symmetricarrangement.Figure 10: Test set up using a single output piezo-shockgeneratorwith seismic mass for impulse reflection.Piezo-stacks with a length up to 200 mm have beentested with a mechanical shock energy content ofmore 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 ratesup to 500.000 m/s² can be expected at the bar'stip.15

Discussion 7Our 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.B High levels of extractable mechanical energy inthe range of 7 J/kg piezo-ceramics are achievable,provided a mechanically high efficient materialis used (like PIEZOMECHANIK's HP-PZT) and thesemi-bipolar operating mode is applied. Then thepower balance exceeds that of standard actuatorPZT by a factor of two.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.E Variation of shock parameters like amplitude andpulse width by electrical means.F Excellent reproducibility of shock profiles.G High repetition rates (in bursts).16

Stand der08Piezo-Multilayer-Technik Outlook1.1 Piezostapel mit “on-stack Isolierung” (osi)The normalized mechanical parameters of a piezostacklike strain, blocking pressures, energy densityare (nearly) independent of the actual dimensions ofthe piezo-element.Therefore piezo-shock-generators can be successfullydesigned 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 applicationwith its very fast rise time, high acceleration level,excellent reproducibility and elevated repetitionrates allow mechanical testing/calibration ofcomponents like acceleration/crash sensors. Newapproaches for quality inspection, mode analysis,structure borne sound evaluation are underinvestigation 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 synchronizingof mechanical shock events with other fastphysical processes.Fixed "Phased Arrays" of piezo-stack based shockgenerators 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 structurehealth monitoring.In an inverted operation mode, the described piezosystemcan be used as shock absorber.17

Shock wave theory in barsand rods (Impact in rods)99.1 Impact propagation in the ideal rodThe 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 notbe 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 stressimpulse instead of shock wave.For the axial propagation of elastic stress impulsesthe theory of thin rods (Graff, Johnson) can beapplied. A stress signal launched from the left in agiven homogenous rod with cross section a will bepropagated to the right with the speed of sound(Figure 12) c given byc = (E/ρ)cSynchronously with the stress impulse sigma (t) thesignal of the particle velocity v(t) propagates alongthe rod (Figure 12) where the fundamental relationholdsσ(t) = -I v(t)A consequence of the 2. law of Newton. I representsthe acoustical impedance given byI = ρ cThe law of Hooke correlates stress σ and strain εaccording toσ = 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 completelywith the frequency of the wave.In general two stress impulses propagate independentlyin both directions. The resulting stress is thesum and given as a function of position and timeσ (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 ofposition and time readsv(x,t) = 1/I (σ right (x,t) - σ left (x,t))Figure 12: Stress impulse (top) and related particle velocity(bottom).cwith the designationstressσparticle velocity vimpulse momentum PenergyWimpedance I18

9.1 Impact propagation in the ideal rodThe impedance of a steel rod is about 4 10 7 kg/(m 2 s)and for a stress impulse of 200 MPa the particlevelocity is about 5 m/s. The impulse energy asmentioned above is to one half elastic which is givenby the integralor the kinetic partThe techniques to measure these two energy contributionsare different. For the elastic part strain gagesare used, for the kinetic energies Laser DopplerAnemometry (LDA) is applied.The propagated impulse momentum P Impulse isdetermined by the integralThe impulse reaching the end of the rod will bereflected. If the end is acoustically free (soft) thestress amplitude changes sign, the particle velocitynot. If the end is acoustically hard (fixed), it is vicaversa. At the end of the rod all energy is in the firstcase kinetic and in the second case completelyelastic.For the change of the rod cross section areas froma 1 to a 2 .Simple transmission - τ and reflection - r rules arevalid for the stress:τ = 2 a 1 /(a 1 + a 2 )r = (a 2 - a 1 )/(a 1 + a 2 )Equivalent rules are given, if the impedances I 1 , I 2changeτ = 2 I 2 /(I 1 + I 2 )r = (I 2 - I 1 )/(I 1 + I 2 )or if both, areas and impedances changeτ = 2 a 1 I 2 /(a 1 I 1 + a 2 I 2 )r = (a 2 I 2 - a 1 I 1 )/(a 1 I 1 + a 2 I 2 )The reflection will be zero if the relation I 2 /I 1 = a 1 /a 2holds.For the particle velocities similar rules are valid, notgiven here (Johnson).The limit of the simple rod theory is reached if thelateral dimension of the rod has to be taken intoaccount leading to additional inertia effects. Thestress state is no longer constant along the crosssection (no plane situation).19

9.2 Stress impulse generation by a ramming rodUsually a stress impulse can be generated by theimpact of two cylindrical rods. An example are theimpacts of flying piston, ram and shaft of the HiltiEP-tool (Electropneumatic Percussion). Here nowthe fundamental difference between the realhammer and the piezo-hammer is based: a hammeris ramming - the piezo-actuator is impacting.If the surfaces of contact are spherical the contacttheory of Hertz (Johnson) can be applied. Thecontact of ideal plane surfaces has to be treatedacoustically or as the numerical limit of very largecontact 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. Forthe effective modulus E eff :E eff = (1 - ν 12 )/E 1 + (1 - ν 22 )/E 2holds. For the effective contact radius r eff theequation hold:1/r eff = 1/r 1 + 1/r 2With E eff und r eff the stiffness constant of theHertzian contact k Hertz is writtenk Hertz = 4/3 E eff r eff1/2The force of contact is given byf contact (w) = k Hertz w 3/2where w is the interpenetration of the two surfaces.Dynamically the Hertzian contact between rod 1und rod 2 with the effective mass m effand the dynamic force1/m eff = 1/m 1 + 1/m 2 )f contact (w(t)) = k Hertz w(t) 3/2lasts during the contact time T contactT contact = 2.9432 (15/16) 2/5 (1/E eff ) 2/5 (m eff ) 2/5 (1/ reff ) 1/5 (1/v) 1/5with v beeing the relative velocity of the two rodsbefore impact.Before impact, the rod (e.g. the flying piston) isflying freely and can be represented in terms of twostress impulses one to the right σ right and a secondto the left σ left both with the same amount σ 0σ 0 = Iv 0 /2but with opposite sign. For the particle velocity therelation holds1/I(σ rechts - σ links ) = v 0During the contact energy and impulse momentumare exchanged.For a dynamic Hertzian contact the force of contactis in good approximation given by an Gaussiansignal form where impulse maximum and impulsewidth depend on the given parameters. In theexample of figure 13 the maximum is assumed to50 kN and the width to 50 µs.20

9.2 Stress impulse generation by a ramming rodForce (kN)50403020100Motive forceStress impulse200180160140Impulse (MPa)402000 10 20 30 40 50 0 10 20 30 40 50Time (µs)Time (µs)1201008060Figure 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 of18 mm diameter corresponds to a stress of about200 MPa (the tension stress of Steel St52 is about520 MPa).Force and stress are related bya is the cross section of the rod. The change ofimpulse momentum Δp:is the integral over the force as a function of time(second law of Newton). Δp = 1.3 kgm/s.The Gaussian signal form can be represented ingood approximation byσ = σ 0 sin(πt/T) 3/2 .and the energy can then be analytically calculatedtoW = 4/3 T/π a/I σ 02It is worthwhile to mention that impact and a singleharmonic wave do not correspond. The Gaussianlike stress impulse needs a Gaussian Fourierdistribution of frequencies in time and space.A stress impulse running for- and backward withinthe rod has only the period in common with thefundamental resonance vibration.21

9.3 Piezo-actuator as active rodThere 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. Theinverse piezo-effect is described by the stateequation in scalar formε = c E σ + d 33 Eε = strainE = electrical fieldc E = elastic constant of compliance at constantelectrical field (e. g. E = 0).σ = stressd 33 is the axial piezo-electrical constantFor σ = 0 the strain is direct proportional to theelectrical field. If the strain is blocked (ε = 0), thenσ = -d 33 /c E Erepresenting a compression. This relation is thebasis of the stress impulse generation using piezoactuators.As the piezo-actuator is a cylindrical rodwe call this special type of rod active.Connecting all piezo-disks of the stack at an instantof time with an electrical voltage (see chapter 5) thepiezo-rod (stack) will react with an instant strain.The form of an active rod can be represented bytwo stress impulses σ right und σ left with equal amountσ 0 and equal sign but opposite propagationdirections. I.e. the particle velocities are also equalin amount but opposite in propagation direction.They are compensating at the beginning, contraryto the free flying rod where the stresses arecompensating.For the stress energy followsV piezoE piezoU 0 /ddW = ½ V piezo E piezo d 332 (U 0 /d) 2volumen of the piezo-stackYoung's modulus of the stackelectrical field strength given by the ratio ofvoltage U 0 and dthickness of an individual piezo-diskIn reality the stress impulse is not ideal square. Dueto finite rise time of the electrical voltage signalconvolution effects are rounding the ideal squarestress impulse form.22