Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 2, pp. 295-306, Warsaw 2009 VERIFICATION OF THE NOMOGRAM FOR AMPLITUDE DETERMINATION OF RESONANCE VIBRATIONS IN THE RUN-DOWN PHASE OF A VIBRATORY MACHINE Grzegorz Cieplok AGH University of Science and Technology, Department of Mechanics and Vibroacoustics e-mail: cieplok@agh.edu.pl The paper contains an experimental verification of a new nomogram for determination of resonance vibration amplitudes in a vibratorymachine drivenbymeans of an inertia vibrator in the run-downphase.Unlike the Katz nomogram, it takes into consideration the interaction between the vibrator and machine body. The verification was performed for a case where the machine body was in curvilinear motion with its trajectory close to circular. Key words: vibratorymachine, transient resonance, run-down 1. Introduction Theproblemofdetermination of thevalues of resonance amplitudesduring the run-up and run-downphases in vibratorymachines driven by inertia vibrators was researched in many works in the scientific literature. First attempts were related to analysis of the equation of motion in the form Mẍ+ bẋ+kx= f(t) (1.1) describing motion of a vibrating mass M supported in flexible and viscous suspension defined by parameters k and b, excited by a force of the given form. In the simplest case, itwasa sinusoidal forcehavingaconstant amplitude and linearly increasing (or decreasing) frequency (Lewis, 1932), or having the amplitude directly proportional to the square of vibrator velocity as in the case described by Katz (1947). The results of the above mentioned approach were also presented in form of nomograms (Goliński, 1979) based on the so- called acceleration factor which combines angular acceleration of the vibrator 296 G. Cieplok with the square of natural frequency of the vibrating machine. In works by Banaszewski and Turkiewicz (1980), Zeller (1949), Fernlund (1963), Fearn andMillsaps (1967), Irretier and Leul (1993), Leul (1994) and others, one can find formulas for resonance amplitudes obtained analogously; i.e. by empirical approximationsofnumerically integrated equation (1.1).Basedon theseworks, one could confirm in general that the resonance amplitude at a constant value of the vibrator angular acceleration is inversely proportional to the square root of that acceleration and the resonance frequency values are varying for the run-up and run-down phases and also depend on the value of vibrator angular acceleration. The most accurate presentation for motion of a mass suspended on a flexible-viscous system and excited by a given force is most probably the work done by Markert and Seidler (2001), who solved equation (1.1) in the case when the force was presented as a linear combination of an arbitrarily selected function and its derivatives in time. However, if the interaction between vibratormotion and the vibrating bo- dy of the machine is not considered, it may lead to gross errors. Michalczyk (1994) pointed out this problem, since it shows a significant effect of the vibra- tion moment on behaviour of the resonance and explained the reasons of the vibrator angular velocity breakdown during the run-down phase in relation to the errors created during reading out theKatz nomogram.Michalczyk (1995) derives formula (1.2) based on the energy balance between the vibrator and themachine body allowing one to evaluate the resonance vibration amplitude Amax from the top levels down for the run-down phase Amax = √ Jzr Mc (1.2) where Jzr – moments of inertia of the vibrator and drive shaft reduced to the rotation axis of the vibrator, Mc – mass of the vibrating part of the machine. The verification of a new nomogram for determination of the amplitude of resonance vibrations for the machine run-down phase presented in this paper is related to motion during which the bodymoves along a circular trajectory. Therefore, it is not possible to directly compare the obtained results with the results of studies of other authors. However, according to Section 4, the nomogram can be indirectly compared with the studies related to rectilinear motion of the machine. The results of comparisons obtained through the application of the no- mogram for the body moving along rectilinear and circular trajectories with Verification of the nomogram for amplitude... 297 more accurate computer simulations (Cieplok, 2008) came out very well, the- reby it can be also expected that the errors at the test stands are of similar values. 2. Equations for a symmetrically supported vibratory machine in relative units. Nomogram Cieplok (2007) analysed a phenomenological model of a vibratory machine illustrated schematically in Fig.1. The machine body, having a mass M is suspended in a flexible viscous system described by constants k and b. The system is excited to vibration by an inertia vibrator characterised by the static unbalance me. Thevibrator inertiamoment combinedwith that of thedriving system was reduced to the coordinate of vibrator rotation and is denoted by Jzr. The vibrator is exposed to action of the driving moment directed along the coordinate ϕ of vibrator angular motion. For this model, equations (2.1) were derived [ Mc 0 0 Mc ][ ẍs ÿs ] + [ b 0 0 b ][ ẋs ẏs ] + [ k 0 0 k ][ xs ys ] = [ Px Py ] [ Px Py ] = [ mesinϕ −mecosϕ ] ϕ̈+ [ mecosϕ mesinϕ ] ϕ̇2 (2.1) Jzrϕ̈−me(ẍsinϕ− ÿcosϕ)=Mel Fig. 1. Model of a vibratorymachine 298 G. Cieplok Upon transformation to the coordinate system 0ξη rotating with the vibrator angular velocity ϕ̇ (Fig.2), these equations assume the following form        Mc 0 Mcη 0 0 0 Mc Mcξ+me 0 0 0 me meξ+Jzr 0 0 0 0 0 1 0 0 0 0 0 1        d dt        vξ vη ω ξ η        = (2.2) =        2Mcωvη− bvξ − (k−ω 2Mc)ξ+bωη+meω 2 −2Mcωvξ − bvη− (k−ω 2Mc)η−bωξ −2mevξω+meηω 2+Mel vξ vη        where Mc =M+m vξ = dξ dt vη = dη dt ω= ϕ̇ (2.3) Fig. 2. Position of the machine bodymass centre in the coordinate systems 0xy, 0ξη The transformation also enabled one to create a definition of relative units and parameters for the machine. Hence, by substituting the following Verification of the nomogram for amplitude... 299 ξr = ξ Au ηr = η Au ωr = ω ω0 σ= m2e2 McJzr q= Mel Jzr 1 ω20 γ = b 2 √ Mck τ = ω0 2π t Au = me Mc ω0 = √ k Mc vξr = dξr dτ vηr = dηr dτ (2.4) set (2.2) may be expressed in the following form        1 4π2 0 − 1 2π ηr 0 0 0 1 4π2 1 2π (1+ ξr) 0 0 0 σ 4π2 1 2π (σξr+1) 0 0 0 0 0 1 0 0 0 0 0 1        d dτ        vξr vηr ωr ξr ηr        = (2.5) =        ω2r − (1−ω 2 r)ξr− γ π vξr + 1 π ωrvηr +2γωrηr −(1−ω2r)ηr − γ π vηr − 1 π ωrvξr −2γωrξr −σ π vξrωr+σηrω 2 r + q vξr vηr        In this way, a set of six physical parameters Mc, me, Jzr, Mel, k, b required for description of themachine dynamics has been reduced to three parameters σ, γ and q. Based on set above (2.5), Cieplok (2008) developed the following: • layer graphs enabling determination of the amplitudemultiplication fac- tor for the run-up phase based on values of the parameters σ, γ and q, • a nomogram (Fig.3) enabling determination of the amplitude multipli- cation factor for the run-down phase based on values of the nomogram parameters σ and γ. 3. Verification of the nomogram In order to verify practical applicability of a new nomogram, an experiment was conducted at the AGH Vibromechanics Laboratory. A machine shown 300 G. Cieplok Fig. 3. Multiplication factor α of the machine body vibration amplitude for the run-down phase; α=A max /A u , where A max – resonance amplitude Fig. 4. Test Stand. 1 – machine body, 2 – inertia vibrator, 3 – spiral spring, 4 – electric motor, 5 – hole for an additional mass in Fig.4 was selected for testing. It consists of a body supported on four symmetrically spaced spiral springs forced to vibrate by means of an inertia vibrator shown in Fig.5. The vibrator is driven by a 4-pole asynchronous electric motor ensuring over-the-resonance work of the machine. The mass of the machine vibrating part Mc was determined based on the change of machine natural vibration frequencies as a result of adding the Verification of the nomogram for amplitude... 301 Fig. 5. Inertia vibrator used for the purpose of experiment Fig. 6. Graph showing natural vibration acceleration of the machine body recorded during the experiment mass md.BasedonFig.6 illustratingmachinenaturalvibrations in thevertical direction, its natural period of vibrations was determined T01 =0.248s (3.1) Upon placing additional masses of total 46.5kg into the open holes shown in Fig.4, the machine new natural vibrations period was determined to be T02 =0.277s (3.2) 302 G. Cieplok Based on the relationship between the vibrating mass natural frequency and the support stiffness, the sought value of mass of the vibrating part was Mc ∼=md 1 ( T02 T01 )2 −1 =187.8kg (3.3) as well as the equivalent spring coefficient k∼= ( 2π T01 )2 Mc =120545 N m (3.4) Next, based on the logarithmic decrement δ of vibration damping (Osiń- ski, 1980), the equivalent viscous damping coefficient of the suspension was determined b= 2Mcδ T01 =136.82 Ns m (3.5) and then the damping factor was obtained γ= b 2 √ Mck =0.0144 (3.6) As now it is possible to notice, the omitting of dissipation in formulas (3.3) and (3.4) does not cause significant errors. The percentage difference between the period of natural undamped and damped oscillations does not exceed (1− √ 1−γ2) ·100%≈ 0.01% Upon identification of geometry of masses creating the active part of the vibrator, its own mass could be determined m=4.7kg (3.7) the radius of unbalance e=0.0156m (3.8) its static unbalance me=0.073kgm (3.9) and themass moment of inertia with respect to the mass centre Jsw =0.00112kgm 2 (3.10) Verification of the nomogram for amplitude... 303 Subsequently, based on the determined vibrator value parameters as well as the catalogue value of the drive motor moment of inertia Jr increased by the drive shaft components inertia Jd, one finally finds Jzr = Jsw+me 2+Jr+Jd =0.00112+0.00115+0.009+0.0025= (3.11) = 0.0138kgm2 Now, from the coefficient σ=m2e2/(McJzr)= 2.1·10 −3 one can calculate the amplitude multiplication ratio α = 10.6 from Fig.3 for γ = 0.0144. It corresponds to the absolute value of resonance amplitude Amax =α me Mc =4.12mm (3.12) Then, from Fig.7 showing a recording of the machine body mass centre displacement during the run-down phase, we can read out the resonance am- plitude Amax =3.72mm (3.13) The percentage difference between the theoretically determined andmeasured values was 10.8%. Fig. 7. Graph of the machine bodymass centre displacement y s during the run-down phase. Experiment 4. Conclusions If we take a practical application into consideration, the obtained result is sufficiently accurate for evaluation of the resonance amplitude for the run- down phase of a machine with the body moving along a circular trajectory. 304 G. Cieplok A better approximation of the resonant vibration amplitude could also be expected for the machine performing a rectilinear trajectory. Although direct comparison of the nomogramwith results of other authors is not possible due to a variety of different models assumed for analysis, however, an indirect comparison may still be possible. The possibility of nomogram adaptation to amachine featuring straight linearmotion of the bodyby applying a twice less value of the parameter σ during readout was indicated in Cieplok (2008). In this way, one could read out a value of the vibration amplitudemultiplication factor to be α≈ 13 for a machine having physical parameters corresponding to the machine used in the experiment and having a rectilinear trajectory of bodymotion. Its value obtained based on below mentioned methods was: • formula (1.2) √ Jzr Mc Mc me =21.6 • Katz nomogram – between 7 and 12, • Fernlund formula – 17.2, • Markert and Seidler formula – 13.8. It should bementioned that for the last three items on the above list, the vibrator angular acceleration was determined based on computer simulation, see Fig.8, by capturing the breakdown shift of the vibrator angular velocity during its passage through the resonance zone. This simulationwas conducted with taking into account idealised friction levels caused by the presence of the machine suspension, which was reflected by a viscous damping effect. Fig. 8. Graph for vibrator angular velocity during run-down phase in the area of resonant frequency. Computer simulation Verification of the nomogram for amplitude... 305 However, we cannot generally have the accurate value of the vibrator an- gular acceleration for the phase of velocity breakdown during resonance and, therefore, the determination of a useful value of the vibration amplitudemul- tiplication factor is not possible. The nomogram presented in this paper only relates to generally accessible machine physical parameters, thus it is a much more convenient and more accurate alternative than those discussed in previous publications. References 1. Banaszewski T., Turkiewicz W., 1980, Analiza wzrostu amplitudy drgań przesiewaczy wibracyjnych podczas rozruchu, Mechanizacja i automatyzacja górnictwa, 144, 11 2. Cieplok G., 2007,Quality analysis of symmetrically supported vibratoryma- chine taking into account a rotor stall in resonance, Machine Dynamics Pro- blems, 31, 1, 14-22 3. Cieplok G., 2008, Ampliduda drgań symetrycznie posadowionejmaszynywi- bracyjnej podczas rezonansuprzejściowego,CzasopismoTechniczne,1-M2008, 37-45,Wydawnictwo Politechniki Krakowskiej 4. Fearn R.L., Millsaps K., 1967, Constant acceleration of an undamped sim- ple vibrator through resonance, Journal of the Royal Aeronautical Society, 71, 680 5. Fernlund I., 1963,Running throughthe critical speed of a rotor,Transactions of Chalmers University of Technology, 277, Gothenburg, Sweden 6. Goliński J.A., 1979,Wibroizolacja maszyn i urządzeń, WNT,Warsaw 7. Hirano I., Matsukura Y., Kiso M., 1968, Behaviour of vibrating system passing through the resonances,Mitsubishi Denki Ghio, 42, 11 8. Irretier H., Leul F., 1993, Näherungsformeln zur Abschätzung der Ma- ximalantwort transient unwuchterregter Rotoren. SIRM II, Schwingungen in Rotierenden Maschinen, Vieweg Verlag, Braunschweig/Wiesbaden, Germany, 207-217 9. KatzA.M., 1947,ForcedVibrationDuring theResonancePassage, Inst.Mekh. Akad. Nauk. SSSR Ing. Spornik III/2 [in Russian] 10. Leul F., 1994, Zum transienten Schwingungsverhalten beim Resonanzdurch- gang linearer Systeme mit langsam veränderlichen a Parametern, Bericht 4/1994 des Instituts furMechanik der Universität-GHKassel, Germany 306 G. Cieplok 11. Lewis F.M., 1932, Vibration during acceleration through a critical speed, Jo- urnal of Applied Mechanics, 54 12. Markert R., Seidler M., 2001, Analytically based estimation of the maxi- mum amplitude during the passage through resonance, International Journal of Solids and Structures, 38 13. Michalczyk J., 1995,Maszyny Wibracyjne. Obliczenia dynamiczne, drgania, hałas, WNT,Warsaw. 14. Michalczyk J., Cieplok G., 1994, Rezonans przejściowy maszyn wirniko- wych – przyczyny błędów oszacowań, Zeszyty Naukowe AGH. Mechanika, 13, 1, Cracow 15. Osiński Z., 1980,Teoria drgań, PWN,Warsaw 16. ZellerW., 1949,Näherungsverfahren zur Bestimmung der beimDurchlaufen der Resonanz auftretenden Höchstamplitude,Motortechnische Zeitschrift, 10 Weryfikacja nomogramu do wyznaczenia amplitudy drgań rezonansowych dla fazy wybiegu maszyny wibracyjnej Streszczenie W pracy poddano weryfikacji doświadczalnej nowy nomogram do wyznaczenia amplitudy drgań rezonansowych maszyny wibracyjnej o napędzie za pomocą wi- bratora bezwładnościowego dla fazy wybiegu. W odróżnieniu od nomogramu Kaca uwzględnia on sprzężenie pomiędzy wibratorem a korpusem maszyny. Weryfikację przeprowadzonodla przypadku,w którymkorpusmaszynywykonuje ruch postępowy o trajektorii zbliżonej do kołowej. Manuscript received September 24, 2008; accepted for print January 29, 2009