Introduction. Methods of vibration control

Transcription

1 ISSN: VOLUME 1, ISSUE DEC-014 Identiication o coulomb, viscous and particle damping parameters rom the response o SDOF harmonically orced linear oscillator Mr.U.L.Anuse. Department o Mechanical Engineering, Solapur University/ WIT/Solapur/ India uttamanuse@gmail.com Pro.S.B.Tuljapure Department o Mechanical Engineering, Solapur University/ WIT/Solapur/India Abstract This paper deals with Theoretical and Experimental methods or identiication o coulomb,viscous and Particle damping parameters rom the response o Single degree o reedom harmonically orced linear oscillator when system damped with more than one type o damping,which parameter is responsible or the control o resonant response o vibrating systems, in experimental method setup have been presented to investigate steady state response amplitude x i or SDOF system or dierent values o amplitude Y i o the base excitation rom this relationship o (X i, Y i ) the values o viscous damping coeicient c and coulomb riction orce F 0,also equivalent viscous damping ratios,have been calculated rom requency response analysis or the systems with viscous damping,viscous and Coulomb riction damping, coulomb riction damping and particle damping by using hal power band-width method and in theoretical studies expression or steady state amplitude X 0 obtained is used to study the eect o requency ratio and coulomb riction parameters on phase angle and amplitude ratio. Keywords: Coulomb Friction parameter; Damping ratio; Hal power band method; Damping Coeicient; Resonant requency; Excitation requency; Particle damping; Viscous damping ; Coulomb damping Relevance In many situations, it is important to identiy damping inormation rom a vibration system with both Coulomb and Viscous sources o damping. Their requent occurrence in practical engineering has aroused or a long time the interest o many researchers in the vibration ield. Friction dampers (with viscous damping as the system damping) are used in gas turbine engines, high speed turbo pumps, large lexible space structures under carriage o railway bogie, vehicle suspension systems etc. These dampers are used to reduce resonant stresses by providing sliding contact between points experiencing relative motion due to vibration, thereby dissipating resonant vibration energy. Introduction Methods o vibration control The problem o reducing the level o vibration in dynamic systems arises mainly due to increase in operating speeds o the machines, large dimensions o constructions, stricter standards and norms by the environmental pollution boards and technological demands placed on keeping vibrations down to accepted levels. The important ways and means o controlling unacceptable vibrations in machines are: System Modiication: Changing the rigidity (stiness) and or inertial parameters (mass, inertia) to modiy natural requency or requencies o the dynamic systems. 1 P a g e

2 ISSN: VOLUME 1, ISSUE DEC-014 Vibration Isolation: Use o isolation techniques in which either the source o vibration is isolated rom the system concerned (orce transmissibility case) or a device is protected rom its point o attachment (motion transmissibility case). The isolation systems can be passive, semi-active or active. Dynamic Vibration Absorber: A Dynamic Vibration Absorber is simplest device used to reduce the steady state vibrations o a system at a certain ixed requency o excitation. It is a passive control device which is attached to a vibrating body subjected to orce or motion excitation. Method o Damping: Damping used is viscous damping viscoelastic damping, electro-rheological damping, magneto-rheological damping, eddy current damping, piezoelectric damping, or Particle damping etc. Passive, semi-active and active damping techniques are common methods o attenuating the resonant amplitudes excited o a structure/machine. Active damping amplitudes o a structure/machine. Active damping techniques are not applicable under all circumstances due to power requirements, cost, environment, etc. Under such circumstances, passive techniques are available alternative. VISCOUS AND COULOMB DAMPING Various orms o passive damping exist, including viscous damping, viscoelastic damping, riction damping, and impact damping etc. Viscous and viscoelastic damping usually have a relatively strong dependence on temperature. Friction dampers, while applicable over wide temperature ranges, may degrade with wear. Den Hartog [5] has presented an exact solution or the steady-state vibration o a harmonically excited oscillator damped by combined dry and viscous riction. The system, as shown in igure (1.1), consists o a orced excited mass with riction orces acting between it and the ground. The several experimental tests to veriy solutions have been perormed to ind out the orced response o a single-degree-o-reedom system with both viscous and dry riction damping. Hundal [6] studied a base-excitation rictional oscillator as shown in igure (1.), in which close orm analytical solutions o the equation o motion were obtained. Results have been presented in non dimensional orm as magniication actors versus requency ratios as unctions o Viscous and Coulomb riction parameters. It has been shown that the mass motion may be continuous or one stop during each cycle, depending upon system parameters. The response o a single degree o reedom spring-mass system with Viscous and Coulomb riction, with harmonic base excitation, has been determined. Levitan [7] analyzed the motion o a system with harmonic displacement o the base, as shown in igure (1.3). The riction orces in his model act between the base and the mass. An analytical solution or the response o the support-excited system has been presented. The solution to the equation o motion has been developed through the application o a Fourier series to represent the rictional orce opposing the relative motion between mass and supporting structure. Perls and Sherrard [8] have extended the results o Den Hartog through the ranges applicable to inertial instruments as accelerometers and jerk meters. They obtained the curves with analog computers or the magniication actor verses requency ratio o second order systems with combined Coulomb and Viscous damping. The igure (1.4) shows a typical vibration instrument with its rame rigidly attached to a sinusoidal vibrating structure having a motion Xcos t. Ferri and Dowell [9] have investigated the vibration response o both single and multi degree-o-reedom systems with combined dry riction and viscous damping. Jacobsen and Ayre [10] have developed an approximate scheme or estimating both viscous and dry riction quantities rom the ree-vibration decrements by noting that the viscous riction dominates in the largeamplitude responses, and that Coulomb riction dominates in the small-amplitude oscillations. As such, they exploited the exponential and linear decay o a ree vibration Viscous or Coulomb-riction damped system. Dimentberg [15] to generate identiication equations. A non-linearly damped Single-Degree-O-Freedom (S.D.O.F. or the short) system under broadband random excitation is considered. A procedure or in-service identiication o the damping characteristic rom measured stationary response is described damping P a g e

3 ISSN: VOLUME 1, ISSUE DEC-014 mechanisms rom measured random vibration data. Extensive results o numerical tests or the procedure have been presented. Marui and Kato [16] have worked out a brie analytical technique or the behavior o the linear orced vibratory system under the inluence o a Coulomb riction orce, as shown in igure (1.5). The analysis has been based on the new simple idea o stopping region. Using this technique, the behavior o the system in the low exciting requency range, where the remarkable inluence o riction easily develops, has been examined and the results have been compared with the experimental ones. Liang and Feeny [19] have proposed a simple identiication algorithm or estimating both Viscous and Dry riction in harmonically orced single degree o reedom mechanical vibration systems. The method has been especially suitable or the identiication o systems or which the traditional ree-vibration scheme is diicult to implement. Numerical simulations have been included to show the eectiveness o the proposed algorithm. A numerical perturbation study has been also included or insight on the robustness o the algorithm. Liang and Feeny [1] have presented a method or estimating Coulomb and Viscous riction coeicients rom responses o a harmonically excited dual-damped oscillator with linear stiness. The identiication method has been based on existing analytical solutions o non-sticking responses excited near resonance. Schematic diagram depicting a SDOF oscillator with viscous, Coulomb riction and base excitation has been shown in igure (1.6). Cheng and Zu [0] have studied a mass-spring oscillator damped with both Coulomb and Viscous riction and subjected to two harmonic excitations with dierent requencies. By employing an analytical approach, closed orm solutions or steady state response have been derived or both non-stop and one-stop motion. From numerical simulations, it has been ound that near the resonance, the dynamic response due to the tworequency excitation demonstrates characteristics signiicantly dierent rom those due to a single requency excitation. In addition, the one-stop motion has been demonstrated peculiar characteristics, dierent rom those in the non-stop motion. Theoretical and Experimental Methods or Identiication o Coulomb, Viscous and Particle Damping Parameters rom the Responses o a SDOF Harmonically Forced Linear Oscillator have been carried out. Fig.1.1 Schematic representation o the system, Fig.1. Schematic representation o the system, analyzed by Den Hartog [5] analyzed by Levitan [7] 3 P a g e

4 ISSN: VOLUME 1, ISSUE DEC-014 Fig.1.3 System with Coulomb riction between mass and ground, with harmonic excitation o the base analyzed by Hundal [6] Fig.1.5 Forced vibratory system Fig.1.6 Schematic diagram depicting a SDOF oscillator with Coulomb riction [16] with viscous, Coulomb riction and base excitation [1] STEADY STATE RESPONSE ANALYSIS Figure (.1) shows a mass-spring-damper system, (SDOF) where a mass m is suspended rom a spring with stiness k and a viscous luid damper with damping coeicient c and Coulomb riction orce F. The mass is subjected to two harmonic excitations P 1 cos ( 1 t+ 1 ) and P cos ( t+ ) with dierent requencies. M 1 (.1) N where M and N are integers and they have no common actors. Thus, the two external harmonic excitations can be regarded as a single nonharmonic, but periodic excitation whose requency is calculated as the greater common divisor o the two original harmonic requencies [0] (.) N Fig..1 A mass spring damper system subjected to excitation P (t) Fig.. Free Body Diagram P cos t P cos t Where P (t) = P a g e

5 ISSN: VOLUME 1, ISSUE DEC-014 Reerring the ree body diagram o igure., the equation o motion o mass m is given as, mx P cos t P cos t cx kx ( N) (.3) Ater writing F N and rearranging, the equation.3 becomes, m x cx kx F P1 cos 1 P cos t 1 (.4) Where, x is the displacement, and is the phase dierence between two harmonic excitations, i.e., 1. Assuming that the response always starts (t=0) at maximum value, the phase dierence between the irst harmonic excitation and the motion is uniquely determined by the unknown phase angle 1. Correspondingly, the time boundary conditions o equation (.4) take on the ollowing simple orm: 0; x x 0 t x 0, t x x 0, x 0 (.5) ; Where x 0 is the amplitude o the vibration. Introducing the ollowing parameters k F p p 1 c n x,, a1 a,, cc m and (.6) m k k, k c n, Dividing both sides by m o Eq. (.4) one obtains, c k F P1 P x x x cos 1 cos t 1 (.7) m m m m m Divide and multiply by k on right side o Eq. (.7) one get, c F k P1 k P k x x n x cos 1 cos t 1 (.8) m k m k m k m Upon substituting parameters rom Eq. (.6) one can write, c x x n x n x a1 n cos 1 an cos t 1 (.9) m The equation o motion, Eq. (.9), then can be rewritten as, c x x n x x n a1 cos 1 a cos t 1.10 m ct m e C1 cos pt C sin pt a1 s1 cos 1 1 s cos t x x (.11) a 1 c c Where, 4 1 si p n tan, i ( i 1, ) m 1 si and C 1 and C are two integration constants. (.1) The two constants C 1 and C, the unknown amplitude x 0 and the phase angle 1 will be determined by using the our conditions in Eq..5, as described in the procedure outlined in Appendix I rom Eq. I-16 onwards. The inal expressions or the amplitude x 0 and phase angle 1 o the steady-state vibration or nonstop motion are obtained as, * * * * b b 4a c x0 * (.13) a 5 P a g e

6 a Where, b c * * * ( ap da) ( bp ea) NOVATEUR PUBLICATIONS ISSN: VOLUME 1, ISSUE DEC-014 ( db aq) ( ap da)( bp ea) ( db aq)( eb bq), ( eb bq), ( AQ BP). (.14) The phase angle 1 is determined as, BR CQ cos 1 AQ BP (.15) Experimental Analysis i) An experimental set up has been designed and developed to obtain input output amplitude relationship or SDOF Harmonically excited system with Viscous, Viscous and Coulomb riction, and Viscous, ii) Coulomb riction and Particle damping. For this purpose, design and development o viscous luid damper, dry riction damper and particle damping system has been carried out. Fig 3.1 Experimental test set-up Parameters Spring stiness k 1 /=1511 N/m Number o turns (N) 14 Wire diameter (d) 4mm Mean diameter (D) 38 mm Outer diameter (D 0 ) 4 mm Inner diameter (D i ) 34 mm Spring index (C) 10 A schematic o the experimental test set-up is shown in the ig. (4.1) o the C rame. The requency response curve o the system is obtained using the accelerometer pick up with its necessary attendant equipment and a FFT analyzer, over a small range o excitation requency.the SDOF system taken or analysis has ollowing parameters. mass m =1.0 kg, spring stiness k = 30.0 N/m, coeicient o viscous damping c =?, Coulomb riction orce F 0 =? Using this experimental set-up, steady state response amplitudes x i o the SDOF harmonically base excited system have been determined by exciting the system near resonance with excitation requency near about Hz. These values are given in Table 4.. Y i x i Y i x i Y 1 =0.701 x 1 = 4.05 Y 1 =0.81 x 1 = 3.15 Y =0.74 x = 4.34 Y =0.986 x = 5.34 Y 3 = x 3 = 4.68 Y 3 =1.01 x 3 = 5.73 Y 4 =0.766 x 4 = 4.8 Y 4 =1.114 x 4 =64.84 Y 5 =0.788 X 5 = 5.04 Y 5 =5.04 X 5 = 8.31 Table 4. the input/output amplitudes o experimental set-up From this (Y i, x i ) relationships, the values o viscous damping coeicient c and Coulomb riction orce F 0 are determined as ollows: i) A straight line relationship between Y 0 and x 0 has been obtained using regression analysis with points (Y i, x i ) taken rom Table 4.. ii) The slope m and intercept z on the ordinate axis have been obtained as, m = and , z = and mm iii) Using the identiication scheme presented in section 3.3o chapter 3, the values o viscous damping coeicient c and Coulomb riction orce F 0 are determined as ollows: 6 P a g e

7 ISSN: VOLUME 1, ISSUE DEC-014 a) Determination o viscous damping coeicient c: The viscous damping ratio ζ is given as Y Y1 ( x x ) 1 1 = 1 slope o line = m (reer Eq. 3.13) From the slope m = and we get, = = But, c/ cc c / mk c viscous coeicient c = 4.65 N-sec/m b) Determination o Coulomb riction orce F 0 : From the intercept, z = and mm we get, Gx z = (reer Eq. 3.8) sinh( ) Gx But, G cosh( ) 1 (reer Eq. 3.14) By putting the value o = determined just above one can get, sinh( ) G cosh( ) 1 G By substituting the value o G one can determine the value o x as, F0 x = But, x k (reer Eq. 3.3) F0 k x Coulomb riction orce F 0 = N Thus the unknown values o viscous damping coeicient c and Coulomb riction orce F 0 have been identiied. Using the experimental set-up, the requency response analysis has been carried out or a harmonically base excited SDOF system with Type o damping r (Hz) X r (microns) 1 S S+C S+P(T:-No balls) S+P(T:-00 balls) S+P(T:-300 balls) S+P(T:-ully illed withballs) S+V S+V+C S+V+C+P (5%ull balls) S+V+C+P (50%ull balls) S+V+C+P (ull balls) Table 4.4 Damping ratio o various damping combination 7 P a g e

8 X (microns) X (microns) NOVATEUR PUBLICATIONS ISSN: VOLUME 1, ISSUE DEC-014 n =8.75 Hz ; 1 = Hz ; = 9.43Hz n =8.75 Hz ; 1 = Hz ; = Hz ξ=0.053 ; X r = µm ξ=0.076 ; X r = µm Fig. 1 Frequency response curve or ξ s Fig. Frequency response curve or ξ s+c n =9.0 Hz ; 1 = Hz ; = Hz n =9.0Hz ; 1 = 7.93 Hz ; = 10.5 Hz ξ=0.11 ; X r =194.5 µm ξ=0.143 ; X r = µm Fig. 3. Frequency response curve or ξ s +v Fig.4. Frequency response curve or ξ s+v+c S+P (no ball ) S+P (00 Balls) (Hz) (Hz) n =7.5 Hz ; 1 = 7.8 Hz ; = 7.9 Hz n =7.5 Hz ; 1 = Hz ; = 8. Hz ξ=0.04 ; X r =6815 µm ξ=0.059 ; X r =635 µm 8 P a g e

9 X (microns) X (microns) NOVATEUR PUBLICATIONS ISSN: VOLUME 1, ISSUE DEC-014 Fig.5. Frequency response curve or ξ s +p Fig.6. Frequency response curve or ξ s+p S+P (300 Balls) S+P(ull balls) RESULT AND DISCUSSION (Hz) (Hz) n =7.5 Hz ; 1 = 7.10Hz ; = 8.63 Hz n =7.5 Hz ; 1 = 6.9 Hz ; = 8.08 Hz ξ=0.063 ; X r =6095 µm ξ=0.0778; X r =4395 µm Fig.7. Frequency response curve or ξ s +p Fig.8. Frequency response curve or ξ s+p RESULT AND DISCUSSION 1. Coulomb riction parameter F/P 1 has pronounced eect on resonant amplitude F/P 1 increases steady state amplitude x 0 decreases at a aster rate.. However, it was not cleared how much contribution o viscous riction in system & in many situations it is required to know content o viscous & Coulomb riction in the system. 3.So procedures is presented or identiication o Coulomb & Viscous riction damping in a harmonically base excited SDOF vibrating system 4. Using experimental set up, steady state response amplitudes X i o SDOF system have been determined by exciting system near resonance about 8.75 Hz. From these (Y i, X i ) relationships, values o Viscous damping coeicient c=4.65 N-sec/m & Coulomb riction orce F 0 =1.06 N were determined. 5. Frequency response analysis has been carried out or various types o damping combination. The amount o content o damping coeicient has been determined by using Hal Power Band Width method. a) s+v increases over s is o the order 1.8 times b) s+c increases over s is o the order times c) s+v+c increases over s is o the order 1.7 times d) s+v+c+p increases over s is o the order 3.46 times CONCLUSIONS 1. From the steady state response analysis carried out or a SDOF with Coulomb and viscous damping and subjected to harmonic excitations (two requency and single requency excitation), it is seen that the Coulomb riction parameter F/P 1 has pronounced eect on the resonant amplitude and as the value o Coulomb riction parameter increases, the steady state amplitude x 0 decreases at a aster rate.. Since the vibrating system has both viscous and Coulomb riction damping, Coulomb riction has pronounced eect on the resonant amplitude. It is not clear rom the analysis how much is the contribution o the viscous riction in the system towards the reduction o resonant amplitude. In many situations, such as control system, it is necessary to know the content o the viscous and Coulomb riction damping in a given vibrating system. For that purpose, the identiication o these two types o riction in a vibrating system will play a signiicant role when the control o vibration is carried out using the method o damping. 9 P a g e

10 ISSN: VOLUME 1, ISSUE DEC-014 REFERENCES 1) J.W. Liang, and B.F. Feeny, 004, Identiying Coulomb and Viscous Friction in Forced Dual-Damped Oscillators, ASME J. Vib. Acoust., 16, pp ) Steven E. Olson, 003, An analytical particle damping model J. Sound Vib. 64, pp ) M.R. Duncan, C.R. Wassgren and C.M. Krousgrill 005, The damping perormance o a single particle impact damper, J. Sound Vib. 86, pp ) Mao, K. M., Wang, M. Y., Xu, Z. W., and Chen, T. N., 004, Simulation and Characterization o Particle Damping in Transient Vibrations, ASME J. Vib. Acoust., 16,pp ) Den Hartog, J. P., 1931, Forced Vibrations with Combined Coulomb and Viscous Friction, Trans. ASME, 53, pp ) Hundal, M. S., 1979, Response o a Base Excited System with Coulomb and Viscous Friction, J. Sound Vib., 64, pp ) E.S. Levitan, 1960, Forced Oscillation o a spring-mass system having combined Coulomb and Viscous damping, Journal o the Acoustical Society o America 3, pp ) Thomas A. Perls and Emile S. Sherrard 1956, Frequency Response o Second Order Systems with Combined Coulomb and Viscous Damping, Journal o Research o the National Bureau o Standards, 57, pp ) J.W.Liang,and.F., Feeny, 1999, The Estimation o Viscous and Coulomb Damping in Harmonically Excited Oscillators, Proceedings o DETC 99, Las Vegas, 10 P a g e