design, development and testing of dual-mass dynamic vibration ...

Journal of Engineering Research and Studies

E-ISSN0976-7916

Research Paper

DESIGN, DEVELOPMENT AND TESTING OF DUAL-MASS DYNAMIC VIBRATION ABSORBER TO CONTROL THE VIBRATIONS IN CANTILEVER BEAM 1

Mr. Kshirsager Prashant R. 2Mr.Kshirsagar Pradip R.

1

Address for Correspondence

Faculty RMCET, Ambav (Devrukh), Ratnagiri, India. 2 Design Engineer (CFD analyst), Engineering Design Services Division, Pune, India ABSTRACT In this work theoretical natural frequencies and mode shape of the cantilever beam type main vibrating system i.e. cutting tool idealized by a cantilever with free ends is determined Dynamic vibration absorber is designed and developed at one of the resonant frequencies of the main vibrating system. Natural frequencies and mode shapes of main vibrating system is determined using FFT analyzer. Also use ANSYS for FEA analysis of cantilever beam type main vibrating system. Experimental set up is developed to test the performance of vibration absorber for the cantilever beam type main system. KEYWORDS: Dynamic Vibration Absorber, Modal Analysis,Dual-mass Absorber system

INTRODUCTION A vibration is a universal phenomenon. It manifests itself in many forms. Some vibrations are blessing some others an unmitigated evil. All engineering systems passing mass and elasticity are capable of vibrations. The effects of these vibrations are excessive variable stress in machine components undesirable noise looseness of parts and failure of the system. The design of such a system requires the study of their dynamic response to various forms of excitations. In order to quench the vibrations a property designed auxiliary system called a dual mass dynamic vibrations absorber is coupled to a main system so that motion from the main system is transferred to the auxiliary system hence protecting main system from harmful effect of vibration. Different types of dynamic vibration absorbers are used to reduce the vibration of single degrees of freedom main system. In this work undamped dynamic vibration absorber is used. In [1, 6] design procedure for cantilever absorber is described. In order to study vibrations of the physical systems such as a cutting tool [9] it is necessary to idealized these, this vibrating system by continuous dynamic system. In this topic the general theoretical background of the design and development of the dynamic absorber system is presented [6]. For theoretical design review [1, 4] of taken. In this work when a structure is undergoing some form of vibration; there are a number of ways in which this vibration can be controlled. Passive control involves some form of structural augmentation or redesign often including the use of springs & dampers that leads control augments the structure with sensors actuators & some form of electronic control system, which specifically aim to reduce the measured vibration levels. Also use of ANSYS for modal analysis of cantilever beam type main vibrating system [10]. The cantilever beam main vibrating system is exited by a harmonic force to design dual mass dynamic absorber system for cantilever beam type main vibrating system, when subjected to harmonic force excitation it necessary to obtain frequency response of systems. For this purpose the general differential equation of motion for this continuous system is obtained. THEORETICAL ANALYSIS [1, 3]: The general differential equation governing transverse vibration of a beam is given as JERS/Vol. III/ Issue II/April-June, 2012/10-13




  ⁄      ⁄   0 Where,



(1)


= modulus of elasticity of the beam, = M I of the beam, ρ= density of the beam, = distance from one of the ends of beam, y = amplitude of vibration. = c/s area of the beam.

Equation is rewritten as, (2)   ⁄    ⁄
   0 Assuming the solution of the equation in the form (3)     And substituting this solution in equation, We get, 4   ⁄
  0 (4) Putting,     ⁄
  0 (5) Where,  = mass per unit length of beam  /" Equation (4) becomes,      0 (6) The four roots of this equation are, #  ,    , %  &,   & Using these values of  the in equation (3) the general solution becomes ,   #  '   ('  %  )'   ()'  0 (7) And writing,  '  cosh  sinh,  ('  cosh  sinh,  )'  cos  &sin,  ()'  cos  &sin. The solution is written in the form as,   1# sin  1 cos  1% sinh  1 cosh (8) This equation was solved for transverse vibration of the beam and natural frequencies of transverse vibration of the beam are given as, 67 23  43 √ : (9) 89 where, 23 = natural frequency in ;? for different modes of vibration for beam changes with B.C.

Journal of Engineering Research and Studies

E-ISSN0976-7916

Density ρ = 7850kgf/m3.M.I. of the beam = I = bd3 / 12= 3.922 x 10-9m4, (l ) = 415mm, (b) = 50mm, (d) = 9.80mm, Material = MS,M.E = 2.038 x 1010kg/m2,  /" = 0.3921kgfs2/m2

Figure.2 First 6 Natural Frequencies for different modes of vibration for cantilever beam Table No.1 Fundamental natural frequencies for first three modes for cantilever beam.[10]

Figure 5 Experimental setup for forced vibration of cantilever beam type main system. From mathematical model of absorber system, Mass of absorbers = ma+mb=0.25x mass or the cantilever main vibrating system = 0.25 x 1.5963 =0.3991 kgf s2/m2 Now, ma = 0.1995 kgf s2/m2 , mb = 0.1995 kgf s2/m2. Absorber spring [ka] = ma x fn12 =0.1995 x 46.422 = kb where fn1 = fundamental natural frequency = 46.42 Hz, Absorber spring length is given as ,ka = p/y = 3EI /la3 ,la = 205.87 mm = lb

FORCED TRANSVERSE RESPONSE [1,4,6]: @

ABC DEF I

GH

J

JF E K LA (MA EK



I

K A A K I E L (M E

 J

I

K A A K A E L (M E

 (10)

Where , β# l = 1.875, β l  4.694 [ consecutive roots ]

a  14.28 ,Po  20.38 kgf ,l=0.415m, A  4.90 x 10( m2

Table 2 Amplitude of Cantilever beam type main system

DESIGN OF DUAL-MASS DYNAMIC VIBRATION ABSORBER [1,3,6]: The dynamic vibration absorbers are designed for cantilever type main vibrating system, when main systems are excited at its fundamental natural frequency. An absorber mass or 25% or the mass or main system [i.e. µ = 0.25] is taken for design. Absorber Arm Specifications Are As Follows: For rod φ = 5mm, Length=400 mm, Material =MS,E= 2.038 x1010 kgf/m2 , ρ = 7850 kgf/m3, M. I .of the beam I = bd3/12 =3.068 x 10-11m4

But as absorber arm is made up of 2 rods of 400mm length as shown in Figure 3. MI of the arm, I = 2I1 = 3.068 x 10-11 x 2 = 6.135923 x 10 11 4 m

Figure 3 The mathematical model of absorber system for cantilever type main system

Figure 4 Experimental setup for free vibration cantilever beam type main system JERS/Vol. III/ Issue II/April-June, 2012/10-13

Table 3 Tuning length for Dual-Mass cantilever absorber

EXPERIMETAL ANALYSIS: Performance testing of the dynamic vibration absorbers designed and developed for cantilever beam type of main system is presented. The testing was carried out on an experimental setup developed and fabricated for the same based on theoretical analysis and design calculations. The dimension of the main and absorber systems were determined and accordingly these systems were fabricated The effectiveness of vibration absorbers for cantilever type main system was tested by following procedure as outlined below: • The natural frequencies and the frequency response of the cantilever beam type main system and absorber system are obtained figure 7(a),table 4. • The tuning lengths of the absorber spring for cantilever beam type main system were determined by exciting this system at their first natural frequency figure 7(c),table 5. • The effectiveness of dual mass dynamic vibration absorber for cantilever beam type main system was carried out by following the procedure as for the cases 1/L = 1, l/L = 0.75, l/L = 0.50 , l/L =0.25 figure 7(c).

Figure. 6 Setup for force vibration of cantilever beam type main system with absorber. RESULTS OF EXPERIMENTAL TESTING: The main system was excited by a impact hammer and response was measured using piezoelectric accelerometer connected to dual channel FFT analyzer testing results are: • The values of natural frequencies of first second and third mode of transverse vibrations of

Journal of Engineering Research and Studies

•

• • •

absorber system are given in table 1,4 ,figure 7 (a) The frequency response of cantilever type main system and absorber system are given in figure 7(b). and Amplitude of vibration for varying absorber spring length are given in figure 7 (c). Comparison between theoretical experimental natural frequencies for cantilever fig.7 (d). Percentage reduction in amplitude response or the main system due o their dynamic absorber are given in Table 6. Table 4 : The value of natural frequencies obtained experimentally

Figure 7. Graphs Table 5: Tuning position cantilever

Table 6: Percentage reduction in amplitude response of cantilever due to absorber

CONCLUSION • The values of natural frequencies of the first three modes of the experimentally are given in Table No. 4 It is seen that these values are differ from those obtained theoretically and obtained from FE analysis are given in Table No.1 due to the fact that the boundary conditions achieved experimentally are not close to the ideal ones figure 7(d) • The deviations in the results of tuning lengths of the absorber springs may be attributed the deviation in the values of the natural frequencies obtained experimentally. • The percentage reduction in the vibration amplitude of the main system was attached to it table 6.from which it is seen that the design absorber for the reduction of the response of main system is satisfactory.

JERS/Vol. III/ Issue II/April-June, 2012/10-13

E-ISSN0976-7916

REFERENCES 1.

Christopher Ting-Kong, South Australia 5005 Thesis, ‘ Design of an adaptive dynamic vibration absorber.’ 1999 P.P.50-77. 2. N. G. Stephen.B.Hunt ‘Reduction of cantilever vibration by a cantilever absorber’1990.p.p.75-79 3. Wang,B.P.,Kitis,L.,Pillkey,W.D.,Palazzolo,(1985),“Syn thesis of dynamic vibration absorber”, Journal of Vibration,Stress and Reliability in Design,Vol.107,Apr,161. 4. Lee-Glauser,G.,Juang,J.N.,Sulla,J.L.(1995), “Optional Active Vibraion Absober Design and Experimental Results”, Journal of Vibration and Acousitcs,Vol.117 5. Yoshida,K., Shimogo,T., Nishimura,H.(1988), “Optional Control of Random Vibration by the Use of Active Dynamic Absorber”, Trans.JSME International Journal,Series III,Vol.31,No.2. 6. Prof.Warburton University Of Nottingham ‘A design procedure for absorbers’ 1979 p.p.68-69, 7. Timoshenko ‘Vibration problems in Engineering’. S Van to strend Co.Inc.New Jersy. 3rd Edition, 1955.p.p.324-368. 8. S.S. Rao, “Mechanical Vibration”, 4th edition prentice, Hall 2003.p.p 611-613 9. G.C. Sen, Bhattacharrya, ‘Principles of Machine Tools’, New Central Book Agency, July 1969 ,vol. 51, No.3, P.P. 313-346. 10. John R.Baker ‘Analysis of a simple cantilevered beam with end load’ 2001-2002.