The Effect of Mass Ratio and Air Damper Characteristics on the Resonant Response of an Air Damped Dynamic Vibration Absorber

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1 Modern Mechanical Engineering, 0,, doi:0.436/mme.0.0 Published Online November 0 ( The Effect of Mass Ratio and Characteristics on the Resonant Response of an Air Damped Dynamic Vibration Absorber Abstract Ranjit G. Todkar, Shridhar G. Joshi Department of Mechanical Engineering, P.V.P. Institute of Technology, Budhagaon, Sangli, India Department of Mechanical Engineering, Walchand College of Engineering, Sangli, India rgtodkar@gmail.com Received October, 0; revised November 5, 0; accepted November, 0 In this paper, it is shown that, a road vehicle DOF air damped quarter-car suspension system can conveniently be transformed into a DOF air damped vibrating system representing an air damped dynamic vibration absorber (DVA) with an appropriate change in the ratio µ of the main mass and the absorber mass i.e. when mass ratio µ >>. Also the effect of variation of the mass ratio, air damping ratio and air spring rate ratio, on the motion transmissibility at the resonant frequency of the main mass of the DVA has been discussed. It is shown that, as the air damping ratio in the absorber system increases, there is a substantial decrease in the motion transmissibility of the main mass system where the air damper has been Maxwell type. Optimal value of the air damping ratio for the minimum motion transmissibility of the main mass of the system has been determined. An experimental setup has been designed and developed with a control system to vary air pressure in the damper in the absorber system. The motion transmissibility characteristics of the main mass system have been obtained, and the optimal value of the air damping ratio has been determined for minimum motion transmissibility of the main mass of the system Keywords: Air Damped Dynamic Vibration Absorber, Motion Transmissibility, Effect of Mass Ratio, Air Damper, Optimization. Introduction Many real engineering systems such as, a road vehicle suspension system, a dynamic vibration absorber system, a vibration isolation system of machinery (where the floor supporting the machine is sufficiently flexible), a double centrifugal pendulum system etc., can be adequately represented as DOF vibrating systems []. As such, in this paper, a DOF air damped vibrating system representing a DOF air damped dynamic vibration absorber has been studied. In this case, the mass ratio µ i.e., ratio of the main mass m to the auxiliary mass m is greater than unity (µ >> ) and is in the range of.5 to 5.0. Also the effect of variation of the mass ratio, air spring rate ratio and air damping ratio on the motion transmissibility of main mass has been discussed. It has been shown that, as the air damping ratio in the absorber system increases, there is a substantial decrease in the motion transmissibility of the main mass system in the neighborhood of resonant frequency for the case where the air damper is Maxwell type [,3]. Optimal value of the air damping ratio for the minimum transmissibility of the main mass system has been determined. An experimental setup has been designed with an air pressure control system for setting the appropriate value of air damping ratio in the system.the motion transmissibility characteristics of the main mass m of the dynamic vibration absorber model have been obtained.. Equations of Motion Equations of motion have been derived and are given respectively in Tables, and 3 for the following [], ) A DOF dynamic vibration absorber system with system and without air damper (Refer Figure in Table ), here after referred as Case. ) A DOF air damped dynamic vibration absorber system with system damping and, Vogit type model for Copyright 0 SciRes.

2 94 R. G. TODKAR ET AL. Table. A general DOF viciously damped vibrating system with system damping. m x (t) Table. DOF Air damped vibrating system using an air damper (Vigot Model), with system damping coefficients c, c and air damper characteristics i) air ddamping. Ratio ζ a and ii) air spring rate ratio k = (k a /k ),where k = stiffness of auxiliary spring and k a = stiffness of air spring. k m c x (t) m Case x (t) k c A general DOF vibrating system (Dynamic Vibration Absorber Model). µ >>. c k k a c a m x (t) where Equations of motion m x k x x c x x k x u c x u mx k xx c x x Mt Mt X A A 00 A 4 3 U B B B B B X A A 0 A 4 3 U B B B B B () () (3) (4) k c DOF air damped vibrating system using an air damper (Vigot Model). (Dynamic Vibration Absorber Model). µ >>. Equations of motion a a k x u c x u m x k k x x c c x x a a (5) m x k k x x c c x x (6) Mt X a a 00 a 4 3 U b b b b b (7) A = ζ, A = v + 4 ζ ζ v, A 00 = (ζ v + ζ v), and A = 4 ζ ζ v, A = ζ v, A = ζ v, A 0 = v and Mt X a a 0 a 4 3 U b b b b b (8) B 4 =, B 3 = (ζ + (ζ /µ) + ζ v ), B = ( + (/µ) + v ), B = (v ζ + ζ v ), B 0 = v air damper (Refer Figure in Table ), hereafter referred as Case. 3) A DOF air damped dynamic vibration absorber system with system damping and with Maxwell type model for air damper (Refer Figure 3 in Table 3), hereafter referred as Case 3. Motion Transmissibility Assuming the steady state solutions in the form x = Xe jwt, x = X e jwt and y = Ye jwt the base excitation as u = Ue jwt and following the usual procedure of solution, the equations of motion have been solved and the expressions for the motion transmissibility Mt (for the main mass ) and Mt (for the auxiliary mass) have been obtained and are given respectively in Equations (3) and (4) for Case in Table and in Equations (7) and (8) for Case in Table and in Equations () and (3) in Table 3. [4]. where a = ζ v, a =(v + 4 ζ ζ v + 4 ζ a k 0.5 ζ v), a 00 = (ζ v + ζ a k 0.5 v + ζ v + k ζ v) and a = 4 ζ ζ v + 4 ζ ζ a k 0.5 v) a = (ζ v + ζ a k 0.5 v + ζ v + k ζ v) a = (ζ v + ζ a k 0.5 v + ζ v + k ζ v), a 0 = v ( + k) and b 4 =, b 3 = (ζ + (ζ / µ) + ζ v + ζ a [k (k 0.5 /µ)]) b = ( + k + 4 ζ v ζ + 4 ζ a k 0.5 v ζ + (/µ) + v + (k/µ)) b = ( v ζ + k v ζ + ζ v + ζ a k 0.5 v ), b 0 = v ( + k) Copyright 0 SciRes.

3 R. G. TODKAR ET AL. 95 Table 3. DOF air damped vibrating system using an air damper (Maxwell Model), with system damping coefficients c, c and air damper characteristics air damping ratio ζ a and air spring ratio k = (k a /k ), where k = stiffness of auxiliary spring and k a = stiffness of air spring. k c m c a x (t) Equations of motion m x k x x c x x ka x y kx u cx u (9) k a a a c y x k yx (0) 0 m y (t) a mx k xx c x x c x y () k c Mt X a a a a a a U b b b b b b b () DOF Air damped vibrating system using an air damper (Maxwell Model) µ >>. dynamic vibration absorber model. where δ= k 0.5 /( ζ a ) Mt 4 3 X a a a 4 0 a a U b b b b b b b a 55 = ζ v, a 44 = [v + 4 δ ζ v + 4 ζ ζ v], a 33 = [ δv + 8 δ ζ ζ v + ζ v + v δ (ζ + ζ ) + ζ v (+k)] a = [δ v (v + 4 ζ ζ )+δ v (4 ζ v + 4 ζ + ζ k )+ v ( + k )], a = δ (4 ζ v + ζ v) + δ(k v + v ), a 00 = δ v and a 4 = 4 ζ ζ v, a 3 = ζ v + 8 δ ζ ζ v + ζ v ( + k), a = δ[4ζ v + ζ v + k ζ v + ζ v] + 4δ ζ ζ v + v [ + k], a = δ v [ζ v + ζ ] + δ v [ + k], a 0 = δ v and (3) b 6 =,b 5 = ( δ + ζ + ( ζ /μ) + ζ v), b 4 = [ + k + ( /μ) + v + (k/μ ) + 4 ζ ζ v] + 4 δ [ζ + (ζ /μ) + ζ v] + δ b 3 = δ[ + 8 ζ ζ v + (/μ) + v + k + (k/μ)] + [ ζ v+ ζ v k + 4 ζ ζ v ] + δ [ ζ + ( ζ /μ) + ζ v], b = δ [+ 4 ζ ζ v + ( /μ) + v ] + δ [4 ζ v + 4 v ζ + ζ v k ] + v (+k), b = δ [ ζ v + 4 v ζ ] + δ v, b 0 = δ v m x (t) m x (t) k c k c c a m x (t) k a k c m y (t) Figure. A general DOF vibrating system (dynamic vibration absorber model). µ >>. k c c m x (t) k a k c a m k c x (t) Figure. DOF air damped vibrating system using an air damper (vigot model). (dynamic vibration absorber model). µ >>. Figure 3. DOF air damped vibrating system using an air damper (maxwell model) µ >>. dynamic vibration absorber model. 3. Motion Transmissibility Mt (μ >> ) For the air damped dynamic vibration absorber system, mass ratio µ has been varied in the range of.5 to 5.0, where λ is the ratio of excitation frequency w to the undamped natural frequency w of the system (m,k )), have Copyright 0 SciRes.

4 96 R. G. TODKAR ET AL. been plotted for Case, Case and Case 3. The peak values of Mt (at resonance) are given in Tables 4, 5 and 6 respectively. 3.. Effect of Variation of Mass Ratio µ The values of µ are varied as µ =.5, µ = 3.3 and µ = 5.0 when ζ = 0., ζ = 0, k = 0. and ζ a = 0.05 with spring rate ratio (k /k ) as Table 4 gives respectively the peak values of Mt at resonant frequencies obtained for Case, Case and Case 3. It is seen that, as the value of µ increases, there is no substantial change in the value of Mt at the first resonant frequencies for the case where the air damper is Maxwell type. Figure 4, Figure 5 and Figure 6 show the corresponding Mt vs λ plots. Peak of Mt Table 4. Peak values of Mt with ζ = 0.00, ζ = 0.0, k = 0. and ζ a = 0.05 and value of mass ratio µ is varied. Case µ =.5 µ = 3.3 µ = 5.0 Vigot Maxwell Case Vigot Model Case Model Case 3 Model Case Maxwell Model Case 3 Case Vigot Model Case Maxwell Model Case 3 st peak nd Peak Mt λ Mt λ Peak of Mt Table 5. Peak values of Mt with µ=3.3, ζ = 0.33, ζ =0.0 and ζ a = 0.05 and value of spring rate ratio k is varied. Case k = k = 0.00 k = 0.50 Vigot Maxwell Case Vigot Model Case Model Case 3 Model Case Maxwell Model Case 3 Case Vigot Model Case Maxwell Model Case 3 st peak nd peak Mt λ Mt λ Table 6. Peak values of Mt with µ = 3.3, ζ = 0.33, ζ = 0.0 and k = 0.0 and value of air damping ratio ζ a is varied. Peak of Mt Case ζ a = 0.05 ζ a = 0.05 ζ a = Vigot Maxwell Case Vigot Maxwell Model Case Model Case 3 Model Case Model Case 3 Case Vigot Model Case Maxwell Model Case 3 st peak nd peak Mt λ Mt λ Copyright 0 SciRes.

5 R. G. TODKAR ET AL. 97 frequencies for the case where the air damper is modeled as a Maxwell type Effect of Variation of Air Damping Ratio ζ a Figure 4. Mt vs λ when ζ = 0., ζ = 0, k = 0. and ζ a = The values of air damping ratio ζ a are varied as ζ a = 0.05, ζ a = and ζ a =0.075 when µ = 3.3, ζ = 0.33, ζ = 0 and k = 0.0 with spring rate ratio (k /k ) as Table 6 gives respectively the peak values of Mt at resonant frequencies obtained for Case, Case and Case 3. It is seen that, as the value of air damping ratio ζ a increases there is a substantial decrease in the value of Mt at the resonant frequencies in the case where the air damper is Maxwell type. 4. Optimal Value ζ aopt of Air Damping Ratio ζ a Figure 5. Mt vs λ when ζ = 0., ζ = 0, k = 0. and ζ a = Figure 6. Mt vs λ when ζ = 0., ζ = 0, k = 0. and ζ a = Effect of Variation of Spring Rate Ratio k The values of k are varied as k = 0.075, k = 0.0 and k = 0.5 when ζ = 0.33, ζ = 0.0, µ = 3.3 and ζ a = 0.05 with spring rate ratio (k /k ) as Table 5 gives respectively the peak values of Mt at resonant frequencies obtained for Case, Case and Case 3. It is seen that, as the value of air damper spring rate ratio k increases, there is a small increase in the peak value of Mt at the resonant The air damping is highly effective when the air damper was modeled as Maxwell type (Case 3). As such, a DOF air damped dynamic vibration absorber system for Case 3 is taken for optimization of air damping ratio ζ a The equation of the motion transmissibility Mt (of the main mass m) is given by Equation (3) of Table 3 for Case 3 when the air damper is Maxwell type model [3]. The value of Mt is affected by the system parameters i.e. mass ratio µ, system damping ratio ζ and the air damper characteristics ) air spring rate ratio k and ) air damping ratio ζ a For minimizing the value of Mt, consider equation (3) for Mt is X Mt U a λ a λ a a a λ a λ b λ b λ b λ b b λ b λ b λ (where constants a 44, a 33, a, a, a 00, b 6, b 5, b 4, b 3, b, b and b 0 have been given in Table 3). The equation for Mt is rearranged in terms of ascending powers of ζ a as X Mt U 4 3 A4a A3a Aa Aa A0 4 3 B B B B B 4 a 3 a a a 4 a 0 (4) where A 4, A 3, A, A, A 0, B 4, B 3, B, B and B 0 are the constants containing system damping ratios ζ, ζ, k, µ, λ and v. The equation (4) for Mt is differentiated w.r.t. ζ a and set equal to zero i.e. (Mt)/ (ζ a ) = 0, a polynomial in terms of ζ a is obtained as h h h h a a a 3 h h h a a a h (5) Copyright 0 SciRes.

6 98 R. G. TODKAR ET AL. where h i s (i = 0,,, 3, 4, 5, 6 and 7) are the constant coefficients containing µ, ζ, ζ,k and λ. The expressions derived for this are very lengthy and have not been included in the body of the write-up. The optimal value ζ aopt of ζ a is obtained by solving the Equation (5) and with the optimal value thus obtained the values of Mt have been determined. 4.. Effect on Optimal ζ aopt and on Mt for Various of Air Spring Rate Ratio k Mt k = 0.05 k = The values of ζ aopt for the air damper Maxwell type model have been obtained for ) k = 0.05, k = 0.05, k = and k = 0. and the results are given in Table 7, ) k = 0.00, k = 0.3,k = 0.4 and k = 0.5, the results are given in Table 8 and 3) k = 0.75, k =, k = and k=3, the results are given in Table 9 (Refer also Figure 7). ζ aopt Figure 7. Mt vs ζ a (Effect of k) for k = 0.05 to Effect of Air Spring Rate Ratio k on Optimal Value ζ aopt of ζ a When µ = 0.335, ζ = 0.33, ζ = 0.0 and λ = Figure 7 shows the effect k on Optimal Value ζ aopt of air damping ratio ζ a. From the results of analysis, it is seen that, as the value of ζ aopt increases with the increase in air spring rate ratio k, the value of Mt increases. Figure 7 shows the variation of the value of Mt with ζ aopt for increasing values of k Effect of Variation of Mass Ratio µ on Optimal Value ζ aopt of ζ a Figure 8 shows the effect of variation of mass ratio µ on ζ aot where air damper is Maxwell type. The value of µ is varied as µ =.5 and µ = 5.0 when ζ = 0.33, ζ = 0, k = and λ = with the spring ratio (k /k ) Table 7. of ζ aopt when air spring rate ratio k is varied. µ = 3.3, ζ = 0.33, ζ = 0.0, λ = modeled as: Maxwell type k = 0.05 k = 0.05 k = k = 0. Mt (min) ζ aopt Table 8. of ζ aopt when air spring rate ratio k is varied. µ = 3.3, ζ = 0.33, ζ = 0.0, λ = modeled as: Maxwell type k = 0. k = 0.3 k = 0.4 k = 0.5 Mt (min) ζ aopt , Figure 8. Mt vs ζ a (Effect of µ ). as It is seen that, as the value of µ increases, there is a significant reduction in the value of ζ aopt and there is also a substantial decrease in the minimum value of Mt. 5. Experimental Setup Figure 9 shows the experimental setup designed and developed for dynamic response analysis of the DOF air damped dynamic vibration absorber system (refer also Plate ). The setup consists of a cam operated mechanism to provide sinusoidal base excitation.the necessary software has been developed to collect and process the dynamic displacements to obtain graphical plots of the input excitation u(t) vs time and the main mass response motion x (t) vs time. The system also incorporates the facility to control the operating air pressure in the system through a computer interfaced system as shown in Figure 9. The values of the main mass and auxiliary mass have been selected in accordance with values reported in the literature. The ratio of main mass m to auxiliary m is about 5 to 0. The DOF air damped vibrating system data selected is as, main mass m = 6.0 kg, auxiliary mass m = 0.85 kg,auxiliary spring rate ratio k = 970 N/M, the spring rate of the spring supporting the main mass m is k = 6300 N / M and mass ratio is μ = (m /m ) is Copyright 0 SciRes.

7 R. G. TODKAR ET AL. 99 Figure 9. Experimental setup for DOF dynamic vibration absorber system (µ >> ). Plate. Experimental detup for an air damped DOF vibration absorber system. Copyright 0 SciRes.

8 00 R. G. TODKAR ET AL. 5.. Specifications of the 5.. Air Pressure Control Using the approach of R.D.Cavanaugh [3] for the design of air damper, following relations have been developed [4,5]. ) ka ns vcpi Nt (5) ) a Q l pipe pi N t d pip e where Q 8 s o π vc nm (6) Using these relations, a cylinder-piston and air-tank type air damper has been developed [3]. The specifications of the developed air damper are : piston diameter d p = 9.85 mm cylinder bore d c = mm., piston rod diameter d r = 0.00 mm, piston length l p = 3.0mm. and height of piston bottom from the cylinder bottom h p = 5.00 mm In the experimental investigation, the first step was to select the value of the air damping ratio ζ a associated with the air spring rate ratio k to be set and the corresponding set of capillary pipe dimensions like pipe diameter d pipe and pipe length l pipe. The ratio (pi/nt), where pi is the operating air pressure and Nt is the ratio (v. t /v c ) is the basis for the selection of the air damping ratio ζ a (also refer Figure 0 and Figure ).The parameters d pipe, l pipe and the ratio (pi/nt) have been varied to change the value of damping ratio ζ a in the system. (pi / Nt)... d c = 0 mm d c = 0 mm. d c = 30 mm. Figure 0. k vs (pi/nt). A computer interfacing system containing the closed loop air pressure control system with a set of two LVDTs to sense the main mass displacement x (t) and base excitation u(t) has been developed.the ratio (pi/nt) plays an important role in controlling the air damping ratio ζ a in the system. The appropriate value of the ratio (pi/nt), depending on the value of ζ a desired in the system can be set by controlling the value of operating air pressure pi for a given value of the ratio Nt= (v t /v c ) or keeping the air pressure in the system at atmospheric pressure and adjusting the value of Nt by adjusting the tank volume v t. 6. Experimental Analyses 6.. Experimental Curves for Motion Transmissibility Mt vs Frequency Ratio λ Using the experimental setup (shown in Figure 9 and Plate ) and by setting the appropriate values of the air spring rate ratio k and the air damping ratio ζ a, the experimental plots of Mt vs λ have been obtained for the following cases ) With µ =.5, ζ = 0.33, ζ = 0 and without air damper.(refer Figure and Table 0 ). ) With µ =.0, ζ = 0.33, ζ = 0 and air damper, with k = 0.43 and ζ a = 0.36 (Refer Figure 3 and Table ) 3) With µ =.5, ζ = 0.33, ζ = 0 and air damper, with k = 0.43 and ζ a = 0.36 (Refer Figure 4 and Table ). 6.. Experimental Curves Mt vs λ for Using Optimal of Air Damping Ratio ζ aopt Table shows the theoretical and experimental peak values of motion transmissibility Mt at resonant frequency with the air damper set for the optimal air damping ratio ζ aopt : k = 0. with ζ aopt = 0.53 and k = 0.4 with ζ aopt = The experimental results have been shown in Figure 5 and Figure 6. ζ a... d pipe =.5 mm d pipe =.0 mm. d pipe =.5 mm for l pipe = 3.0 m (pi / Nt ) Figure. ζ a vs (pi/nt). Figure. Mt vs λ for Case 6. (i). Copyright 0 SciRes.

9 R. G. TODKAR ET AL. 0 Table 0. Peak values of Mt for Case 6. (i). Peak of Mt Theoretical Results Experimental Results st peak Mt λ Figure No. Table. Peak values of Mt with optimal valve of optimal air damping ratio ζ aopt with µ =.5, ζ = 0.33, ζ = 0.0. Theoretical Experimental Theoretical Experimental Peak of Mt k =0., ζ aopt = 0.53 k = 0.4, ζ aopt =0.68 st peak Mt λ Figure No. 5 6 Figure 3. Mt vs λ for 6. (ii), µ =.0, k = 0.43 and ζ a = Table. Peak values of Mt for the Case 6. (ii) and for the Case 6. (iii) with ζ = 0.33, ζ = 0.0. Figure 5. Mt vs λ. For µ =.5, ζ = 0.33, k = 0., ζ aopt = Peak of Mt st peak µ =.0, k = 0.43 and ζ a = 0.36 Theoretical Experimental µ =.5, k = 0.43 and ζ a = 0.36 Theoretical Experimental Mt λ Figure No. 3 4 Figure 6. Mt vs λ. µ =.5, ζ = 0.33, k = 0.4, ζ aopt = Conclusions Figure 4. Mt vs λ for 6. (iii), µ =.5, k = 0.43 and ζ a = In this paper, the effect of mass ratio and the air damper characteristics on the resonant response of an air damped DOF vibrating system representing an air damped dynamic vibration absorber model have been studied with the air damper Maxwell type. There is no substantial change in the value of Mt with the increase Copyright 0 SciRes.

10 0 R. G. TODKAR ET AL. in the value of mass ratio µ. However, with the increase in the value of the air spring rate ratio k there is a considerable increase in the value of the Mt at the resonant frequency where the air damper is Maxwell type. It is seen that, with the increase in the value of the air damping ratio ζ a there is a considerable decrease in the value of the Mt at the resonant frequency where the air damper is Maxwell type. Further it is seen that as the value of the air spring rate ratio k increases, the value of the optimum value ζ aopt of the air damping ratio ζ a increases with increase in the value of motion transmissibility Mt. It is also observed that there is a considerable reduction in the value of ζ aopt with the increase in the value of the mass ratio µ, in the range µ =.5 to µ = 5.0. An experimental setup has been developed with an appropriate air pressure control system. A cylinder-piston and air-tank type air damper has been designed and developed to obtain the desired value of the air damping ratio ζ a from the air damper. From the results of the experimental analysis shown in Figure 3 and Figure 4, it is seen that the experimental peak values of Mt are close to the corresponding theoretical peak values of Mt obtained from the theoretical analysis where the air damper is Maxwell type. From the Figure 5 and Figure 6, it is seen that the theoretical and experimental values of Mt for ζ aopt = 0.53 with k = 0. and ζ aopt = 0.68 with k = 0.40 are in good agreement. From the theoretical and experimental investigations carried out, it is seen that the addition of the air damping in the absorber system (m,k ) improves substantially the motion transmissibility characteristics of the main mass of the DOF dynamic vibration absorber model over a range of excitation frequencies in the region of resonance. 8. Acknowledgements The first author Prof. R.G.Todkar acknowledges with thanks the authorities of P.V.P.Institute of Technology, Budhgaon, Dist. SANGLI (Maharashtra) INDIA for their support and encouragement during the period of this work. 9. References [] R. A. Williams, Electronically Controlled Automotive Suspension Systems, Computing and Control Engineering Journal, Vol. 5, No. 3, 994, pp doi:0.049/cce: [] T. Asami and Nishihara, Analytical and Experimental Evaluation of an Air Damped Dynamic Vibration Absorber: Design Optimizations of the Three-Element Type Model, Transaction of the ASME, Vol., 999, pp [3] R. D. Cavanaugh, Hand Book of Shock and Vibration, Air Suspension Systems and Servo-controlled Isolation Systems, Chapter 33, pp. -6. [4] R. G. Todkar and S. G. Joshi, Some Studies on Transmissibility Characteristics of a DOF Pneumatic Semiactive Suspension System, Proceedings of International Conference on Recent Trends in Mechanical Engineering, Ujjain Engineering College, Ujjain, 4-6 October 007, pp [5] P. Srinivasan, Mechanical Vibration Analysis, Tata Mc-Hill Publishing Co., New Delhi, 990. Copyright 0 SciRes.

11 R. G. TODKAR ET AL. 03 Nomenclature k k m m stiffness of spring for absorber mass stiffness of spring for main mass absorber mass main mass µ mass ratio = (m /m ) w (k /m ) / w (k /m ) / v natural frequency ratio = (w /w ) ζ system damping ratio for main mass system ζ system damping ratio for auxikary mass system w applied frequency λ frequency ratio = (w/w ) d p piston diameter d c cylinder bore l p length of the piston height of piston from bottom of the cylinder h p d pipe l pipe inside diameter of the capillary pipe length of the capillary pipe µ o viscosity of air n index of expansion of the air k a stiffness of air spring k spring rate ratio = (k a /k ) w a (k a /m ) / c a ζ a coefficient of viscous damping of the air damper damping ratio provided by the air spring ζ aopt optimal value of air damping ratio u(t) base excitation x (t) dynamic displacement response auxiliary mass m x (t) dynamic displacement response of main mass m Mt motion transmissibility of the auxiliary mass m Mt motion transmissibility of the main mass m Copyright 0 SciRes.