Variable stiffness nonlinear isolator: Design, analysis and simulation

lethanhdanh@iuh.edu.vn ABSTRACT It is difficult for a conventional linear isolator including an elastic element in parallel with a damper to prevent the low frequency vibration band. Hence, this work will design, analysis and simulate a variable stiffness nonlinear isolator (VSNI) with air spring. The main feature of the VSNI is that the isolated object is supported by a mechanism including wedge-roller-air spring (named the main mechanism-MM) meanwhile the model of VSNI can obtain the lower resonance frequency and the higher vibration attenuation than the equivalent linear isolation model (ELIM) but guarantee the load bearing capacity through introducing a cam-roller-air spring (named auxiliary mechanism-AM). Because the pressure in air springs is the key parameter which is used to adjust the stiffness of the MM and AM, the influences of the pressure on the restoring force and the dynamic response of the VSNI are presented. Furthermore, the effect of the isolated load on the isolation response of the VSNI is investigated. For this purpose, the complex stiffness of the air spring will be analyzed. Next, the motion equation of the system will be built. The numerical simulation of the vibration transmissibility of the proposed model will be performed through fourth-order Runge-Kutta algorithm. Keywords: Nonlinear isolation, Air spring, Low frequency, Stiffness correction Cite this Article: T.D. Le, Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation. International Journal of Mechanical Engineering and Technology 10(11), 2019, pp. 36-51. 1. INTRODUCTION As commonly known, reducing the stiffness of a vibration isolator would produce the low resonance frequency that extends the isolation band toward low frequency. However, this way is difficult to carry out for a traditional linear isolator including a spring connecting with a damper in parallel, because a reduction in stiffness will result in a large deflection and low load capacity. This is also a major limitation of the traditional linear isolator for applying widely in engineering practice such as vehicle suspension or protection of machinery, equipment. Especially, high precision equipment because it is easily sensitive to external Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 37 editor@iaeme.com vibration, shock for example instrumentation. Recently, the passive isolation method with low static and high dynamic stiffness (LSHD) or quasi-zero stiffness (QZS) has been researched in deep to improve the isolation effectiveness as well as to broaden the isolation range in literature [1]. I. Kovacic et al. [2] analyzed the effects of the static force on the dynamic response of the quasi-zero-stiffness system and the stability of the steady-state response. An isolation model using the negative stiffness structure for vehicle seat was analyzed and simulated numerically by Le et al. [3], showing that this design model outperforms in comparison with the equivalent linear model. Then, the experimental investigation of which confirmed the theoretical model as shown in [4]. X.Ch. Huang et al. [5] studied the dynamic response and stability of a high-static and low dynamic stiffness isolator including an Euler beam formed negative stiffness corrector paralleled with a conventional linear isolator. By applying time-delayed active control strategy, the performance of the isolator with quasi-zero stiffness would be improved as researched by X. Su et al. [6]. A novel dynamic model with stable-quasi-zero stiffness which was constructed by a positive stiffness component and a pair of inclined linear springs providing negative stiffness was suggested by Hao et al. [7]. In order to widen previous studies of the isolator with the characteristic of high-static and low- dynamic stiffness, Shaw et al. [8] had indicated that simple changes in the shape of the force- displacement curve can have large effects on the amplitude and frequency of peak response, and can even create unbounded response at the certain levels of excitation. An archetypal dynamic model with quasi-zero stiffness which comprises a lumped mass denoting the isolated object and a pair of the horizontal springs providing negative stiffness in parallel with a vertical linear spring to bear the load was studied by Z. Hao et al. [9]. A multi-Direction Quasi-Zero-Stiffness vibration isolator with time-delayed active control, which can be realized excellent vibration isolation in three directions simultaneously was suggested and analyzed by Xu et al. [10]. The effects of the equilibrium position on the dynamic response of the isolation system using negative stiffness structure were analyzed by Le et al. [11]. In addition, by integrating the linear mechanical spring and magnets, the high-static-low- dynamic stiffness isolator was proposed and analyzed in [12-13]. Besides, Q. Le et al. [14] introduced a vibration isolator which is the combination between the magnetic spring and rubber membranes to attain low natural frequency. One of the main issues of the isolation method with the LSHD is the adjustment of the stiffness according to the change of the isolated load. It can be realized by replacing the spring or regulating the configurative parameters of the system. This adjustment or replacement may cause difficulties for applying in practice. Hence, this paper will introduce an innovative variable stiffness nonlinear isolator with air spring. The stiffness of the proposed model can be adjusted via controlling the pressure in air springs so that VSNI can remain both the desirable low stiffness at the wanted static equilibrium position and load capacity as well as reduction in the static deformation. The proposed model can be used in the precision fabrication field, instrumentation, etc. In addition, this isolation way can be employed in other engineering fields such as the space antennae, satellites, isolation platforms, etc. The rest of the paper is organized as follows. The configuration of the proposed system is presented in section 2. Mechanical-pneumatic coupling model of the VSNI is analyzed in section 3. Dynamic modeling of the VSNI is obtained and then the response simulation is carried out in section 4.Finally, some conclusions are drawn in section 5. 2. CONFIGURATION OF THE VSNI As shown in Fig. 1(a), the wedges (9) fixed on the vertical bars (8) along with rollers (3) work as wedge mechanism. This mechanism combining with two air springs (1) is used to support the load plate (4), which is named “main mechanism-MM” indicated by dashed-line rectangle. Besides, the auxiliary mechanism-AM plotted by the dot-line rectangle, including T.D. Le 38 editor@iaeme.com semicircular surfaces (6) fixed on the vertical bars (8), rollers (5) and two air springs (2), is introduced to modify the dynamic stiffness of the VSNI. Herein, the semicircular surface and roller are considered as the cam mechanism. The load plate only moves in vertical direction through the guide (7). During operation, the air springs are always compressed, it means that the main mechanism offers positive stiffness, whereas, the auxiliary mechanism has the vertically negative stiffness. The pressure in the air springs (1) can be adjusted to remain the designed static equilibrium position when the weight of the isolated object is changed. However, this adjustment may produce the negative or positive stiffness of the VSNI. To remain low positive stiffness of the VSNI, the pressure of the springs (2) can be properly regulated through a relationship between the pressure in the springs (1) and (2) which will be discussed in next section. Fig. 1(b) presents the photograph of the VSNI. In order to reduce the effect of the friction on the dynamic response, the linear bearings are installed in the model. Figure 1. (a) Physical model of the VSNI; (b) Photograph of the VSNI 3. MECHANICAL-PNEUMATIC COUPLING MODEL OF VSNI 3.1. Analysis of the Air Spring Figure 2. Modeling of the air spring Uncompressed state dEil, Til, Gil dEol, Gol Compressed State P,V,T x Fair hd hair Dh Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 39 editor@iaeme.com The air spring model is considered as in Fig. 2. Without the heart change, the thermodynamic equation in the air spring is described as following: il ch ae oldE dE dE dE   (1) in which Eil and Eol are the air energies of input and output line, Ech is the air energy in spring, Eae is the work of air expansion. These energies are given as following: il p il il ol p ol ch v air v air ae dE C T G dE C T G dE C m dT C Tdm dE PdV      (2) where Cp and Cv are specific heat capacities at constant pressure and volume, respectively. Til is the temperature of air at the inlet; mair, T and P are the mass, the temperature and pressure of the air in the air spring, V is the volume of the air spring. Gil and Gol are mass low rates at inlet and outlet. From the ideal air equation, we have: air gas gas air air gas PV m R T PdV VdP R Tdm m dT R     (3) Substituting Eq.(2-3) into Eq.(1), the air spring internal pressure equation is expressed as below:  il il ol gas ol n P G RT G R T PV V    (4) where Rgas is the gas constant, n=Cp/Cv is the ratio of specific heat capacity. Considering that the charging and discharging processes are not to occur, changing the pressure in the air spring is given by: P P n V V   (5) The restoring force created by the air spring is obtained as following:  air atmF P P A  (6) Here in A is effective area of the air spring, Patm is the ambient pressure By differentiating Eq.(6) versus the deformation of x, the stiffness of the air spring is obtained as below:  airair atm dF nAP dV dA K P P dx V dx dx      (7) T.D. Le 40 editor@iaeme.com Let hd be the design height of the air spring. Adh, Vdh and Pdh are the effective area, volume and pressure at the design height hd, respectively. Taking the linearization of the Eq. (7), around this height, the stiffness of the air spring is approximated as following:  hd hd Vair dh atm A hd nA PdF dF K P P dx dh V        (8) here, ; d d V A h h h h dV dA dh dh       The linearized restoring force of the air spring can be rewritten as following:  air air dh atm dhF K h P P A  D   (9) with  air dh h h xD    , hair is the height of the air spring at the uncompressed state, Pdh and Adh are the pressure and the effectiveness area of the air spring at the design height, respectively, 3.2. Restoring force and stiffness of the VSNI In order to analyze the restoring force of the VSNI, the model of force acting on the wedge and cam mechanisms is shown in Fig. 3. The dot line presents the initial position of the system, meaning that at this position, air springs 1 and 2 are uncompressed. As the load plate moves vertically down an amount of y, the result is that the both air springs are compressed horizontally by an amount of x1 and x2 given by Eq. (10), respectively. Figure 3. Modeling of force acting on the wedge and cam mechanisms Ho r x2 y x1 a R y A ir s p ri n g 1 A ir s p ri n g 2 Fair1 Fair2 Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 41 editor@iaeme.com   1 2 2 2 2 2 tan (a) (R r) ( ) (R r) (b)o o x y x H y H a        (10) in which R and r are the radii of the cam and roller, Ho is maximum displacement of the load plate in vertical direction, a is the inclined angle of the wedge. Consider a small vibration of the load plate around the design static equilibrium position (shorten design position-DP) at which the load plate only is acted by the vertical restoring force created by the air springs 1, the design height of the air springs 1 and 2 is:   1 1 0 2 2 2 2 tan (a) R r (R r) (b) d air d air o h h H h h H a        (11) where subscripts „1‟ and „2‟ are designated for the air springs 1 and 2, respectively. By using the Eq. (9), the restoring forces of the main mechanism FMM and the auxiliary mechanism FAM are expressed as following: 12 1 12 tan 2 tan h MM air airF K y Fa a D  (12)       2 2 2 2 2 2 22 2 (R r) 2 1 2 (R r) (R r) o h o AM air o air o o H H y F K H y F H y H y D                  (13) herein 11 h airF D and 12 h airF D are the forces of the air springs 1 and 2 at the initial position (y=0) obtained as following:  1 1 1 1 1 1( ) h air dh atm dh air air dF P P A K h h D     (14)  2 2 2 2 2 2( ) h air dh atm dh air air dF P P A K h h D     (15) in the analysis above, 2 2 1 1 2 2tan ; R r (R r)air d o air d oh h H h h Ha        By letting ou H y  and introducing dimensionless parameters as below:     1 1 2 1 1 1 2 2 2 2 2 21 1 1 1 1 22 2 2 2 2 ˆ ˆ ˆˆ; ; ; ; ; ( ) ( ) (R r) (R r) 2 ;B 2 ; V 1 tan ˆ1 1 C 2 (R r) h h d oMM MM MM MM o air air d d V d atm d d L d d atm d d d d d d d o d P HF F u F F u H K R r K R r P nA A P A V A K nA P A V P nA V A A H V       a   D D                                   22 2 21 1 1 1 1 ˆ2 1 1 2 ; ; (R r) 1 tan atm d o atm d L Ld d atm d d dh P A H P A D K KnA V P A V P    a               T.D. Le 42 editor@iaeme.com Then, Eq. (12-13) can be written in form of dimensionless as below:   1ˆ ˆ ˆˆ 2 tanhMM o MMF H u F aD   (16) 2 2 2 2 2 2 ˆ ˆ1 1 ˆ1ˆ ˆ ˆ1 1 ˆ ˆ ˆ ˆ1 1 1 1 o o AM H H u F B C u A u D u u u u                              (17) The stiffness of the MM and AM can be attained by differentiating of Eq. (16-17) versus the dimensionless displacement uˆ , we obtain the dimensionless dynamic stiffness in vertical direction as following: Kˆ 1MM  (18)         2 2 2 2 3 2 3 2 2 2 2 2 2 2 2 3 2 3 2 2 ˆ ˆˆ 1 1 ˆ 1 Kˆ A 1 ˆ ˆ1 1ˆ ˆ1 1 ˆ ˆˆ1 1 ˆ1 1 ˆ ˆ1 1ˆ ˆ1 1 o o AM o o u H H u D u uu u H u H u B B C C u uu u                                           (19) Furthermore, to remain the design position, it is necessary to adjust the pressure in the air springs 1 according to the change of the mass (M) of the isolated object as following 1 12 tan dh atm dh Mg P P A a   (20) in which g is gravity acceleration 3.3. Analysis of the restoring force Considering the MM with the effectiveness area Adh1of the air springs 1 listed in table 1, Fig. 4 presents the relation between the isolated mass (M), the inclined angle (α) of the wedge and pressure Pdh1 for which the VSNI would achieve the design equilibrium position. It can be seen that in order to remain the DP, the inclined angle of the wedge or the pressure in the air springs 1 should be increased or decreased in according with the rise or fall in the isolated load, respectively. However, compared with the regulation of the inclined angle of the wedge, the adjustment of the pressure in the air spring is easier to realize. In addition, note that the load capacity of the isolated model is large for the low pressure and the large inclined angle of the wedge. But in this case if the inclined angle is less than 10 o , to support the heavy load, the pressure in air springs of the MM must be very large that may cause a difficulty for applying in practice. However, the case of the inclined angle of the wedge larger than 45 o results in the larger deformation of the air spring than the vertical deformation of the load plate as given in Eq. (10a), which may cause a deformation exceeding the limitation of the air spring. Overall, it is necessary for the wedge to select its inclined angle to appropriate to a specially practical application including the maximum deformation of the air spring, maximum working pressure and the mass of the isolated object. For analysis purpose about Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 43 editor@iaeme.com the effects of the pressure in air springs on the stiffness of the MM and AM as well as the dynamic response of the VSNI, the inclined angle of the wedge is chosen at value of 37 o . Figure 4. Relationship between the inclined angle α of the wedge, the mass M and pressure Pdh1 From Eq. (16), it is noted that the dimensionless restoring force of the main mechanism is a linear function versus the displacement of the isolated object and is influenced by the pressure in the air springs 1 at the design height. This is simulated in Fig. 5 for Pdh1=1.3, 1.5, 1.97, 2.5, 3.0, 3.5, 4.0 bar, the other parameters as listed in table 1. As observed, the slope of the dimensionless restoring force of the ˆ MMF which is defined as the ratio of the change in the restoring force to the corresponding change in the displacement is the stiffness of the MM as given by Eq. (18). The ˆ MMF is varied according to the change of the pressure Pdh1 but the slope of the curve of the ˆ MMF is remained at the value of unity. This means that the pressure Pdh1 can be used to regulate the restoring force of the MM so that the system always obtains the design position as there is a change in the isolated mass. In addition, the curve of the ˆ MMF for Pdh1=1.97 bar plotted by the solid line divides the plane  ˆ ˆ,MMF y into two regions. In the upper one, the ˆ MMF is always positive, whilst, in the lower region, the ˆ MMF has negative value when the vertical displacement begins from 0 to the specific position depending on the Pdh1 if exceeding this position, ˆ MMF will offer the positive value. For instance, Pdh1=1.3, ˆ MMF <0 within from 0 to 0.152 and out of this range ˆ MMF >0. In order to guarantee the load bearing function, the ˆ MMF is always larger than zero in the vertical displacement region from 0 to ˆ2 oH . Therefore, the pressure Pdh1 in the air springs of the MM must be calculated to obtain the positive restoring force of the ˆ MMF in the displacement area within from 0 to ˆ2 oH . In order to guarantee this requirement, the restoring force of the MM at the initial position given in Eq. (14) must be larger than zero. P d h 1 ( b a r) M (Kg) a (Deg ree) T.D. Le 44 editor@iaeme.com Table 1 The parameters of simulation Parameters Original values R 60 r 20 1, 2,A A  -0.11 1, 2,V V  0.0114 Adh1, Adh2 0.0105 m 2 Figure 5. Dimensionless restoring forces of the MM simulated by Eq.(16) for various values of Pdh1 (detailed notations of line types are seen in figure), another parameters given in table 1. Fig. 6 shows the dimensionless restoring force curve  ˆAMF of the auxiliary mechanism through Eq. (17) for Pdh1=2.3 bar, the various values of the ratio () of the pressure Pdh2 to Pdh1 (seeing details for notations of the types of lines and chosen values of  in Figure), the same other parameters as in Fig. 5, that are parameters given in table 1. It is interesting to observe that at the design position, the ˆ AMF is always equal to zero. At the DP, the slope of the curve of the ˆ AMF exhibited by the dashed line is minus one for =1.53. As known, the slope of ˆ AMF is the equivalent the dimensionless stiffness of the auxiliary mechanism given by Eq. (19), hence, with this value of  the dimensionless stiffness of the AM is also equal to minus one. If the value of  exceeds 1.53, for instance, =1.8, the slope of the dimensionless force curve of the AM denoted by the dashed-dot-dot line is smaller than -1, meanwhile, the slope of which is quasi-zero for =0.42 (expressed by the solid line) and larger than zero for =0.35 (denoted by the dot line). In addition, Fig. 7(a) and (b) present the dimensionless restoring force curves of the auxiliary mechanism for various values of  within 0.42<≤1.53. Herein, the chosen parameters and the notations of the curves are given in detail in upper-right panel of each figures, the other parameters are the same as in Fig. 6. As observed, around the design position, the restoring force of the auxiliary mechanism is decreased according to increasing the vertical displacement, meaning that auxiliary mechanism offers the negative stiffness around this position. However, with a smaller value of  the AM will retain negative stiffness 0 0 0.8 -0.2 Pdh1=1.97 bar Pdh1=1.3 bar Pdh1=1.5 bar Pdh1=2.5 bar Pdh1=3.0 bar Pdh1=3.5 bar Pdh1=4.0 bar yˆ ˆ M M F 0.152 0 ˆ2H Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 45 editor@iaeme.com over a smaller range of the vertical displacement as shown in Fig. 7(a). Furthermore, if the value of Pdh2 is much greater than 0.42 for instance =1.4, 1.5, 1.53 the auxiliary mechanism almost achieves the negative stiffness within 0 to ˆ2 oH as plotted in Fig. 7(b). In addition, the larger the value of  is, the lower the slope of the restoring force of AM is. Consequently, the dynamic stiffness of the AM is decreased in accordance with the upturn in the pressure ratio. Based on these analysis results, the dimensionless stiffness of the MM is always equal to one regardless the change in the pressure of the air springs 1 but the stiffness of the AM is always varied according to the change in the pressure of the air springs 2. This indicates that the stiffness of the VSNI can be controlled to obtain the design position and the wanted low stiffness at this position as the isolated load is varied. However, due to MM in parallel with AM, the stiffness of the VSNI is linear combination of the stiffness of the MM and AM. Reducing the dynamic stiffness of the AM would result in a decline of the dynamic stiffness of the VSNI. Hence, to guarantee the load capacity, the dimensionless stiffness of the AM must be larger than or equal to minus one. In the case of the dimensionless stiffness value of AM being -1, the dynamic stiffness of the VSNI at the design position is equal to zero. Figure 6. The displacement-force ˆ AMF relation for the different values of  (detailed annotation of varied types of lines and selected values of  are presented in upper-right corner of the figure) Figure 7. Restoring force curves of the auxiliary mechanism: (a) for =0.50, 0.55, 0.60, (b) for =1.40, 1.50, 1.53 (detailed annotations of various types of lines are presented in upper-right corner of each figure) -0.4 0.0 0.4 =1.53 =0.42 =1.80 =0.35 Designed position yˆ ˆ A M F 0 0 ˆ2H 0 ˆyˆ H -0.3 0.0 0.3 =1.53 =1.50 =1.40 yˆ ˆ A M F 0 0 ˆ2H (a) (b) -0.032 0.000 0.032 =0.50 =0.55 =0.60 yˆ ˆ A M F 0 0 ˆ2H Negative stiffness region Negative stiffness region T.D. Le 46 editor@iaeme.com 4. DYNAMIC MODELING AND RESPONSE SIMULATION OF THE VSNI As mentioned above, the restoring forces of the MM and AM as well as stiffness of which depend on the pressure of the air springs 1 and 2, showing that the changes in the pressure will inevitably affect on the dynamic response of the VSNI. Furthermore, the restoring force of the AM offers strongly nonlinear characteristic. Hence, this section will express an accurate dynamic model of the VSNI. Then, the complex dynamic response of which will be performed through numerical simulation by using a fourth-order Runge-Kutta algorithm with a fixed time step of 1/100 of the harmonic excitation period. The data of maximum amplitude are sampled via using Poincaré sections [15]. 4.1. Dynamic Modeling Figure 8. Schematic diagram of VSNI. Specifically, the dashed-line rectangle is presented for the main mechanism. The dot-line rectangle is representative for the auxiliary mechanism In general, the VSNI is a one degree of freedom isolated model and shown in term of simplified model as in Fig. 8. Herein, the isolated object (M) is supported by the main mechanism with positive stiffness (KMM) denoted the dashed-line rectangle which is connected in parallel with the auxiliary mechanism offering the negative stiffness (KAM) exhibited by the dot-line rectangle. The stiffness of both mechanisms can be adjusted through controlling the pressure of the air springs. Besides, C is the coefficient of damping to decay the free vibration. Consider a harmonic excitation Ze from the base as shown in Fig. 8 in which Z is the absolute displacement of the isolated object. Then, by applying the Lagrange equation, the dynamic modeling of the proposed system is written as following: 24 tan MM AM e e Mu C u F F Mg Mz z u z a          (21) M KMM K AM Z Ze(t) C Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation 47 editor@iaeme.com 4.2. Dynamic Response a. The effect of the pressure ratio  on the amplitude-frequency response Consider the system given by Eq. (21), in which the pressure Pdh1 of the air springs 1 is 2.3 bar, the pressure ratio  is varied from 0.42 to 1.53, other configurative parameters of the VSNI are given in table 1. For these values, in order to vibrate around the design position the load plate should support an isolated object having mass=102.86 Kg. The result is that the vibration transmissibility curve of the model is plotted in Fig. 9 based on the integration of Eq. (21) for the excitation amplitude of 10 mm and the excitation frequency  swept from 0 to 50 rad/s. The detailed notations of the types of lines are presented in right-corner of figure. It confirms that when the value of  is reduced from 1.48 to 0.75, meaning a corresponding enhancement in the slope of the restoring force curve of the AM as shown in Fig. 6, the resonance frequency and amplitude are increased. This result indicates that the VSNI will offer a large isolation region corresponding to a great value of . Besides, the vibrated attenuation of VSNI is upgraded according to the growth of . Figure 9. Transmissibility of absolute displacement for M= 102.86, Pdh1=2.3 bar, various values of  (see more detailed notations for lines and selected values of  in right-corner of figure) Especially, if =1.53, for which the slope of the restoring force curve of the AM at the equilibrium position is -1 (seeing in Fig. 6), the isolated region and the vibration attenuation rate of the VSNI denoted in Fig. 10(a) are larger than that of one considered in Fig. 9. In addition, for this case, the peak amplitude of the VSNI is also reduced remarkably. But the same excitation and simulation parameters as in Fig. 9 that are Pdh1, M, α, Ze, the isolated effectiveness of the equivalent linear isolation model (ELIM) which is set by removing the auxiliary mechanism from the VSNI in Fig. 1 is poorer than that of the VSNI as shown in Fig. 10(b). It can be observed that, the frequency for the vibration attenuation of the ELIM is bigger than approximately 28 rad/s, whilst, the vibration attenuation of the VSNI having =1.53 begins from 5 rad/s. 10 20 30 40 50 0 1 2 3 4 5 6 7 8 =1.42 =1.20 =0.98 =0.75 =1.48 De cr ea se o f  Excitation frequency (rad/s) |Z |/| Z e | T.D. Le 48 editor@iaeme.com Figure 10. The comparison of the vibration transmissibility of the VSNI having =1.53 (a) and ELIM (b). b. The effect of the mass on the dynamic response As analyzed above, it is evident that the advantages of the VSNI in the case of the accuracy of the isolated mass, indicating that the VSNI achieves the design equilibrium position. However, in fact, the weight of the isolated object can be changed or inaccurate. This can lead to the effect on the isolation effectiveness of the VSNI. Hence, this sub-section will be taken account into the dynamic response when the VSNI is not to obtain the design equilibrium position. By integrating Eq. (21) for the pressure ratio =1.31, M= 82.28 and 123.4 kg, other parameters and the excitation are the same as in Fig. 9. The result is to obtain the vibration transmissibility curve as shown in Fig. 11(a) for M=82.28 kg and in Fig. 11(b) for M=123.4 kg. It is interesting to observe that the frequency at which the vibration attenuation begins to occur is approximately 17 rad/s for M=82.28 kg and 14 rad/s for M= 123.4 Kg. However, it may exist a disadvantage that in isolated region, it appears a small area denoted by dashed- line ellipse called the amplified region in which the displacement of the isolated object can be increased compared with the excitation. This is verified through an excitation from the base with the amplitude of 10 mm, other configurative parameters of the VSNI are remained as above. The first case, the isolated load and the excitation frequency are 82.28 kg and 28 rad/s, respectively by which the time history of the absolute displacement and the phase orbit are shown in Fig. 12(a) and 12(c), meanwhile, Fig. 12(b) and 12(d) show the displacement of the isolated object and the phase trajectory for the second case in which the former is at 123.4 kg and the later is at 23 rad/s. It can be seen that both of cases of the excitation, the displacement of the isolated object is bigger than the excitation. As observed, the amplitudes of the isolated object are approximately 26 mm and 34 mm for the 1 st and 2 nd cases, respectively. Furthermor