A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat

Journal of Sound and Vibration 330 (2011) 6311–6335

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A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat Thanh Danh Le a, Kyoung Kwan Ahn b,n a b

Graduate school of Mechanical and Automotive Engineering, University of Ulsan, Republic of Korea School of Mechanical and Automotive Engineering, University of Ulsan, Daehakro 93, Nam-gu, Ulsan 680-764, Republic of Korea

a r t i c l e i n f o

abstract

Article history: Received 16 June 2011 Accepted 29 July 2011 Handling Editor: L.N. Virgin Available online 31 August 2011

This paper designs and fabricates a vibration isolation model for improving vibration isolation effectiveness of the vehicle seat under low excitation frequencies. The feature of the proposed system is to use two symmetric negative stiffness structures (NSS) in parallel to a positive stiffness structure. Here, theoretical analysis of the proposed system is clearly presented. Then, the design procedure is derived so that the resonance peak of frequency–response curve drifts to the left, the load support capacity of the system is maintained, the total size of the system is reduced for easy practical application and especially, the bending of the frequency–response curve is minimized. Next the dynamic equation of the proposed system is set up. Then, the harmonic balance (HB) method is employed to seek the characteristic of the motion transmissibility of the proposed system at the steady state for each of the excitation frequency. From this characteristic, the curves of the motion transmission are predicted according to the various values of the configurative parameters of the system. Then, the time responses to the sinusoidal, multi frequency and random excitations are also investigated by simulation and experiment. In addition, the isolation performance comparison between the system with NSS and system without NSS is realized. The simulation results reveal that the proposed system has larger frequency region of isolation than that of the system without NSS. The experimental results confirm also that with a random excitation mainly spreading from 0.1 to 10 Hz, the isolation performance of the system with NSS is greatly improved, where the RMS values of the mass displacement may be reduced to 67.2%, whereas the isolation performance of the system without NSS is bad. Besides, the stability of the steady-state response is also studied. Finally, some conclusions are given. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction As is well known, the vibration at low excitation frequencies (0.5–5 Hz) are the main risk factors for lumbago or backache, which seriously affect to mental and physical health of drivers and passengers and reduce their working efficiency studied by Paddan [1]. In order to upgrade the ride comfort and safe for drivers and passengers, these vibrations should be suppressed. Thus, currently vibration isolation approaches have been studied in great depth to upgrade the suppression of the vehicle seat vibration. The model of a single degree of freedom (system without NSS) that is often used to present vibration isolation. It consists of a linear stiffness spring in parallel to a damper given by Thorby [2] and Kelly

n

Corresponding author. Tel.: þ82 52 259 2282; fax: þ82 52 259 1680. E-mail addresses: [email protected] (T.D. Le), [email protected] (K.K. Ahn).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.07.039

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Nomenclature a b C d

D F Fap Fh Fv g hid K Kv Kh Kap KSEP L Lh Lo M nr Q t T Ta u U v V x zm ze Zm

length of bar in millimeter distance from wall to the mass in millimeter damping coefficient in N/m/s range of the displacement from the static equilibrium position over which the dynamic is always smaller the static stiffness in millimeter dissipation function restoring force of system in N approximate restoring force of system in N force of horizontal spring in N force of vertical spring in N acceleration of gravity in m/s2 initial deformation of the vertical spring in millimeter dynamic stiffness of system in N/m stiffness of vertical spring in N/m stiffness of horizontal spring in N/m approximate dynamic stiffness of system N/m dynamic stiffness of system at static equilibrium position in N/m length of the road segment in meter length of horizontal spring in arbitrary position in millimeter original length of horizontal spring in millimeter mass in kg reference spatial frequency in cycle/m general external force in N time in s kinetic energy absolute vibration transmissibility displacement of the mass relative to the base in millimeter amplitude of steady-state response in millimeter velocity in m/s potential energy displacement of mass from initial position to arbitrary position in millimeter absolute displacement response of mass in millimeter excitation signal in millimeter amplitude of mass absolute displacement in millimeter

Ze

amplitude of displacement of the excitation in millimeter

Greek letters

a b

g 1, g 2 d x

Z t n no j o on of O

spring ratio angle of bar to horizon in degree configurative parameters of the proposed system (g1 ¼a/Lo; g2 ¼b/Lo) nonlinear parameter in Duffing equation dimensionless viscous damping coefficient linear parameter in Duffing equation dimensionless time (ont) disturbance signal in millimeter initial disturbance value in millimeter phase difference between excitation and response of system in radiant excitation frequency in rad/s natural frequency of the system without NSS in rad/s fundamental temporal frequency ratio of excitation frequency to natural frequency of the system without NSS vibrating system (o/on)

Subscripts a ap e f h id m n o r SEP v

absolute approximation excitation fundament horizontal initial deformation mass nature origin reference Static equilibrium position vertical

Superscripts ^

 0

denotes dimensionless quantity denotes time derivative denotes dimensionless time derivative

pffiffiffi [3]. The vibration attenuation is only obtained for pffiffiffithe input frequencies greater than 2 times the natural frequency of the isolation system. For input frequencies below 2 times the natural frequency and especially those close to the natural frequency, the vibration level of the isolated equipment is actually increased compared with that of the base. Hence, the system without NSS can only offer a good effectiveness for the high excitation frequencies. To solve this problem, many papers have given solutions to expand the frequency region of isolation such as a thin strip is bent such that the two ends are brought together and clamped to form a teardrop shape studied by Virgin et al. [4,5]. It acts as a nonlinear spring for supporting the load and mitigating the transmissibility of a dynamic vertical excitation. By increasing the loop length, the loop becomes less stiff vertically. The result is to achieve low resonance frequency. However, the load bearing capacity of the system is reduced. Chin et al. [6] showed that by implementing anti-spring technique the Euler spring vibration isolators was significantly improved. It means that the resonance frequency of Euler spring can be reduced. Paut et al. [7] have been studied application of pairs of pre-columns bonded with a viscoelastic filler

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as vibration isolator. Hostens [8] analyzed the vibration isolator that includes the air spring and the air damping. The result is to obtain vibration attenuation for low input frequencies and no linkage friction exists. But a disadvantage of the system is that the extra air volume is too large. In addition, du Plooy [9] developed a tunable vibration absorbing isolator. In this system, a liquid inertia vibration eliminator, which employs hydraulic fluid as the vibration absorber mass, is incorporated in the isolator. The stiffness, absorber mass, and port and reservoir geometry can be used to tune the absorber Besides, Carrella et al. [10–12] proposed the useful vibration isolation models, which comprises the negative stiffness structure in parallel to the positive stiffness structure. Due to opposite negative and positive stiffness structure, the result is to achieve low dynamic stiffness. This leads to the resonance peak of the frequency–response curve drifting to the left. Based on this idea, we would like to design another vibration isolation structure for improving vibration isolation effectiveness of the vehicle seat under low excitation frequencies. Because the characteristic of the proposed system is Duffing oscillation hence the frequency–response curve is bended to the left or the right depending on the softening or hardening system investigated fully by Brennan [13]. This leads to a range of excitation frequency, in which there are three steady-state solutions. One of them is unstable and the other two are stable. The actual response of the system depends on the initial condition. Unfortunately, among two stable states, only one state is suitable for isolating vibration. Therefore, in order to cope with this phenomenon, here we give a design procedure to minimize the bending of the frequency–response curve and reduce total dimension of the system for easy practical application. Then, isolation performance of the proposed system is simulated under various excitation conditions. In addition, in this paper, an experimental apparatus is made to verify the vibration isolation capability of the proposed system. Remainder of this paper is organized as follows. The description of the proposed system is presented in Section 2. The mechanical characteristic of the system is analyzed in Section 3. From this analysis, a design procedure of the system is suggested. In Section 4, the exact and approximate dynamic equation is set up and the characteristic of the vibration transmissibility of the system is derived. The frequency and time responses of the system to the sinusoidal, multi frequency and the random excitations are presented and analyzed in Section 5. Besides, the experimental model is shown in Section 6. Then, the experimental results are shown in Section 7. The stability of the steady-state response is presented in Section 8. Finally, some conclusions are drawn in Section 9.

2. Description of the proposed model The vehicle seat simple model comprises an isolation system, a seat frame and human body as shown in Fig. 1. In this paper, the weigh as well as the elasticity of the seat cushion and the stiffness, the damping of the components inside of human body such as spine are neglected. We only focus to study the vibration isolation system of the seat. Here, the weight of the seat frame and human body is referred as mass The proposed model of the seat isolation system including two symmetric negative stiffness structures in vertical direction, a damper and a load supporting spring (vertical spring) is shown in Fig. 2. Each negative stiffness structure, which is configured by a horizontal spring having the stiffness Kh in series with a bar having the length a, is connected with the vertical spring in parallel. One end of the horizontal spring is fixed to the wall while the other end is connected to one side of part named as the slide guide block, which is free to slide in horizontal plane on two parallel rails. Another side of the slide block is pined to one end of the bar through a hinge joint. Another end of the bar is connected to the isolating equipment (mass) having the weight of M by another hinge joint. The bar can rotate around the hinge joint. Based on the simple configuration, the isolating equipment is allowed to move along the sliding way in the vertical plane. The dynamic stiffness can be adjusted by regulating the distance b and the length of the bar a, or changing the ratio a which is defined to be the ratio between the stiffness of the horizontal spring and that of vertical spring. Initially, at the static equilibrium position presented by the dashed line, the mass is held in equilibrium by the compression force of the vertical spring Fv and the gravity force Mg which is opposite of the force Fv. Therefore, the load supported capacity in this system depends only on the stiffness of the vertical spring and its initial deformation.

Seat Isolation system Cabin floor Fig. 1. Vehicle seat model.

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F Initial position

Mass b

x hid M

zm

u



Kv

C

Lh

Sliding block

ze = Zesin(t+)

Fig. 2. Model of the suspension seat using the proposed isolation system (a) and the schematic representation of isolation system (b).

The driving process is based on the condition that three springs are always in compression. When the base is excited leading to the vibration, vibration is transmitted from the base to the isolating equipment through the vertical spring and the damper. The vibration level of the mass depends on the dynamic stiffness of the isolation system.

3. Mechanical characteristic of the proposed model 3.1. The negative stiffness structure The negative stiffness structure is considered as in Fig. 1, but the vertical spring and the damper are removed. Here, the weight of the isolated equipment is ignored. As displayed in Fig. 1, the mass is displaced downward an amount x from the

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initial position by the force F which oppose the displacement. The results are that two horizontal springs are compressed and generates two vertical restoring force acting on the mass. The total virtual work done on the isolated equipment in the vertical direction is derived as follows:

dU ¼ F dx2Fh tanðbÞdx

(1)

Applying the principle of virtual work, the following equation can be derived: F dx2Fh tanðbÞdx ¼ 0

(2)

where Fh ¼Kh(Lo  Lh) is the horizontal spring force, b is the angle of the bar to horizon in degrees, and Lo and Lh are the original length and the length at an arbitrary position of the horizontal springs, respectively, and here, the length of the sliding block is neglected. At an arbitrary position, the angle b can be determined as tanðbÞ ¼

hid x bLh

(3)

here qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lh ¼ b a2 ðhid xÞ2

hid ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ðbLo Þ2

(4)

(5)

Substituting Eqs. (3)–(5) and the expression of the horizontal spring force, Fh, into Eq. (2), the restoring force of the system can be derived as 0 1 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Lo b 2 (6) F ¼ 2Kh B  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Að a ðbLo Þ xÞ a2 ð

a2 ðbLo Þ2 xÞ2

a2 ð

a2 ðbLo Þ2 xÞ2

It is convenient to define the following dimensionless parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  F x a b a b , x^ ¼ , g1 ¼ , g2 ¼ , h^ id ¼  1 ¼ g21 ðg2 1Þ2 F^ ¼ Kh Lo Lo Lo Lo Lo Lo where F^ is the dimensionless restoring force, x^ is the dimensionless displacement, g1 and g2 are the configurative parameters, h^ id is the dimensionless initial deformation of the vertical spring, a is the length of the bar, and b is the distance from the wall to the mass. In terms of these dimensionless parameters, the dimensionless restring force can be derived from Eq. (6) as 0 1 1 g2 B C F^ ¼ 2@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1Aðh^ id x^ Þ 2 2 ^ ^ 2 2 g1 ðhid x^ Þ g1 ðhid x^ Þ

(7)

Eq. (7) shows the parametric dependence between the dimensionless restoring force F^ and the dimensionless displacement x^ . Fig. 3 shows the dimensionless force-deflection characteristic for the various configurative parameters given in Table 1. Here, the range of g1 and g2 includes g2 o1, 1r g2 o1þ g1 and g2 Z1þ g1. From Fig. 3 it can lead to some predictions in the design of a negative stiffness structure. The first case is shown in Fig. 3(a), if value g2 o1 the restoring force curves always decrease as the displacement increases. In this case, the structure derives the negative stiffness but this condition will not be considered for designing a negative stiffness structure of the proposed model. This reason will be clearly explained in next section. If the values 1r g2 o1.75, the restoring force exists the maximum and minimum forces at the location of the mass x^ 1 and x^ 2 , respectively. If the mass displacement is within between x^ 1 and x^ 2 , the dimensionless restoring force decreases according to the increase of the dimensionless mass displacement, whereas if the mass position is out of this range the restoring force increases according to the increase of the mass dimensionless displacement. It implies that in this case, the structure has two different stiffness values according to the displacement of the mass. For instance the values of g2 ¼ 1.2, the stiffness in the displacement region I is negative and one is positive if the displacement of the mass is in region II as shown in Fig. 4. For the values of g2 Z1.75, the dimensionless restoring force always increases. It means that the stiffness is always positive. Hence, this mechanism cannot treat as a negative stiffness structure. The second case is shown in Fig. 3(b), for the values of g1 Z0.2, it is similar to the first case of the values 1r g2 o1.75, the system also has displacement intervals with negative stiffness. When the values of g1 o0.2, the system cannot work as the negative stiffness structure, due to the stiffness of the structure is always positive.

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Dimensionless force F

1.2

γ2 = 1.75

0.8

γ2 = 1.6

0.4

γ2 = 1.44

0.0

γ2 = 1.375 γ2 = 1.2

-0.4

γ2 = 1.08

-0.8 -1.2 0.0

γ2 = 1.0 γ2 = 0.92 0.2

0.4

0.6

0.8

1.0

1.2

1.4

Dimensionless force F

1.6 1.2

γ1 = 0.2

0.8

γ1 = 0.4

0.4 0.0

γ1 = 0.6

-0.4

γ1 = 0.8

-0.8

γ1 = 1.0

-1.2

γ1 = 1.2

-1.6 0.0

0.4 0.8 1.2 1.6 2.0 Dimensionless displacement x

2.4

Fig. 3. Dimensionless force–deflection characteristic. (a) For various values of g2 and (b) for various values of g1. Table 1 The configurative parameters of the negative stiffness structure. Parameter Case 1: change of the value of g2

Case 2: change of the value of g1

g1

0.75

g2

0.92, 1.0, 1.08, 1.2, 1.37, 1.44, 1.6 and 1.75

0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 1.2

3.2. Design procedure for propose system From above analysis, the negative stiffness structure can be connected in parallel with a positive stiffness spring to achieve a low equivalent stiffness (dynamic stiffness) as shown in Fig. 2. When the mass moves downward x amounts from the initial position, the result is that the mass is applied by three compressive forces including two restoring forces generated by two horizontal springs as above analysis and a restoring force of the vertical spring. Thus, in this case the vertical restoring forces of the system can be obtained by adding the restoring force of the vertical spring on the left side of Eq. (7) as follows: 0 1 1 g2 B C F^ ¼ x^ þ2a@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ1Aðh^ id x^ Þ g21 ðh^ id x^ Þ2 g21 ðh^ id x^ Þ2

(8)

Let u^ ¼ h^ id x^ , which is the dimensionless displacement of the isolated equipment relative to the base and Eq. (8) can be derived as 0 1 1 g2 B C F^ ¼ h^ id u^ þ 2a@qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1Au^ 2 2 2 2 g1 u^ g1 u^

(9)

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1.2 Region I Dimensionless force F

0.8

Region II

0.4 0.0 -0.4 Region II

-0.8 x1 -1.2 -0.2

0.0

x2 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Dimensionless displacement x Fig. 4. Region of negative and positive stiffness.

Dimensionless equivalent stiffness K

2.0

γ2 = 1.75

1.5

γ2 = 1.6

1.0

γ2 = 1.44

0.5

γ2 = 1.375

0.0

γ2 = 1.2

-0.5

γ2 = 1.08

-1.0

γ2 = 1.0

-1.5

γ2 = 0.92

-2.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Dimensionless displacement u

0.8

Fig. 5. Dynamic stiffness curves for a ¼ 1 and the various values of g2.

By differentiating Eq. (8) with respect to the dimensionless displacement, x^ , and then substituting u^ ¼ h^ id x^ , the dimensionless dynamic stiffness of the system is obtained as 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 ð1g2 Þ þ g21 u^ C ^ ð g 1Þ u B 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10)  K^ ¼ 1þ 2a@ A 2 ðg21 u^ Þ3=2 g21 u^ 2 where a ¼ Kh =Kv is the spring ratio. By substituting u ¼0 into Eq. (10), the dimensionless equivalent stiffness at the static equilibrium position ðK^ SEP Þ is given by   g g 1 K^ SEP ¼ 1 þ 2a 2 1 (11)

g1

Eq. (10) defines the dependence of the static displacement, the configurative parameters g1, g2 and the spring ratio a on the dynamic stiffness. These influences are displayed in Figs. 5 and 6 with respect to the various values of the physical parameter given in Table 2. As shown in Fig. 5, for the value of g2 41 and a ¼1, the dynamic stiffness curve is the concave parabola, on which the equivalent stiffness is minimum at the static equilibrium position and always greater than  1. From the above analysis, it is understood that the isolation system with negative stiffness cannot support the load. Therefore, the physical parameters should be properly selected so that the dynamic stiffness of system at the static equilibrium position is zero or larger than zero. For instance, g2 ¼1.08 and a ¼1, as shown in Fig. 6(a), the value of K^ SEP is –0.78. By decreasing only value of a to 0.56 or 0.428 the value of K^ SEP reaches to zero or 0.23, respectively.

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Dimensionless equivalent stiffness K

1.2 0.8 α = 0.428

0.4

α = 0.56

0.0

α = 0.714

-0.4

α = 0.928

-0.8 -1.2 -0.8

α = 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Dimensionless equivalent stiffness K

0.8 α = 0.428

0.4

α = 0.571 0.0

α = 0.714

-0.4

α = 0.857

-0.8 α = 1.0

-1.2 -1.6

-2.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Dimensionless displacement u

0.8

Fig. 6. Dynamic stiffness curves for the various values of a. (a) For g2 ¼ 1.08 and (b) for g2 ¼ 0.92.

Table 2 The physical parameter of the proposed model. Parameter

Origin value

g1 g2 a

0.75 0.92, 1.0, 1.08, 1.2, 1.375, 1.44, 1.6 and 1.75 0.428, 0.56, 0.571, 0.714, 0.857, 0.982 and 1.0

When g2 o1 the dynamic stiffness curve of the system is the convex parabola, on which the stiffness achieves the maximum value at the static equilibrium position as shown in Fig. 5. The peak of this curve can move up according to the decrease of the spring ratio a. If the value of a reaches to a small enough value, the dynamic stiffness of the system around the static equilibrium position can be larger than zero as shown in Fig. 6(b); but, the stiffness of the system will be negative when the isolated equipment locates away from the static equilibrium position with a large enough distance and hence, the system is easily to become unstable. Besides, it is also verified that the dynamic stiffness becomes constant and is only dependent on the spring ratio a without regarding to the location of the isolated equipment in case of g2 ¼1. For instance, g2 ¼1 and a ¼ 1 the dynamic ^ of the displacement from the stiffness is equal to  1 as shown in Fig. 5. In this case, the mass has the infinite range (d) static equilibrium position over which the dynamic is always smaller than the static stiffness. However, if the value of g2 is ^ is reduced and determined by larger than 1, this range (d) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2=3 2 ^d ¼ g2  2ag1 ðg2 1Þ (12) 1 ^ 2a þ K1

Dimensionless dynamic stiffness

T.D. Le, K.K. Ahn / Journal of Sound and Vibration 330 (2011) 6311–6335

2.0

γ2 = 1.75

1.6

γ2 = 1.6

1.2

γ2 = 1.375

6319

γ2 = 1.2

0.8

γ2 = 1.08 0.4

γ2 = 1.0

0.0 -0.4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Dimensionless displacement u

0.8

Fig. 7. Dynamic stiffness characteristic with the zero condition at the equilibrium position.

From Eq. (11), the relationship between the physical parameters of the system, for which the dynamic stiffness at the equilibrium position is zero, is obtained by enforcing KSEP ¼0 as follows:

g2 ¼ 1þ

2a1 g 2a 1

(13)

Combining of Eqs. (10) and (13), the zero dynamic stiffness characteristic at the static equilibrium position is plotted in Fig. 7 for the various values of g2. It is seen that in case of the values g2 ¼1 and a ¼ 0.5, the dynamic stiffness is always zero ^ of displacement is at any location of the mass. However, according to the increase of the values of g2, the range (d) decreased. As known, the system, which has the largest range of the displacement, can realize the best performance of isolation. Thus, g2 ¼1, a ¼0.5 is the best value for designing an isolation system. But in practical applications, these values are difficult to obtain due to the existence of the errors of the design, fabrication and assembly, etc. Consequently, in order to obtain low dynamic stiffness small size of the system is ensured, when designing a vibration isolating system, the basic physical parameters of the system consisting of the spring ratio a, the configurative parameters g1 and g2 must satisfy the following conditions: 8 g  1 but g2 41 ðaÞ > > < 2 g2 1 o g1 ðbÞ (14) > > : a o 2ð1 þgg1 g Þ ðcÞ 1

2

^ of displacement for which the dynamic Condition (14) has revealed that the proposed system has the largest range (d) stiffness of the system is nearly zero and the size of the system is the smallest. And in addition, the bending of the frequency–response curve is effectively reduced. This will be explored clearly in Sections 4 and 5. 4. Setting up the differential equation of the motion and the approximate solution Considering a vertical excitation from the base, ze, is transmitted to the isolated equipment, M, as shown in Fig. 1. The displacement of the isolated equipment relative to the base is u, and zm is the absolute displacement of the isolated equipment in space. The restoring force in Eq. (9) is rewritten in the dimension form as follows: 0 1 B C 1 g2 C F ¼ Kv x þ 2Kh B @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   þ 1Aðhid xÞ

g21 

hid x Lo

2

g21 

hid x Lo

2

^ v Lo . with F ¼ F=K When the system is deflected from the initial position an amount, x, its potential energy, V, can be derived as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1  2  h2 1 x h x 2 A V ¼ Kv x þ2Kh hid  x þ2Kh ð1g2 ÞLo @ g21  id  g21  id 2 2 Lo L2o

(15)

(16)

The dissipation function is defined as D¼

1 Cðz_ m z_ e Þ2 2

(17)

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The kinetic energy, T, in the system is given by T¼

1 M z_ 2m 2

(18)

Then, applying Lagrange’s equation that it is expressed by   d @T @T @V @D  þ þ ¼Q dt @z_ m @zm @zm @zm

(19)

where Q¼  Mg is the generalized external force in vertical direction. By substituting Eqs. (16)–(18) into Eq. (19), the equation of the motion of the isolated equipment can be derived as follows: 0 1 B C ð1g2 Þ C M z€ m þ Cðz_ m z_ e ÞKv x2Kh B @1þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Aðhid xÞ þ Mg ¼ 0

g21 

hid x Lo

2

(20)

The relative displacement, velocity, and acceleration of the isolated equipment are determined as below (21)

u ¼ hid x, u_ ¼ z_ m z_ e ,

and u€ ¼ z€ m z€ e

(22)

By substituting Eqs. (21) and (22) into Eq. (20), the motion equation is rewritten as follows: 0 1 u€ þ

C Kv K B ð1g2 Þ C u_ þ u2 h @1þ qffiffiffiffiffiffiffiffiffiffiffiffiffi Au ¼ z€ e M M M g2  u2 1

(23)

L2o

where a superscript dot denotes the time (t) derivatives, o is the frequency of excitation, and C is the viscous damping coefficient. Considering the steady-state vibration around the static equilibrium position with a small displacement, the exact restoring force, which is expressed by Eq. (9), can be approximated by expanding power series around the static equilibrium position (u ¼0) as   1g2 þ g1 g 1 3 F^ ap ¼ h^ id þ 12a (24) u^ þ a 2 3 u^ þ OðuÞ

g1

g1

where F^ ap is the approximate restoring force and O(u) expresses the higher order term. Then, Eq. (24) can be rewritten in term of dimension as below   1g2 þ g1 g 1 Fap ¼ hid Kv þKV 12a u þ aKV 23 2 u3 g1 g1 Lo

(25)

The potential energy, V, is derived as below # " #  " 1 1g2 þ g1 u2 h2id 1 g 1 u4 h4id 12a   V ¼ Kv hid ðhid uÞ þ þ a 23 2 2 4 g1 Lo 4 g1 2 2 4

(26)

From Lagrange’s Eq. (19), the approximate dynamic equation of the mass at the steady state is derived as follows:   C Kv 1g2 þ g1 g 1 Kv 3 12a u ¼ z€ e (27) uþa 2 3 u€ þ u_ þ M M g1 g1 ML2o Here, a sinusoidal excitation is considered as ze ¼ Ze cosðotÞ

(28)

By putting Eq. (28) into Eq. (27) the approximate dynamic equation at steady state around the static equilibrium position is derived as follows:   C Kv 1g2 þ g1 g 1 Kv X 2 3 12a uþa 2 3 u ¼ Ze o2 cosðotÞ (29) u€ þ u_ þ M M g1 g1 ML2o By introducing the dimensionless parameters as 



o C Kv 1g2 þ g1 12a , x¼ , Z¼ 2 , 2M on g1 on on M u_^ u€^ K X 2 g 1 u 00 d ¼ a 2v 2 2 3 , u^ ¼ , u^ 0 ¼ and u^ ¼ 2 , Z2 on on MLo g1 on

t ¼ on t, O ¼

where on ¼

pffiffiffiffiffiffiffiffiffiffiffiffi Kv =M is the un-damped natural frequency of the system without NSS.

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In term of these dimensionless parameters, Eq. (29) can be rewritten as below 00 0 3 u^ þ 2xu^ þ Zu^ þ du^ ¼ O2 cosðOt þ jÞ

(30)

where a prime denotes the dimensionless time (t) derivative and the unknown phase difference j between the excitation, ^ tÞ, is added into the excitation. and response, uð Eq. (30) is the approximate dynamic equation that expresses the steady-state vibration of the isolated equipment around the equilibrium position with the small amplitude. It is well-know nonlinear vibration equation ‘‘called Duffing’s equation’’. Here the nonlinear parameter d in Duffing equation is always positive hence the frequency–response curve of the Duffing’s equation is bended to the right as presented by Brennan et al. [13]. The bending level depends on the value of d. In terms of isolation, this phenomenon is not useful. Therefore, when designing isolation system the value of d selected should be the smallest. The solution of Eq. (30) consists of two parts: a particular solution and a homogeneous solution (free vibration term). Here, positive damping, hence the free vibration term will die away in time. The resulting response is called the steadystate response and it only consists of the particular solution. The steady-state response has the same frequency as the excitation, but that phase is shifted from that of the excitation an amount that depends on the damping. Therefore, in this paper, in order to find the steady-state response of the system for each of the certain excitation frequency, the Harmonic balance (HB) method is a useful approach. Because the HB method has some merits given in Ref. [14] such as not restricted to the weakly nonlinear problem, and has high accuracy. The single-mode HB approximation for the steady-state response may be presented as ^ cosðOtÞ u^ ¼ U ^ is the dimensionless amplitude of the steady-state relative response. where U Then, substituting Eqs. (31) into Eq. (30) and neglecting the third harmonic, we obtain  3 ^3 ^ U ^ O2 ÞcosðOtÞ2xU ^ O sinðOtÞ ¼ O2 cosðOtÞcosðjÞO2 sinðOtÞsinðjÞ dU þ ZU 4

(31)

(32)

Equating the coefficients of the same cosines and sines from both sides of Eq. (32), the steady-state condition can be obtained as following: 8  ^ 3 þ ZU ^ U ^ O2 ¼ O2 cosðjÞ ðaÞ < 3 dU 4 (33) : 2xU ^ O ¼ O2 sinðjÞ ðbÞ Squaring and adding Eqs. (33)(a) and (b) leads to a frequency–amplitude relationship as below  2 3 ^3 ^ U ^ O2 þ4x2 U ^ 2 O2 ¼ O2 dU þ ZU 4 ^ is expressed as Then, solving for O2 as a function of U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 2 ð3dU ^ 2 þ 4ðZ2x2 ÞÞ 7 ðð3dU ^ 3 þ 4ZUÞ ^ 2 64U ^ 6 x2 Þ ^ 4 x2 ðZx2 Þ48dU U 2 O ¼ ^ 2 1Þ 4ðU

(34)

(35)

Then, the absolute vibration transmissibility (Ta) is defined as the ratio of the magnitude of the absolute displacement of the isolated equipment to the base as ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ^ 3 3 2 9zm 9 2U U 2 þ Ze2 þ2UZe cos j 2 ^ ^ ^ ^ Ta ¼ ¼ ¼ U þ1þ 2 dU þ ZUU O , (36) Ze 9ze 9 O 4 When g2 ¼1 and a ¼0.5, the nonlinear vibration equation (23) becomes the linear equation. Thus, the vibration transmissibility can be derived as following: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ta ¼

1

O2

2

4x þ O2

(37)

It can be seen from Eq. (37) that the vibration transmissibility Ta is always less than one for any excitation frequency. It is independent of the configurative parameters (b, a) of the system 5. Numerical simulation 5.1. Frequency responses to the sinusoidal excitation The motion transmissibility characteristic is one of the indices for evaluating the isolation performance of the system at the steady state for each of the certain excitation frequency. The physical parameters are used in the numerical calculation

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Table 3 The physical parameter of the system used for the frequency response simulation. Parameter

Original value

g1 g2

0.2, 0.4, 0.6 and 0.75 1.375, 1.44, 1.6 and 1.75 0.02 0.5 and 1.0

x

a

3.0

Vibration transmissibility Ta

2.5 2.0 1.5 1.0

2 = 1.75

2 = 1.44

0.5

2 = 1.6

2 = 1.375

0.0 0

1

2

3

4

5

6

Frequency (Hz) Fig. 8. Vibration transmissibility curve versus the excitation frequency for difference values of g2.

listed in Table 3. The input signal is a sinusoid having the amplitude value of 10 mm and the excitation frequency changing from 0 to 6 Hz. Consider the case of the configuration parameters a ¼1, g1 ¼0.75 and changing the value of g2 from 1.375 to 1.75. As mentioned above, according to the decrease of the value of g2 from 1.75 to 1.375, the dynamic stiffness of the system is also reduced. The result is that frequency region for isolating vibration is expanded as shown in Fig. 8. The minimum input frequency for attenuating vibration is decreased from 4.0 to 0.3 Hz and the rate of the vibration attenuation is increased. Especially, according to the decrease of the value of g2, the resonance peak of the transmissibility curve and the size of the system are also decreased. Besides, as the value of g2 reduces, the load support capacity of the system is upgraded. For instance, if g2 ¼1.75 the weight of the mass is 2.14 kg, or g2 ¼1.375 the weight of the mass is 11.36 kg. If g2 ¼ 1.75, the dynamic stiffness of the system is always larger than the static stiffness as shown in Fig. 5. The result is that the system with NSS offers the same frequency region of isolation as the system without NSS as displayed in Fig. 9. The solid line exhibits for the system with NSS and the dashed line denotes for the system without NSS. Another case of configuration parameters is a ¼0.5 and g2 ¼1. As above analysis, this is the best case for isolating, because, the system with NSS can isolate for any input frequency as displayed in Fig. 10.

5.2. Time responses to a sinusoidal wave excitation Above sub-section, the vibration amplitude of the system with NSS at the steady state is predicted by the approximate dynamic equation for each of the given excitation frequency. In this section, the numerical simulation with respect to time is realized through the exact dynamic equation by using the fourth-order Runge–Kutta method. The input signal is also a sinusoid with the amplitude of 10 mm and the excitation frequency to be 1.0 and 1.5 Hz. The physical parameters are used in the numerical calculation given in Table 4. The responses of the system with NSS in terms of the absolute displacement–time dependent for the excitation frequency to be 1.0 Hz are given shown in Fig. 11, the solid line presents for the system with NSS and the dashed line denotes for the system without NSS. It is observed that the displacement response of the mass using the system with NSS is remarkably reduced compared with the excitation as shown in Fig. 11(a), whereas the vibration level of the system without NSS is higher than the excitation as shown in Fig. 11(b). The results are evident, as shown in Fig. 8 with the isolation system with NSS having g2 ¼1.375, the excitation frequency 1.0 Hz is in the isolation range. p Whereas, as is known, ffiffiffi with the system without NSS, the input frequency for attenuating vibration is greater than 2on ¼ 1:76 Hz (here

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Vibration transmissibility Ta

10 System with NSS System without NSS

8

6

4

2

0 0

1

2

3 4 5 Frequency (Hz)

6

7

8

Fig. 9. Vibration transmissibility comparison between the proposed system and that without NSS.

Vibration transmissibility Ta

1.0

0.8

0.6

0.4

0.2

0.0 0

1

2

3 4 Frequency (Hz)

5

6

Fig. 10. Vibration transmissible characteristic of the proposed system for g2 ¼1 and a ¼ 0.5. Table 4 The physical parameter of the system used for the time response simulation. Parameter

Original value

g1 g2

0.75 1.375 0.02 1 11.36

x

a M

on ¼7.85 rad/s¼1.25 Hz). Also the root mean square (RMS) values of the mass displacement are given in Fig. 12. It can be seen from this figure that using the system with NSS the RMS values are decreased by 95.39%. But the same at the excitation condition, the RMS values of the mass displacement are increased by 183.2% for the system without NSS. In another case, the excitation frequency of 1.5 Hz is applied to the isolation system. From the simulation result shown in Fig. 13, it is observed that the vibration attenuate rate for the system with NSS is higher than that in case of the excitation frequency 1.0 Hz, where the RMS values of the mass displacement is reduced to 97.62% as given in Fig. 14. Otherwise, using the system without NSS, the vibration level is increased compared with the excitation with the increase in the RMS values of the displacement to be 136.74%.

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50

16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12

Excitation Response of the system

40 30 Displacement (mm)

Displacement (mm)

6324

20 10 0 -10 -20 -30 -40 -50

0

5

10

15

20 25 30 Time (s)

35

40

45

50

0

5

10

15

20

25 30 Time (s)

35

40

45

50

Fig. 11. Displacement response for input frequency 1.0 Hz. (a) System with NSS and (b) system without NSS.

21 20.025

RMS of displacement (mm)

18 15 12

+183.2%

9 6

7.071 -95.39%

3

0.326

0 Excitation

System without NSS

System with NSS

Fig. 12. RMS value of the mass displacement for the excitation frequency 1.0 Hz.

5.3. Time response to a multi frequency wave excitation Consider the excitation having following form: ze ¼ 5 sinð2p  0:56tÞ þ3:5 sinð2p  1:6tÞ þ 2cosð2p  0:9tÞ þ4 cosð2p  1:8tÞ

(38)

In this case, the displacement history versus time of the excitation is displayed in Fig. 15. The displacement response comparison between the system with NSS denoted by the solid line and that without NSS plotted by the dashed line is presented in Fig. 16 and the RMS values of the displacement are given in Fig. 17. It can be concluded from these figures that the isolation performance of the system with NSS is remarkably improved with the RMS values reducing to 77.22% for the displacement. Oppositely, the vibration level of the system without NSS is increased compared with the excitation since the RMS values of the mass displacement are increased by 34.34%.

5.4. Time responses to a random excitation The dynamic response of the proposed system to a random base excitation is predicted through an excitation source that it simulated by a vehicle traveling on the actual road. The spectrum of a geometrical road profile is given by Nayfeh

16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12

6325

60 Excitation Response of the system

40 Displacement (mm)

Displacement (mm)

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20 0 -20 -40 -60

0

5

10

15

20 25 30 Time (s)

35

40

45

50

0

5

10

15

20 25 30 Time (s)

35

40

45

50

Fig. 13. Displacement response for input frequency 1.5 Hz. (a) System with NSS and (b) system without NSS.

18 16.74

RMS of displacement (mm)

15

12 +136.74% 9

6

7.071

-97.62%

3

0.168

0 Excitation

System without NSS

System with NSS

Fig. 14. RMS value of the mass displacement for the excitation frequency 1.5 Hz.

[15] as following:  w1 8 > , < Pðnr Þ nnr  w2 PðnÞ ¼ > n : Pðnr Þ n , r

if n r nr (39) if n 4 nr

where the value P(nr) provides a measure for the roughness of the road. Frequency w1 ¼2.0, w2 ¼1.5 and nr ¼1/2p cycle/m is a reference spatial frequency. More specifically, if the vehicle is assumed to travel with a constant horizontal speed, v, over a given road, the force resulting from the road irregularities can be described by following series: ze ¼

N X

Zie sinðiof t þ ji Þ

(40)

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Zie ¼ 2PðiDnÞDn is the amplitude of the excitation harmonics. They are evaluated from the road spectra selected. Dn ¼2p/L and L is the length of the road segment. The phase ji are treated as random variable. In addition, the value of the

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20

Displacement (mm)

10

0

-10

-20 0

5

10

15

20

25 30 Time (s)

35

40

45

50

Fig. 15. Time history of the multi frequency excitation.

25 Response of the system without NSS Response of the system with NSS

20

Displacement (mm)

15 10 5 0 -5 -10 -15 -20 0

5

10

15

20

25 30 Time (s)

35

40

45

50

Fig. 16. Displacement response comparison between the system with NSS and without NSS.

fundamental temporal frequency of is determined as below

of ¼

2p v L

(41)

The values are used for simulating the road profile given in Table 5. Fig. 18 describes the power spectral densities (PSD) of the road profile. It can be seen that the frequency region of input signal is mainly spread over frequency range from 0.02 to 10 Hz. As shown the result in Fig. 19(a), the displacement response of the system with NSS denoted by the solid line is decreased compared with the excitation exhibited by the dashed line. But the same as the excitation condition, the displacement performance of the system without NSS is increased as shown in Fig. 19(b). In addition, as given in Fig. 20, the root mean square (RMS) values of the mass displacement are 0.484 mm (reducing to 88.87%) and 12.231 mm (increasing to 180.37%) using the system with NSS and the one without NSS, respectively.

6. Experimental apparatus In this section, an experimental apparatus is built to test the isolation performance of the proposed system having the physical parameters listed in Table 6. According to the values of these parameters, the dynamic stiffness curve of the proposed system is predicted as in Fig. 21. It can be seen that around the static equilibrium position, the dynamic stiffness

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8 7.187

RMS of displacement (mm)

7 +34.34% 6 5

5.35

4 -77.22% 3 2 1.218

1 0 Excitation

System without NSS

System with NSS

Fig. 17. RMS value of the mass displacement for the multi frequency excitation. Table 5 The physical parameters of the system used in the road profile simulation. Parameter

Value

P(nr) v L N Dt

16  10–6 m3/cycle 10 m/s 100 m 100 0.001 s

108 107

PSD of ze (mm2/Hz)

106 105 104 103 102 101 100 10-1 10-2 10-3 10-2

10-1

100

101

Hz Fig. 18. PSD of the random excitation.

value of the system is low and the dimensionless dynamic stiffness value is about 0.2. As above analysis, the benefit of low stiffness is to minimize the attraction of the vibration of the system and to have the large frequency range of isolation. Fig. 22 shows the experimental setup of the proposed system. The mass is supported by the vertical spring and moves in vertical direction through two guide rods. In order to eliminate the effect of the friction phenomenon between sliding surface of the mass and the guide rods, two commercially linear bushing is attached between them. Two negative stiffness structures, each of which comprises a horizontal spring and a bar in series as shown in Fig. 22(b), connected in parallel with the vertical spring. At the static equilibrium position, the horizontal spring and the bar are aligned. Initially, the

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30

12 Profile of the rough road Response of the system

10

20

6

Displacement (mm)

Displacement (mm)

8

4 2 0 -2 -4 -6

10 0 -10 -20

-8 -30

-10 0

5

10

15

20

25 30 Time (s)

35

40

45

50

0

5

10

15

20

25 30 Time (s)

35

40

45

50

Fig. 19. Simulation result of the mass displacement versus time under the random excitation. (a) System with NSS and (b) system without NSS.

13

12.231

12 RMS of displacement (mm)

11 10 9 180.37%

8 7 6 5

4.356

4 3 88.87%

2 1

0.484

0 Excitation

System without NSS

System with NSS

Fig. 20. RMS values of the mass displacement for random excitation.

Table 6 The physical parameter for setting the experimental apparatus. Parameter

Original value

a b Kh Kv M

187 mm 350 mm 640 N/m 800 N/m 5 kg

compressed deformation of the horizontal spring is regulated by changing the distance b through the distance adjusting part. This part can slide freely on the guide slot. The base is shaken in vertical direction by using a vibration exciter. In this experiment, the servo hydraulic vibration system (Vibmaster), which is manufactured by Park electric, is used as vibration exciter. This servo system consists of a power unit, a hydraulic cylinder that is controlled by servo valve with a nominal operational frequency range of 0–50 Hz, and a controller. The hydraulic cylinder with diameter of 3.5 cm and stroke of 720 mm, which is fitted with low friction Teflon seals to reduce nonlinear effects, is connected to inside frame by a pivot hinge. This frame is treated as a base and guided in vertical direction by four guide rods.

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8

Dimensionless stiffness

7 6 5 4 3 2 1 0 -0.6

-0.4

-0.2 0.0 0.2 Dimensionless displacement

0.4

0.6

Fig. 21. Dimensionless dynamic stiffness predictive curves.

Negative stiffness structure

Main spring

Mass

Damper

Vibration exciter

Base

Adjusting part Rail

Hinge joint

C

Horizontal spring

Outside frame

Bar

Displacement sensor

Negative stiffness structure Main spring

Damper

Base

Vibmaster

Fig. 22. Experimental setup. (a) Experimental system; (b) detail of negative stiffness structure; and (c) photograph of the experimental apparatus.

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A laser sensor (sensor model RF 603) with the range of measure 90–190(mm) is installed to measure the vertical displacement of the isolating equipment. A PCI 1711 card treated as A/D converter which is a trademark of Advantage Co. Ltd. IBM to extract the output signal of the displacement sensor. A PC is used to handle all the I/O data operation for whole system. 7. Experimental result This section presents the isolation performance comparison between the system with NSS and the system without NSS. Initially, the mass is at the static equilibrium position, at which the bar and the horizontal spring of the negative stiffness structure are aligned. Then, the base is excited by the Vibmaster. At the same time, the mass displacement is collected by laser sensor installed at the top of outside frame. First, the frequency response, which is defined to be the ratio of the mass displacement amplitude 9Zm9 to excitation 9Ze9, is tested as shown in Fig. 23. In this case, the excitation signal is sinusoid having the amplitude 10 mm and the frequency changing from 0.1 Hz to 3 Hz. From this figure, it revealed that with the system with NSS, the resonance phenomenon will almost not occur. By contrast, that of the system without NSS is high and appears at about 2.1 Hz. It can be also seen that the input frequency for attenuating vibration is larger than 0.6 Hz and 2.9 Hz for the system with NSS and system without SNN, respectively. Second, time response of the system is verified by a random excitation having the frequency spectrum mainly spread within 0.9 and 10 Hz displayed in Fig. 24. The time history of the excitation is presented in Fig. 25. The experimental result 4.0 System with NSS System without NSS

Vibration transmissibility

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Excitation frequency (Hz) Fig. 23. Frequency response comparison between the system with NSS and without NSS.

106

PSD of ze (mm2/Hz)

105 104 103 102 101 100 10-1 10-2 10-3 10-1

100

101 Hz

Fig. 24. Power spectrum density of the random excitation.

102

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15 12

Displacement (mm)

9 6 3 0 -3 -6 -9 -12 -15 0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Time (s)

Fig. 25. Time history of the random excitation.

15 12

Displacement (mm)

9 6 3 0 -3 -6 -9 -12 -15 0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

60 50

Displacement (mm)

40 30 20 10 0 -10 -20 -30 -40 -50 -60 7 8 9 10 11 12 13 14 15 Time (s)

Fig. 26. The mass position response versus time. (a) System with NSS and (b) system without NSS.

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RMS of Displacement (mm)

6332

34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

32.118

+268.54%

8.715 -67.2% 2.858 Excitation

System without NSS

System with NSS

Fig. 27. RMS value of the mass displacement for the random excitation.

PSD of mass displacement ((mm)2/Hz)

109 System without NSS System with NSS

107 105 103 101 10-1 10-3 10-2

10-1

100 Hz

101

102

Fig. 28. PSD of the mass displacement.

is shown in Fig. 26. The values RMS of the mass displacement by using the system with NSS and the system without NSS are given in Fig. 27 for comparison. From these figures it can be concluded that using the system with NSS, the amplitude of the oscillation around the static equilibrium position of the mass is smaller than the excitation as shown in Fig. 26(a) with the RMS value reducing to 67.2%, whereas the mass displacement response of the system without NSS is actually increased compared with the excitation as displayed in Fig. 26(b) with RMS value increasing to 268.54%. In addition, Fig. 28 shows comparison between PSD of the mass displacement using the system with NSS and that using the system without NSS. The solid-line plots for the system with NSS, the dashed-line denotes for the system without NSS. This result confirms that when using the system with NSS, the isolation performance is improved for larger excitation frequency than 0.6 Hz. Because, in this area the PSD value of the mass displacement by using the system with NSS is smaller than that of the excitation. As above analysis, when a negative stiffness structure is added in the isolation system, its dynamic stiffness is reduced. As known, the reduction in stiffness will minimize the attraction of the vibration. In addition, from Fig. 28 again it confirms that the system without NSS the resonance peak appears at the frequency about 2.1 Hz, because at this frequency the power spectrum density of the mass displacement is the biggest (larger than107 mm2/Hz) while that of the excitation is only about 105 mm2/Hz. However this phenomenon is not to appear for the system with NSS. The result is evident as shown in Fig. 23 that the system with NSS the frequency resonance does not exist. And it can be observed that the vibration attenuation rate increases according to the increase in the excitation frequency.

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8. Stability of the steady-state response By the HB method, the steady-state behavior of the vibration transmissibility is obtained at steady state. In this section, the stability of the response is considered. Its stability is usually investigated by superposing a small dimensionless ^ cosðOtÞ, that is ^ tÞ on u^ ¼ U perturbation, vð ^ cosðOtÞ þ vð ^ tÞ u^ ¼ U

(42)

^ satisfies Eq. (35). By substituting Eq. (42) into Eq. (30) and combining the steady-state conditions (35) leads to the where U evolution equation for the dimensionless disturbance n^ ðtÞ, as 00 v^ þ mv^ þðp þ 2q cosð2iÞÞv^ ¼ 0,

(43)

letting

i ¼ Ot,p ¼

Z þ 3d2U^ O

2

2

,

q¼

^2 3dU 4O2

,

m¼

2x

O

,

(44)

where the prime denotes the derivative with time i. Eq. (43) is called the damped Mathieu equation. p(q) represents the transition curve that separates the pq-plane into the regions of stability and instability. The leading term approximation of the transition curve for Eq. (43) is given by Nayfeh [15] p ffi 17 ðq2 m2 Þ1=2

(45)

If each of the points, (p,q), which are calculated by Eq. (44), is between the transition curves, the steady-state solution is unstable for the small disturbances. Thus, the unstable condition can be found by ( p 4 1ðq2 m2 Þ1=2 (46) p o 1 þ ðq2 m2 Þ1=2 By substituting Eq. (44) into condition (46), then the unstable condition in terms of U 4 , O, d, Z and x is obtained as ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 2 4 > < Z þ 3d2U^ 4 O2  9d16U^ 4x2 O2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (47) 2 4 > : Z þ 3dU^ 4 O2 þ 9d2 U^ 4x2 O2 2 16 This condition is plotted in Fig. 29 for the configurative parameters of the system g1 ¼ 0.75 and g2 ¼1.44. The dot curve ^ OÞ is inside the region of instability, conditions (47) are satisfied. This is the locus of Eq. (45). It is seen that if each of set ðU, indicates that the disturbance will develop with respect to time. ^ ¼ 7 and O ¼0.51, it is seen that this point is in the unstable region, as in Consider a point on the plane pq having U Fig. 29. Thus, for a small initial disturbance, no ¼0.1 mm, it will oscillate unstably with respect to time, as in Fig. 30(a). At ^ ¼ 9 and O ¼0.54, which is also in the range of instability, the disturbance also increases with respect to another point U ^ ¼ 4:5 and O ¼0.5, or U ^ ¼ 2 and O ¼0.6, they are out of the unstable region as shown time in Fig. 30(b). On the contrary, if U

Dimensionless relative amplitude U

10 9 8 7 6

Unstable area

5 4 3 2 1 0 0.0

0.2

0.4

0.6 0.8 1.0 Frequency ratio Ω

1.2

1.4

Fig. 29. Presentation of the unstable region for the steady-state solution for g1 ¼ 0.74 and g2 ¼1.44.

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^ ¼ 7, O ¼ 0.51 and (b) for U ^ ¼ 9, O ¼0.54. Fig. 30. Time histories of the disturbance for the initial disturbance no ¼ 0.1. (a) For U

in Fig. 28. Thus, the disturbance will die away in time for any value of the initial disturbance, as shown in Fig. 31(a) and (b), respectively.

9. Conclusions The concept of using a negative stiffness structure connected to a positive stiffness spring in parallel was studied. The static and dynamic analysis of the proposed model was considered. Then, the design procedure of the system is suggested to obtain low dynamic stiffness and the small size of the structure. This paper considered the forced vibration of the proposed system under the harmonic excitation from the base. The approximate dynamic equation that was described by Duffing’s equation was obtained. The harmonic balance solution to the Duffing equation was effectively employed to find the response of the system for each of the given excitation frequency. From this result, the vibration transmissibility curve is predicted with respect to the excitation frequency. Then, the time responses of the system for the sinusoidal, multi frequency and random excitations are investigated. The simulation results showed that the proposed system can offer a good effectiveness of isolation for low excitation frequencies. Especially, with g2 ¼ 1 and a ¼0.5, the proposed system can isolate vibrations for any excitation frequency and the size of the structure is remarkably reduced. In addition, the experiment is carried out to verify the isolation performance of the proposed system. Experimental results showed that with the frequency of the random excitation mainly spread from 0.1 to 10 Hz, the isolation performance of the proposed system is remarkably enhanced, where the root mean square (RMS) values may be reduced to 67.2%; whereas, the displacement response of the mass using the system without negative stiffness structure is actually increased compared with the excitation, where the RMS value of the mass displacement may be increased to 268.54%.

T.D. Le, K.K. Ahn / Journal of Sound and Vibration 330 (2011) 6311–6335

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^ ¼ 4:5, O ¼ 0.5; (b) for U ^ ¼ 2, O ¼0.6. Fig. 31. Time histories of the disturbance for the initial disturbance no ¼0.1. (a) For U

Finally, the region of instability of the steady-state response was studied. If the steady-state response is inside the unstable region, the result is that the disturbance will increase with respect to time. The disturbance will be decayed when the steady-state response is out of this region. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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