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Based on the parallel mechanism theory, a new vibration-isolating platform is designed and its kinetic equation is deduced. Taylor expansion is used to approximately replace the elastic restoring force expression of vibration-isolating platform, and the error analysis is carried out. The dynamic-displacement equation of the vibration-isolating platform is studied by using the Duffing equation with only the nonlinear term. The dynamic characteristics of the vibration-isolating platform are studied, including amplitude-frequency response, jumping-up and jumping-down frequency, and displacement transfer rate under base excitation. The results show that the lower the excitation amplitude, the lower the initial vibration isolation frequency of the system. The influence of the platform damping ratio

The existing research has proved that the parallel mechanism has the advantages of compact structure, good stiffness, and strong bearing capacity and can be applied to multidimensional vibration isolation [

All in all, researchers at home and abroad obtained large number of achievements related to vibration isolation system related to parallel mechanism on both the vibration response and the control strategy. But most researches of the multidimensional vibration isolation conducted by the scholars were based on the Steward platform; the vibration isolation characteristics of the non-Steward platform was rarely to be seen. The studies above were based on the active control and there were no variable stiffness and variable damping researches based on passive vibration isolation technology in the field.

Based on previous studies, this paper presents an integrated innovation and puts forward the design theory of variable damping and variable stiffness vibration-isolating platform based on parallel mechanism, which makes the platform have the operating characteristics of “large damping in low-frequency large-amplitude movement and small damping in high-frequency small-amplitude movement” and “high static stiffness and low dynamic stiffness.” Through the structural design and spatial layout of the multibar mechanism (as shown in Figure

Schematic diagram of prototype design. In the figure: 1, bearing platform; 2, base; 3, damper; 4, fixed hinge; 5, down limit pad.

The new vibration-isolating platform based on parallel mechanism, designed by our research team, is shown in Figure

Vertical upward jump of prototype.

Vertical downward jump of prototype.

By adjusting the initial compression amount of the three springs in the upper end, the whole platform can have quasi-zero stiffness in the vertical direction. If the quasi-zero stiffness characteristic can not be pursued, the platform can be equated with the parallel mechanism-based vibration-isolating platform developed by Professor Ma et al., but it can only isolate the intermediate-frequency and high-frequency vibrations with an excitation frequency greater than

The simplified vertical mechanical model of the vibration isolation platform is shown in Figure

Vertical mechanical model of the vibration isolation platform.

As shown in Figure

So (

With the vertical displacement

When both sides of (

If we take the derivative of

When a set of coefficients are given, the force-displacement curve of the vibration isolation platform can be got, as is demonstrated in Figure

Force-displacement curve of the platform under selected parameters.

It is necessary to simplify (

According to (

Inclusion of only the third derivative in (

Error between approximate force-displacement curve and accurate one.

It can be seen from Figure

Supposing that the vibration isolation platform slightly vibrates in small range near point A, where the stiffness tends to zero, the simplified dynamic model of the platform is shown in Figure

Dynamic model of platform.

Under the simple harmonic excitation condition, the approximate force-displacement equation of the platform can be presented as follows:

Furthermore, (

By using the harmonic balance method, the same order harmonic term on both sides of the differential equation has the same coefficient. In the meanwhile, only the predominant excitation frequency is considered, ignoring higher harmonic terms [

Upon substitution of (

On elimination of phase

With the stiffness ratio

Amplitude-frequency characteristic curve of platform.

As can be seen in Figure

In summary, only when the external excitation frequency is larger than

In general, to make the solution of its dynamic differential equation tend to the nonresonant branch with small amplitude, it is required that the excitation frequency

Set base displacement excitation

Schematic diagram of system displacement transfer rate.

The relative displacement between vibration isolation object and base is

Conduct nondimensional processing on formula (

Set the system displacement response as

By the constant variation method,

Formula (

Therefore,

Substitute

According to the definition of average method, the right side of formula (

Let the left be zero; integrate the above formula:

Add the squares of the above two formulas to obtain the amplitude-frequency response function shown in formula (

This expression expresses that the amplitude

Let

In order to ensure that the response of the system is bounded, namely, to ensure that the resonance curve has a stable section, formula (

Substitute

Set the displacement response of the vibration isolation object as

According to the definition of displacement transfer rate, the displacement transfer rate of the platform is

When considering only the vertical spring in the mechanism, the mass of vibration isolation object is only carried by a vertical spring. The vibration-isolating platform becomes a traditional linear vibration-isolating platform. Through the analysis above, it is easy to get the displacement transfer rate of the linear system shown in formula (

Let stiffness

The relationship between damping ratio and displacement transfer rate.

Let damping ratio

The relationship between displacement excitation amplitude and displacement transfer rate.

For different excitation amplitudes, when the external excitation frequency is large enough, the displacement transfer rate curve of the vibration isolation system and its corresponding linear system will tend to coincide, indicating that the system has the same vibration isolation capacity in the high-frequency band as in the linear vibration-isolating system. But in the low-frequency region, the vibration isolation effect of the system is obviously better than that of the corresponding linear system. With the reduction of the displacement excitation amplitude, the difference between the two becomes more and more obvious.

With the decrease of displacement excitation amplitude

Based on the parallel mechanism theory, a new vibration-isolating platform is designed, and the dynamic characteristics of the vibration-isolating platform are studied, including amplitude-frequency response, up and down hopping frequency, and displacement transfer rate, providing some thinking and methods to optimize this kind of vibration-isolating platform. The study obtained the following conclusions:

The effect of platform damping ratio

The effect of base displacement excitation amplitude on displacement transfer rate is related to the size of excitation amplitude

The authors declare that they have no conflicts of interest.

The authors would like to thank the financial supports of the Natural Science Foundation of Anhui Province (Grant nos. 1508085ME70 and 1708085ME127), of the National Natural Science Foundation of China (Grant no. 51575001), of Anhui University scientific research platform innovation team building projects (2016–2018), of Anhui Provincial Education Department Natural Science Foundation (Grant no. KJ2016A799), and of Wuhu Science and Technology Research Project (Grant no. 2014cxy 07).