Abstract
Aimed to solve the problem of weak antiresonance performance and poor working stability in current antiresonance vibrating machines, this paper presents a nonlinear dynamic model that reflects the actual working state of the antiresonance machine. Under the material mass fluctuation condition, the dynamic response of the antiresonance vibrating system has been discussed, and the dynamic parameters selection problem of the antiresonance vibrating system has been analyzed which could be used to improve working performance stability. In the paper, the influence of nonlinear factors of the antiresonance vibrating machine on the driving body and the working body has been analyzed under the material fluctuation conditions also. The results can provide the theoretical support and experimental basis for improving the design and working performance of the antiresonance vibrating machine.
1. Introduction
Vibration machine is a kind of mechanical equipment which used to carry out all kinds of technological processes by vibration motion, such as vibration conveyor, vibration screen, vibration dryer and so on [1]. To design the vibration machine, two problems must be considered that the first is the vibration frequency and the vibration response amplitude must meet the requirement of the vibration system; the next is to reduce the vibration action on the foundation and to reduce the working noise pollution to the environment. Above, the two problems look like paradoxical, but can be effectively solved by using the antiresonance theory [24]. In the literature [5], the viewpoint that to design the vibration machine based on the antiresonance principle had been put forward, and the two kinds of antiresonance vibrating machines include the origin antiresonance machine and the crosspoint antiresonance machine had been presented. The vibration exciters of the two types of antiresonance machines were not installed on the working body of the vibrating machine, but to install on the driving body, namely installed on the lower body of the vibrating machine. So the excitation source of the two kinds of antiresonance machine did not participate vibrating motion with the vibration system, which made the structure of the working body greatly simplified, and the mass of vibration can be reduced about 30 %50 % [68]. Because of the exciting force did not directly act to the working body of the two kinds of antiresonance machine, the machine life can be improved obviously. And because the lower body of the vibrating machine was almost no vibration, to guarantee the stiffness and strength requirements of the vibrating machine in the design process was easier, and the excitation device can be designed according to the static load, the noise of the whole machine can be significantly reduced.
However, the excited frequency of the antiresonance vibration machine is requested particularly stable in the practical application, or the antiresonance point of the antiresonance vibrating machine may be deviate because of the fluctuation of the material mass, and the amplitude of the vibration response will change also, which will reduce the working performance of the vibrating machine [9, 10]. Therefore, it is very important to improve the vibration amplitude stability of the upper body and lower body by revealing the relationship between the resonance point, the antiresonance point and the mass ratio, and to select the appropriate dynamic parameters. Know from the relevant literatures that can be consulted [1113], that the design of antiresonance vibrating machine have a certain blindness currently, because of a large number of dynamic parameters have been related to the antiresonance vibrating machine, and there are no research literatures have solved the difficult problem that the dynamic parameters how to influence the mechanical characteristics of antiresonance vibrating machine. In the paper, by multiple scales method and based on the existing antiresonance test prototype, the problem that how to select the dynamic parameters to make the working body amplitude of the antiresonance vibrating machine to stabilize at the design value and the driving body amplitude is the smallest under the conditions of material mass fluctuation had been researched. And the influence of different systemic parameters and material mass fluctuation on the vibration response of the antiresonance vibrating machine has been discussed. All the research results are helpful to improve the working performance and design level of this kind of antiresonance vibrating machine.
2. Dynamic model of nonlinear antiresonance vibrating machine
The dynamic model that shown as Fig. 1 abstract from the existing antiresonance test prototype in laboratory that shown as Fig. 2. In which, ${m}_{1}$ is the working body mass of the antiresonance vibrating machine which include the upper body mass of vibrating machine and the load material mass, ${m}_{2}$ is the driving body mass of the antiresonance vibrating machine which include lower body mass of vibrating machine and the exciters mass (the reverse synchronous rotary inertia vibration exciters were fixed on the lower body), ${f}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\Omega}t\right)$ is the excited force which shown as Fig. 1, and $\mathrm{\Omega}$ is the excited frequency.
Fig. 1Dynamic model of nonlinear antiresonance vibrating system that driven by dual rotors
Based on D’Alembert’s principle, the dynamic equations of the nonlinear antiresonance vibrating system that shown as Fig. 1 can be expressed as:
where, ${y}_{1}$ and ${y}_{2}$ are vibration response displacement of upper body and lower body of the vibrating machine. According to the design and operational requirement of this kind of antiresonance vibrating machine, to select the rubber spring to be the main vibration spring and the supporting spring, to set ${k}_{1}$ and ${c}_{1}$ is the linear dynamic stiffness coefficient and the equivalent damping coefficient of main vibration spring in $y$ direction, to set ${k}_{2}$ and ${c}_{2}$ is the linear dynamic stiffness coefficient and the equivalent damping coefficient of supporting spring in $y$ direction, and to set ${\beta}_{1}$ and ${\beta}_{2}$ is the three times nonlinear stiffness coefficient of main vibration spring and supporting spring in $y$ direction (where, ${\beta}_{1}$ and ${\beta}_{2}$ are constant which related to the spring material ) [14, 15].
In order to decouple Eq. (1), to select the vibration modal matrix $\left[\begin{array}{ll}{\eta}_{11}& {\eta}_{12}\\ {\eta}_{21}& {\eta}_{22}\end{array}\right]$ to make $\left[\begin{array}{l}{\widehat{y}}_{1}\\ {\widehat{y}}_{2}\end{array}\right]=\left[\begin{array}{ll}{\eta}_{11}& {\eta}_{12}\\ {\eta}_{21}& {\eta}_{22}\end{array}\right]\left[\begin{array}{l}{y}_{1}\\ {y}_{2}\end{array}\right]$. And then to orthogonal process Eq. (1), to transform the physical coordinate quantity of Eq. (1) into the modal coordinate quantity:
$+\left[\begin{array}{c}{\widehat{a}}_{11}{\widehat{y}}_{1}^{3}+{\widehat{a}}_{12}{\widehat{y}}_{2}^{3}+{\widehat{a}}_{13}{\widehat{y}}_{1}^{2}{\widehat{y}}_{2}+{\widehat{a}}_{14}{\widehat{y}}_{1}{\widehat{y}}_{2}^{2}\\ {\widehat{a}}_{21}{\widehat{y}}_{1}^{3}+{\widehat{a}}_{22}{\widehat{y}}_{2}^{3}+{\widehat{a}}_{23}{\widehat{y}}_{1}^{2}{\widehat{y}}_{2}+{\widehat{a}}_{24}{\widehat{y}}_{1}{\widehat{y}}_{2}^{2}\end{array}\right]=\left[\begin{array}{l}{\widehat{F}}_{1}\\ {\widehat{F}}_{2}\end{array}\right],$
where, ${\omega}_{1}$ and ${\omega}_{2}$ are the natural frequency of the antiresonance vibrating system that shown as Fig. 1.
Set ${\widehat{\xi}}_{ij}=\epsilon {\xi}_{ij}$ ($i=\mathrm{}$1, 2 and $j=\mathrm{}$1, 2) and ${\widehat{a}}_{ik}=\epsilon {a}_{ik}$ ($i=\mathrm{}$1, 2 and $k=\mathrm{}$1, 2, 3, 4), where $\epsilon $ is the small parameters. Then set:
where, ${T}_{0}=\tau $, ${T}_{1}=\epsilon \tau $.
Based on the multiple scales method of nonlinear vibration systems [16, 17], to perturbation analyze the modal equations Eq. (2), then the displacement response expression of Fig. 1 antiresonance vibrating system can be obtained as:
where:
${\sigma}_{1}$ is the tuning parameter of the multiple scales method:
$+\frac{{a}_{13}{A}_{1}^{2}{A}_{2}}{{\left(2{\omega}_{1}+{\omega}_{2}\right)}^{2}{\omega}_{1}^{2}}{e}^{i\left(2{\omega}_{1}+{\omega}_{2}\right){T}_{0}}+\frac{{a}_{14}{A}_{1}{A}_{2}^{2}}{{\left({\omega}_{1}+2{\omega}_{2}\right)}^{2}{\omega}_{1}^{2}}{e}^{i\left({\omega}_{1}+2{\omega}_{2}\right){T}_{0}}+cc,$
${\zeta}_{21}=\frac{3{a}_{21}{A}_{1}^{2}{\stackrel{}{A}}_{1}2{a}_{24}{A}_{1}{\stackrel{}{A}}_{2}{A}_{2}2i{\xi}_{21}{\omega}_{1}{A}_{1}}{{\omega}_{1}^{2}{\omega}_{2}^{2}}{e}^{i{\omega}_{2}{T}_{0}}+\frac{{a}_{22}{A}_{2}^{3}}{8{\omega}_{2}^{2}}{e}^{3i{\omega}_{1}{T}_{0}}$
$+\frac{{a}_{21}{A}_{1}^{3}}{9{\omega}_{1}^{2}{\omega}_{2}^{2}}{e}^{3i{\omega}_{2}{T}_{0}}+\frac{{a}_{23}{A}_{1}^{2}{A}_{2}}{{\left(2{\omega}_{1}+{\omega}_{2}\right)}^{2}{\omega}_{2}^{2}}{e}^{i\left({\omega}_{2}2{\omega}_{1}\right){T}_{0}}$
$+\frac{{a}_{24}{A}_{1}{A}_{2}^{2}}{{\left({\omega}_{1}+2{\omega}_{2}\right)}^{2}{\omega}_{2}^{2}}{e}^{i\left({\omega}_{1}+2{\omega}_{2}\right){T}_{0}}+\frac{{\eta}_{12}}{2\epsilon \left({\omega}_{1}^{2}{\omega}_{2}^{2}\right)}{f}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\Omega}t\right)+cc,$
where, $cc$ is the complex conjugate of the first two terms.
3. Perturbation analysis of material mass fluctuation
Using the random perturbation analysis method to research the dynamic response of the antiresonance vibrating machine that shown as Fig. 1 under the material mass fluctuation condition. Set the mass of the upper body is:
where, ${m}_{1o}$ is the deterministic mass part of the upper body, ${m}_{1r}$ is the zero mean random mass part of material, $\epsilon $ is the small parameter, $\alpha $ is the zero mean random variable, $\epsilon {m}_{1r}\left(\alpha \right)$ is the zero mean random fluctuation of material mass.
To analyze the vibrating responses of the nonlinear antiresonance system which described by Eq. (10) and Eq. (11), the physical meaning of the analysis results were not obvious. So in order to discuss the influence factors of vibration response of the vibrating system under the antiresonance state and the condition of material mass fluctuation, based on the design parameters of the existing antiresonance test prototype, and the general practice for dynamic parameters selection of antiresonance vibrating system [18], set ${\beta}_{1}={\beta}_{2}\approx 0$, and let them into the nonlinear antiresonance equations that described by Eq. (1). Then lead Eq. (1), and Eq. (3)Eq. (8) into Eq. (10) and Eq. (11), the approximate linear vibration response of Eq. (1) vibrating system can be gotten as:
where, ${B}_{1o}$ is the determining part of the amplitude ${B}_{1}$ of the working body of the antiresonance vibrating system that shown as Fig. 1, ${B}_{2}$ is the amplitude of the driving body of the antiresonance vibrating system. Aimed to the fluctuation mass of the material on the upper body, set the standard deviation is ${\sigma}_{m1}$, then the random partial variance of working body amplitude under the condition of material mass fluctuation can be expressed as:
where, ${\phi}_{1}$ is the vibrating response phase of the working body:
Therefore, by discussing the amplitude ${B}_{1}$ (including ${B}_{1o}$ and the random partial variance of the working body amplitude) and the amplitude ${B}_{2}$, the paper can be used to research the dynamic response problem of the nonlinear antiresonance vibrating machine that shown as Fig. 1, under the condition of the random fluctuation of material mass.
4. Response analysis and dynamic parameters selection
In the design process of the antiresonance vibrating machine, the working body mass ${m}_{1o}$, the driving body mass ${m}_{2}$, the spring dynamic stiffness coefficient and the equivalent damping coefficient would be determined by the design conditions and working conditions, which were difficult to be adjusted during the working process of vibrating machine. So in order to ensure the amplitude stability of the working body and driving body of the antiresonance vibrating machine, and the amplitude of the driving body is the least, the paper will select different dynamic parameters combinations of the upper body and the lower body, to research the effects on the vibration response of the antiresonance vibrating system under the conditions of material mass fluctuation.
Then introducing the following symbols to nondimensional treat Eq. (13) and Eq. (14): ${\omega}_{1}=\sqrt{{k}_{1}/{m}_{1}}$ is natural frequency of the working body, which is defined as the antiresonance frequency of the antiresonance vibrating system, ${\omega}_{2}=\sqrt{{k}_{2}/{m}_{2}}$ is natural frequency of the driving body, $\mu =({m}_{1o}+\epsilon {m}_{1r}\left(\alpha \right))/{m}_{2}$ is the mass ratio of the working body and the driving body, $\gamma =\epsilon {m}_{1r}\left(\alpha \right)/{m}_{1o}$ is the mass ratio of the material fluctuation mass and deterministic mass part of the working body, $z=\mathrm{\Omega}/{\omega}_{1}$ is the frequency ratio, $\chi ={\omega}_{1}/{\omega}_{2}$ is the antiresonance frequency ratio, ${\xi}_{1}={c}_{1}/2{m}_{1}{\omega}_{1}$ is the damping ratio of working body, ${\xi}_{2}={c}_{2}/2{m}_{2}{\omega}_{2}$ is the damping ratio of driving body, ${B}_{0}={f}_{2}/{k}_{2}$ is the support spring static deformation of driving body, ${\rho}_{1}={B}_{1}/{B}_{0}$ is the dynamic amplification factor of working body, ${\rho}_{2}={B}_{2}/{B}_{0}$ is the dynamic amplification factor of driving body.
Based on the dual rotors antiresonance test prototype in laboratory that shown as Fig. 2, to select the structural parameters as follows:
Fig. 2Dual rotors antiresonance test prototype
Set ${m}_{1o}=$ 160 kg, ${m}_{2}=$ 128 kg, $\mu =$1.25, and the material mass fluctuation range $\mathrm{\Delta}{m}_{1r}=$±32 kg, ${k}_{1}=$ 300 kN/m, ${k}_{2}=$ 700 kN/m, ${\xi}_{1}={\xi}_{2}=$0.55, ${f}_{2}=$200 N. According to design principles of antiresonance vibrating machine, to set the excited frequency $\mathrm{\Omega}$ locate in the vicinity of antiresonance frequency of the vibrating system. During the course of the vibrating machine working, the material mass fluctuates, which will cause the drift of antiresonance frequency point of the antiresonance system. So to set the target working amplitude of the working body is 5.0 mm. Then to research the vibration response of the upper body and the lower body of the antiresonance vibrating machine that shown as Fig. 2, when the antiresonance frequency ratio $\chi \mathrm{}=$2, 2.5 and 3.
According to the experimental data of the antiresonance vibrating machine that shown as Fig. 2, to calculate the mechanical characteristic parameters which expressed in Table 1.
Table 1Dynamic parameters table of antiresonance vibrating system
Parameters  $\chi =\text{2}$  $\chi =\text{2.5}$  $\chi =\text{3}$  
Before material mass fluctuation  After material mass fluctuation  After excited frequency adjustment  Before material mass fluctuation  After material mass fluctuation  After excited frequency adjustment  Before material mass fluctuation  After material mass fluctuation  After excited frequency adjustment  
${m}_{1}$/ kg  160  192  192  160  192  192  160  192  192 
$\mu $  1.250  1.38  1.38  1.31  1.44  1.44  1.50  1.66  1.66 
${\omega}_{1}$ / (rad/s)  43.30  40.57  40.57  43.30  40.57  40.57  43.30  40.57  40.57 
$\chi $  2.0  1.944  1.944  2.50  2.601  2.601  3.0  2.877  2.877 
$\mathrm{\Omega}\mathrm{}$(rad/s)  83.773  83.773  76.967  83.773  83.773  74.902  83.773  83.773  73.714 
${\rho}_{1}$  0.419  0.407  0.422  0.221  0.197  0.202  0.183  0.179  0.184 
${\rho}_{2}$  0.0377  0.0619  0.0355  0.0284  0.0401  0.0290  0.0139  0.0242  0.0143 
${B}_{1}$ / mm  5.0  4.47  4.92  5.0  4.59  5.02  5.0  4.53  4.96 
${B}_{2}$ / mm  0.735  0.899  0.740  0.730  0.902  0.737  0.731  0.910  0.739 
Analyze the Table 1 to know that, when the load of the working body fluctuates from no load to full load, the amplitude of working body decrease change, but the decreased amplitude is less than or equal to 0.55 mm, namely the reduction ratio less than 10 %, which meet the initial design goal conditions ${B}_{1}=\mathrm{}$5.0 mm of the antiresonance vibrating machine that show as Fig. 2, namely the amplitude remains almost unchanged under the material mass fluctuation. So, when selecting the mechanical characteristic parameters from Table 1, the working state of the vibrating machine is hardly affected by the material mass fluctuation. Table 1 also reflects that, the amplitude of the lower body increase change, and the increased amplitude is less than or equal to 0.25 mm, but which cause the dynamic load that pass to the foundation increases significantly. To calculate the dynamic load value to know that the increase value of dynamic load is less than or equal to 775 N, and comparing with design and using conditions of this kind of vibrating machine [4, 18] which is in the scope of allow.
During the experiment, to adjust the rotational speed of driving system in the real time to make the excited frequency back to the near zone of the antiresonance frequency, and then know from the Table 1 that, the vibrating response of upper body and lower body come back to almost the same state as system no load (Refer the excited frequency adjustment in Table 1). Therefore, during the actual use of the antiresonance vibrating machine, in order to achieve the amplitude stability of working body, the rotational speed of excited motors must be adjusted, to make them track the drift of antiresonance frequency point.
Now set the antiresonance frequency ratio $\chi =\mathrm{}$2.5 of antiresonance test prototype that shown as Fig. 2, and refer to the Table 1 to select the dynamic parameters of the antiresonance test prototype, then the working process state of the test prototype can be detected, and the antiresonance response photo under noload condition can be expressed as Fig. 3(a). Know from the vibrating response curves that shown as Fig. 3(b), that the working body can maintain steady state vibration with vibrating amplitude of approximately 5.0 mm and the driving body can maintain quiver motion with vibrating amplitude of approximately 0 mm (the actual measured value of vibrating amplitude ${B}_{2}=$0.7 mm, which far less than the vibrating amplitude of the driving body of the traditional vibrating machine [4, 18]). All indicate that the antiresonance test prototype shown as Fig. 2 has good antiresonance effect with the dynamic parameters shown as Table 1 under noload conditions.
Fig. 3Vibrating response signal of the antiresonance test prototype under noload conditions
a) Vibrating response signal of working body
b) Vibrating response signal of driving body
Fig. 4Vibrating response signal of the antiresonance test prototype under 20 % load increase conditions
a) Vibrating response signal of working body
b) Vibrating response signal of driving body
When the test prototype was exerted 20 % loads which shown as Fig. 4(a), the actual measured vibrating response curves could be obtained as Fig. 4(b). The vibrating amplitude of working body reduced to about 4.5 mm, and the vibrating amplitude of driving body increased to about 1 mm. Now compare the vibrating response curves with the literature [1] to know that, although the material mass fluctuations have a great influence on the working state of the antiresonance vibrating system, the antiresonance vibrating machine that shown as Fig. 2 still has good antiresonance working state. Comparing the curves in Fig. 3(b), Fig. 4(b) and the analysis data in Table 1 to know that, the experimental results can perfectly agreement with the theoretical analysis, and the correctness of the theoretical deduction conclusion in the paper can be verified.
In order to discuss the effect of material mass change on the amplitude response of the antiresonance vibrating machine, according to Table 1, to select the common used characteristic parameters $\chi =$ 2.5 and $\mu =$ 1.44 under the actual working conditions, to calculate the relationship between the dynamic amplification factors ${\rho}_{1}$ and ${\rho}_{2}$ of working body and driving body and the fluctuations of material mass when $\mathrm{\Delta}{m}_{1}$ fluctuates from –32 kg to 32 kg, namely the working body mass fluctuate within the range ±20 %. The analysis results curves were shown as Fig. 5.
Fig. 5Relationship between the dynamic amplification factors and the fluctuations of material mass
a) Dynamic amplification factor of working body
b) Dynamic amplification factor of driving body
Know from Fig. 5(a) that, ${\rho}_{1}$ presented a decreasing and tending to remain stable. It indicated that the positive fluctuation of the material mass average value could help to cut the influence on working body vibration response of antiresonance vibrating machine which caused by material mass fluctuation. With the positive increase of the material mass fluctuation, the dynamic amplification factor ${\rho}_{1}$ of working body decreased. Conversely, the influence on the dynamic amplification factor ${\rho}_{1}$ of the working body increased gradually, because of the negative material mass fluctuated. Hence, in order to ensure the working body had the stable desired amplitude, the designed mass of material should be small and it was limited in the appropriate range, when to design this kind of antiresonance vibrating machine that shown as Fig. 2. Know from Fig. 5(b) that, whether the material mass was forward fluctuating or negative fluctuating, the material mass fluctuation would enhance the influence on the vibrating response amplitude of driving body, which could cause the dynamic amplification factor ${\rho}_{2}$ of driving body to increase change. It would decrease the vibration isolation effect of the antiresonance vibrating machine. So the fluctuate range of material mass of antiresonance vibrating system should be limited. And the computer control should be introduced, to adjust the excited frequency of driving system realtime to track the antiresonance frequency point of the antiresonance vibrating machine according to the fluctuation of material mass, which would make the vibrating amplitude response of driving body to be the smallest [19, 20].
Although during the practical applications of most antiresonance vibrating machines, the nonlinear factors influence may be ignored, the nonlinear response phenomenon do exist, in some cases they are not negligible. So, according to the displacement response Eq. (10) and Eq. (11), the amplitude frequency response curves of Fig. 2 antiresonance vibrating machine with different nonlinear stiffness coefficients have been researched which shown as Fig. 6.
Analyze Fig. 6 to know that, when ${z}^{\mathrm{\text{'}}}<$0, the vibrating response of the antiresonance vibrating system should follow the linear change law, namely when analyzing the vibrating response of the antiresonance system, the three nonlinear term influencing factors of the spring could be ignored. When ${z}^{\mathrm{\text{'}}}=$0, if the antiresonance frequency of the vibrating system equals to excited frequency, the amplitude of driving body is zero (which shown as Fig. 6(b)). When ${z}^{\mathrm{\text{'}}}>$0, the nonlinear influencing factors enhanced significantly, the vibrating response amplitude of working body and driving body emerged bifurcation phenomenon, and it enhanced with the increase of excited frequency. Know from Fig. 6 that, when ${z}^{\mathrm{\text{'}}}>$0, whether the mean value of material mass fluctuation increase or decrease, the bifurcation mutation amplitude response of working body and driving body were much larger than the condition of material mass was stable. Therefore, aiming to the nonlinear antiresonance vibrating machine in case of fluctuation of material mass, the working excited frequency should be make to stabilize in a small area near the antiresonance frequency of the antiresonance vibrating machine. And the excited frequency fluctuation should be tried to keep located in the left area of antiresonance frequency, to avoid the bifurcations phenomenon of the amplitude response, because which should affect the vibrating response stability of the antiresonance system seriously [21, 22]. In Fig. 6, with the increasing change of material mass, the mass of upper body would be changed, which caused the amplitude of working body and driving body increasing slightly, but the antiresonance frequency would decrease also. Which lead to the minimum of driving body vibrating response curve that shown as Fig. 6(b) moved to left. Consequently, under the premise of meeting the antiresonance requirements during working, the excited frequency of the vibrating machine should be decreased, to keep the amplitude of working body stable and the amplitude of driving body was least.
Fig. 6Effect of the nonlinear spring stiffness coefficients on the amplitude frequency response of the antiresonance vibrating system
a) Effect of the nonlinear spring stiffness coefficients on working body
b) Effect of the nonlinear spring stiffness coefficients on driving body
5. Conclusions
This paper presented a nonlinear dynamic model of antiresonance vibrating machine under the condition of material mass fluctuation. The multiscale method and perturbation analysis were used to analyze the nonlinear dynamic behavior of the antiresonance vibrating machine in the case of three term nonlinear spring stiffness. And the relationship among the dynamic parameters of the antiresonance vibrating machine had been simplified analyzed also.
The fluctuation of material mass can lead to a drift of antiresonance point of the antiresonance vibrating machine. When the material mass increases, the antiresonance frequency is smaller than the original antiresonance frequency. On the left side of antiresonance frequency, the vibration response of the antiresonance vibrating system follows the linear changing law so that the nonlinear influence can be ignored. However, on the right side of antiresonance frequency, as the frequency increases, the nonlinear influence will increase significantly, and the vibrating response amplitude will emerge bifurcation phenomenon. In addition, with the increasing of the excited frequency, the amplitude at the bifurcation point will also increase. Since the change of the material mass may lead the vibration response of the antiresonance system to vibrate between linear and nonlinear performance, resulting in abnormal fluctuations of the amplitude of working body and driving body, the material mass should be kept stable during the working process of the antiresonance machine.
In order to keep the vibration response stability of the antiresonance vibrating system, the excited frequency should be limited into a little bit less than the original antiresonance frequency, so that antiresonance vibrating machine could have a good antiresonance performance. Otherwise, the vibrating system will show a strong nonlinear response, which may seriously affect the working performance of the antiresonance system. The research results have huge guidance significance for design and application of the antiresonance vibrating machines, and it is much helpful to improve current working performance of the antiresonance vibrating machine.
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The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant No. 51105066), and the Fundamental Research Funds for the Central Universities (Grant No. N130403011).