Double-layer vibration suppression bilinear system featuring electro-rheological damper with optimal damping and semi-active control

3.1. Research of Skyhook damping control strategy

Skyhook damping control strategy was put forward by D. Karnopp [13]. This kind of vibration suppression model was based on a single degree of freedom. In practice, it is not possible to connect the damping with a fixed reference plane, at this time the damping force being provided is proportional to the absolute speed of the mass. This is a kind of ideal condition, called “ceiling” damping, damping coefficient as $c_{sky}$. Suppose that the speed of incentive input layer is ${\dot{x}}_2$, the speed after vibration suppression is ${\dot{x}}_1$. The actual control algorithm based on the “skyhook damping” is as follows:

The numerical simulation was performed based on this system under this control strategy. Taking system damping ratio $\zeta =\mathrm{}$0.7, the simulation was programmed using Matlab under the condition of the single frequency excitation signal four times of the system natural frequency. The controllable damping value is determined by the Eq. (10). As shown in Fig. 5, the output force switch between zero and skyhook damping force under the condition of high frequency input excitation signals. As shown in Fig. 5, the suppression effects of the semi-active control for speed $v_1$ and displacement peak $x_1$ are obvious, the control effect of high frequency part has been increased significantly.

Fig. 5Steady-state response of skyhook damping semi-active control systems under sine excitation

3.2. Research of ODSAC strategy

Due to the limitations of the passive vibration suppression technology, the effect of vibration suppression under the condition of more complex signal frequency components of the input force is bad, the better effect of vibration suppression technology is needed to be found. The vibration suppression effect can be achieved by means of real-time adjusting the mass, stiffness and damping of the semi-active vibration suppression system. Now more mature methods are used to change the damping coefficient, so far, the methods of changing the damping values of damper have already a lot of ready-made solutions, such as the commonly used ones with electro-rheological and magneto-rheological damper, etc. So the methods of damping coefficient change of semi-active control system are used to achieve the vibration suppression.

3.2.2. Theoretical derivation of optimal damping non-differentiable control curve

(1) $\delta$ function:

The function defined in the real domain $\mathbb{R}$ is called as $\delta$ function which satisfy the following conditions:

22

${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(t\right)\delta \left(t\right)dt=f\left(0\right).$

When $t\ne$ 0, $\delta \left(t\right)=t_0$, $f\left(t\right)$ is an arbitrary function which is continuous at $t=$ 0.

Due to Eq. (22), $\delta \left(t-t_0\right)$ can be defined as:

23

${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(t\right)\delta (t-t_0)dt=f\left(t_0\right).$

Among above equations, when $t\ne t_0$, $\delta \left(t-t_0\right)=$ 0, $f\left(t\right)$ is an arbitrary function which is continuous when time $t=t_0$.

(2) The definition of variational derivative [43]:

For the inner product space $\left(S,⟨\cdot ,\cdot \cdot ⟩\right)$, mapping: $F$: $S$ →$(\mathbb{K}=\mathbb{R}$ or $\mathbb{C})$, $F$ is called as a functional on $S$. If $\varphi \in S$, for $\forall \delta \varphi \in S$, $\lambda \in \mathbb{K}$, if exist $A_{\varphi }\in S$ such that:

24

$F\left(\varphi +\lambda \delta \varphi \right)-F\left(\varphi \right)=\lambda ⟨A_{\varphi },\delta \varphi ⟩+o\left(\lambda \right),$

namely:

25

$\underset{\lambda \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{F(\varphi +\lambda \delta \varphi )-F\left(\varphi \right)}{\lambda }=⟨A_{\varphi },\delta \varphi ⟩.$

$A_{\varphi }$ denotes the derivative of $F$ at $\phi$. If you take $S=\mathbb{R}$ or $\mathbb{C}$. Inner product $⟨a,b⟩=ab$, the above derivative defined is consistent with the definition of ordinary function, and it is also consistent to generalize to ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$.

Taking $S$ as the generalized function space, then inner product $⟨f,g⟩=\underset{\mathrm{\Omega }}{\int }f\left(x\right)g\left(x\right)dx$, hence, according to this definition and Eq. (24), we can obtain:

26

$F\left(\varphi +\lambda \delta \varphi \right)-F\left(\varphi \right)=\lambda \underset{\mathrm{\Omega }}{\int }{A}_{\varphi }\left(x\right)\delta \varphi \left(x\right)dx+o\left(\lambda \right),$

where $o\left(\lambda \right)$ is dimensionless, it can be transformed into:

27

$\underset{\lambda \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{F(\varphi +\lambda \delta \varphi )-F\left(\varphi \right)}{\lambda }=\underset{\mathrm{\Omega }}{\int }{A}_{\varphi }\left(x\right)\delta \varphi \left(x\right)dx.$

Denote $A_{\varphi }=\delta F\left(\varphi \right)/\delta \varphi$, $A_{\varphi }$ is still the generalized function, its values at $x$ are usually written as:

28

$A_{\varphi }\left(x\right)=\frac{\delta F\left(\varphi \right)}{\delta \varphi }\left(x\right)=\frac{\delta F\left(\varphi \right)}{\delta \varphi \left(x\right)}.$

(3) The definition of the functional:

Denote $F_y\left(f\right)=f\left(y\right)$, thus there are:

29

$\underset{\lambda \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{F_y(f+\lambda \delta f)-F_y\left(f\right)}{\lambda }=\underset{\lambda \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f\left(y\right)+\lambda \delta f\left(y\right)-f\left(y\right)}{\lambda }=\delta f\left(y\right).$

According to the Eq. (23), there is:

30

$\delta f\left(y\right)=\underset{\mathrm{\Omega }}{\int }\delta \left(x-y\right)\delta f\left(x\right)dx.$

From the Eq. (27), one can get to know:

31

$\underset{\lambda \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{F_y(f+\lambda \delta f)-F_y\left(f\right)}{\lambda }=\underset{\mathrm{\Omega }}{\int }{A}_{\varphi }\left(x\right)\delta f\left(x\right)dx.$

According to the Eq. (28), there are:

32

$A_{\varphi }\left(x\right)=\frac{\delta F_y\left(f\right)}{\delta f}\left(x\right)=\frac{\delta F_y\left(f\right)}{\delta f\left(x\right)}.$

Comparing Eq. (30) with Eq. (29), Eqs. (31-32), one can obtain:

33

$\delta \left(x-y\right)=\frac{\delta F_y\left(f\right)}{\delta f}\left(x\right)=\frac{\delta F_y\left(f\right)}{\delta f\left(x\right)}=\frac{\delta f\left(y\right)}{\delta f\left(x\right)}.$

Given the objective function Eq. (11), solving its variation gradient, variational derivative is the most critical item among the results, namely:

34

$\frac{2x_2\delta x_2}{\delta c}.$

Making use of the Eqs. (1-2), solving the variational derivative on the damping $c\left(t\right)$, yields:

35

$-k_1\left(\frac{\delta x_1}{\delta c}-\frac{\delta x_2}{\delta c}\right)-\delta (t-\tau )({\dot{x}}_1-{\dot{x}}_2)-c\left(\frac{\delta {\dot{x}}_1}{\delta c}-\frac{\delta {\dot{x}}_2}{\delta c}\right)=m_1\frac{\delta {\ddot{x}}_1}{\delta c}-k_2\frac{\delta x_2}{\delta c}$
$+k_1\left(\frac{\delta x_1}{\delta c}-\frac{\delta x_2}{\delta c}\right)+\delta \left(t-\tau \right)\left({\dot{x}}_1-{\dot{x}}_2\right)+c\left(\frac{\delta {\dot{x}}_1}{\delta c}-\frac{\delta {\dot{x}}_2}{\delta c}\right)=m_2\frac{\delta {\ddot{x}}_2}{\delta c}.$

Among them, $\delta (t-\tau )$ is given by Eq. (33), namely $\delta (t-\tau )=\delta c\left(\tau \right)/\delta c\left(t\right)$, ${\dot{x}}_1$, ${\ddot{x}}_1$, ${\dot{x}}_2$, ${\ddot{x}}_2$ all represent solving first derivative and second derivative of the displacement of the sprung mass m1 and unsprung mass $m_2$ versus time $t$, its numerical solution of the $\delta x_2/\delta c$ can be worked out using the Eq. (35). Here, the relationship between $\tau$ and $t$ is more easily confusing. In fact, both of them all denote time, but the difference is that $\tau$ is a certain point of time, and $t$ is a time to traverse. It can be understood that each variable of motion Eqs. (1-2) in each certain time point $\tau$ all has a certain status, while solving variational derivative with respect to $c\left(t\right)$ in Eq. (35) is equivalent as to add a “tiny perturbation function” on every moment $\tau$, that is $\delta c\left(t\right)$. This is the meaning in physics of Eq. (35). At the same time, due to the introduction of the impulse function $\delta (t-\tau )\text{,}$ it can also be understood in mathematics that the influence of continuous non-differentiable points on the optimum control curve is considered.

Now making use of the gradient descent method to calculate the optimal control curve is described below:

Fig. 6Flow chart of optimal damping solved by using numerical method

Taking an initial $c\left(t_0\right)$ at first, using the optimal passive damping $c\left(t_0\right)=\text{0.7}{c}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and then $c\left(t\right)$ is fixed, $x_1$ and $x_2$ are obtained by numerical solutions of the original problems, and then carrying out numerical solution of Eq. (35), $\delta x_2/\delta c$ can be got, the gradient direction of $c\left(t\right)$ can be obtained by substituting $\delta x_2/\delta c$ into Eq. (35), and then update $c\left(t\right)$, the above steps are repeated, until meeting the optimization condition. Fig. 6 shows the flow chart of optimal damping solved by using numerical method. Among in Fig. 6, $\alpha$ is the coefficient factor, taking $\alpha =\mathrm{}$10¹³ as the initial value of $\alpha$. $\mu$ is a variable to change $\alpha$, taking $\mu =\mathrm{}$0.7 at first, then adjust parameter values of $\mu$ according to the experimental situation.

4.1. Research on vibration suppression effect of impact and sine mixed input signals

The transient system response can generally be ignored under harmonic excitation and periodic excitation force, due to the role of damping. However, in some cases, although the transient process time that the system being suffered by incentives (such as the impact force or the machine running a mutation) is very short, but the vibration amplitude is great, the phenomenon of the short-term overstressing will be produced in some parts inside the machine, these have greater destructive power. Therefore, the study of transient impact incentive signal is very necessary.

In the actual project, the frequency components included in input force are more complex. Now more common case of input force for impact and sine mixed signal will be simulated in the following context.

A given input force is an impact and sine mixed signal input force. The impact force can be expressed as follows:

It can be seen from Fig. 7 that the output force time domain curves obtained by semi-active optimal damping control compared to several other control curves are more “smooth”, that is to say, under the impact and sine mixed signal input, the effects of optimal control damping are better, so the best effects of vibration suppression can be gotten. Fig. 7 also shows that the amplitude of the minimum damping vibration suppression control are increasing from small to large, reaching the maximum 77 N to the steady status after four seconds, while the effects of the semi-active optimal damping vibration suppression are better than the other four control methods.

Although five control methods all have not obvious isolation effects, but the amplitudes of the semi-active optimal damping vibration suppression are smaller than that of the other four curves, the “overshoot” of semi-active optimal damping vibration suppression is relatively smaller, so the best comprehensive performance indexes can be obtained.

The frequency-spectrum graph of the five kinds of damping control methods is shown in Fig. 8. The first-order and second-order natural frequencies of vibration suppression system are close to 2 and 5.7 Hz. The frequency-spectrum value of the minimum passive damping vibration suppression is 16.69 dB at resonance frequency of $f=$ 2 Hz, it is the largest frequency-spectrum value of the five kinds of damping control methods. The frequency-spectrum value varied sharply to –23.92 dB at frequency $f=$ 5.75 Hz. The frequency-spectrum value of the optimal damping semi-active control arrives to its maximum peak value 7.07 dB at frequency of $f=$ 2 Hz, it is the smallest frequency-spectrum value of the five kinds of damping control methods at frequency of $f=$ 2 Hz. And then the value reaches the local minimum of –41.11 dB at frequency of $f=$ 5.12 Hz, go to –35.09 dB at frequency of $f=$ 5.75 Hz, and then to the second peak value –7.63 dB at frequency of $f=$ 6 Hz.

The optimal control damping of output force is shown in Fig. 9. This illustrates that the damping values change from starting point 1610, then drop to minimum value point 402.4 from 0.2 seconds, then to the maximum value point 1610 again, and then period circle at the intervals of every 0.2 seconds. So we can take either maximum damping control at the origin point or minimum damping control after every almost 0.2 second time period, and then run in accordance with period circles.

Fig. 7Output force under impact and sine mix signal incentive

Fig. 8Frequency-spectrum graph of output force under impact and sine mix signal incentive

Fig. 9Optimal control damping of output force under impact and sine mixed signals incentive

Fig. 10Output force under random and sine mix signal incentive

4.2. Research on isolation effect of random and sine mix input signal

Now more common case of input force for random and sine mixed signal will be simulated in the following context. Now a given input power is a uniform distribution random signal and sine mixed signal, the maximum amplitude is 10.

Due to the vibration suppression system is mainly used in the mechanical system; the limit of general mechanical system vibration frequency is 50 Hz, so that the frequency range was chosen as 0.1 to 50 Hz. Because of the frequency components of the random signal are as much as all frequency components, including the frequency components after 50 Hz, so a low pass filter is needed to add, filtering out the frequency components of a random signal after 50 Hz.

It can be seen from Fig. 10 that the amplitude of the output force time domain curves obtained by semi-active optimal damping control compared to several other control curves is the smallest, so that the best effects of vibration suppression can be gotten under the random and sine mixed input signal.

The amplitude of the minimum damping control vibration suppression is shown in Fig. 10, it is increasing from 28.43 to 70.39, reaching the local maximum 80 N to the relatively steady status after 4.5 seconds.

The frequency-spectrum graph of the five kinds of damping control methods is shown in Fig. 11. The frequency-spectrum value of the minimum passive damping vibration suppression is 16.33 dB at resonance frequency of 2Hz and the frequency-spectrum value varied sharply to –23.56 dB at 5.7 Hz. The frequency-spectrum value of the optimal damping semi-active control reaches 6.76 dB in its maximum peak value at 2 Hz, it is also the most smallest frequency-spectrum value of the five kinds of damping control methods at frequency $f=$ 2 Hz. And then it is passing the local minimum –38 dB at 5.87 Hz, and then to the second peak value –12.54 dB at 6 Hz. So the frequency spectrum characteristic of the optimal damping control is the best of the five kinds of control methods at the first-order and second-order natural frequencies.

Fig. 12 shows the optimal control damping of output force. This illustrates that the damping values change from starting point 1127, then rise to 1610, persist 1610 for 0.15 seconds, then drop to middle transition value point 1437, 1015, 693, 402, then rise up to 1610, persist for 0.2 seconds, then drop down to 402.4 from 0.44 second, then to maximum value point 1610 again, and then repeat period circle at the intervals of every 0.2 seconds. So the maximum damping control can mainly be taken and minimum damping control can be adopted in smaller portions.

Fig. 11Frequency-spectrum graph of output force under random and sine mix signal incentive

Fig. 12Optimal control damping of output force under random and sine mixed signals incentive

4.3. Research on isolation effect of random and impact mix input signal

Now the random and pulse input force mixed signal will be simulated in the following context. Now a given input power is a uniform distribution random signal and pulse mixed signal, the maximum amplitude is 10.

It can be seen from Fig. 13 that the amplitude of the output force time domain curves obtained by semi-active optimal damping control compared to several other control curves is the smallest, that is to say, under the random and impact mixed input signal, the best effects of vibration suppression can be gotten. Fig. 13 shows that the amplitude of the skyhook damping vibration suppression is the biggest, its overshoot is obvious, while the amplitude of the semi-active optimal damping vibration suppression is smaller than the other four control methods. Five control methods all have obvious effects of vibration suppression.

The frequency-spectrum graph of five damping control methods is shown in Fig. 14. The frequency-spectrum value of the minimum passive damping vibration suppression is –11.42 dB, namely the maximum value of five control methods at a resonance frequency of 2 Hz and the frequency-spectrum value varied sharply to –22.12 dB at 5.7 Hz. The frequency-spectrum value of the optimal semi-active control damping reaches –19.49 dB at 2 Hz, namely the minimum frequency spectrum value of five control methods and the local minimum of –44.33 dB at 3.38 Hz then to the value –30.23 dB at 5.75 Hz.

The optimal control damping of the output force is shown in Fig. 15. This illustrates that the damping values change from starting point 1610, then drop to middle transition value point 1192, then rise up to 1610, then drop down to minimum value 402.4 from 0.08 seconds, then to maximum value point 1610 again, and then oscillating during 0.2 seconds, the percentages of the maximum damping values are much more than the percentages of the minimum damping values.

So we can take mainly maximum damping control and a small part of minimum damping control according to the curves of damping control.

Fig. 13Output force under random and impact mix signal incentive

Fig. 14Frequency-spectrum graph of output force under random and impact mix signal incentive

Fig. 15Optimal control damping of output force under impact and random mixed signals incentive

The exact root mean square (RMS) values of the output force of the five kinds of damping control methods under impact and sine mixed signals, random and sine mixed signals, and random and impact mixed signals incentive are listed in Table 1. The isolation effect of optimum semi-active damping is superior to that of the minimum passive damping isolation and that of the optimal passive damping, it is close to that of the maximum passive damping vibration suppression and the skyhook damping control vibration suppression, so that the isolation effect of the semi-active optimal damping control is the best of the five kinds of damping control strategies under three kinds of mixed input signal incentive. The vibration suppression effect of the impact and random mixed input signals is remarkable.

5.1. Experimental equipment

The experiment of the double-layer vibration isolation of bilinear system was proceeded in the Institute of Intelligent Mechatronics Research (IIMR) of Shanghai Jiao Tong University. The main equipments used in the experiments are that the Pulse analyzer of Denmark B&K company, ten piezoelectric type acceleration sensors of B&K, Charge amplifier, Sony digital signal recorder, two electro-magnetic vibrators for modal analysis, two PCB modal force hammers, four PCB micro ICP acceleration sensors, double-layer vibration isolation experiment platform, and so on.

The ERF damper with single damping duct developed by our laboratory is shown in Fig. 16.

Fig. 16ERF damper with single damping duct

a) Schematic configuration

b) Photograph

5.2. Properties of ERF damper

The schematic diagram of the bypass ERF damper with single damping duct used in this study is shown in Fig. 16. The damper consists of a hydraulic cylinder, which is divided into two working chambers by a piston. The bypass, fitted to the side of the hydraulic cylinder, comprises two concentric tubular electrodes and an annulus through which the ER fluids flow. The positive voltage produced by a high voltage supply unit is applied to the inner electrode, while the negative voltage is connected to the outer electrode. In defect of electric fields, the ERF damper produces the viscous damping force only by the fluid-flowing resistance. However, if a certain level of the electric voltage is supplied to the ERF damper, additional damping force due to the yield stress of the ER fluid would be produced. This damping force of the ERF damper can be continuously tuned by changing the voltage applied to the damper. Based on the Bingham constitutive model of ER fluids, which is sufficiently accurate for design calculation although it does not capture details of the deformation behavior of an actual damper, the approximation of the damping force in the bypass damper is obtained as follows:

where:

where $c_0$ is no-applied voltage damping coefficient, which is determined by the plastic viscosity of ER fluids and the geometry of the manufactured damper, $u$ is the controllable yield stress damping force generated by the applied voltage, ${\alpha }_0$, ${\alpha }_1$ and ${\alpha }_2$ are the intrinsic parameters of the ERF damper and can be experimentally determined, $U$ is the voltage; $\dot{x}$ is the velocity of piston motion, and $\mathrm{s}\mathrm{g}\mathrm{n}\left(\mathrm{}\right)$ is a signum function.

Fig. 17 reports the measured damping force with respect to the piston velocity at various voltages. It is obtained by calculating the maximum damping force at each velocity. The piston velocity increases from 15 to 100 mm/s gradually, while the excitation amplitude is maintained to be constant at 60 mm. Such a plot is frequently employed to evaluate the level of damping performance in the damper manufacturing industry. For the proposed damper with single damping duct the damping force increased with the applied voltage, as expected. For instance, the damping force increases up to 227 N at a piston velocity of 100 mm/s and voltage of 5 kV. This Figure agrees fairly well with the presented damping model given by Eq. (46).

Chart of real-time controlling module is shown in Fig. 18.

Fig. 17Under various voltages damping force of ERF damper with single damping duct versus piston velocity, U= 0.0 kV (dot line), U= 1.4 kV (dash dot line), U= 2.5 kV (dash line), U= 3.5 kV (solid line), and U= 5.0 kV (thick line)

Fig. 18Chart of real-time controlling module

5.3. Measurement experiment of transmissibility and result analysis under condition of no vibration isolator

The vibrator working at a fixed frequency is used in the measuring experiments. The parameters of the force vibration isolation testing bench are as follows:

The sprung mass $m_1=\mathrm{}$60 kg, the unsprung mass $m_2=\mathrm{}$16 kg, the stiffness of the sprung mass $k_1=\mathrm{}$33000 N/m, the stiffness of the unsprung mass $k_2=\mathrm{}$185000 N/m. The undamped force transfer rate of this test bed can be calculated according to the theoretical formula, the transmissibility curve is shown in solid line of Fig. 19. When the ERF vibration isolator is not installed in a force vibration isolation test-bed, the measured force transmissibility curve of the double layer vibration isolation (square icon shown in Fig. 19) is consistent with the theoretical curve; this illustrate that the measurement method is reliable. As shown in Fig. 19, the first and the second-order natural frequency of the vibration isolation system is 3.4 Hz and 18.64 Hz, respectively. The maximum transmissibility of the first-order and the second-order system are 57 and 38.71, respectively.

Fig. 19Transmissibility of double-layer force isolation testing bench with no damper

Fig. 20Force transmissibility of double-layer vibration suppression bilinear system with ODSAC law with single damping orifice ERD

5.4. Experiment of ERF damper with single damping orifice under ODSAC of double layer vibration isolation bilinear system

In order to further validate the control effect of actual vibration isolation of bilinear system under the action of the ODSAC which is designed in this paper, the experiment of ERF damper with single damping duct is carried out in the above experiment platform of the force vibration isolation system. Where $a_0=\mathrm{}$10.23 N, $a_1=\mathrm{}$6.38 N/kV, $a_2=$ 2.59 N/kV2, no-applied voltage viscous damping coefficient $c_0=\mathrm{}$1013.4 N∙m^-1∙s. The sampling frequency of a randomly profiled roadway input excitation is 300 Hz. The signal frequency is 2 Hz, 2.8 Hz, 3 Hz, 3.2 Hz, 3.5 Hz, 3.8 Hz, 4 Hz, 5 Hz, 7 Hz, respectively. The transmissibility of double-layer vibration isolation bilinear system under different frequency randomly profiled roadway input excitation signals at a different voltage are listed in Table 2. The frequency domain curve of the transmissibility of the system can be drawn from this data, which is shown in Fig. 20. It can be seen from the diagram that the first order resonance peak frequency of the practical system is about 2.8 Hz, the vibration amplitude of the upper quality is the largest in here. It is observed that the first order resonant frequency of the actual system is different from the theoretical calculation value, this is because that the quality itself of spring and damper of the actual system will affect the vibration isolation system. The different voltages (0 KV, 2 KV, 4 KV) being applied in ERF damper in the resonance peak play an obviously different vibration reducing effects. The greater the applied voltage is, the lower the transmissibility is. This is completely consistent with the theoretical analysis.

It can also be seen from the Fig. 20 that the applied voltage can make the transmissibility increase instead when the excitation frequency is greater than 3.5 Hz, namely the crossover frequency of system. The transmissibility curve of the passive damping control of vibration isolation as damping ratio $\xi =\mathrm{}$0.1 and the optimal damping semi-active control (ODSAC) of vibration isolation can be seen from Fig. 20.

The results show that the vibration isolation effect of the double-stage force vibration isolation bilinear system under the action of the ODSAC is better than that of the passive damping vibration isolation control system. Although there are some deviations, compared with the results of experimental test, but the trend is consistent with the measured transmissibility curve, this shows that the proposed control method is feasible and effective.

5.5. Experiment of ERF damper under recorded signal of randomly profiled roadway input excitation

It can be seen from Fig. 21 that the amplitude of the output force time domain curves obtained by ODSAC compared to several other control curves is the smallest, so that the best effects of vibration suppression can be gotten under a recorded signal of a randomly profiled roadway input excitation.

Fig. 21Output force under recorded signal of randomly profiled roadway input excitation

Fig. 22Frequency-spectrum graph of output force under recorded signal of randomly profiled roadway input excitation

The frequency-spectrum graph of the five kinds of damping control methods is shown in Fig. 22. The frequency-spectrum value of the minimum passive damping vibration suppression is 0.26 dB at resonance frequency of 3.4 Hz, and the frequency-spectrum value varied sharply to –23.66 dB at 19.5 Hz. The frequency-spectrum value of ODSAC arrive at –11.48 dB in its maximum peak value at 3.4 Hz, it is also the smallest frequency-spectrum value of the five kinds of damping control methods at frequency $f=\mathrm{}$3.4 Hz. And then it is passing to the local minimum of –49.08 dB at 19.3 Hz. So the frequency spectrum characteristic of ODSAC is the best of the five kinds of control methods at the first-order and second-order natural frequencies.

Fig. 23 shows the optimal control damping of output force. This illustrates that the damping values change from starting point 562.8, then drop to minimum value point 140.7, then to maximum value point 562.8 again, and then repeat period circle.

Fig. 23Optimal control damping of output force under recorded signal of randomly profiled roadway input excitation

The exact RMS values of the output force of the five kinds of damping control methods under a recorded signal of randomly profiled roadway input excitation are listed in Table 3. The isolation effect of ODSAC is much better than that of the minimum passive damping isolation and that of the optimal passive damping, so that the isolation effect of the ODSAC is the best of the five kinds of damping control strategies under a recorded signal of a randomly profiled roadway input excitation.