Linear stiffness vibration isolator with D-EMSD
A displacement-dependent electromagnetic shunt damper is devised, with the damping force varying as the relative displacement from the loading plate to bottom plate. The structural sketch of a linear stiffness vibration isolator with D-EMSD is given in Fig. 1. The linear spring provides linear stiffness and structural damping, while the linear bearing guides the movement vertically and reduces friction. The permanent magnet group moves vertically with the upper load plate via the connecting arm. The electromagnetic coil is mounted on the electromagnetic coil base, which is fixed to the lower base. When the upper plate and lower base move relative to one another, the permanent magnet group and electromagnetic coil move as well. According to the electromagnetic induction theorem and Lenz’s law, there will be a damping force between them to prevent their relative movement, which is electromagnetic force.
The two permanent magnets are joined by a spacer in the initial configuration, and the permanent magnet group is positioned in the center of the electromagnetic coil31. The electromagnetic coils are positioned symmetrically along the spacer’s center and connected to the external circuit. Through the variable resistance base, the electromagnetic coil base is fixed to the changeable resistance in the external circuit. By moving the brush, which is attached to the upper load plate via the connecting arm, it can alter the resistance in the circuit. The equivalent voltage source can be used to simplify the closed electromagnetic coil.
In the initial state, the permanent magnet group is set in the middle of the electromagnetic coil, and the two permanent magnets are connected together by a spacer31. The electromagnetic coils are distributed symmetrically along the center of the spacer and connected to the external circuit. The variable resistance in the external circuit is fixed to the electromagnetic coil base through the variable resistance base. It can change the resistance into the circuit by adjusting the position of the brush, which is connected to the upper load plate through the connecting arm. The circuit diagram of the electromagnetic coil and the external circuit is shown in Fig. 2. The enclosed electromagnetic coil can be reduced to the equivalent voltage source \({V_e}\), equivalent resistance \({R_e}\), and equivalent inductance \({L_e}\).
Connection diagram of the coil and Rs.
Faraday’s law states that an electric potential will exist when the permanent magnet group and the coil have a relative velocity \({v_{\text{r}}}\), which results in an electric potential \({V_e}\), which yields
$${V_e}={C_e}{v_{\text{r}}}.$$
(1)
In Eq. (1), \({C_e}\) denotes the coupling coefficient of electromagnetism. The structure parameter values of D-EMSD are as follows: the inner radius and outer radius of PM are 3 mm and 14 mm respectively, and the height is 10 mm, the residual flux density is 1.45T, H is 13 mm, respectively. Thus, it can be stated as a function of the relative displacement between the electromagnetic coil and the permanent magnet, its expression is defined as31
$${C_{\text{e}}}={c_1}x+{c_3}{x^3},$$
(2)
where, \({c_1}= – 2.8987 \times {10^3}\), \({c_3}=2.6617 \times {10^7}\), and x is the relative displacement of the electromagnetic coil to the permanent magnet.
When the structural dimensions of the electromagnetic coil and the permanent magnet are fixed, the relationship between the electromagnetic force of the electromagnetic coil \({F_e}\) and the current I in the electromagnetic coil can be expressed as
where, \({C_m}\) is known as the electromechanical coupling coefficient, and \({C_m}={C_e}\).
As a result, the electromagnetic force \({F_e}\) is inversely proportional to the total impedance in the circuit that is connected to the electromagnetic coil, or proportionate to the current I in the electromagnetic coil. The internal resistance \({R_e}\), the electromagnetic coil’s inductance \({L_e}\), and the changeable resistance \({R_s}\) in the external circuit make up the total impedance, as shown in Fig. 2. Therefore, the resistance value of the resistor can alter the total impedance when the structural dimensions of the electromagnetic coil are fixed. Using the brush as the boundary line, \({R_s}\) is separated into upper resistor \({R_1}\) and lower resistor \({R_2}\), which are linked in parallel. The overall length is 2L and the total resistance value is 2R. The resistance value is at its maximum at the initial equilibrium position, i.e. \({R_1}={R_2}\). While the resistance value decreases as the system displacement increases, the electromagnetic force grows. Furthermore, the resistance value Rs of can be written as
$${R_s}=\left( {1 – \frac{{{x^2}}}{{{L^2}}}} \right)\frac{R}{2}.$$
(4)
QZS vibration isolator with D-EMSD
A QZS vibration isolation system is constructed by adding the cam with roller follower to the linear stiffness vibration isolator with D-EMSD. The horizontal springs, the vertical spring, and the ball contacting pair, which consistently maintains alignment with the center of the horizontal face, are the main parts of the cam-roller type QZS. Figure 3 displays the structural sketch of the QZS isolation system coupled with D-EMSD. Cams and rollers make up the ball contact pairs. A level adjusting mechanism is added to modify the degree of pre-compression of the horizontal springs, while the two rollers and cams are always moving in a horizontal and vertical direction, respectively.
Structural sketch of cam-roller type QZS isolator coupled with D-EMSD (1.load, 2. support platform, 3. pre-compression spring, 4.roller, 5. cam, 6.spring, 7.D-EMSD, 8. foundation support).
The stiffness of horizontal and vertical springs is represented as \({k_{\text{h}}}\) and \({k_{\text{v}}}\), separately. The roller and cam have respective radii of r1 and r2. When the main system is in static equilibrium, The horizontal springs’ compression is δ, and the displacement of the vibration-isolated object is x. Hence, the total restoring force of the system can be expressed as:
$${f_{\text{r}}}\left( x \right)={k_{\text{v}}}x – 2{k_{\text{h}}}x\left[ {1+\frac{{\delta – \left( {{r_1}+{r_2}} \right)}}{{\sqrt {{{\left( {{r_1}+{r_2}} \right)}^2} – {x^2}} }}} \right].$$
(5)
Correspondingly, the stiffness of the system can be attained as
$$K=\frac{{{\text{d}}{f_r}}}{{{\text{d}}x}}={k_{\text{v}}} – 2{k_{\text{h}}}\left[ {1+\frac{{\left[ {\delta – \left( {{r_1}+{r_2}} \right)} \right]{{\left( {{r_1}+{r_2}} \right)}^2}}}{{{{\left[ {{{\left( {{r_1}+{r_2}} \right)}^2} – {x^2}} \right]}^{\frac{3}{2}}}}}} \right].$$
(6)
Through the dimensionless transformation of Eqs. (5) and (6), the following can be obtained:
$${F_{\text{r}}}\left( X \right)=X – 2\nu X\left[ {1+\frac{{\overline {\delta } – 1}}{{\sqrt {1 – {X^2}} }}} \right],$$
(7a)
$$\bar {K}=1 – 2\nu \left[ {1+\frac{{\overline {\delta } – 1}}{{{{\left( {1 – {X^2}} \right)}^{\frac{3}{2}}}}}} \right],$$
(7b)
where, \({F_{\text{r}}}\left( X \right)=\frac{{{f_{\text{r}}}\left( x \right)}}{{{k_{\text{v}}}\left( {{r_1}+{r_2}} \right)}}\), \(X=\frac{x}{{{r_1}+{r_2}}}\), \(\nu =\frac{{{k_{\text{h}}}}}{{{k_{\text{v}}}}}\), \(\bar {\delta }=\frac{\delta }{{{r_1}+{r_2}}}\),\(\bar {K}=\frac{K}{{{k_{\text{v}}}}}\).
For the convenience of calculation, Eq. (7) can be expanded by 13th-order Taylor expansion as:
$$\begin{gathered} {F_{\text{r}}}\left( X \right)=\left( {1 – 2\nu \bar {\delta }} \right)X+\nu \left( {1 – \bar {\delta }} \right){X^3}+\frac{3}{4}\nu \left( {1 – \bar {\delta }} \right){X^5}+\frac{5}{8}\nu \left( {1 – \bar {\delta }} \right){X^7} \hfill \\ +\frac{{35}}{{64}}\nu \left( {1 – \bar {\delta }} \right){X^9}+\frac{{63}}{{128}}\nu \left( {1 – \bar {\delta }} \right){X^{11}}+\frac{{231}}{{512}}\nu \left( {1 – \bar {\delta }} \right){X^{13}}+{\text{O}}({X^{15}}), \hfill \\ \end{gathered}$$
(8a)
$$\begin{gathered} \bar {K}=\left( {1 – 2\nu \bar {\delta }} \right)+3\nu \left( {1 – \bar {\delta }} \right){X^2}+\frac{{15}}{4}\nu \left( {1 – \bar {\delta }} \right){X^4}+\frac{{35}}{8}\nu \left( {1 – \bar {\delta }} \right){X^6}+ \hfill \\ \frac{{315}}{{64}}\nu \left( {1 – \bar {\delta }} \right){X^8}+\frac{{693}}{{128}}\nu \left( {1 – \bar {\delta }} \right){X^{10}}+\frac{{3003}}{{512}}\nu \left( {1 – \bar {\delta }} \right){X^{12}}+{\text{O}}({X^{14}}). \hfill \\ \end{gathered}$$
(8b)
Since the corresponding linear stiffness of a QZS system should be zero, the horizontal spring’s stiffness ratio in relation to the vertical spring must satisfy
$$\nu =\frac{1}{{2\bar {\delta }}}.$$
(9)
Due to errors in processing or assembly, it is challenging to guarantee that the primary system’s vertical equivalent linear stiffness is precisely zero in real-world engineering applications. The stiffness ratios of the vertical and horizontal stiffnesses are thus chosen as \(\nu =0.96/2\bar {\delta }\), and a tiny amount of linear stiffness is kept. Let r1 = 0.0035 m, r2 = 0.0045 m, δ = 0.0048 m, i.e. \(\bar {\delta }=0.6\),the fitting curves between the exact values of the system restoring force and stiffness as they change with deformation and the 13th-order approximate values can be drawn as shown in Fig. 4. It is discovered that, within the specified range, the approximate value and the actual value of the restoring force and stiffness agree quite well. Moreover, within a certain range, the system stiffness exhibits the characteristics of quasi-zero stiffness.
Comparison force and stiffness curves with deformation. (a) Force curves with deformation, (b) stiffness curves with deformation.
The critical position value that the cam-roller mechanism can provide vertical force is expressed as \({x_{\text{b}}}\), as shown in Fig. 5. Only when the absolute value of displacement is less than \({x_{\text{b}}}\) can the cam-roller mechanism provide vertical component force. Therefore, this work only considers the case that the displacement of the cam-roller mechanism is within the critical range. And the critical cam-roller contact condition can be found as follows:
$${x_{\text{b}}}=\sqrt {{r_2}({r_2}+2{r_1})}$$
(10)
Critical position of cam-roller.
The physical model of cam-roller type QZS isolator coupled with D-EMSD under harmonic excitation is shown in Fig. 6, where m stands for the mass, \({f_{\text{r}}}\) denotes the nonlinear elastic restoring force, c denotes the linear damping coefficient,\({F_{\text{e}}}\)denotes the electromagnetic shunt damping force. Additionally,\(F\cos \omega t\) is the external force excitation.
System model of QZS isolator coupled with D-EMSD.
Accordingly, the dynamics differential equations of the QZS isolator coupled with D-EMSD can be established as
$$m\ddot {x}+c\dot {x}+{f_r}\left( x \right)+{F_e}=F\cos (\omega t),$$
(11a)
$${L_e}\ddot {q}+{R_e}\dot {q}+(1 – \frac{{{x^2}}}{{{L^2}}})\frac{R}{2}\dot {q} – {c_1}x\dot {x} – {c_3}{x^3}\dot {x}=0.$$
(11c)
In order to simplify the calculation process and facilitate the parameter analysis, the following parameters are introduced to make the governing equations dimensionless:
\(Q=q/{q_0}\), \({\omega _1}=\sqrt {{k_{\text{v}}}/m}\), \(\Omega =\omega /{\omega _1}\), \(\tau ={\omega _1}t\), \({\zeta _1}=c/\left( {2m{\omega _1}} \right)\), \(\mu ={c_1}{q_0}/\left( {m{\omega _1}} \right)\), \(\kappa ={c_3}{q_0}{({r_1}+{r_2})^2}/\left( {m{\omega _1}} \right)\), \(\bar {\delta }=\delta /\left( {{r_1}+{r_2}} \right)\), \({f_{\text{e}}}={F_{\text{e}}}/{q_0}\left( {{r_1}+{r_2}} \right)\), \({\gamma _1}=1 – 2\nu \bar {\delta }\), \({\gamma _3}=\nu \left( {1 – \bar {\delta }} \right)\), \({\gamma _5}=3\nu \left( {1 – \bar {\delta }} \right)/4\), \({\gamma _7}=5\nu \left( {1 – \bar {\delta }} \right)/8\), \({\gamma _9}=35\nu \left( {1 – \bar {\delta }} \right)/64\), \({\gamma _{11}}=63\nu \left( {1 – \bar {\delta }} \right)/128\), \({\gamma _{13}}=231\nu \left( {1 – \bar {\delta }} \right)/512\), \(f=F/{k_v}({r_1}+{r_2})\), \({\zeta _{\text{2}}}={R_{\text{e}}}/2{L_{\text{e}}}{\omega _1}\), \({\zeta _3}=R/2{L_{\text{e}}}{\omega _1}\), \(\rho ={({r_1}+{r_2})^2}/{L^2}\), \(\eta ={c_1}{({r_1}+{r_2})^2}/{L_e}{q_0}{\omega _1}\), \(\sigma ={c_3}{({r_1}+{r_2})^4}/{L_e}{q_0}{\omega _1}\), \((\cdot )^{\prime}={\text{d}}(\cdot )/{\text{d}}\tau\),\((\cdot )^{\prime\prime}={{\text{d}}^2}(\cdot )/{\text{d}}{\tau ^2}\). Where, \({\omega _{\text{1}}}\)and\({\zeta _{\text{1}}}\)represent the natural frequency and damping ratio of the system respectively. \({\gamma _n}\)represents the nonlinear stiffness coefficient, where n is 1, 3, 5, 7, 9, 11, 13 respectively. f and fe respectively represent the dimensionless excitation amplitude and electromagnetic shunt damping force. Therefore, Eq. (11) can be converted into
$$\begin{gathered} X^{\prime\prime}+2{\zeta _1}X^{\prime}+{\gamma _1}X+{\gamma _3}{X^3}+{\gamma _5}{X^5}+{\gamma _7}{X^7}+{\gamma _9}{X^9}+{\gamma _{11}}{X^{11}}+{\gamma _{13}}{X^{13}} \hfill \\ +\mu XQ^{\prime}+\kappa {X^3}Q^{\prime}=f\cos (\Omega \tau ), \hfill \\ \end{gathered}$$
(12a)
$$Q^{\prime\prime}+2{\zeta _2}Q^{\prime}+(1 – \rho {X^2}){\zeta _3}Q^{\prime} – \eta XX^{\prime} – \sigma {X^3}X^{\prime}=0.$$
(12b)