## 5 Equivalent Circuit of the Coupled Mechanical Oscillators
### Equivalent Circuit Diagram of \(\beta_{i}(\tau)\)  
From the Equation (9b), we can re-write the expression of the non-dimensional stiffness coupling of the first oscillator as \(\beta_{1}(\tau)=\left[\left(\frac{\beta+1}{2}\right)-\left(\frac{\beta-1}{2} \right)\cos(2\omega_{\text{nd}}\tau)\right]\left(y_{1}-y_{2}\right)\). If we insert the value of the \(\beta=11.11\), we get, \(\beta_{1}(\tau)=(6.055-5.055\cos(2\omega_{\text{nd}}\tau))\)\((y_{1}-y_{2})\). Similarly, from the Equation (10b), considering the value of \(\mu\) as \(1.1938\), we get, \(\beta_{2}(\tau)=(7.2284-6.0346\cos(2\omega_{\text{nd}}\tau))(y_{2}-y_{1})\). The Figure 15 is the corresponding circuit diagram of the term \(\beta_{1}(\tau)\). From the expression of \(\beta_{1}(\tau)\), first we implement the \((-5.055\cos(2\omega_{\text{nd}}\tau))\) term. In the Figure 15, \(V_{\text{in}}\) is a sine wave having an amplitude and frequency \(V\) and \(f\), respectively. So, \(V_{\text{in}}\) can be expresses as, \(V_{\text{in}}=V\sin(2\pi ft)\). We pass this sine wave through a differentiator op-amp OA11. The output the differentiator will be \(V_{\text{ext}}=-2\pi Vf\cos(2\pi ft)\), which is equivalent to the expression \((-5.055\cos(2\omega_{\text{nd}}\tau))\). So, \(-2\pi Vf\) = \(-5.055\), which gives,  
\[V=\frac{5.055}{2\pi f} \tag{14}\]  
As, as the non-dimensional time can be expressed as, \(\tau=\frac{t}{CR}\), we can write, \(2\omega_{\text{nd}}\tau=2\pi ft\), which yields,  
\[f=\frac{\omega_{\text{nd}}}{\pi CR} \tag{15}\]  
The output of the op-amp OA11, \(V_{\text{ext}}\), is added with a dc voltage, \(V_{\text{DC}}\) (having the value \(6.055\)) using an inverting adder op-amp OA12. The expression of the output becomes \(-V_{\text{DC}}-V_{\text{ext}}\). This output will be in phase with \(V_{\text{ext}}\) after passing through a unity gain inverting op-amp OA13. So, the output of OA13 becomes \((6.055-5.055\cos(2\omega_{\text{nd}}\tau))\). Figure 14: The equivalent circuit diagram of the force due to the magnetic spring \(f_{\text{S2}}(y_{2})\). The resistance values are: \(R_{11}=14.91\) k\(\Omega\), \(R_{12}=2.1\) k\(\Omega\), \(R=100\) k\(\Omega\). \(OA10\) is the Op-Amp, \(M_{9}\) and \(M_{1}0\) are the two multipliers. (Color Online). Figure 15: The equivalent circuit diagram of the non-dimensional spring force term \(\beta_{1}(\tau)\). The resistance values are: \(R=R_{9}=R_{10}=R_{11}=R_{13}=R_{12}=R_{19}=R_{20}R_{21}=100\) k\(\Omega\). The capacitor \(C=1\) nF OA11, OA12, OA13, and OA14 are the four Op-Amps, \(M_{1}1\) is the multiplier. \(V_{\text{in}}\) is the sine wave originating from a function generator. \(V_{\text{DC}}\) is a dc voltage source of value \(6.055\) V. (Color Online). The op-amp OA14 adds the two inputs \(-y_{1}\) and \(y_{2}\) and gives the output in the form of \((y_{1}-y_{2})\). The multiplier \(M11\) multiplies the output of the two op-amps OA13 and OA14, and creates \(\beta_{1}(\tau)\) as an output. In order to make the equivalent circuit of the term \(\beta_{2}(\tau)\), we have chosen the same approach. The only difference from \(\beta_{1}(\tau)\) is that, an extra \(\mu\) term is multiplied with the \(\beta_{1}(\tau)\) term, and the sign of the two inputs are interchanged, i.e., \(y_{1}\) and \(-y_{2}\). The expression of the amplitude of the input ac sine wave will be,  
\[V=\frac{6.0346}{2\pi f} \tag{16}\]  
The expression of the linear frequency \(f\) will remain the same as calculated in the Equation (15). The circuit diagram is shown in Figure 16.  
Up to this, we have explained the electronic circuit analog of the forces due to the dry-friction, magnetic spring, and the non-dimensional springs, respectively of the coupled system. Now, we shall explain the main circuit diagram using the Equations (11) and (12). Figure 17 depicts the equivalent circuit diagram of the considered coupled mechanical oscillators. The outputs of the op-amps OA20, OA24, OA26, OA30 are the four state-variables of the system, \(y_{1}\), \(z_{1}\), \(y_{2}\), and \(z_{2}\), respectively.