# Nonlinear Dynamics of an Electromagnetically Actuated
## 6 Analysis of the Stability for the Primary Resonance
### Circuit Model Verification
To avoid some confusion, it should be mentioned that \(Z\) (input of AD633) from the last relationship  
Figure 17: Four types of op-amp configurations that implement different mathematical operations: (**a**) inverse proportional operation circuit, (**b**) summation and difference operation circuits, (**c**) differentiating circuit, and (**d**) integrating circuit. described has nothing to do with \(Z\) from the following, which represents the movement variable from Equation (17). The first block of the circuit in Figure 18 has the \(Z\) function at the input and raises the third power of the input signal at the output. The first floor implements the square of the \(Z\) function divided by 10 and realizes the input and output offset compensation (POTI, \(R_{8}\), and \(R_{9}\), similar to the second stage). After the second stage, \(Z\) is raised to the third power divided by 100 (and this stage compensates for the input and output offset), and the third stage (non-inverting operational amplifier) divides the input by 100 because at the output \(Y\) is used to find \(Z^{3}\). For the \(Z^{1}\) case, the first block is completely removed. The case with \(Z^{3}\) is described in this section. For the case of \(Z^{5}\), an AD633 multiplier is interspersed between the second stage, and the last one, which has the following inputs: at \(X_{1}\), the output of the second floor (\(Z^{3}\)) and at \(Y_{1}\), the output of the first floor (\(Z^{2}\)) (the other inputs are at ground). Thus, after changing the value of the feedback resistor of the last stage (op-amp) to 10 M\(\Omega\), the output \(Y=Z^{5}\) is obtained. A similar way is used for case \(Z^{7}\). Figure 19 shows how the functions \(Z\) and \(\overline{\mathsf{Z}}\) were derived. Each block also contains an inverting op-amp since the first stage has the derived function with negative polarity at its output. Signal summation is performed in the block from Figure 20.  
The expression of the final \(\widetilde{\alpha}_{1}\) function is implemented with the circuit in the last block (Figure 21). The first stage performs a multiplication and the second circuit in the block implements the integration function with an op-amp. Figure 19: The circuits that implement (**a**) the derivative of function \(Z\) and (**b**) the derivative of \(\overline{\mathsf{Z}}\). Figure 18: First block—raising the \(Z\) function to a power by multiplication (with 3 stages). literature, our technique is based on the presence of some auxiliary functions and optimal convergence-control parameters, which assure the high accuracy of the approximate solutions in comparison with numerical integration results. These parameters are determined using a rigorous mathematical procedure. The initial and the first iterations are determined by an original technique, and the initial nonlinear governing equation is reduced to two linear differential equations. OAFM is applied to obtain an approximate analytical solution in the neighborhood of the primary resonance. This method is easy to apply, producing very accurate results using only the first iteration. By applying OAFM, we reduced the complexity of the problem while maintaining accuracy, which is an advantage over traditional methods like perturbation techniques or the method of multiple scales. In future work, the investigation should be enlarged to other varieties of dynamic conditions, including, for example, Van der Waals and Casimir forces. The analysis of the local stability of the present problem was made by means of the Routh-Hurwitz criteria. The eigenvalues of the Jacobian matrix depend on the trace of the Jacobian matrix and on the discriminant of the characteristic equation. The Hopf bifurcation and saddle-node points do not exist. Global stability is analyzed by means of [PERSON]'s direct method and [PERSON]'s invariant principle. The Lyapunov function depends on the approximate solution obtained through our procedure with the help of a control variable and [PERSON]'s principle.