Let's break down the problem step-by-step: --- ## **Step 1: Write the Electrical Equation** The armature circuit is described by: \[ e_{in}(t) = R I_L(t) + \frac{d\lambda}{dt} \] where: - \( e_{in}(t) \) = input voltage - \( R \) = resistance - \( I_L(t) \) = solenoid current - \( \lambda \) = magnetic flux linkage From the problem: \[ I_L(\lambda) = a_3 \lambda^3 + a_1 \lambda \] So, \[ I_L(t) = a_3 \lambda^3(t) + a_1 \lambda(t) \] --- ## **Step 2: Write the Mechanical Equation (Newton's Second Law)** The forces on the mass \( m \) are: - Electromagnetic force: \( F_{em}(\lambda) \) - Spring force: \( -kx \) - Damping force: \( -b\dot{x} \) - Inertia: \( m\ddot{x} \) So, \[ m\ddot{x} + b\dot{x} + kx = F_{em}(\lambda) \] From the problem: \[ F_{em}(\lambda) = c_6 \lambda^6 + c_4 \lambda^4 + c_2 \lambda^2 \] --- ## **Step 3: Complete Mathematical Model** Combine all equations: ### **Electrical Subsystem:** \[ e_{in}(t) = R I_L(t) + \frac{d\lambda}{dt} \] with \[ I_L(t) = a_3 \lambda^3(t) + a_1 \lambda(t) \] ### **Mechanical Subsystem:** \[ m\ddot{x} + b\dot{x} + kx = c_6 \lambda^6 + c_4 \lambda^4 + c_2 \lambda^2 \] --- ## **Step 4: Coupling Between Electrical and Mechanical Systems** The flux linkage \(\lambda\) depends on the position \(x\), but the problem does not provide a direct relationship here. If needed, the coupling would be provided as \(\lambda = \lambda(x, I_L)\), but for this problem, \(\lambda\) is treated as an independent variable. Assume the spring is at equilibrium ( \( x = 0 \) ) when \( F_{em} = 0 \). --- ## **Step 5: Final Model Summary** ### **1. Electrical Equation:** \[ e_{in}(t) = R [a_3 \lambda^3(t) + a_1 \lambda(t)] + \frac{d\lambda}{dt} \] ### **2. Mechanical Equation:** \[ m\ddot{x}(t) + b\dot{x}(t) + kx(t) = c_6 \lambda^6(t) + c_4 \lambda^4(t) + c_2 \lambda^2(t) \] where \( \lambda \) is the magnetic flux linkage (state variable), and \( x \) is the displacement of the armature mass. --- ## **Final Answer (Boxed):** \[ \boxed{ \begin{align*} &\text{Electrical:} \quad e_{in}(t) = R [a_3 \lambda^3(t) + a_1 \lambda(t)] + \frac{d\lambda}{dt} \\ &\text{Mechanical:} \quad m\ddot{x}(t) + b\dot{x}(t) + kx(t) = c_6 \lambda^6(t) + c_4 \lambda^4(t) + c_2 \lambda^2(t) \end{align*} } \] This is the **complete mathematical model** of the electromechanical actuator as described in the problem.