2 Figure P3.32 shows a schematic diagram of a solenoid actuator. Energizing the coil (armature circuit)creates the electromagnetic force Fem that acts on the armature-valve mass m. Through experimentation Vaughan and Gamble [6] developed the following nonlinear equation for solenoid current: I_{L}(\lambda)=a_{3} \lambda^{3}+a_{1} \lambda(\mathrm{amps}, \mathrm{A}) where à is the magnetic flux linkage (Wb) and az and a, are constants. This nonlinear current model accounts for back emf and the variable inductance because of the displacement of the armature mass.Vaughan and Gamble also developed the empirical equation for electromagnetic force: F_{\mathrm{cm}}(\lambda)=c_{6} \lambda^{6}+c_{4} \lambda^{4}+c_{2} \lambda^{2}(\mathrm{~N}) where c6, C4. and c, are empirical constants from experiments. Derive the complete mathematical model of the electromechanical actuator. Assume that the return spring is undeflected when xr 0 and the electromagnetic force is zero.

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