Modern mechanical systems, such as machine tools and automatic inspection machines, often require high- speed/high-accuracy linear motions. Linear motors essentially work the same as rotary motors, only opened up and laid out flat (see Fig 1). Each motor is made of two parts - a permanent magnet assembly and a coil assembly. The coil assembly has copper windings that conduct current (1). The magnet assembly generate magnetic flux density (B). When the current and flux density interact, force (F) is generated in the direction shown in Fig 1. FLUX DENSITY (B) CURRENT (1) COIL ASSEMBLY FORCE (F) MAGNET ASSEMBLY MAGNETIC ATTRACTION (Fa) Ptg 1 A DC linear motor driving an inertia load is shown in Fig 2, in which E is the voltage source used to drive the coils to control the motion of the load; R, is the armature resistance, Lis the armature inductance and is the armature current; E = Kex is the back EMF, where K, is the back EMF coefficient and x represents the displacement of the load. The motor produces a force F = K,I,where K, is the force constant. The motor and load have a total mass M and a damping coefficient of b Fig 2 Find the equations of motion for the mechanical and electrical systems. Hint: Your electrical system has one loop with just the resistor, inductor, input voltage and back emf. The mechanical system has a damping force and the motor force (F) acting on it.