Solenoid s Mass M Milan K SIDING *** K STRAIGHT TRAY A railway point's actuator has been constructed as shown above. The linear movement on the changeover tray is accomplished by movement of the solenoid S of inductance, L and series internal resistance, R. Two springs of coefficient k return the tray to a central position if no force from the solenoid operates. The damper C is installed to control the motion by avoiding bounce and percussive damage. The mass of the solenoid core is M [kg] and the Force derived from current flow is ai. (a) Derive a system differential equation relating the voltage input e to the movement x of the core of the solenoid. (b) Derive a relationship between M, C and K for the mechanical part of the circuit only to be critically damped. Prove that the transfer function of the system is (c) X(S) E(S) LM3+(RM+LC)2+(RC+2LK)s+2RK (d) [Hint: Relate e to i and i to x separately in LAPLACE TRANSFORM then combine.] The normal separation of tray rail to fixed rail is 0.2m, the spring coefficients are rated at 1500 N.m each, the attractive force coefficient of the solenoid, a is 15 N/A, and its internal resistance, R is 0.51. What is the magnitude of the voltage step required if the design criterion on the force holding the tray to the fixed rail is chosen to be 100 N