LVDT (Linear Voltage Differential Transformer) need a complex excitation circuit and signal conditioning to work. Fortunately, Analog Devices provide AD598 for complete solution of LVDT signal excitation and conditioner to ease your life. AD598 can be operated in single supply mode, great for  battery-powered LVDT Application. Here is the schematic diagram of the circuit:

LVDT Signal Conditioner Design Procedure AD589 Single Supply circuit schematic

The design procedur for determining the components values can be obtained in the datasheet of AD598 from Analog Devices.