Impulse response function single coil
Consider all of the variables in play with a shaker like the one you have shown.

The inductance of the coil varies depending on the position of the "target."  As the target moves closer to the "electromagnet" the inductance will increase.  This is true when the target is not moving, but when it is moving, the energized, magnetized coil generates its own voltage, similar to the back emf of a motor.  How quickly the target is moving depends both on the mass of the target, which must be accelerated by the "resultant force,"(F=MA) after subtracting the force of the spring that is part of the shaker to provide return to the start position when the current is removed.  The spring rate determines the instantaneous spring force, which is dependent on its position and preload.  The resultant force is not constant for a given current, but changes in a non-linear fashion depending on the distance between the electromagnet and the target.  When you finally add in the non-linearity of the core material, you can see that the response of the device must be dependent on a large number of initial conditions: notably target mass, position, spring force, and the amplitude of the input.  The system does not lend itself to the calculation of an impulse response that will be useful over a range of conditions and input amplitudes.

You could measure a step or sine response for a given mass and input amplitude from a static position using an accelerometer on the target mass, and create a table of responses for different masses at different input amplitudes.  This at least would give you a feel for the range of inputs for a given mass where the shaker response was somewhat proportional to the input.