what is the difference between dynamic stiffness and Quadrature Dynamic
Stiffness?

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I guess these are not very common terms. Most of the original literature that mentions these terms is associated with Donald E. Bently and Agnes Muszynska.

These are two slides from a presentation Don delivered in 1998 that may help.

Regards- Ali M. Al-Shurafa

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Dynamic stiffness has been around for may years.  That the original literature is from the authors you mention is suspect.  Quadrature is a term they liked, perhaps there is an electrical connection.  On the DVFIIs, there was output on the back label quadrature - if I remember correctly.  One needed to connect the outputs correctly to generate polar plots on an x-y recorder.

One could think of this as a complex stiffness.

Important for dynamic stiffness is that these are frequency dependent (in general).  One force acts in the same direction as the displacement, and one force acts orthogonal to the displacement. If you like to look at velocity or acceleration, remember that velocity leads displacement by 90 degrees (at a single frequency) and acceleration leads velocity by 90 degrees.

An analogy that some may find useful to understand the concept of dynamic stiffness is the tringle of AC power: Active, Reactive and Apparent.

Regards- Ali M. Al-Shurafa

One example of dynamic stiffness (without damping) is K - omega^2 M.  at a natural frequency (omega) there is no stiffness.

Often we are interested in the inverse of dynamic stiffness, compliance.  In balancing on used influence coefficients (sometimes units vary, e.g. with fixed radius, but the units are basically displacement/force or in terms of velocity or acceleration - the speed squared term to give force is implicit and not explicit when we use it.  However, one must consider the speed [which is frequency, too, for balancing] when using influence coefficients or balancing.)

Together with the velocity and acceleration items, these are transfer functions, which depend upon frequency.