Consistent rather than heuristic nondimensionalization of the fluid and oscillator dynamics in fluid-structure interaction, leads to decoupling of amplitude from frequency response. Further, recognizing that the number of governing dimensionless parameters should decrease, rather than increase, due to the fluid-structure synergy at the interface, an eigen-relation is revealed for a cylinder in Flow Induced Oscillations (FIO), including VIV and galloping: mA/mbod = CA/m* = 1/f*2-1. It shows that, for a given dimensionless oscillation frequency f*, the ratio of real added-mass to oscillating-mass is fully defined. Amplitude decoupling and the eigen-relation, lead to explicit expressions for coefficients, phases, and magnitudes of total, added-mass, and in-phase-with-velocity forces; revealing their dependence on the generic Strouhal number (Stn = fn*), damping, and Reynolds. Heuristic dimensionless parameters, (mass-damping, reduced velocity, mass-ratio, force coefficients) used in VIV data presentation are not needed. Theoretical derivations and force reconstruction match nearly perfectly with extensive experimental data collected over a decade in the Marine Renewable Energy Laboratory (MRELab) at the University of Michigan using four different oscillator test-models. Beyond the single frequency response model, the residuary force is derived by comparison to experiments. Established facts regarding VIV and galloping and new important observations are readily explained: (1) The effects of Strouhal, damping-ratio, mass-ratio, Reynolds, reduced velocity, and stagnation pressure. (2) The cause of expansion/contraction of the VIV range of synchronization. (3) The corresponding slope-change in oscillation frequency with respect to the Strouhal frequency of a stationary-cylinder. (4) The critical mass-ratio m* implying perpetual VIV. (5) The significance of the natural frequency of the oscillator in vacuo. (6) The effect of vortices on VIV and galloping. (7) The magnitude of vortex forces. (8) The indirect and direct vortex effects. (9) The unification of VIV and galloping onset. (10) Defining the next step in higher order theories for VIV and galloping beyond the eigen-relation.