text: The theory of bimodal AFM operation encompasses several aspects. Among them, the approximations to express the Euler-Bernoulli equation of a continuous cantilever beam in terms of the equations of the excited modes, the type of interaction forces acting on the tip, the theory of demodulation methods or the introduction of finite-size effects.
  In a nutshell, the tip displacement in AFM is approximated by a point-mass model,
  where , , , , , and  are, respectively, the driving frequency, the free resonant frequency, the quality factor, the stiffness, the driving force of the i-th mode, and the tip–sample interaction force. In bimodal AFM, the vertical motion of the tip (deflection) has two components, one for each mode,
  with , , , as the static, the first, and the second mode deflections; ,  and  are, respectively, the amplitude, frequency and phase shift of mode i.
  The theory that transforms bimodal AFM observables into material properties is based on applying the virial  and energy dissipation  theorems to the equations of motion of the excited modes. The following equations were derived
  where  is a time where the oscillation of both modes are periodic;  the quality factor of mode i. Bimodal AFM operation might be involve any pair of eigenmodes. However, experiments are commonly performed by exciting the first two eigenmodes.
  The theory of bimodal AFM provides analytical expressions to link material properties with microscope observables. For example, for a paraboloid probe (radius ) and a tip-sample force given by the linear viscoelastic Kelvin-Voigt model, the effective elastic modulus  of the sample, viscous coefficient of compressibility , loss tangent  or retardation time  are expressed by
  For an elastic material, the second term of equation to calculate  disappears because  which gives . The elastic modulus is obtained from the equation above. Other analytical expressions were proposed for the determination of the Hamaker constant and the magnetic parameters of a ferromagnetic sample.