2.3.4 Cantilever small oscillations in a force field

Consider the case when in addition to the driving force (see (1) in chapter 2.3.3), an external force acts on an oscillator. The equation of motion in this case is written as


Because depends only on spatial coordinates, the qualitative feature of oscillations is the same as in (6) chapter 2.3.3. Force action results in the change of the oscillator equilibrium position about which the oscillations occur. For small oscillations we can take Taylor of at point corresponding to the equilibrium position:


where is expressed through and as follows:


and is determined from the following condition


Changing in equation (1) by in accordance with (3) and taking into account (4), we get


where , , , – damping factor determined in chapter 2.3.2.

As can be seen, equation (5) is identical to (5) in chapter 2.3.3. These equations differ in the introduction of the other spring stiffness and new equilibrium position . Note that we can neglect the second order and higher terms in equation (2) only if the following condition is met


where – oscillations amplitude at frequency determined by formula (7) below. However, there are cases when and the higher order terms in (2) are to be taken into consideration.

Similarly to formulas (7, 8) in chapter 2.3.3 and (8) in chapter 2.3.2, oscillations amplitude and phase shift in the presence of external forces gradient is given by



where – oscillations amplitude at resonant frequency .

Thus, the force gradient results in an additional shift of a vibrating system amplitude-frequency characteristic (AFC) and phase-frequency characteristic (PFC). Fig. 1 shows AFC and PFC at different values of .



Fig. 1.  AFC – (a) and PFC – (b) at different values .

Resonant frequency in the presence of external force can be written, by analogy with formula (10) in chapter 2.3.3, as


Hence, the additional shift of the AFC is


If we can take Taylor of the radicand in formula (10) to obtain:


From expression (8) it follows that the force gradient results in the PFC shift so that its inflection point, at which the phase value is , corresponds to the frequency




If condition is met, formula (13) is the same as formula (11).

Let us determine the phase shift if there is force gradient present. If oscillations occur under the driving force at frequency , the phase shift is . In case of the force gradient presence, the phase shift in accordance with (8) becomes:


If we can make a Taylor's expansion of expression (14) as follows


Hence, the additional phase shift due to the force gradient is (Fig. 2)


Fig. 2.  Variation of the oscillations phase
with resonant frequency.

Fig. 3.  Variation of the oscillations amplitude
with resonant frequency.

Consider now the amplitude change under the force gradient (Fig. 3).

The maximal change of in case of the resonant frequency (9) variation, takes place at certain frequencies of the driving force. To these frequencies corresponds the maximum slope of the tangent to the AFC curve (linear portion of AFC):


The change in oscillations amplitude (7) at frequency (Fig. 3) due to the force gradient in accordance with formulas (7) and (11) is given by


The considered oscillations mode is widely used in AFM. To study the force interaction of an oscillating cantilever with a sample, in particular, one can examine the change in resonant frequency (11), amplitude (18) and phase (16) of oscillations and then, according to obtained data, render the force value (see, for example, chapter 2.7.1).


  • External force action (if condition (6) is met) results only in the change of the oscillator effective stiffness (resonant frequency) and equilibrium position about which the oscillations occur.
  • The system oscillations obey the law which is the same as in the absence of the external force.
  • The change in resonant frequency, phase and amplitude of oscillations is proportional to the external force gradient and is determined by formulas (11), (16), (18).