Response Curve

 

In this section we focus on understanding the hysteresis found at the main resonance of the Duffing oscillator because hysteresis at the main resonance and at some secondary resonances is easy to observe experimentally. The results in this section can also be derived by the method of harmonic balance by taking m = n = 1 in equation (3.35); however, we will use a more general method that is computationally a little simpler.

We generally expect that in a nonlinear system the maximum response frequency will be detuned from its natural frequency. An estimate for this detuning in the undamped, free cubic oscillator,

 

is obtained by studying this equation by the method of slowly varying amplitude  [16]. Write

 

and substitute equation (3.37) into equation (3.36) while assuming varies slowly in the sense that . Then equation (3.36) is approximated by

 

where we ignore all terms not at the driving frequency. Equation (3.38) has a steady-state solution in A, denoted by . In this case,

 

since A' = 0, and [17]

 

To first order, the strength of the nonlinearity increases the normalized frequency by an amount depending on the amplitude of oscillation and the nonlinearity parameter. This approximate value for a Duffing oscillator is consistent with the results found in Appendix B where exact solutions for a cubic oscillator are presented.

Hysteresis is discovered when we apply the slowly varying amplitude approximation to the forced, damped Duffing equation,

 

Substituting equation (3.37) into equation (3.41), and again keeping only the terms at the first harmonic, we arrive at the complex amplitude equation

 

which in steady-state (A' = 0) becomes

 

To find the set of real equations for the steady state, write the complex amplitude in the form

 

where both a and are real constants. Then equation (3.43) separates into two real equations,

 

and

 

which collectively determine both the phase and the amplitude of the response. Squaring both equations (3.45) and (3.46) and then adding the results, we obtain a cubic equation in ,

 

illustrated in Figure 3.10, which is known as the response curve . In this approximation the steady-state response is given by

 

where a is the maximum amplitude of the harmonic response determined from equation (3.47) and is the phase shift determined from equations (3.45 and 3.46).