of its static response. When the mass point is released from a deformed state, 

 it will oscillate about the reference state. Without damping, the oscillation 

 will continue indefinitely with a period which is dependent on the magnitude 

 of the initial displacement. The natural period for this system is 0.2639 

 seconds for an initial displacement of 20 cm. If the mass point is forced at 

 some other frequency with a force magnitude equal to that required to produce 

 the initial static defection, some interesting things occur. Two examples are 

 shown in Figures 2 and 3. Excitation below the natural frequency induces a 

 response similar to the linear case where the impressed frequency is dominant 

 and the response amplitude approximates the static response. The little 

 ripple in the response is at the natural frequency and would be expected to 

 die out in the presence of appropriate damping. Excitation above the natural 

 frequency introduces a new phenomenon. As before, there are two frequencies 

 present: one at the imposed frequency and the other at a varying frequency. 

 The variable frequency response appears as a damped transient because of the 

 numerical damping that was included in the integrator. The varying frequency 

 is a direct result of the geometric nonlinearity which causes the natural 

 frequency to be a function of the amplitude of the response. An important 

 aspect of this behavior is that the decay of the transient is long compared 

 to the period of the excitation. 



Figure 4 shows the results of an attempt to force the system at the 

 apparent natural frequency. In this solution there is no damping in the model 

 nor is there any intentionally in the numerical integrator. Although the plot 

 is a crude one which attempts only to show the peaks and valleys, it shows 

 behavior not found in the linear problem. The response is not unbounded and is 

 quite complicated in form. There is not a single amplitude, and for the most 



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