As you said harmonic and random analysis are linear so can not have/include nonlinearities (like the nonlinear contacts you describe, e.g., frictionless and frictional).
Transient dynamic analysis is needed here. In this type of analysis, one can have a swept-sine signal (https://www.ni.com/docs/en-US/bundle/labwindows-cvi/page/advancedanalysisconcepts/lvac_frequency_sweep.html)
All the best
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