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The majority of microscale and nanoscale mechanical systems encountered in the world are stiff. One example of such a system is a microbead trapped in an optical tweezer. For this example, the ratio of a body's mass to the viscous damping coefficient is negligible. Apart from being stiff, this system is also stochastic meaning that the differential equations representing the system include a random variable. This type of model can be solved with existing stochastic differential equation (SDE) solvers. Addressing the stochastic nature of the model usually requires averaging the results of multiple simulations. The computational time required to run the hundreds of simulations that may be necessary to obtain a useful average can be prohibitive. This paper presents a scaling approach that significantly reduces the computational time required to obtain these averages. However, this scaling changes the model and therefore the power spectral density (PSD) of the predicted motion. This work provides a general analysis of a mass-spring-damper model, such as an optical tweezer, that clearly shows the effect of scaling on the frequency content of the stochastic system. This analysis shows that the scaling technique can be used effectively without much loss of information. An experimental dataset of a 2 μm diameter microbead falling into an optical trap is compared with the simulation for validating the scaling method.

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