Energy Approximate Analysis Method To Solve The Free Vibration Of Carbon Nanotubes

Carbon nanotubes (CNTs) have excellent mechanical properties, and how to quicklyestimate the natural frequency of vibration is of great significance.Based on the research of how the additional boundary effect caused by small scaleinfluences the vibration of single-walled nanotubes (SWCNTs) builded on Eringen’selasticity theory and Euler beam model, a general method, namely, energy approach toanalysis the free vibration of CNTs under typical boundaries is proposed. The approximatenatural frequency expressions of CNTs are derived based on energy function. The researchachievements are as follows:1. The work done by the shear force and bending moment caused by small scale in theboundary is offset each other in the case of no initial stress; Elastic medium has noeffect on modal shape. For the cantilever beam, it also has no effect on the criticalscale value that leads to no dynamic vibration behavior.2. For SWCNTs without elastic medium, when the scale parameter is less than thecritical percentage of85%,48%,75%,82%,83%,88%, respectively, the predictederror of first to sixth order frequency parameters is less than5%; For SWCNTsembedded in Winkler elastic medium, the derived formulas are suitable for clamped-clamped supported (CC), clamped-simply supported (CS) and simply-simplysupported (SS) beams. When the coefficient of elasticity reaches to GPa, the formulasfor the cantilever beam also have good prediction accuracy.3. The prediction error of the out-of-phase vibration frequency for double-wallednanotubes with typical boundary conditions, different small scale and initial stress byenergy method is less than2%; The prediction error of the in-phase vibrationfrequency is less than3%for CC, CS, SS beams; When small scale is large,increasing the pre-stress may leads to low order frequency is greater than the high order frequency.4. The predicted accuracy of fluid-conveying cantilever beam by energy methoddecreases with the increase of the flow rate. The calculation for high order frequencyis more accurate than low order; When the small scale is less than0.3, and the flowrate is less than2000m/s, the real component prediction error is less than5%, theimaginary part prediction error is less than7%.This paper proposes a rapid effective method to calculate the natural frequency ofCNTs, and presents approximate natural frequency expressions of CNTs by energyfunction. The attractiveness of the derived results lies in their simplicity. It is usefull fordesigners to estimate the frequencies of CNTs.