Discussion of the Improved Methods for Analyzing a Cantilever Beam Carrying a Tip-Mass under Base Excitation

Table 2

Characteristic properties of the cantilever beam including tip-mass.

Physical properties	Beam	Tip-mass
Young’s modulus, 𝐸 (GPa)	69	—
Density, 𝜌𝑠 or 𝜌𝑚 (kg/m^3)	2630	7753
Length, 𝐿 or 𝐿𝑚 (mm)	120.4	15.6
Thickness, 𝑡𝑠 or 𝑡𝑚 (mm)	4.9	29.5
Width, 𝑤 (mm)	10.2	10.2
Total mass, 𝑀 (g)^a	—	33.83
Offset, 𝑟𝑐 (mm)^b	—	8.01
Rotary inertia, 𝐽 (kg·m^2)^b, c	—	3.33 × 10^−6

aThe total mass of the tip-mass including the elongated part of beam was calculated by
𝑀=𝐿_𝑚𝑡_𝑚𝑤𝜌_𝑚+𝐿_𝑠𝑡_𝑠𝑤(𝜌_𝑠−𝜌_𝑚), where 𝐿_𝑠= 10 mm.
^bThe offset distance and the rotary inertia were calculated according to the geometry and material properties of the beam and tip-mass using equations:
𝑟_𝑐=(𝜌_𝑚𝑡_𝑚𝐿²_𝑚−𝜌_𝑚𝑡_𝑠𝐿²_𝑠+𝜌_𝑠𝑡_𝑠𝐿²_𝑠)/(2[𝜌_𝑚𝑡_𝑚𝐿_𝑚+𝐿_𝑠𝑡_𝑠(𝜌_𝑠−𝜌_𝑚)]), 𝐽=𝜌_𝑚𝐿_𝑚𝑡_𝑚𝑤[(1/12)(𝐿²_𝑚+𝑡²_𝑚)+((1/2)𝐿_𝑚−𝑟_𝑐)²]+𝐿_𝑠𝑡_𝑠𝑤(𝜌_𝑠−𝜌_𝑚)[(1/12)(𝐿²_𝑠+𝑡²_𝑠)+(𝑟_𝑐−(1/2)𝐿_𝑠)²].
^cThe corresponding parameters 𝜙, 𝑘/𝐿, and 𝑟_𝑐/𝐿 were calculated as follows:
𝜙=𝑀/(𝜌_𝑠𝑡_𝑠𝑤)=2.1377, 𝑘/𝐿=𝐽/𝑀/𝐿=0.0824, and 𝑟_𝑐/𝐿=0.0665.