Frequency Equation of Flexural Vibrating Cantilever Beam Considering the Rotary Inertial Moment of an Attached Mass

Table 1

The first five natural frequencies of the cantilever beam considering the bending moment.

Frequency number (i)	Nondimensional natural frequency
	= 100; = 1	= 10; = 1	= 1; = 100	= 1; = 10	= 1; = 1	= 1; = 0.1	= 1; = 0.01	= 1; = 0.001	= 1; = 0	= 1; (ref [11])
1	0.1749603	0.6054297	2.950242	2.91158	2.55044	1.70226	1.571098	1.55867	1.557298	1.557298
2	22.30366	22.47413	23.93841	23.93129	23.86031	23.17405	19.13831	16.57872	16.25009	16.25009
3	61.66046	61.83822	63.43232	63.42955	63.40190	63.1286	60.72695	53.50208	50.89584	50.89584
4	120.90684	121.0850	122.7265	122.7251	122.71062	122.5668	121.2014	113.2826	105.1983	105.1983
5	199.86943	200.0480	201.7208	201.71991	201.71107	201.6229	200.7639	194.2422	179.2320	179.2320

Notes. The values of this column were calculated by authors using the frequency equation from [11].
And the results were carefully checked by comparing the values of (using the character in this reference).