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This note is concerned with the natural vibration problem of a mechanical system, consisting of a fixed-free axially vibrating elastic rod which is restrained by a linear spring in-span. The frequency equation of the system is derived first. Then the mode shapes are given and finally a sensitivity formula is established
Contexts in source publication
Context 1
... problem to be investigated in the present note is the natural vibration problem of the system shown in Figure 1. It consists of a fixed-free axially vibrating elastic rod, which is restrained by a linear spring in-span. ...
Context 2
... equation (8) is the frequency equation of the system shown in Figure 1. Its numerical solution yields the dimensionless eigenfrequency parameters b , which then given via (5) the unknown eigenfrequencies v of the system. ...
Context 3
... the non-dimensional position co-ordinate x¯ = x/L is introduced. The final aim is to give in the following a formula for the sensitivity of the eigenfrequencies of the system in Figure 1 with respect to small changes in the location of the in-span spring attachment point around its nominal position, i.e., the rate of change of the eigenfrequencies with respect to the location parameter h. To this end, the frequency equation (8) has to be differentiated partially with respect to h. ...
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... first two dimensionless eigenfrequency parameters b 1 and b 2 are given in Table 1 for various values of the location and stiffness parameters h and a k , respectively, which include a sufficiently great range of practical applications. For further numerical applications, the following values are chosen for the non-dimensional data of the mechanical system in Figure 1: h = 0·5 and a k = 1. The selected data means that T 1 ...
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... first two dimensionless eigenfrequency parameters of the system in Figure 1 for 4·845239 5·112311 5·366411 5·379986 5·086985 4·787422 4·734118 5·048169 5·393809 5·354032 5·00 1·624742 1·743339 1·907960 2·119308 2·380644 2·677605 2·920212 2·957131 2·829218 2·653662 4·868896 5·177490 5·483805 5·515368 5·163306 4·804168 4·739828 5·139881 5·542558 5·454354 ...
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Citations
... The exact characteristic equation of a rod with a single in-span spring has been derived in Refs. [20,21]. ...
... To derive Eq. (20), the continuity of the displacement eigenfunction (8) at ξ¼ξ j has been taken into account, see previous point (i); also, it has been considered that the left and right hand limits of all Dirac's deltas in Eq. (4a) are equal to zero, i.e. lim ξ-ξ À j δðξ À ξ k Þ ¼ 0 and lim ξ-ξ þ j δðξ Àξ k Þ ¼ 0 for k¼1,2,…N, including k¼ j. Because the displacement eigenfunction (8) is always continuous at ξ¼ξ j whether the ESD is activated or not, Eqs. ...
... Because the displacement eigenfunction (8) is always continuous at ξ¼ξ j whether the ESD is activated or not, Eqs. (20) show that the tangent lines to the stress eigenfunction (12), on the left and right sides of the ESD location ξ j , will always exhibit the same slope. ...
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