Unexpected result with bar fixed to spring with periodic loading

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The discussion revolves around a steady-state problem involving a bar fixed on one side and subjected to periodic loading on the other. The expected displacement expression based on Hooke's law was not obtained; instead, an additional term dependent on frequency appeared in the solution. This term, which modifies the spring constant, raises questions about its physical significance in the context of a rigid bar. The model used aligns with the longitudinal displacement equation for bars, suggesting that the extra term may relate to rotary inertia effects. The conclusion indicates that this additional term is valid and can be understood through the theory of single-degree-of-freedom vibrating systems.
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Hi, I solved a steady state problem involving a bar fixed to string in the left side and pulled periodically on the right side f(x,t)=P_0sin(wt). To check the solution i made E (young's modulus) go to infinity, essentially making the bar rigid. the expression i expected to receive is:

u(x,t) = \frac{P_0sin(wt)}{k}

which is hookes law.

but the expression i received was:

u(x,t) = \frac{P_0sin(wt)}{k-ρ_{1D}Lw^{2}}

the density is one dimensional and L is the bar length.
this expression has an extra term that depends on the frequency which subtracts from the spring constant.
i checked the units and my calculations and they seem to add up. i can't visualize the effect of frequency on the displacement field for a rigid bar. Does this term really "exist" or is this some kind of error?

BTW, the model i used for the bar is the longitudinal displacement equation for bars:
(AEu)''+f(x,t)=ρ\stackrel{..}{u}
 
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The "extra" term looks like the rotary inertia of the beam, and a term something like that should be there.

Look up the theory of single-degree-of-freedom (SDOF) vibrating systems.
 
Thank you, I see it know.
 
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