h. HARMONIC FORCING AT AN INTERMEDIATE POINT 



When the external harmonic force F and moment G act at an intermediate 

 point instead of at the end of the uniform beaun, the problem is more 

 complicated but can be handled by a double application of the equations 

 for forcing at one end. 



First the fundajnental beam equations are generalized so as to 

 include distributed forces and moments of respective magnitudes F and G 

 per unit length, acting on the beam in the same principal plane; see 

 Figure 1. On a slice dx thick there is then a net force dP + F dx and a 

 net moment dM + G dx so that, after dividing through by dx, the equations 

 of motion become in place of Equations [2a,b]: 



9 V _ 9P_ 9M 



% 2 - 3x * ^1 ; 9x = - P - ^1 [l6a,b] 



Rotary inertia is omitted here. Equations [2c, d] and the elastic analysis 

 leading to them require no change. 



Now let the external forces and moments act only on a very thin 

 slice of the beam. See Figure 2, where the thickness of the slice is 

 greatly exaggerated for clarity and at its faces, the common device of 

 drawing the beam as separated is adopted. Let the values of the shear 

 force P and moment M in the beam be P' and M' at the left side of the 

 slice (toward negative x) and P", M" at the right side. Then the 

 associated reactions on the slice are -P' and -M' at the left side but P" 

 and M" at the right. 



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