5. HARMONIC END FORCING OF A UNIFORM BEAM 

 HAVING EXTERNAL AND INTERNAL DAMPING 



End forcing of an undamped uniform beam was considered previously in 

 this report. The same problem will now be attacked with the beam subjected 

 to both external and internal damping. 



Let the external damping be due to a uniform force per unit length of 

 magnitude -cvi3v/3t, with p denoting the mass per unit length of the beam, 

 V its displacement parallel to a principal plane, and t the time. Here 

 for convenience cu is written instead of the usual c. 



As in Reference 13, pp. 6 and 7, the internal damping is assumed to 

 be due to a resistance to variation of the bending stress such that the 

 instantaneous value of the stress is o = E(e + n3e/3t), where E denotes 

 Yoiong's modulus, n a damping constant, and e the bending strain. (in 

 Reference 13, Eq was denoted by v . ) Thus, the moment M due to bending is: 



3 



% = 



= EI 



/sfv ^ ^ 3 v\ 

 \9x^ 3t3x/ 



* From elementary beam theory, if c is the distance from the neutral 



surface to the fiber whose strain is 5 and p is the radius of curvature 

 of the elastic curve at the section for which the bending moment is M, 

 then 



M = ?i- = i E (c + n 1^) = EI ^ + b -^ 



b = Enl 



because 











c p 



2 

 . 3 V 



2 ' 

 3x 



3 

 1 3e 3 V 

 c 8t 2 ' 

 3t3x 



27 



