Table El. Normal Modes t//,(z) and Corresponding 

 Values of /,~'^^ for Various Structural Forms 



I. Pivoted Rigid Rod 



Normal modes: <//] = z/L, i//,;^i = 



lr\l^ ■ : (5/3)'^= = 1.2910 



11. Taut String 



Normal modes: i/», = smivz/ L 



ir^'^ : (4/3)'/' = 1.1547 



III. Pinned-Pinned Beam 

 Normal modes: »//, = sini-n-z/ L 



/-'/2 : (4/3)'/== 1.1547 



IV. Cantilevered Beam 



Normal modes; i//, = cosh \,z/L - cos X,z/L - a,(sinh kiz/L - sin X/z/L) 



i A., a, /,""•= 



1 1.87510410 0.73409550 0.6525 



2 4.19409113 1.01846644 0.7494 



3 7.85475743 0.99922450 0.7686 



V. Clamped-Clamped Beam 



Normal modes; i//, = cosh K,z/ L — cos \,z/ L — u,(sinh k,z/ L — sin \,zl L) 

 i \, a, /,-"^ 



1 4.75300408 0.98250222 0.7348 



2 7.8532046 1.00077731 0.7694 



3 10.9956078 0.99996645 0.7817 



VI. Clamped-Pinned Beam 



Normal modes: i|/, = coshX,z/L - cos \,z/L - a,(sinh X;z/L - sinX,z/L) 

 / \, a, /, "' 



1 



3.92660230 



1.00077730 



0.7694 



2 



7.06858275 



1.00000144 



0.7891 



3 



10.21027613 



1.00000000 



0.7972 



VII. Free-Pinned Beam 

 Normal modes: i//, = cosh X,(z/L) + cos X,(z/L) - a,[sinh X,(z/L) + sin X,(z/L)] 



/ \, a, J,-'" 



1 3.92660 



1.000773 



0.7628 



2 7.06858 



1.000000 



0.7890 



3 10.2102 



1 .000000 



0.7972 



174 



