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Natural frequencies of double section beam, with mass at the top

suspended mass

position of suspended mass from the top

bottom column cross section area

bottom column cross section inertia

bottom column lenght

top column cross section area

top column cross section inertia

top column lenght

Hooke modulus

density

 <p></p> 0.998648648648649

ELEMENTS STIFFNESS MATRIX

SYSTEM STIFFNESS MATRIX

function to assembly matrix

 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10 mat

CONSTARINED SYSTEM STIFFNESS MATRIX

 10

ELEMENTS MASS MATRIX

SYSTEM MASS MATRIX

function to assembly matrix

 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10 mat

CONSTARINED SYSTEM MASS MATRIX

 10

DYNAMICAL MATRIX

EIGENVALUE EVALUATION

 # # 0.0567 0.0002 1.5588 10 5 - ^ * 2.1646 10 6 - ^ * 1.5921 10 6 - ^ * - 1.9853 10 7 - ^ * 1.1 10 14 - ^ * 3.2695 10 16 - ^ * 8 1 mat

eigenvalues

 0.0594 0.3876 0.7642 - 0.4958 - 0.019 - 0.0813 1.7029 10 7 - ^ * - 1.7636 10 6 - ^ * - 0.0306 0.1583 0.1268 - 0.3112 0.0173 - 0.908 6.5592 10 7 - ^ * - 6.7949 10 6 - ^ * - 0.2154 0.8428 0.0845 - 0.4867 0.1408 0.2773 - 1.3133 10 7 - ^ * 1.4177 10 6 - ^ * 0.0522 0.0544 0.4042 0.3519 - 0.2257 0.1454 9.5712 10 7 - ^ * - 9.8481 10 6 - ^ * - 0.6841 0.0649 - 0.0354 - 0.0257 0.0339 0.0111 - 3.4654 10 7 - ^ * 7.2783 10 6 - ^ * 0.0689 0.2267 - 0.3366 - 0.3841 0.6804 0.188 - 9.1124 10 7 - ^ * - 7.1422 10 6 - ^ * - 0.6848 0.0672 - 0.0388 - 0.0296 0.0407 0.013 - 0.0051 - 0.0671 - 0.0689 0.2267 - 0.3366 - 0.3841 0.6804 0.188 - 1 - 0.9977 8 8 mat

eigenvector matrix

NATURAL FREQUENCIES

 0.6686 11.1367 40.3105 108.1762 126.1343 i * - 357.1928 1.5175 10 6 ^ * 8.802 10 6 ^ * 8 1 mat

CHECKING VALUE: EXACT VALUE OF NATURAL FREQUENCIES(UNIFORM CROSS SECTION BEAM and WITHOUT SUSPENDED MASS OF THE TROLLEY

 2.4227 Hz * % 262.3252 - 15.1837 Hz * % 36.3388 - 42.519 Hz * % 5.4788 -

MODE SHAPES

 x w.first x w.second x w.third

MODE SHAPES and NATURAL FREQUENCIES

Deformed shape at the second frequency

 x w.first UNDEFORMD 2 1 sys 0.669 Hz *

Deformed shape at the second frequency

 x w.second UNDEFORMD 2 1 sys 11.137 Hz *