Terjemahan dari buku Dynamics of Structure, Anil K. Chopra. Penerjemahan tidak bertanggung jawab pada kebenaran isinya karena ini hanya untuk tujuan perkuliahan.Full description
dinamika struktur
Terjemahan dari buku Dynamics of Structure, Anil K. Chopra. Penerjemahan tidak bertanggung jawab pada kebenaran isinya karena ini hanya untuk tujuan perkuliahan.
tugas analisa dinamika struktur UNISSULADeskripsi lengkap
Full description
dinamika strukturDeskripsi lengkap
Menentukan sketsa getaran dengan persamaan umum getaran. Lebih disarankan didownload dalam format office word (.doc)
pengantar kuliah daerah aliran sungaiFull description
Deskripsi lengkap
T.sipil. tentang SDOF
T.sipil. tentang SDOFDeskripsi lengkap
PENGANTAR
BAYZONI
Multidegree-of-Freedom Systems • A structure can be modeled and its response analyzed using a SDOF model if the mass is essentially concentrated at a single point that can move, translate, or rotate only in one direction, or if the system is constrained in such a way as to permit only a single mode of displacement. In general, the mass of a larger building or structure is distributed thro throug ugho hout ut the the struc structu ture re and and can can move move in many many ways. • A real realis istic tic descr descrip ipti tion on of the the dyna dynami mic c resp respon onse se of such uch systems tems gen generall rally y requi equirres the the use use of a number number of independe independent nt displace displacement ment coordinat coordinates, es, and and mode modeli ling ng of the the sy syst stem em as a mult multid ideg egre reee-of of-freedom (MDOF) system. • Dynamic analysis of such MDOF systems is discussed in the following sections.
Multidegree-of-Freedom Systems • A structure can be modeled and its response analyzed using a SDOF model if the mass is essentially concentrated at a single point that can move, translate, or rotate only in one direction, or if the system is constrained in such a way as to permit only a single mode of displacement. In general, the mass of a larger building or structure is distributed thro throug ugho hout ut the the struc structu ture re and and can can move move in many many ways. • A real realis istic tic descr descrip ipti tion on of the the dyna dynami mic c resp respon onse se of such uch systems tems gen generall rally y requi equirres the the use use of a number number of independe independent nt displace displacement ment coordinat coordinates, es, and and mode modeli ling ng of the the sy syst stem em as a mult multid ideg egre reee-of of-freedom (MDOF) system. • Dynamic analysis of such MDOF systems is discussed in the following sections.
Equations of Motion •
•
The MDOF analysis procedure is illust illustrat rated ed by exami examinin ning g the dynami dynamic c respo respons nse e of the the idea ideali lize zed d thre threes esto tory ry buil buildi ding ng shown shown in figu figure re belo below. w. The The mass of the structure is assumed to be concentrated at the floor levels, which are are furth urther er assu assume med d to be rigi rigid d and and displa displace ce in one transla translatio tional nal direct directio ion n only. Thus, the dynamic behavior of this structu structure re is compl complete etely ly defin defined ed by the three-story displacements u1(t), u2(t) and u3(t). The equati equation on of of motio motion n of any any story story can can be deri derive ved d fro from the the expr expres essi sion on of dynamic equilibrium of all of the forces acting on the story mass, including the inertia, damping, and elastic forces that result from the motion, and the externally applied force. The equations of equilibrium for the two stories can be written as follows (usi (usin ng notation analogous to the SDOF case):
MULTI DEGREE OF FREEDOM • Model 3 derajat kebebasan x1
x2 F1(t)
m1 k 1
x3 F2(t)
m2 k 2
F3(t) m3
k 3
• Keseimbangan Gaya
1 k 1. x1 k 2 . x2 x1 F 1 (t ) 0 m1. x
2 k 2 . x2 x1 k 3 . x3 x2 F 2 (t ) 0 m2 . x 3 k 3 . x3 x2 F 3 (t ) 0 m3 . x
• Dalam bentuk Matrik
K . X F t M . X • Dalam hal ini: m1 M 0 0 x1 X x 2 x3
0
0 m2 0
0 m 3
x 1 X x 2 x 3
k 1 k 2 K k 2 0
k 2 k 2 k 3 k 3
F1 ( t ) Ft F2 ( t ) F3 ( t )
• Dalam hal terdapat redaman maka: C . X K . X F t M . X
k 3 k 3 0
• Keterangan:
GETARAN BEBAS • Getaran Bebas Tanpa Redaman
[]{̈} + []{} = 0
Solusi dari persamaan di atas adalah:
̇ (())=− =.cos+ .sin 2 . s i n +2.cos ̈ () = − .cos − . sin
Sehingga diperoleh persamaan:
−2[]{} + []{} =0
[] − 2[]{} = 0
Persamaan di atas dapat ditulis:
{} = ‖[] − 02[]‖
Dengan aturan Cramer solusi dari persamaan di atas:
‖[] −2[]‖ =0
Pemecahan non-trivial dimungkinkan
Persamaan ini disebut persamaan frekuensi sistem, dengan memperluas determinan akan diperoleh persaman aljabar berderajat N dalam parameter 2 untuk sistem yang mempunyai B derajat kebebasan. 2 disebut “eigen-value”
DETERMINANT
CONTOH: Asumsi:
m3=1 m2=1
m1=1
K3= 3
K2= 4 K1= 5
• Lantai kaku • Tidak ada deformasi aksial • Semua massa terkumpul pada lantai
Developing a Way To Solve the Equations of Motion • This will be done by a transformation of coordinates from normal coordinates (displacements at the nodes) To modal coordinates (amplitudes of the natural Mode shapes). • Because of the orthogonality property of the natural mode shapes, the equations of motion become uncoupled, allowing them to be solved as SDOF equations. • After solving, we can transform back to the normal coordinates.
Solutions for System in Undamped Free Vibration (Natural Mode Shapes and Frequencies)
Solutions for System in Undamped Free Vibration (continued)
Mode Shapes for Idealized 3-Story Frame
Concept of Linear Combination of Mode Shapes (Transformation of Coordinates) U=ΦY
