# Partial Differential Equation Course (ins 214E) Term Homework Free Vibrations of Simply Supported Beam

 Date 08.05.2017 Size 96.83 Kb.
Partial Differential Equation Course (INS 214E) Term Homework

Free Vibrations of Simply Supported Beam

Consider the following linear fourth-order partial differential equation with the boundary conditions   (initial displacement) t > 0 and 0 < x < L

  (initial velocity) t > 0 and 0 < x < L

where   and   are given functions and v is vertical displacement (or deflection) of the beam and c is constant and called as the wave speed. This equation corresponds to free vibrations of simply supported beam with L length. The equation for wave speed is where EI is flexural rigidity of the section of the beam with length L,   is density and A is area of cross-section of the beam. The load is suddenly removed the beam will vibrate freely.

1. Obtain the expression for the free undamped transverse vibration of the simply supported beam and natural frequency expression   of the beam by using the Method of Separation of Variables.

2. Obtain expression of free vibration for simply supported beam if P (w, Q) is removed suddenly at time t = 0.

3. Draw the modes of deflection along the beam and modes of deflection with time along the beam (for three different   values).

4. Obtain and draw the solution by using the Finite Difference Method as a numerical solution approach (based on Excel).

In the problem, the load will be assumed to be

Group I:
 

B

A  P=300 N      L x v

EI

Figure1: Simply supported beam with L length and intermediate concentrated load P x2

w=200 N/m G        
roup II:

B

A    x EI L

v

Figure2: Simply supported beam with L length and partially distributed uniform load   Q=350 N/m

G
A   roup III:

B    x EI  L v x1

x2 In order to get the deflection due to the static load, also assume that the deflected shape is represented by half range Fourier series: and the differential equation relating the deflection (displacement) and the load is Use the following values to draw the mode shapes of the deflection:  Student Groups (with respect to the initial letter of first names):

GROUP I: Abdullah – Barış: CASE I (File Color: Black)

GROUP II: Burak – Merve: CASE II (File Color: Red)

GROUP III: Miraç – Yücel: CASE III (File Color: Blue)
 x1 and x2 distance values x1 (m) x2 (m) Group No: Abdullah Bingöl 0.30L - I Ahmet Ceyhun Uluğer 0.35L - I Ahmet Doğan Baygeldi 0.40L - I Ahmet Sadık Özbek 0.45L - I Ahmet Yapa 0.50L - I Ali Asker Tatar 0.55L - I Alper Can Özdemir 0.60L - I Alper Selçuk İmren 0.65L - I Anıl Acar 0.70L - I Aslı Dikicioğlu 0.75L - I Ayşen Çırakoğlu 0.80L - I Bahattin Berk Ayraçma 0.85L - I Barış Çelebi 0.20L - I Barış Önen 0.25L - I Barış Umut Çakıcı 0.15L - I Burak Akıllı 0.20L L II Caner Demir 0.25L L II Cansın Görkem Özyurt 0.30L L II Cihan Özgören 0.35L L II Çağdaş Çolak 0.40L L II Deniz Abdikoğlu 0.45L L II Derya Alparslan 0.50L L II Doğan Çevik 0.55L L II Ekin Şimşek 0.60L L II Ercan Çelik 0.65L L II Erkan Küçük 0.70L L II Erkan Lafcı 0.75L L II Faik Alper Kanbur 0.80L L II Fatih Yeşilyurt 0 0.50L II Gökalp Kara 0 0.55L II Gökçe Aras 0 0.65L II Hakkı Deniz Gül 0 0.45L II Halil Murat Turan 0 0.65L II Hüseyin Oğuz 0 0.75L II İsmail Taş 0 0.40L II Levent Keser 0 0.80L II Mehmet Emin Atak 0 0.35L II Melih Aytaççık 0 0.85L II Merve Usta 0 0.90L II Miraç Kara 0 L III Muhammed Çağrı Abdullah 0.10L L III Mustafa Hakan Şanal 0.20L L III Mustafa Yetim 0.25L L III Oğuzcan Kal 0.30L L III Ömer Faruk Gençay 0.35L L III Ragibe Ece Yükselen 0.40L L III Recep Uçar 0.45L L III Samet Güleryüz 0.50L L III Sedat Kömürcü 0.55L L III Serhan Özer 0.65L L III Serkan Kaplan 0.70L L III Süleyman Nazif Durmaz 0.75L L III Şükrü Kürşat Gökçay 0 0.50L III Şükrü Tercan 0 0.55L III Ünal Atalay 0 0.65L III Yemliha Yalçın 0 0.45L III Yiğit Uğur 0 0.40L III Yücel Erdem Atalay 0 0.35L III Özgün Özeren 0 0.70L III Ömer Faruk Halıcı 0 0.30L III

Due to: Final Exam of Partial Differential Equation(INS 214E)