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



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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



and initial 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)



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