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Problems Problem :1 Equation of the lateral vibration of a 100 story building.

Consider the lateral vibration of a 100 story building. The mass are assumed to be concentrated at the floors. The walls constitutes springs. This can be models as a 1-D system. (See figure) 100th floor 99th floor 98th floor

M100 K100

X100 K100

X99

M99 K99

K99

X98

M98

2nd floor

M2 K2

X2 K2

M1

1st floor K1

X1 K1

i) Derive the equation of motion of 1st floor. ii) Derive the equation of motion of 100th floor. iii) Derive the generalized equation of motion of nth floor ( 2  n  99 )

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Problem 2 : Generating code to calculate vibration modes of a 100 story building In this problem, your task is to generate MATLAB code to calculate vibration mode of a 100 story building. In order to simplify the problem, it is assumed that M1  M 2  "  M 99  M100  3kg and k1  k2  "  k99  k100  8N / m are given. ‘Building_your_kerberos_name’ should be function name (as well as m-file name) and upload it to 2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments. Function has following structure. function v= Building_your_kerberos_name(nvm) mode number you want to generate

nvm: vibration

v: eigen vector for nvm-th

vibration mode (Hint: Define the mass and the stiffness matrices first, and then solve eigenvalue problem.)

Problem 3 :

Plotting some vibration modes of a 100 story building

With m-code you generated in problem 9.2, print a single figure for each sub-question where x axis represents the eigenvector value, and y axis represents the building floor. Use different line types for different vibration modes. Legend for the vibration modes should be provided. i) The first four vibration modes ii) The 25th, 50th, 75th, and 100th vibration modes

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Solutions Problem 1 : i)

Equation of the lateral vibration of a 100 story building.

K2 (X1 - X2 )

K2 (X1 - X2 ) M1

K 1X1

=

2K 2  X 1  X 2   2K1 X 1  M1 X˙

M1

K 1X1

M 1X1˙

1

M1 X˙ 1  2K 2  X1  X 2   2K1 X1  0 ii)

K100 (X100  X99 )

=

M100 K100 (X100  X99 )

M100

M 100 100˙ X

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2K100  X100  X 99   M100 X˙ 00  2K 100  X100  99X M 100X100˙

1  0

iii) Derive the generalized equation of motion of nth floor ( 2  n  99 ) Kn+1(Xn - Xn+1)

Kn+1(Xn - Xn+1)

Mn Kn (Xn  Xn-1)

Kn-1(Xn  Xn-1)

2K n1

=

 Xn

 X n1

M n X˙n  2K n1

Mn

  2K n  X n

 Xn

 X n1

 X n1

  M n X ˙n

  2K n  X n

 X n1

 0

Problem 2 : Generating code to calculate vibration modes of a 100 story building

First, the mass matrix can be calculated, based on equations in problem 9.1.

Next, stiffness matrix can be also obtained, based on equations in problem 9.1.

M nXn˙

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The above equation is represented as the form of eigen values problem. Now, we can applied MATLAB function eig to it. The m-codes are shown as below.

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function v=building(nvm) % function v=building(nvm) %

nvm: vibration mode number you want to generate

%

v: eigen vector for nvm-th vibration mode

% Define several parameters

m=300; %

Mass for each floor k=800; % Spring constant for each wall % Number of floors in the building % Initialize mass and stiffness matrix M=eye(n); K=zeros(n); % Generating mass matrix

M=m*M;

% Genrating stiffness matrix % [2k_1+2k_2 -2k_2]

K(1,1:2)=2*k*[1

0]+2*k*[1 -1]; for i=2:n-1 % [2k_n-1 2k_n-1+2k_n -2k_n]

n=100;

Matlab Assignment Help K(i,i-1:i+1)=2*k*[-1 1 0]+2*k*[0 1 -1]; end % [-2k_99 2k_100] K(n,n-1:n)=2*k*[-1 0]+2*k*[0 1]; % Obtain eigen value and eigen vector matrix [V,D]=eig(K,M); % Extract eigen vector for nvm-th vibration mode v=V(:,nvm); end Problem 3 : Plotting some vibration modes of a 100 story building i) Following in-line commands generate plots for the first 4 fundamental modes. >> h=plot(building(1),1:100,'r-',building(2),1:100,'b--', … building(3),1:100,'g-.',building(4),1:100,'m:'); >> axis tight; grid on; >> set(h,'LineWidth',5) >> set(gca,'FontSize',16); >> xlabel('\bfNormalized Amplitude (a.u.)'); >> ylabel('\bfFloor Number of the Building'); >> title('\bfFirst 4 fundamental mode of 100 story buliding') >> legend('\bf1st mode','\bf2nd mode','\bf3rd mode','\bf4th mode');

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First 4 fundamental mode of 100 story buliding 100

Floor Number of the Building

90

80

70

60

50

40

30 -8

-6

20

-4

-2

0

2

Normalized Amplitude (a.u.)

4

6

8 x 10

-3

10

ii)

Following in-line commands generate plots for 25th, 50th, 70th, and 100th fundamental modes.

1st mode 2nd mode 3rd mode 4th mode

>> h=plot(building(25),1:100,'r-',building(50),1:100,'b--', … building(75),1:100,'g-.',building(100),1:100,'m:'); >> axis tight; grid on; >> set(h,'LineWidth',2) >> set(gca,'FontSize',16); >> xlabel('\bfNormalized Amplitude (a.u.)'); >> ylabel('\bfFloor Number of the Building'); >> title('\bfSelected 4 different modes of 100 story buliding') >> legend('\bf25th mode','\bf50th mode','\bf75th mode','\bf100th mode');

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Seleted 4 different modes of 100 story buliding

90

25th mode 50th mode 75th mode 100th mode

Floor Number of the Building

80

70

60

50

40

30

20

10 -8

-6

-4

-2

0

2

Normalized Amplitude (a.u.)

4

6

8 x 10

-3