One very important application of the eigenvalue problem is in the theory of vibra-tions. Consider the two-degrees-of-freedom problem shown in Figure 3.8.
The governing differential equations describing the motion of the two masses are m x1 1�� =k x2( 2-x1)-k x1 1 (3.11) m x2 2�� = -k x2( 2-x1)-k x3 2 (3.12) We wish to determine the modes of oscillation such that each mass undergoes har-monic motion at the same frequency. To obtain such a solution, set
x1=A1exp(i tw ) (3.13)
x2=A2exp(i tw ) (3.14)
Substituting Equations (3.13) and (3.14) into Equations (3.11) and (3.12) gives k k
Equations (3.15) and (3.16) are two homogeneous linear algebraic equations in two unknowns. There is a theorem in linear algebra that says that the only way for two
x1
Figure 3.8 Two-degrees-of-freedom vibration system.
homogeneous linear algebraic equations in two unknowns to have a nontrivial solu-tion is for the determinant of the coefficient matrix to be zero.
k k
The equation for l becomes
l2-(a11+a22)l+(a a11 22-a a12 21)=0 (3.17) The solution of Equation (3.17) gives the eigenvalues, l1and l2, which are the square of the two natural frequency of oscillations for this system. The ratio of the amplitudes of the oscillation of the two masses can be obtained by substituting the values of l into Equation (3.15) or Equation (3.16); that is,
A
The eigenvector, V1, associated with l1 is A
and the eigenvector, V2, associated with l2 is A
Since A1 is arbitrary, we can select A1 = 1, then
V1 a11 1 a12
= 1-
-
( l )/ (3.18)
and
V2 a11 2 a12
= 1-
-
( l )/ (3.19)
If [1] V is an eigenvector of a matrix a corresponding to an eigenvalue l, so is bV with any b ≠0.
MATLAB’s eig Function
MATLAB has a built-in function that gives the eigenvalues of a square matrix X.
MATLAB’s description of the function follows:
E = eig(X) is a vector containing the eigenvalues of a square matrix X.
[V,D] = eig(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D.
For the problem under discussion, matrix a replaces matrix X. Thus, the statement [V, D] = eig(a) gives the eigenvectors associated with l1 and l2. V(:,1) is associated with l1 and V(:,2) is associated l2.
Example 3.8
Suppose in Figure 3.8 the following parameters were given:
m1=m2=1500kg k, 1=3250N m k/ , 2=3500N m k/ , 3=3000N m/ A program that will determine
1. The eigenvalues of the system by both Equation (3.17) and by MATLAB’s eig function
2. The eigenfunctions by both Equations (3.18) and (3.19) and MATLAB’s [V,D] function
follows.
Example 3.9
% eigen2.m
% Eigenvalues and eigenvectors.
% E = eig(a) is a vector containing the eigenvalues of a square matrix a.
% [V,D] = eig(a) produces a diagonal matrix D of eigenvalues and a
% full matrix V whose columns are the corresponding eigenvectors so
% that a*V = V*D.
% Units are in SI units.
clear; clc;
k1=3250; k2=3500; k3=3000; m1=1500; m2=1500;
a(1,1)=(k1+k2)/m1;
a(1,2)=-k2/m1;
a(2,1)=-k2/m2;
a(2,2)=(k2+k3)/m2;
% Lamda^2-(a(1,1)+a(2,2))*Lamda+(a(1,1)*a(2,2)-a(1,2)*a(2,1))=0 b=-(a(1,1)+a(2,2)); c=a(1,1)*a(2,2)-a(1,2)*a(2,1);
Lamda1=(-b-sqrt(b^2-4*c))/2;
Lamda2=(-b+sqrt(b^2-4*c))/2;
E=eig(a);
fprintf(‘Lamda1=%7.5f Lamda2=%7.5f \n’, Lamda1,Lamda2) fprintf(‘E(1)= %7.5f E(2)=%7.5f \n’,E(1),E(2));
v1=[1;-(a(1,1)-Lamda1)/a(1,2)]
v2=[1;-(a(1,1)-Lamda2)/a(1,2)]
[V,D] = eig(a);
V1=V(:,1) V2=V(:,2) D
The following results were obtained:
Lamda1=2.08185 Lamda2=6.75149 E(1)=2.08185 E(2)=6.75149 v1 =
1.0000 1.0364 v2 =
1.0000 –0.9649 V1 =
–0.6944 –0.7196 V2 =
–0.7196 0.6944 D =
2.0818 0
0 6.7515
Examining the results, it can be seen that Lamda1 = E(1) and Lamda2 = E(2). Also, V1 is a scalar multiple of v1 and V2 is a scalar multiple of v2.
Exercises
Exercise 3.1
Use pencil and paper and the Gauss Elimination method to solve the following system of equations:
a. 2x1 +3x2 – x3 = 20 4x1 – x2 + 3x3 = –14 x1 + 5x2 + x3 = 21 b. 4x1 + 8x2 + x3 = 8
–2x1 – 3x2 + 2x3 = 14 x1 + 3x2 +4x3 = 30 c. 2x1 + x2 + x3 – 11x4 = 1
5x1 – 2x2 + 5x3 – 4x4 = 5 x1 – x2 + 3x3 – 3x4 = 3 3x1 + 4x2 – 7x3 + 2x4 = – 7
Projects
Project 3.1
For the following truss structure (see Figure P3.1) write a MATLAB program that will determine the internal forces in the structural members by the method described in Example 3.3. Print out the reactions, the coefficient matrix, the mem-bers’ internal forces, and a check on the solution.
B D F
A
C E G I K
12'
L 6 bays at 9 ft each
4kN
4kN 2kN
2kN 4kN 6kN
F2 F4
F1
F3
F8 F12
F9 F11
F5
F7
F16 F20
F17 F19 F21 F15
F10 F14 H J
F6 F18
Figure P3.1 Truss structure for Project 3.1.
Project 3.2
For the truss structure shown in Figure P3.2 write a MATLAB program that will determine the internal forces in the structural members by the method described in Example 3.3. Print out the reactions, the coefficient matrix, the members’ internal forces, and a check on the solution.
Project 3.3
An automobile suspension system is simulated by two springs connected by a bar supporting the automobile’s weight as shown in Figure P3.3a. We shall assume that the ends of the spring are at the same elevation and X is measured downward from this position (Figure P3.2b). The automobile weight is applied on the bar at point G. The equilibrium position of the bar is shown in Figure P3.3c. The governing equations at the equilibrium position follow:
Fx i k X L k X L W
If the system is disturbed and released, it will vibrate at its natural frequencies. The system has two degrees of freedom, resulting in a vertical and a rotational vibration.
B
Figure P3.2 Truss structure for Project 3.2.
Let x be measured from the equilibrium position, then X = X0 + x and �� ��X =x
The governing equations describing the vibrating system follow:
M x�� = -k X1[ 0+ -x L1(ϑ0+ϑ)]-k X2[ 0+ +x L2(ϑ0+ϑ)]+W (P3.3c) Iz z��ϑ=k X1[ 0+ -x L1(ϑ0+ϑ)]L k X1- 2[ 0+ +x L2(ϑ0+ϑ)]L2
(P3.3d) Mx�� = -k X1( 0-L1 0ϑ )-k X2( 0-L2 0ϑ )+W k x L- 1( - 1ϑ)-k x2( +LL2ϑ) (P3.3e)
Unstretched position
(d) (c) (b) Gφ (a)
X
Unstretched position Equilibrium position X0
θ0
θ0 θ Unstretched position
Equilibrium position x x
x = x0 + x
X0 l1
k1 k2
l2
Figure P3.3 Automobile suspension system.
By Equation (P3.3a), the sum of the first three terms on the right-hand side of Equation (P3.3e) is zero. Thus,
M x�� = -k x L1( - 1ϑ)-k x L2( + 2ϑ) (P3.3f) Similarly, Equation (P3.3d) can be rewritten as
Iz z��ϑ=k X1( 0-L1 0ϑ )L k X1- 2( 0+L2 0ϑ )L2+k x L L1( - 1ϑ) 1-kk x L2( + ϑ2 )L2 (P3.3g)
By Equation (P3.3b), the sum of the first two terms on the right-hand side of Equation (P3.3g) is zero. Thus,
Izz��ϑ=k x L L k x L1( - 1ϑ) 1- 2( + 2ϑ) L2 (P3.3h) We wish to determine the modes of oscillation such that the vertical and rotational vibrations are at the same frequency. To obtain such a solution, set
x A= exp(i tw ) (P3.3i)
ϑ= Bexp(i tw ) (P3.3j)
Substituting Equations (P3.3i) and (P3.3j) into Equations (P3.3f) and (P3.3h), respectively, gives
k k
M A k L
M k L
M B
1+ 2- 2 2 2 1 1 0
-
-
=
w (P3.3k)
k L k L
I A k L k L
I B
z z z z
2 2 1 1 2 22
1 12
2 0
-
+ +
-
=
w (P3.3l)
Using MATLAB’s eig function, determine the natural frequencies of oscillation for the system. Use the following variable values:
k1 = 35 kN/m, k2 = 38 kN/m, L1 = 1.4 m, L2 = 1.7 m, M = 1500 kg, Iz z = 2170 kg-m2.
Project 3.4
Suppose a manufacturer wishes to purchase a piece of equipment that costs
$40,000. He plans to borrow the money from a bank and pay off the loan in 10 years in 120 equal payments. The annual interest rate is 6%. Each month the
interest charged will be on the unpaid balance of the loan. He wishes to determine what his monthly payment will be. This problem can be solved by a system of linear equations.
Let xj = the amount in the jth payment that goes toward paying off the princi-pal. Then the equation describing the jth payment is
jth payment M xj P x In
n n j
= = +
-
=
∑
= -11
(P3.4a)
where
M = the monthly payment.
P = the amount borrowed.
I = the monthly interest rate = annual interest rate/12.
The total number of unknowns is 121 (120 x values and M).
Applying Equation (P3.4a) to each month gives 120 equations. One additional equation is
P xn
n n
=
=
∑
= 1 120(P3.4b)
Develop a computer program that will
1. Ask the user to enter from the keyboard the amount of the loan (P), the annual interest rate, I, and the time period, Y, in years.
2. Set up the system of linear equations, using An,m as the coefficient matrix of the system of linear equations. The n represents the equation number and m represents the coefficient of xm in that equation. Set x121 = M.
3. Solve the system of linear equations in MATLAB.
4. Print out a table consisting of four columns.The first column should be the month number, the second column the monthly payment, the third column the amount of the monthly payment that goes toward paying off the princi-pal, and the fourth column the interest payment for that month.
Reference
1. Kreyszig, E., Advanced Engineering Mathematics, 8th Ed., John Wiley & Sons, 1999.
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