1. Chapter 4 Multiple Degree of Freedom Systems
The Millennium bridge required
many degrees of freedom to model
and design with.
Extending the first 3 chapters to more then one degree of freedom
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The first step in analyzing multiple degrees of freedom (DOF) is to look at 2 DOF
DOF: Minimum number of coordinates to specify the position of a system
Many systems have more than 1 DOF
Examples of 2 DOF systems
car with sprung and unsprung mass (both heave)
elastic pendulum (radial and angular)
motions of a ship (roll and pitch)
Fig 4.1
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3. 4.1 Two-Degree-of-Freedom Model (Undamped)
A 2 degree of freedom system used to base
much of the analysis and conceptual
development of MDOF systems on.
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4. Free-Body Diagram of each mass
Figure 4.2
k2(x2 -x1)
k1 x1 m2 m1 x1 x2
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5. Summing forces yields the equations of motion:
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6. Note that it is always the case that
A 2 Degree-of-Freedom system has
Two equations of motion!
Two natural frequencies (as we shall see)!
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The dynamics of a 2 DOF system consists of 2 homogeneous and coupled equations
Free vibrations, so homogeneous eqs.
Equations are coupled:
Both have x1 and x2.
If only one mass moves, the other follows
Example: pitch and heave of a car model
In this case the coupling is due to k2.
Mathematically and Physically
If k2 = 0, no coupling occurs and can be solved as two independent SDOF systems
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8. Mechanical Engineering at Virginia Tech
Initial Conditions
Two coupled, second -order, ordinary differential equations with constant coefficients
Needs 4 constants of integration to solve
Thus 4 initial conditions on positions and velocities
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9. Solution by Matrix Methods
The two equations can be written in the form of a
single matrix equation (see pages if matrices and vectors are a struggle for you) :
(4.4), (4.5)
(4.6), (4.9)
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Initial Conditions
IC’s can also be written in vector form
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11. The approach to a Solution:
For 1DOF we assumed the scalar solution aelt Similarly, now we assume the vector form:
(4.15)
(4.16)
(4.17)
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This changes the differential equation of motion into algebraic vector equation:
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The condition for solution of this matrix equation requires that the the matrix inverse does not exist:
The determinant results in 1 equation
in one unknown w(called the characteristic equation)
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14. Back to our specific system: the characteristic equation is defined as
(4.20)
(4.21)
Eq. (4.21) is quadratic in w2 so four solutions result:
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Once  is known, use equation (4.17) again to calculate the corresponding vectors u1 and u2
This yields vector equation for each squared frequency:
Each of these matrix equations represents 2 equations in the 2 unknowns components of the vector, but the coefficient matrix is singular so each matrix equation results in only 1 independent equation. The following examples clarify this.
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16. Examples 4.1.5 & 4.1.6:calculating u and 
m1=9 kg,m2=1kg, k1=24 N/m and k2=3 N/m
The characteristic equation becomes w4-6w2+8=(w2-2)(w2-4)=0 w2 = 2 and w2 =4 or
Each value of w2 yields an expression or u:
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17. Computing the vectors u
2 equations, 2 unknowns but DEPENDENT!
(the 2nd equation is -3 times the first)
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Only the direction of vectors u can be determined, not the magnitude as it remains arbitrary
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19. Likewise for the second value of w2:
Note that the other equation is the same
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20. What to do about the magnitude!
Several possibilities, here we just fix one element:
Choose:
Choose:
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21. Thus the solution to the algebraic matrix equation is:
Here we have introduce the name
mode shape to describe the vectors
u1 and u2. The origin of this name comes later
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22. Return now to the time response:
We have computed four solutions:
(4.24)
Since linear, we can combine as:
(4.26)
Note that to go from the exponential
form to to sine requires Euler’s formula
for trig functions and uses up the
+/- sign on omega
determined by initial conditions.
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23. Physical interpretation of all that math!
Each of the TWO masses is oscillating at TWO natural frequencies w1 and w2
The relative magnitude of each sine term, and hence of the magnitude of oscillation of m1 and m2 is determined by the value of A1u1 and A2u2
The vectors u1 and u2 are called mode shapes because the describe the relative magnitude of oscillation between the two masses
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What is a mode shape?
First note that A1, A2, f1 and f2 are determined by the initial conditions
Choose them so that A2 = f1 = f2 =0
Then:
Thus each mass oscillates at (one) frequency w1 with magnitudes proportional to u1 the 1st mode shape
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25. A graphic look at mode shapes:
If IC’s correspond to mode 1 or 2, then the response is purely in mode 1 or mode 2. x1 x2 k1 k2 m1 m2
Mode 1:
x2=A
x1=A/3 x1 x2 k1 k2 m1 m2
Mode 2:
x2=A
x1=-A/3
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26. Example 4.1.7 given the initial conditions compute the time response
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At t=0 we have
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4 equations in 4 unknowns:
Yields:
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The final solution is:
(4.34)
These initial conditions gives a response that is a combination of modes. Both harmonic, but their summation is not.
Figure 4.3
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30. Solution as a sum of modes
Determines how the second
frequency contributes to the
response
Determines how the first
frequency contributes to the
response
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31. Mechanical Engineering at Virginia Tech
Things to note
Two degrees of freedom implies two natural frequencies
Each mass oscillates at with these two frequencies present in the response and beats could result
Frequencies are not those of two component systems
The above is not the most efficient way to calculate frequencies as the following describes
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32. Some matrix and vector reminders
Then M is said to be positive definite
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33. 4.2 Eigenvalues and Natural Frequencies
Can connect the vibration problem with the algebraic eigenvalue problem developed in math
This will give us some powerful computational skills
And some powerful theory
All the codes have eigen-solvers so these painful calculations can be automated
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34. Some matrix results to help us use available computational tools:
A matrix M is defined to be symmetric if
A symmetric matrix M is positive definite if
A symmetric positive definite matrix M can be factored
Here L is upper triangular, called a Cholesky matrix
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35. If the matrix L is diagonal, it defines the matrix square root
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A change of coordinates is introduced to capitalize on existing mathematics
For a symmetric, positive definite matrix M:
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How the vibration problem relates to the real symmetric eigenvalue problem
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38. Important Properties of the n x n Real Symmetric Eigenvalue Problem
There are n eigenvalues and they are all real valued
There are n eigenvectors and they are all real valued
The set of eigenvectors are orthogonal
The set of eigenvectors are linearly independent
The matrix is similar to a diagonal matrix
Window 4.1 page 285
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39. Mechanical Engineering at Virginia Tech
Square Matrix Review
Let aik be the ikth element of A then A is symmetric if aik = aki denoted AT=A
A is positive definite if xTAx > 0 for all nonzero x (also implies each li > 0)
The stiffness matrix is usually symmetric and positive semi definite (could have a zero eigenvalue)
The mass matrix is positive definite and symmetric (and so far, its diagonal)
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40. Normal and orthogonal vectors
(4.43)
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41. Normalizing any vector can be done by dividing it by its norm:
(4.44)
To see this compute
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Examples through 4.2.4
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The first normalized eigenvector
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Likewise the second normalized eigenvector is computed and shown to be orthogonal to the first, so that the set is orthonormal
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45. Modes u and Eigenvectors v are different but related:
(4.37)
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46. This orthonormal set of vectors is used to form an Orthogonal Matrix
called a matrix of eigenvectors (normalized)
P is called an orthogonal matrix
(4.47)
P is also called a modal matrix
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47. Example 4.2.3 compute P and show that it is an orthogonal matrix
From the previous example:
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48. Mechanical Engineering at Virginia Tech
Example Compute the square of the frequencies by matrix manipulation
In general:
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Example 4.2.5
The equations of motion:
Figure 4.4
In matrix form these become:
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50. Next substitute numerical values and compute P and 
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51. Next compute the eigenvectors
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52. Next check the value of P to see if it behaves as its suppose to:
Yes!
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A note on eigenvectors
Does this make any difference?
No! Try it in the previous example
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All of the previous examples can and should be solved by “hand” to learn the methods
However, they can also be solved on
calculators with matrix functions and
with the codes listed in the last section
In fact, for more then two DOF one must
use a code to solve for the natural
frequencies and mode shapes.
Next we examine 3 other formulations for
solving for modal data
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55. Mechanical Engineering at Virginia Tech
Matlab commands
To compute the inverse of the square matrix A: inv(A) or use A\eye(n) where n is the size of the matrix
[P,D]=eig(A) computes the eigenvalues and normalized eigenvectors (watch the order). Stores them in the eigenvector matrix P and the diagonal matrix D (D=L)
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More commands
To compute the matrix square root use sqrtm(A)
To compute the Cholesky factor: L= chol(M)
To compute the norm: norm(x)
To compute the determinant det(A)
To enter a matrix: K=[27 -3;-3 3]; M=[9 0;0 1];
To multiply: K*inv(chol(M))
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57. An alternate approach to normalizing mode shapes
From equation (4.17)
(4.53)
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58. There are 3 approaches to computing mode shapes and frequencies
Is the Generalized Symmetric Eigenvalue Problem
easy for hand computations, inefficient for computers
(ii) Is the Asymmetric Eigenvalue Problem
very expensive computationally
(iii) Is the Symmetric Eigenvalue Problem
the cheapest computationally
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