Download - ME 563 Mechanical Vibrations Lecture #12
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ME 563 Mechanical Vibrations
Lecture #12 Multiple Degree of Freedom Free Response + MATLAB
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Free Response 1
We can solve for the homogeneous solution to a coupled set of equations in a multiple degree of freedom linear system by:
- Identifying the initial conditions on all the states - Assuming a solution of the form {x(t)}={A}est
What does this last assumption imply about the response?
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Two DOF System 2
Consider the two degree of freedom system of equations:
If we make a solution of the form, , as we did for the single DOF case, we obtain:
Non-trivial solutions satisfy:
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Ms2 + 2Cs+ 2K( ) ⋅ Ms2 + Cs+ K( ) − Cs+ K( )2 = 0
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Characteristic Equation 3
There are four solutions that satisfy the characteristic equation and these solutions are expressed as follows when the modal frequencies are underdamped:
The solution to the homogeneous equation is then written as follows (for the first two complex conjugate roots):
s1=σ1+jω1, s2=σ1-jω1, s3=σ2+jω2, and s4=σ2-jω2
Real part: decay rate Imaginary part: oscillation frequency
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Free Response Form 4
The free response is usually written in the following form for a multiple degree of freedom system:
Four constants Four initial conditions are required.
Modal vector (can be scaled)
Decaying co-sinusoid (common to both degrees of freedom)
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Example Solution 5
For K=1 N/m, C=0.1 Ns/m, and M=1 kg, the solution is given by:
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MATLAB Solution 6
To obtain solutions for the free response in MATLAB, the procedure explained before for a single DOF system is used:
Or
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MATLAB Solution 7
We use the following formulation to define the outputs, {y}, if we are only interested in the displacement variables:
Or:
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y{ }2×1 = C[ ]2×4 q{ }4×1 + D[ ]2×2 f (t){ }2×1
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Free Response (Eigen-Analysis) 8
We can also solve the homogeneous equations of motion by:
- Identifying the initial conditions on all the states - Identifying the modal frequencies, s, and vectors, {X}, using eigen-analysis.
Do you remember what an eigenvalue problem looks like?
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Eigenvalue Problem 9
Consider the case when we have no damping:
We can write this set of equations of motion as follows:
Now assume a solution of the form {x(t)}={A}est
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Eigenvalue Problem 10
Consider the case when we have no damping:
This last equation is the standard eigenvalue problem. Matlab solves this equation with the command “eig(A)”:
[v,d]= eig(A);
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Matlab eig 11
How do we interpret what Matlab gives us with this command?
is a diagonal matrix of -s1,22
is a matrix of modal vectors
How do we find the modal frequencies and vectors?
d v
d =
2.6180 0 0 0.3820
v =
-0.8507 -0.5257 0.5257 -0.8507
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Matlab eig 12
First, we take the negative square root of the diagonal entries of the d matrix providing the modal frequencies in rad/s:
Second, we can scale the columns of v to produce easy to interpret modal vectors:
√-d =
0+1.6180i 0 0 0+0.6181i
v =
1.0000 0.6180 -0.6180 1.0000
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Matlab Form of Response 13
The form of the free response can then be written in the form:
In summary, the eigenvalue problem for an undamped system is given by:
Where does the damping fit?
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Damped Eigenvalue Problem 14
To obtain solutions for the free response in a damped system, the state variable form of the equations of motion are used:
and then the eigenvalues and eigenvectors of the state matrix are calculated using eig. This approach works because the assumed solution {q}est is also used for the 1st order system:
[v,d]= eig(A);
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Matlab eig 15
How do we interpret what Matlab gives us with this command in the first order form of the state variable model?
is a diagonal matrix of modal frequencies s1,2 and s3,4 (1:2,1:4) is a matrix of modal vectors {X}1 and {X*}1 d v
diag(d) =
-0.1309 + 1.6127i -0.1309 - 1.6127i -0.0191 + 0.6177i -0.0191 - 0.6177i
s3,4
s1,2
{X}2 and {X*}2
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Matlab eig 15
Eigenvectors are not the modal vectors of the 2nd order system – eigenvectors involve both position and velocity states. We look to the position states to obtain the scaled modal vectors.
v = [ -0.0362 - 0.4457i -0.0362 + 0.4457i 0.0224 + 0.2755i 0.0224 - 0.2755i 0.7236 0.7236 -0.4472 + 0.0000i -0.4472 - 0.0000i
-0.4472 + 0.0000i -0.4472 - 0.0000i -0.7236 -0.7236 0.0085 - 0.2763i 0.0085 + 0.2763i 0.0138 - 0.4470i 0.0138 + 0.4470i
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Damping in MDOF Systems 16
Note that for the previous example, the damping matrix [C] corresponded to a proportionally viscously damped system:
For this type of damping, note that the modal vectors were entirely real, i.e., the phase between DOFs was either 0 or 180o.
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Damping in MDOF Systems 16
What if the system is not proportionally damped? The modal vectors in this case are complex but the response is not:
For this type of damping, note that the modal vectors are complex, so the phase between the DOFs is not 0 or 180o.
What does this mean?
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Damping in MDOF Systems 17
What if the system is not proportionally damped? The modal vectors in this case are complex but the response is not:
Real
Imag
Mode #2
Mode #1
2nd order EOMS cannot be uncoupled using real modal vectors!