Finding eigenvalues for 4 DOF system

In summary, the conversation is about finding the eigenvalues of a fourth order dynamic system and reducing it to a first order system using auxiliary variables. The focus is on finding or qualifying the eigenvalues of the resulting matrix.
  • #1
Sirsh
267
10
Hey all,

I've derived a fourth order dynamic system as represented by the following:
Capture.jpg

I need to determine the eigenvalues for this system to check whether they're purely real with no imaginary components.

How should I go about doing this? I have done eigenvalue problems in the past, but not to this extent. Would I just determine the eigenvalues for each matrix then do the addition of them?

Note: all the variables (excluding theta's and x's) have constant values.

Any help would be appreciated, thanks!
 
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  • #2
Here's how I would approach it, first introduce auxiliary variables and reduce the system to first order. For example replace [itex] \ddot{\theta}_2 \to \dot{\omega}_2[/itex] and [itex]\dot{\theta}_2 \to \omega[/itex] with the additional equation [itex]\dot{\theta}_2 = \omega_2[/itex]. You'll end up with an 8x8 system but it will be first order, of the form:

[tex] \dot{\mathbf{u}} = A \mathbf{u} + \mathbf{b}[/tex]
With [itex] \mathbf{u} = ( \theta_2,\theta_3, x_2, x_5, \omega_2,\omega_3, v_2,v_5)^T[/itex].

You can then focus on finding or qualifying the eigen-values of the matrix [itex]A[/itex].
 
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Likes Sirsh
  • #3
jambaugh said:
Here's how I would approach it, first introduce auxiliary variables and reduce the system to first order. For example replace [itex] \ddot{\theta}_2 \to \dot{\omega}_2[/itex] and [itex]\dot{\theta}_2 \to \omega[/itex] with the additional equation [itex]\dot{\theta}_2 = \omega_2[/itex]. You'll end up with an 8x8 system but it will be first order, of the form:

[tex] \dot{\mathbf{u}} = A \mathbf{u} + \mathbf{b}[/tex]
With [itex] \mathbf{u} = ( \theta_2,\theta_3, x_2, x_5, \omega_2,\omega_3, v_2,v_5)^T[/itex].

You can then focus on finding or qualifying the eigen-values of the matrix [itex]A[/itex].

Thank you for your help Jambaugh!
 

1. What is the purpose of finding eigenvalues for a 4 DOF system?

The eigenvalues of a 4 degree of freedom (DOF) system represent the natural frequencies of the system. They are important in understanding the dynamic behavior of the system and can help to determine the stability of the system.

2. How do you find the eigenvalues for a 4 DOF system?

The eigenvalues of a 4 DOF system can be found by solving the characteristic equation of the system, which is obtained by equating the determinant of the system's equations of motion to zero. This can be done using various methods such as the direct numerical method or the modal analysis method.

3. What factors can affect the eigenvalues of a 4 DOF system?

The eigenvalues of a 4 DOF system can be affected by factors such as the mass and stiffness properties of the system, as well as any external forces or damping present. Changes in these factors can alter the natural frequencies and stability of the system.

4. How do the eigenvalues of a 4 DOF system relate to its modes of vibration?

The eigenvalues of a 4 DOF system are directly related to its modes of vibration. Each eigenvalue represents a unique natural frequency of the system and corresponds to a specific mode of vibration. The mode shapes can also be determined from the eigenvectors associated with the eigenvalues.

5. Why is it important to consider eigenvalues when analyzing a 4 DOF system?

Considering the eigenvalues of a 4 DOF system is crucial in understanding its dynamic behavior and stability. By knowing the natural frequencies, one can predict the response of the system to different excitations and make necessary adjustments to ensure its stability and performance.

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