Spring constant matrix and normal modes (4 springs and 3 masses)

In summary, the conversation discusses the process of finding the normal modes of a system using direct matrix methods. The system consists of springs with stiffness k1, k2, k3, k4 and block masses m1, m2, m3. The goal is to find the real angular frequencies by solving for the determinant of a matrix. However, the method used results in complex angular frequencies, leading to the question of what went wrong. The conversation ends with the suggestion to try the same method on a simpler system and to note any discrepancies in the signs.
  • #1
LCSphysicist
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Homework Statement
ALl below
Relevant Equations
ALl below
We need to find the normal modes of this system:
1596888481855.png

Well, this system is a little easy to deal when we put it in a system and solve the system... That's not what i want to do, i want to try my direct matrix methods.

We have springs with stiffness k1,k2,k3,k4 respectively, and block mass m1, m2, m3
And we need to annul the determinant of
1596888665443.png
, not just, i supposed the displacement Z as complex, so we need real angular frequencies.

To mount the matrix K, we need to know all Kij, being Kij the spring stiffness equivalent if we move just block j and analyze the motion of block i, j can be equal i.

So making all the things, we end with:

1596889004699.png


Apparently this is wrong, i found just complex w, what is the problem?
 

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  • #2
Try the same method on a much simpler arrangement, one you can easily check from first principles. Note where the signs disagree.
 
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  • #3
haruspex said:
Try the same method on a much simpler arrangement, one you can easily check from first principles. Note where the signs disagree.
Yeh that's enough... thank
 

Related to Spring constant matrix and normal modes (4 springs and 3 masses)

1. What is a spring constant matrix?

A spring constant matrix is a mathematical representation of the relationship between multiple springs and masses in a system. It is used to calculate the forces and displacements of the system based on the stiffness of each spring and the mass of each object.

2. How is a spring constant matrix calculated?

A spring constant matrix is calculated by arranging the stiffness coefficients of each spring and the mass of each object into a matrix, and then using mathematical operations to determine the relationships between the forces and displacements in the system.

3. What are normal modes in a system with 4 springs and 3 masses?

Normal modes in a system with 4 springs and 3 masses refer to the different patterns of motion that the system can exhibit. These modes are determined by the stiffness of the springs and the mass of the objects, and can be calculated using the spring constant matrix.

4. How does changing the spring constants affect the normal modes in a system?

Changing the spring constants in a system can affect the normal modes by altering the stiffness of the springs and therefore changing the patterns of motion that the system can exhibit. This can result in different frequencies and amplitudes of oscillation for the system.

5. Can a spring constant matrix be used to analyze systems with more than 4 springs and 3 masses?

Yes, a spring constant matrix can be used to analyze systems with any number of springs and masses. However, the calculations become more complex as the number of components in the system increases.

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