Find Practical Resonance Frequencies in Linear Differential Equations

In summary, the equation for determining the practical resonance frequencies for a second order linear differential equation with damping is as follows: -mathbf{M}^{-1} \mathbf{K} x= \omega_n^2 x.
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Kostas1335
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I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations.
Hi all,

I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations.

First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order linear system of differential equations with damping, exhibits in its solution. In other words the practical resonance frequency approaches the natural frequency, if the damping on the system approaches zero.

Now with that clear, let's look a the math:

We have the following differential equation with its coefficients for mass ##\mathbf{M}## and stiffness ##\mathbf{K}## being ##n\times n## matrices:
$$ \mathbf{M} \ddot{x} + \mathbf{K} x = \mathbf{F_0} $$
From the above, we know that the system's natural frequency for every mass element in the mass matrix is the square root of the generalised eigenvalues:
$$-\mathbf{M}^{-1} \mathbf{K} x= \omega_n^2 x$$
where it is visible that the eigenvalues ##\lambda = \omega_n^2##. Consequently we solve to find ##\lambda## in the equation above, and take its square root to determine the natural frequencies.

My question lies on how does the above eigenvalue problem gets expressed in order to find the practical frequency this time, for the case where damping is taken into account. So the system of equations would now be:
$$ \mathbf{M} \ddot{x} + \mathbf{C} \dot{x} + \mathbf{K} x = \mathbf{F_0} $$
I know that in the case were we are not dealing with a system of equations, but with coefficients with one value only, we can find the practical frequency as:
$$\omega_r = \sqrt{ \frac{k}{m} - \frac{c^2}{2m^2} }$$
However I cannot seem to be able to understand how to generalise the above for the case for when I have a system of equations.

Your help on this one would be greatly appreciated. As you understand I am just now starting to learn about this equations.

Kind Regards,

Kostas.
 
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FAQ: Find Practical Resonance Frequencies in Linear Differential Equations

What is the purpose of finding practical resonance frequencies in linear differential equations?

The purpose of finding practical resonance frequencies in linear differential equations is to identify the specific frequencies at which a system will exhibit resonance, or a large amplitude response, when subjected to an external force. This information can be useful in understanding the behavior of various physical systems and can aid in designing and optimizing systems for specific applications.

How do you determine the practical resonance frequencies in a linear differential equation?

To determine the practical resonance frequencies in a linear differential equation, you must first solve the equation for its characteristic equation. Then, use the roots of the characteristic equation to find the natural frequencies of the system. The practical resonance frequencies can be calculated by adding or subtracting the damping ratio from the natural frequencies.

What is the significance of the damping ratio in determining practical resonance frequencies?

The damping ratio is a measure of the system's ability to dissipate energy, and it plays a crucial role in determining the practical resonance frequencies. A higher damping ratio will result in a lower amplitude response at resonance, while a lower damping ratio will result in a higher amplitude response. Therefore, the damping ratio helps to determine the stability and behavior of the system at resonance.

Can practical resonance frequencies be applied to real-world systems?

Yes, practical resonance frequencies can be applied to real-world systems. Many physical systems, such as bridges, buildings, and electronic circuits, exhibit resonance behavior and can be analyzed using linear differential equations. By finding the practical resonance frequencies, engineers and scientists can design and optimize these systems to avoid unwanted resonance effects and improve their overall performance.

What are some common applications of practical resonance frequencies in science and engineering?

Practical resonance frequencies have a wide range of applications in science and engineering. Some common examples include designing and optimizing mechanical systems, such as bridges and buildings, to withstand external forces and vibrations; analyzing and improving the performance of electronic circuits; and studying the behavior of biological systems, such as the vibrations of vocal cords in speech production. They are also used in fields such as acoustics, optics, and seismology to understand and manipulate the behavior of waves and oscillations.

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