Damping conditions for multi degrees of freedom

In summary, the equations for damped vibration modes in a system with one mass and one spring and damper are d^2<4mk for underdamped, d^2>4mk for overdamped, and d^2=4mk for critically damped. However, if the system has two masses, three springs, and three dampers, the equations of motion are more complex and depend on the configuration of the system. In the most general case, the damped vibration modes are the eigenvalues and vectors of a quadratic eigenproblem. The solutions are of the form x(t) = X e^(-σ + iω)t, where X is a complex eigenvector, -σ + iω is the complex
  • #1
gopi9
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0
I know that d^2<4mk for underdamped, d^2>4mk for overdamped and d^2=4mk for critically damped. This is true if there is only 1 mass and spring and damper. How to use these equations if I have 2 mass, 3 spring and 3 dampers. That is d,m,k are in 2x2 matrices. Please some one help me with this.
 
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  • #2
You should specify the configuration exactly and then we can write the equations of motion!
 
  • #3
In the most general case, the damped vibration modes of the system are the eigenvalues and vectors of the quadratic eigenproblem ##(m + \lambda d + \lambda^2k)x = 0##. The solutions will be of the form ##x(t) = X e^{(-\sigma + i \omega) t} ## where ##X## is a complex eigenvector, ##-\sigma + i \omega## is the complex eigenvalue, and the physical solution ##x(t)## is the real part of the right hand side of the equation. ##\sigma## represents the amount of damping, and ##\omega## the damped oscillation frequency (which is 0 for an overdamped system)

If you get lucky, the eigenvectors will be real, and the same as the eigenvectors of the undamped system ##(m - \omega_u^2 k)x = 0## (but of course the eigenvalues will be different, and the undamped frequency ##\omega_u## is not equal to the damped frequency ##\omega##).
 

1. What are damping conditions for multi degrees of freedom systems?

Damping conditions for multi degrees of freedom systems refer to the characteristics of the damping forces that are applied to a system with multiple independent degrees of freedom. These conditions help to control the motion and stability of the system by dissipating energy and reducing vibrations.

2. Why are damping conditions important in multi degrees of freedom systems?

Damping conditions are important in multi degrees of freedom systems because they help to prevent excessive vibrations and oscillations, which can lead to instability and damage to the system. They also improve the overall performance and accuracy of the system.

3. How are damping conditions determined for multi degrees of freedom systems?

Damping conditions for multi degrees of freedom systems are typically determined through mathematical models and simulations. These models take into account factors such as the stiffness of the system, the mass of the components, and the type of damping forces being applied.

4. What are the types of damping forces used in multi degrees of freedom systems?

The most common types of damping forces used in multi degrees of freedom systems are viscous damping, Coulomb damping, and Hysteresis damping. Viscous damping is based on the velocity of the system, Coulomb damping is based on the displacement of the system, and Hysteresis damping is based on the energy dissipation within the system.

5. How can damping conditions be optimized for multi degrees of freedom systems?

Damping conditions for multi degrees of freedom systems can be optimized by adjusting the damping coefficients and tuning the damping forces to match the natural frequencies of the system. This can be done through experimentation and analysis to find the most effective damping conditions for a specific system.

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