MDOF Vibrations Modal Coordinates

[JohnSimpson]

suppose you have a system

Ma + Kx = 0 for some nxn matricies M and K

assuming a harmonic solution of the form a = -w^2*x will, in principle, allow calculation of the natural frequencies based on the characteristic equation, and calculation of the corresponding mode shapes.

It's my understanding that to solve for the actual response of the system x(t) to some initial conditions, you have three options

1. Transfer Functions, which I don't really like frankly
2. State space x_dot = A*x, which is solvable by matrix exponentiation
3. "Modal coordinates", where the response of the system is written as a linear combination of the mode shapes

Could someone explain modal coordinates in detail? I can't find any decent online references that do a good job of explaining the concept and how you follow through with it and solve for x(t)

 

[Valerie Park]

given initial conditions. Modal coordinates is a method used to solve for the response of a system to some initial conditions. It is based on the idea of expressing the response as a linear combination of the system's mode shapes. In other words, the response is modeled as the sum of the contributions from each of the system's modes. To do this, we first need to solve for the system's mode shapes, which are the solutions of the characteristic equation (M + Kx = 0). Once we have the mode shapes, we can then write the response as a linear combination of them. The coefficients of this linear combination will be determined by the initial conditions of the system. So, for example, if our system has two modes, the response x(t) can be written as:x(t) = c1*x1(t) + c2*x2(t)where c1 and c2 are the coefficients determined by the initial conditions, and x1(t) and x2(t) are the two mode shapes. Then, to solve for the response of the system, all we need to do is solve for the coefficients c1 and c2 using the initial conditions. This can be done by solving a set of linear equations, or by using the inverse of the matrix of mode shapes. Once we have the coefficients, we can then plug them into our equation for x(t) to get the complete solution.

 

[charng]

Sure, I would be happy to explain modal coordinates in more detail. Modal coordinates are a method of representing the response of a dynamic system in terms of its natural modes of vibration. This approach is often used in structural and mechanical engineering to analyze the behavior of complex systems.

To understand modal coordinates, it is helpful to first understand what we mean by natural modes of vibration. In simple terms, these are the different ways in which a system can vibrate or oscillate without any external forces acting on it. Each natural mode has a unique frequency and corresponding mode shape, which describes the spatial distribution of the vibration within the system.

Now, let's consider the system described in the content above, where Ma + Kx = 0. This is a common equation used to model the behavior of a dynamic system, where M is the mass matrix and K is the stiffness matrix. The solution to this equation will give us the natural frequencies and corresponding mode shapes of the system.

To solve for the response of the system x(t), we can use modal coordinates. This involves expressing the displacement vector x(t) in terms of the mode shapes of the system. This can be written as:

x(t) = Σq_i(t)Φ_i

where q_i(t) are the modal coordinates and Φ_i are the mode shapes. This equation essentially says that the displacement of the system at any given time is a linear combination of the mode shapes, with the modal coordinates acting as coefficients.

To determine the values of the modal coordinates, we can use the initial conditions of the system and the characteristic equation derived from Ma + Kx = 0. This will allow us to solve for the modal coordinates and ultimately determine the response of the system x(t).

One of the advantages of using modal coordinates is that it simplifies the analysis of complex systems. By expressing the response of the system in terms of its natural modes, we can easily identify which modes contribute the most to the overall response. This can be particularly useful in identifying potential problem areas in a system and determining the best course of action to address them.

In summary, modal coordinates are a method of representing the response of a dynamic system using its natural modes of vibration. This approach simplifies the analysis of complex systems and can provide valuable insights into their behavior. I hope this explanation helps to clarify the concept of modal coordinates for you.