How to calculate the Consistent Mass Matrix for a 6DOF's Frame

[Kelles]

Homework Statement

Hello everyone, as you can see from the picture below this is a 2 storey frame with 6DOF's (2 translational DOF's = u1 and u2, 4 rotational DOF's = u3, u4, u5 and u6).

i would like to determine the consistent mass matrix (6x6 and symmetric) of the frame without carrying individual calculations for each element and then superimposing them. I would like to setup the matrix by just using the relevant coefficients directly.

The procedure that i am following to setup the matrix is: a unit of acceleration is applied at each DOF successively while the other DOF are kept at zero and then determine the inertia forces that are generated at all other DOF so to keep the system in equilibrium. For example, the mass influence coefficient m12 is the external force generated at DOF 1 due to a unit acceleration at DOF 2.

mc is the mass per unit length of the column
h is the height of the column

mb is the mass per unit length of the beam
l is the length of the beam

The Attempt at a Solution

As you can see below i have already started putting the matrix together but there are certain coefficients for which i am not sure about and i would like your help if anyone has come upon this before.

The coefficients and equations that i am using within the matrix i have found them in both the Chopra and Clough Dynamics of Structures books.

Highlighted in yellow are the two coefficients for which i am not sure about, the other ones i am mostly sure they are correct

I am having a really hard time figuring out the coefficients for the 1st row and 1st column which are supposed to be the same since the matrix is symmetric. I would really appreciate it if you could at least help me complete the first column.

The first column of the matrix is determined by assuming a unit acceleration at DOF u1 as shown in the picture below and then determining the coefficients for all other DOF

 

[Kelles]

If anyone thinks that i am missing any useful information please let me know

 

[SteamKing]

The highlights didn't come through.

 

[Kelles]

The highlights didn't come through.

Hello, i didnt really understand what you mean by that.

 

[SteamKing]

Hello, i didnt really understand what you mean by that.

In your OP, the matrix initially didn't appear at all. Now it's there with the highlighted terms indicated.

In getting back to your original question, I think you have overlooked the connectivity matrix of this frame in setting up your mass matrix. For example, joint #1 is influenced only by joints #1 and #3. Similarly, joint #2 is influenced by joints #2 and #4. In general, for joint #n, you will have an influence from all of the members directly connected to that joint. All other influences are zero.

 

[Kelles]

Thanks a lot for you reply...however i think you have misunderstood the concept here. You are referring to joints while i am referring to DOF's. I can see what you mean but i think you are wrong but no worries

There are 6DOF's in this structure...2 horizontal and 4 rotational...shown on the first figure at the top

therefore in order to complete the first column of the Matrix you need to move the structure in the direction of the DOF (Degree of Freedom) m11 as shown in the last figure and determine the reaction at each DOF due to the movement.

The degree of freedom m11 is the horizontal direction (to the right) at the first story of the structure as shown on the last figure

Next in order to complete the second column you need to apply a movement at DOF m22 which is exactly the same as m11 but on the top storey.

Its not as complicated as it looks, i know how to do it for just one storey because i have seen an example but for this one i am not really sure how the structure reacts...i know the coefficients to be used but as i said i do not know how the structure reacts in the case of mass matrix...i know how to do the stiffness matrix of this one but mass is different.

And to complete columns 3 4 5 6 you need to apply a rotational movement at each DOF and determine the reaction

This is called the consistent mass matrix, its different from the ordinary mass matrix because the mass of the structure is assumed be lumped at the nodes and mass is also assigned to the rotational degrees of freedom. this matrix is used in finite element analysis. It should provide more accurate results but its more complicated during the dynamic analysis