# How do I calculate modal mass with combined distributed and lumped masses?

• thethinwhiteduke
In summary, the conversation discussed the methods for calculating modal mass and stiffness and how to model both distributed and lumped masses together. The equation for modal mass was provided, and an example was given for a tower with only a distributed mass. The issue of combining the two types of masses for the calculation was mentioned, but the solution was eventually found.
thethinwhiteduke

## Homework Statement

The question is featured in the image below. I know the methods to calculate modal mass, stiffness etc, I'm just not too sure how to model both the distributed and lumped masses together.

## Homework Equations

Modal Mass, M* = ∫m.φ^2(x) dx

## The Attempt at a Solution

Say for instance, that the tower had no lumped mass at the top, so we only had to consider the distributed load (m = 3000 kg/m), the solution would be:
M* = m∫φ^2(x) dx (limits of integration are from 0 to L(=80m))
∫φ^2(x) dx = ∫(1-cos((πx)/(2L))^2 dx = L - 4L/π + L/2
M* = 3000 x (80 - 320/π +40) = 54423kg

My issue is not knowing how to combine the distributed and lumped masses together for this calculation. Any help is greatly appreciated

I've managed to solve the problem now. Thank you for the concern though!

## 1. What is MDOF earthquake response?

MDOF (Multi-Degree of Freedom) earthquake response is a method used to analyze the behavior of structures during an earthquake. It takes into account the multiple modes of vibration that can occur in a structure, as well as the interaction between these modes.

## 2. How is MDOF earthquake response different from SDOF response?

MDOF earthquake response is different from SDOF (Single Degree of Freedom) response in that it considers the effects of multiple modes of vibration, while SDOF only looks at one mode of vibration. MDOF is a more complex and accurate method of analysis, but it also requires more computational resources.

## 3. What factors are considered in MDOF earthquake response analysis?

MDOF earthquake response analysis takes into account factors such as the stiffness and mass distribution of the structure, the characteristics of the earthquake ground motion, and the damping properties of the structure.

## 4. How is MDOF earthquake response used in structural design?

MDOF earthquake response is used in structural design to ensure that a structure can withstand the forces and deformations caused by an earthquake. It helps engineers to determine the optimal design of a structure to improve its earthquake resistance.

## 5. What are the limitations of MDOF earthquake response analysis?

MDOF earthquake response analysis has some limitations, such as the assumption of linear behavior of the structure, which may not always be accurate. It also requires significant computational resources and can be time-consuming. Additionally, it may not be suitable for highly complex or irregular structures.

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