What does distributed-mass cantilever beam mean in this book?

Question

What does “distributed-mass” mean in this cantilever beam?

Does it have the same meaning as uniformly distributed load or other meaning?

Answer 1

Distributed mass means exactly what the word says - the mass is distributed, whether or not uniformly, over the length of the cantilever beam. A tapered cantilever beam is an example of having linearly decreasing mass distribution.

Answer 2

There is a lack of context, but the diagrams seem to be contrasting different approaches to modelling deflection of a cantilever beam. The parameter 't' seems suggestive of a dynamic analysis of an oscillating beam. It isn't clear without further context why they have included the term 'mass' in description for fig(a) but not the subsequent figures, however they are likely considering point masses in the other two figures. Either way, deflection of the beam will be due to imposed forces, either self weight(due to mass*gravity) distributed along its length or through application of external point or distributed forces (or in the case of an oscillating beam, stored internal elastic potential energy being converted to kinetic energy).

The continuous model in fig(a) would be represented by an equation or differential equation which could model any point along the beam.

Models (b) and (c) are models which hold only at the points indicated on the length of the beam, and are therefore discretisations of the problem, similar to the type of discretisation you would find in a finite element analysis.

Answer 3

By the context it means a beam with an infinite number of degrees of freedom, which means a full continuum solution for the deflection equation.

You can work this out by the fact that the first drawing shows the deflection as $v(x,t)$, the second drawing (a 1-dof model) shows the deflection as $v(t)$ (and there is an underlying interpolation function implicit in the black line) and the third drawing shows the deflection as $v_1(t)$, $v_2(t)$ and $v_3(t)$.

These types of discretizations are common when working on frame theory. Perhaps during your studies you've encountered a situation like the picture below.

This is a 4 dof discretization of a beam element. If you fix the left end and condense the rotation into the deflection you'll end up with the second case on your picture. These are all idealizations of the real behavior of the beam, which is the distributed mass case shown in the first picture.

Edit: I forgot to add an example of a distributed mass beam you may have encountered in classes or introductory literature. Euler-Bernoulli beam theory states the deflection of a beam is given by $$ \frac{d^2}{dx^2}\left(E(x)I(x)\frac{d^2v}{dx^2}\right)=q(t) $$ If you solve that differential equation (with appropriate boundary conditions) you'll get the continuous solution for displacement, which would be the $v(x,t)$ shown in the first picture.