Problem 1 - Concepts and Applications of Finite Element Analysis

Question

A differential element having mass \(d m\) and velocity \(v\) has kinetic energy \(v^{2} d m / 2\). Show that the kinetic energy of a finite element is therefore \(\\{\mathbf{d}\\}^{T}[\mathbf{m}]\\{\mathbf{d}\\} / 2\), where \([\mathrm{m}]\) is the consistent element mass matrix and \(\\{\mathbf{d}\\}\) is the vector of element nodal d.o.f.

Step-by-Step Solution

Answer

The kinetic energy of a finite element as \(\{d\}^{T}[m]\{d\}/2\), is obtained by using the principle kinetic energy equation with matrix multiplication rule. The final expression represents that the kinetic energy is a quadratic form of the displacement with the mass matrix acting as a multiplier or weighing factor.

Step 1: Understand basic equations

One of the forms to express kinetic energy (K.E) is \(\frac{1}{2} m v^{2}\) where \(m\) is mass and \(v\) is velocity. For an infinitesimally small element with mass \(d m\) and velocity \(v\), the kinetic energy can be written as \(d K = \frac{1}{2} v^{2} d m\).

Step 2: Incorporate the definition and notation for kinetic energy

The kinetic energy of a finite element can be expressed as \(\frac{1}{2} m v^{2}\) where \(m\) is the consistent mass matrix of the element and \(v\) is the velocity vector. Substitute d.o.f vector \(\{d\}\) for velocity \(v\). This leads to \(K.E = \frac{1}{2} m \{d\}^{2}\).

Step 3: Apply the matrix multiplication rule

In order to express the kinetic energy in the desired form, the rule of matrix multiplication is used. The expression \(\{d\}^{2}\) would in essence manifest as the transpose of \(\{d\}\) i.e. \(\{d\}^{T}\) multiplied by \(\{d\}\). Therefore, \(K.E = \frac{1}{2} m \{d\}^{T}\{d\}\).

Step 4 : Reorganize the equation

Reorganize the above equation to conform to the format in the statement of the problem i.e. \(K.E = \frac{1}{2} \{d\}^{T}[m]\{d\}\). Here \([m]\) is the consistent element mass matrix and \(\{d\}\) is the vector of element nodal d.o.f.

Key Concepts

Differential Element Mass

The concept of 'differential element mass' is fundamental in finite element analysis when dealing with dynamic problems, such as determining the kinetic energy of a structure. Imagine breaking down a structure into tiny pieces, or 'elements', each so small that their properties can be assumed to be uniform. The mass of each of these diminutive elements is represented by the term 'differential element mass', often denoted as \( dm \).

The importance of this lies in its role in calculating dynamic quantities. For example, in our exercise where the kinetic energy of a finite element is sought, each of these tiny masses will move with its velocity, contributing a small part to the total kinetic energy. By integrating (mathematically summing up) these differential contributions over the whole structure, we can obtain the total kinetic energy, ensuring precision that originates from the detailed subdivision of the structure.

Consistent Mass Matrix

The 'consistent mass matrix', denoted as \([m]\) in our exercise, is a crucial element when determining the kinetic energy of a structure modeled with finite elements. This matrix represents how mass is distributed across the finite element, considering the shape and size of the element and how that mass participates in the dynamic behavior of the whole system.

If we think of our structure as a mesh of interconnected points, or 'nodes', the consistent mass matrix incorporates the mass linked to these nodes and the connections between them. The mass isn't concentrated at a single point; it's spread out in a way that reflects the actual distribution of material in the element. Moreover, the matrix is 'consistent' because it is derived from the same shape functions used to define the element's geometry and displacement field, ensuring that the mass distribution aligns with the element's physical properties.

Nodal Degrees of Freedom

When we refer to 'nodal degrees of freedom' (d.o.f.), we're talking about the number of independent movements that a node in a finite element system can exhibit. These are the basis for how we understand and characterize the motion or displacement of the structure at specific points.

In our exercise, the vector \(\{d\}\) represents the displacements or rotations at the nodes of a finite element. The degrees of freedom at each node might include translations in the x, y, and z directions, as well as rotations about these axes. By knowing these nodal degrees of freedom, we can start to visualize and compute how the entire structure deforms under loads, as each node’s response contributes to the overall behavior. They are fundamental in the formation of the equations that describe the dynamic characteristics of the entire system, including calculating the kinetic energy.

Matrix Multiplication

Matrix multiplication is a key operation in many mathematical and engineering disciplines, including the computational methods used in finite element analysis. When we calculate kinetic energy, as shown in the exercise, we use matrix multiplication to combine the mass matrix with the nodal degree of freedom vectors.

In the given equation, we see the mass matrix \([m]\) multiplied by the vector of nodal degrees of freedom \(\{d\}\). To do this correctly, one must follow the rules of matrix multiplication, where the number of columns in the first matrix must match the number of rows in the second matrix. In this case, the multiplication involves the transposition of the nodal degrees of freedom vector (denoted as \(\{d\}^{T}\)), resulting in a scalar value that then becomes part of the kinetic energy formula. This matrix operation is essentially the mathematical way of distributively linking the kinetic contributions of all the nodes, according to their mass and displacement, to obtain the total kinetic energy of the structure.