Multi-Degree-of-Freedom

Question 1

How does the inclusion of a second (or more) degree of freedom affect the solution?

Answer 1

Equations use vectors and matrices instead of individual values

Question 2

How do you create the mass matrix from 2 independent masses A and B?

Answer 2

[ A 0 ]
[ 0 B ]

Question 3

How do you create the stiffness matrix where spring A connects mass A to a surface and spring B connects mass B to mass A?

Answer 3

[ kA + kB -kB ]
[ -kB kB ]

Question 4

If Ax=0, where A is a matrix and x is a vector, what does the nature of A mean for the solution x?

Answer 4

If A is not singular, x = 0, if A is similar, x has infinite solutions

Question 5

What must be true for (K-wn^2M)X=0 to have a non trivial solution?

Answer 5

The determinant of the term inside the brackets must be zero

Question 6

How is the eigenproblem typically simplified?

Answer 6

Lambda = omega n ^2

Question 7

In an eigenproblem, what is true of the natural frequencies?

Answer 7

sqrt(k/m) will not necessarily give the solution

Question 8

What is the size of the stiffness and mass matrices for an N degree of freedom problem?

Answer 8

NxN

Question 9

How do you solve a 3 DoF problem?

Answer 9

Say one of the solutions is equal to an arbitrary value alpha, then solve for the other two in terms of alpha

Question 10

What is the Rayleigh quotient?

Answer 10

rho(X) = XtKX/XtMX

Question 11

If there is Rayleigh Damping then what is the diagonal matrix of C (Dc)?

Answer 11

Dc = alpha Dk + beta Dm

Question 12

What is the procedure for modal analysis?

Answer 12

Form eigenmatrix X
Transform the governing equation in x(t) to g(t)
Determine the initial conditions
Solve the diagonalised equation for g(t)
Convert back to x(t)

Question 13

What is the rule of thumb for the number of rigid body modes in a system?

Answer 13

The number of degrees of freedom - the number of springs

Question 14

What is a rigid body mode?

Answer 14

A movement that can be applied to a system which does not cause any oscillations

Question 15

In vibration, what is a node and what is an anti node?

Answer 15

Nodes are points of minimal displacement, anti nodes have the most displacement

Question 16

How is stability contrasted to equilibrium?

Answer 16

Equilibrium is when the net force acting on an object is zero
Stability is when the energy of an object (including potential) is at minimum

Question 17

If comparing two 2x1 eigenvectors how do you tell which has the LOWER frequency?

Answer 17

Both eigenvector terms have the same sign

Question 18

If a pendulum is free to move in a circle but remains at a fixed distance from its suspension point, how many degrees of freedom does it have?

Answer 18

2 degrees of freedom

Question 19

What is an important property of large orthogonal matrices?

Answer 19

The inverse matrix is equal to the transpose

Question 20

When is Rayleigh Quotient used?

Answer 20

When solving an eigenvector problem gives multiple different answers

Question 21

If a question asks for the lowest natural frequency of a system with no fixity, what is the likely answer?

Answer 21

The lowest natural frequency is 0