Multi-Degree-of-Freedom Flashcards
How does the inclusion of a second (or more) degree of freedom affect the solution?
Equations use vectors and matrices instead of individual values
How do you create the mass matrix from 2 independent masses A and B?
[ A 0 ]
[ 0 B ]
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?
[ kA + kB -kB ]
[ -kB kB ]
If Ax=0, where A is a matrix and x is a vector, what does the nature of A mean for the solution x?
If A is not singular, x = 0, if A is similar, x has infinite solutions
What must be true for (K-wn^2M)X=0 to have a non trivial solution?
The determinant of the term inside the brackets must be zero
How is the eigenproblem typically simplified?
Lambda = omega n ^2
In an eigenproblem, what is true of the natural frequencies?
sqrt(k/m) will not necessarily give the solution
What is the size of the stiffness and mass matrices for an N degree of freedom problem?
NxN
How do you solve a 3 DoF problem?
Say one of the solutions is equal to an arbitrary value alpha, then solve for the other two in terms of alpha
What is the Rayleigh quotient?
rho(X) = XtKX/XtMX
If there is Rayleigh Damping then what is the diagonal matrix of C (Dc)?
Dc = alpha Dk + beta Dm
What is the procedure for modal analysis?
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)
What is the rule of thumb for the number of rigid body modes in a system?
The number of degrees of freedom - the number of springs
What is a rigid body mode?
A movement that can be applied to a system which does not cause any oscillations
In vibration, what is a node and what is an anti node?
Nodes are points of minimal displacement, anti nodes have the most displacement
How is stability contrasted to equilibrium?
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
If comparing two 2x1 eigenvectors how do you tell which has the LOWER frequency?
Both eigenvector terms have the same sign
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?
2 degrees of freedom
What is an important property of large orthogonal matrices?
The inverse matrix is equal to the transpose
When is Rayleigh Quotient used?
When solving an eigenvector problem gives multiple different answers
If a question asks for the lowest natural frequency of a system with no fixity, what is the likely answer?
The lowest natural frequency is 0
