Multi-degree of freedom systems – review
Free vibration analysis
· No damping
Equation of motion:
We know from theory of differential equations that the solution is:
Thus we need to find the frequency ω and the amplitudes A and B.
Characteristic equation:
w2 eigenvalues of [M]-1[K]
eigenvectors of [M]-1[K].
w: natural frequencies (number = number of d.o.f.)
mode shapes (number = number of d.o.f.).
General expression for free vibration:
Conclusions:
· Vibration=mode1*c1+mode2*c2
· Participation of each mode depends on initial conditions
Steps in free vibration analysis
1. Find natural frequencies and mode shapes
2. Express displacements in terms of natural frequencies and mode shapes:
3. Find constants and phase angles from initial conditions
Example
Second natural frequency:
Mode shape for above frequency:
Usually assume first entry is 1. Therefore, mode shape is:
Masses move in opposite directions
First natural frequency:
Mode shape for above frequency:
Masses move together.
Displacements of two masses are superposition of displacements in the two modes:
Find c1, c2 and , from initial conditions
Orthogonality of mode shapes and decoupling of equations of motion
Mode shapes are orthogonal wrt the mass and stiffness matrices:
Therefore:
where [M'] and [K'] are diagonal matrices.
We can use the orthogonality property of the mode shapes to uncouple the equations of motion through the following transformation:
This transformation expressed the unknown displacements in terms of the basis vectors and . If we substitute the expression for the displacement into the equations of motion then we have two new equations wrt the unknown coordinates .
These equations are equivalent to the original system but they can be easily decoupled.
[M'] and [K'] are diagonal matrices, which means that we have two uncoupled equations with unknowns coordinates p1(t) and p2(t).
Therefore, the equations of motion can be written wrt to the new coordinate system can be written as follows:
In some cases it is better to solve these two uncoupled equations than the coupled equations. To solve these equations we need the initial conditions for coordinates p1(t) and p2(t). These are obtained as follows:
Procedure for calculating free vibration response by uncoupling the equations of motion
1. Determine matrices:
2. Determine initial values of coordinates p1(t) and p2(t) and initial values of the derivatives.
3. Solve uncoupled equations for p1(t) and p2(t):
where m'ii and k'ii are the diagonal elements of matrices M' and K', using the initial conditions found in step 2.
4. Recover displacements x1(t) and x2(t) using the following equations: