Multi-Degree-of-Freedom System Response to Multipoint Base Excitation
By Tom Irvine
Email:
October 16, 2013
______
Introduction
Figure 1.
Figure 2.
The free-body diagram is given in Figure 2.
The system has a CG offset if .
The system is statically coupled if .
The rotation is positive in the clockwise direction.
The variables are
y / is the base displacementx / is the translation of the CG
/ is the rotation about the CG
m / is the mass
J / is the polar mass moment of inertia
ki / is the stiffness for spring i
zi / is the relative displacement for spring i
Sign Convention:
Translation: upward in vertical axis is positive.
Rotation: clockwise is positive.
Figure 3. Figure 4.
Sum the forces in the vertical direction for mass 1.
(1)
(2)
(3)
(4)
Sum the forces in the vertical direction for mass 2.
(5)
(6)
(7)
(8)
Sum the forces in the vertical direction for mass 3.
(9)
(10)
(11)
(12)
(13)
Sum the moments about the center of mass.
(14)
(15)
(16)
(17)
(18)
The equations of motion are
(19)
Variables
M / Mass matrixK / Stiffness matrix
I / Identity matrix
/ Transformation matrix
U / Displacement vector
ud / Displacements at driven nodes
uf / Displacements at free nodes
The equation of motion for a multi-degree-of-freedom system is
(20)
(21)
Partition the matrices and vectors as follows
(22)
(23)
where
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
Create a transformation matrix such that
(32)
(33)
(34)
Premultiplyby ,
(35)
Again, the partitioned equation of motion is
(36)
Transform the equation of motion to uncouple the stiffness matrix so that the resulting stiffness matrix is
(37)
(38)
(39)
(40)
(41)
(42)
Let
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
By similarity, the transformed mass matrix is
(52)
(53)
(54)
The equation of motion is thus
(55)
The final displacements are found via
(56)
(57)
APPENDIX A
Example
Figure A-1.
Consider the system in Figure A-1. Assign the following values. The values are based on a slender rod, aluminum, diameter =1 inch, total length=24 inch.
Table A-1. ParametersVariable / Value
/ 100 lbm
/ 100 lbm
/ 18.9 lbm
J / 907 lbm in^2
/ 20,000 lbf/in
/ 20,000 lbf/in
/ 8 in
/ 16 in
Let
where the amplitude is in units of G and time t is in seconds.
Note that the mass values for m1 and m2 are actually arbitrary since these degrees-of-freedom are driven.
The following parameters were calculated for the sample system via a Matlab script.
enforced_acceleration
enforced_acceleration.m ver 2.3 October 16, 2013
by Tom Irvine
Enter the units system
1=English 2=metric
1
Assume symmetric mass and stiffness matrices.
Select input mass unit
1=lbm 2=lbf sec^2/in
2
stiffness unit = lbf/in
Select file input method
1=file preloaded into Matlab
2=Excel file
1
Mass Matrix
Enter the matrix name: mm
Stiffness Matrix
Enter the matrix name: kk
Input Matrices
mass =
0.2591 0 0 0
0 0.2591 0 0
0 0 0.0490 0
0 0 0 2.3497
stiff =
20000 0 -20000 -160000
0 20000 -20000 320000
-20000 -20000 40000 -160000
-160000 320000 -160000 6400000
Select modal damping input method
1=uniform damping for all modes
2=damping vector
1
Enter damping ratio
0.05
number of dofs =4
Enter the starting time (sec)
0
Enter the end time (sec)
1
Enter the sample rate (samples/sec)
4000
Enter the number of dofs with enforced acceleration. (maximum = 4)
2
Each input file must have two columns: time & acceleration
Enter the first dof
1
Enter the applied acceleration input matrix name for this dof.
a130
Enter the second dof
2
Enter the applied acceleration input matrix name for this dof.
a150
MT =
0.2849 0.0068 0.0326 0.0979
0.0068 0.2686 0.0163 -0.0979
0.0326 0.0163 0.0490 0
0.0979 -0.0979 0 2.3497
KT =
1.0e+06 *
-0.0000 0.0000 0.0000 0
0.0000 -0.0000 0.0000 0
0.0000 0.0000 0.0400 -0.1600
0 0 -0.1600 6.4000
Natural Frequencies
No. f(Hz)
1. 1.2038e-07
2. 6.5509e-06
3. 140.91
4. 271.55
Modes Shapes (column format)
ModeShapes =
1.8735 0.0116 -0.5798 0.1089
0 1.9292 -0.2099 -0.3008
0 0.0000 4.6346 1.0430
0 0.0000 0.1565 -0.6438
Mwd =
0.0326 0.0163
0.0979 -0.0979
Mww =
0.0490 0
0 2.3497
Kww =
40000 -160000
-160000 6400000
Natural Frequencies
No. f(Hz)
1. 133.79
2. 267.93
Modes Shapes (column format)
ModeShapes =
4.4005 1.0291
0.1486 -0.6352
Participation Factors
part =
0.5645
-1.4422
Output arrays:
ea_disp - displacement
ea_vel - velocity
ea_acc - acceleration
Figure A-2.
Figure A-3.
1