MODULE 08
MULTIDEGREE OF FREEDOM SYSTEMS
MODAL SUPERPOSITION METHOD
Structure vibrating in a given mode can be considered as the Single Degree of Freedom (SDOF) system. Structure
can be considered a series of SDOF. For linear systems the response can be found in terms of the behavior in each
mode and these summed for the total response. This is the Modal Superposition Method used in linear dynamics
analyses.
A linear multi-DOF system can be viewed as a combination of many single DOF systems, as can be seen from the
equations of motion written in modal, rather than physical, coordinates. The dynamic response at any given time is
thus a linear combination of all the modes. There are two factors which determine how much each mode
contributes to the response: the frequency content of the forcing function and the spatial shape of the forcing
function. Frequency content close to the frequency of a mode will increase the contribution of that mode. However,
a spatial shape which is nearly orthogonal to the mode shape will reduce the contribution of that mode.
2
MODAL SUPERPOSITION METHOD
The response of a system to excitation can be found by summing up the response of multiple Single Degree of Freedom
Oscillators (SODFs). Each SDOF represents the system vibrating in a mode of vibration deemed important for the vibration
response.
3
MODAL SUPERPOSITION METHOD VS DIRECT INTEGRATION
Cost
Step-by-step solution
Modal solution
Results of modal analysis
are required as a prerequisite for modal solution
Number of
time steps
Cost of modal solution vs. Step-by-step solution
4
ELBOW
Fixed support to
the back
Model file
ELBOW.SLDPART
Material
Al2014
Supports
Fixed to the back face
Loads
Harmonic force excitation
Damping
2% modal
Objectives:
• Time Response analysis
• Frequency Response analysis
• Modal mass participation
• Comparison between Static and Dynamic stress results
• Comparison between Time Response and Frequency Response results
Harmonic load
Constant amplitude 25000N
Frequency range 0-500Hz
5
ELBOW
Mode 1
96Hz
Mode 2
103Hz
Mode 3
247Hz
Mode 4
380Hz
Results of modal analysis
6
ELBOW
Mode No.
Freq (Hertz)
X direction
Y direction
Z direction
1
96.03
0.491
0.116
0.000
2
103.94
0.065
0.326
0.244
3
247.60
0.003
0.024
0.000
4
381.71
0.137
0.217
0.231
5
615.47
0.062
0.080
0.020
0.757
0.762
0.495
SUM
Modal mass participation
7
ELBOW
Mesh control
Finite element mesh; use default element size and apply mesh control 5mm to the round fillet
8
ELBOW
Maximum static
stress 18.4MPa
Results of static analysis
9
ELBOW
Sensor to monitor
displacements
Sensor to monitor
stresses
Location of sensors
10
ELBOW
mm
3
3
2
UZ displacement amplitude frequency
response.
2
Modes 2, 3, 4 show
1
Mode 1 does not show because it has 0
mass participation in Z direction
1
0
0
100
200
300
400
500
Hz
2
mm
UX displacement amplitude frequency
response.
Modes 1, 2, 4 show
Mode 3 does no show because it has almost)
zero mass participation in X direction
1
0
0
100
200
300
400
500
Hz
11
ELBOW
MPa
500
400
300
Von Mises stress frequency response
200
Mode1 and mode 2 are indistinguishable
100
0
0
MPa
100
200
300
400
500
Hz
500
400
Von Mises stress frequency response
300
In he range 90-115Hz shows the effect of
mode 1
200
100
0
90
95
100
105
110
115
Hz
12