Upload
bertram-lesley-palmer
View
217
Download
0
Embed Size (px)
DESCRIPTION
3 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. MODAL SUPERPOSITION METHOD
Citation preview
MODULE 08
MULTIDEGREE OF FREEDOM SYSTEMS
2
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.
MODAL SUPERPOSITION METHOD
3
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.
MODAL SUPERPOSITION METHOD
4Cost of modal solution vs. Step-by-step solution
Number of time steps
Cost
Modal solution
Step-by-step solution
Results of modal analysis are required as a pre-
requisite for modal solution
MODAL SUPERPOSITION METHOD VS DIRECT INTEGRATION
5
Model file ELBOW.SLDPARTMaterial Al2014Supports Fixed to the back faceLoads Harmonic force excitationDamping 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
Fixed support to the back
ELBOW
6
Mode 1 96Hz Mode 2 103Hz
Mode 3 247Hz
Results of modal analysis
Mode 4 380Hz
ELBOW
7
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
SUM 0.757 0.762 0.495
ELBOW
Modal mass participation
8Finite element mesh; use default element size and apply mesh control 5mm to the round fillet
Mesh control
ELBOW
9Results of static analysis
Maximum static stress 18.4MPa
ELBOW
10Location of sensors
Sensor to monitor displacements
Sensor to monitor stresses
ELBOW
11
0
1
1
2
2
3
3
0 100 200 300 400 500
UZ displacement amplitude frequency response.
Modes 2, 3, 4 show
Mode 1 does not show because it has 0 mass participation in Z direction
mm
Hz
Hz
0
1
2
0 100 200 300 400 500
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
mm
ELBOW
12
0
100
200
300
400
500
0 100 200 300 400 500
Von Mises stress frequency response
Mode1 and mode 2 are indistinguishable
MPa
Hz
0
100
200
300
400
500
90 95 100 105 110 115 Hz
MPa
Von Mises stress frequency response
In he range 90-115Hz shows the effect of mode 1
ELBOW