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