MODAL ANALYSIS OF DISCRETE SDOF SYSTEMS

1

Linear spring400000N/m

Model file 1DOF.SLDASM

Material AISI 1020

Restraints Fixed base

Restraints preventing RBMs

Loads None

Objective:

• Modal analysis

Mass 10kg

1DOF

1DOF.SLDASM

2

Restraints definition.

These restraints are required to make the first mode correspond to the mode of vibration of SDOF

1DOF

Restraints defined in local cylindrical system, only axial displacement component allowed

Fixed restraint to base

3

Analytical solution

1DOF

4

We may assume solution as:

then

Equation of motion of free undamped vibration:

Results of modal analysis

Mode 1

Numerical result: 32.36Hz

Analytical result: 31.8Hz

Mode 2

Numerical result: 12084Hz

This is not a SDOF mode

1DOF

5

Model file SWING ARM.SLDASM

configuration 01

Material 1060 Alloy

Restraints Fixed base

Fixed hinge to arm

Loads none

Objective:

• Modal analysis

SWING ARM

Linear spring2000N/m

m = 0.56kg

SWING ARM.SLDASM

6

L1=0.2m

kL=2000N/m

L2=0.1m

m = 0.56kg

Mass of beam is negligible

SWING ARM

Analytical solution 7

SWING ARM

Mode 1

Numerical result: 4.64Hz

Analytical result: 4.75Hz

Mode 2

Numerical result: 23.05.Hz

This is not a SDOF mode

8

Model file SWING ARM.SLDASM

configuration 02

Material 1060 Alloy

Restraints Fixed base

Fixed hinge to arm

Loads none

Objective:

• Modal analysis

SWING ARM

Linear spring2000N/m

m = 0.09kg

SWING ARM.SLDASM

9

kL=2000N/m

L=0.1m

m = 0.09kg

SWING ARM

Analytical solution 10

11

Mode 1

Numerical result: 43.4Hz

Analytical result: 41.1Hz

Mode 2

Numerical result: 2154.Hz

This is not a SDOF mode

SWING ARM

SWING ARM

Element size 8.3mm Element size 3mm

Moving beam

Element size 1.5mm

Moving beam

Element size 1.5mm

Moving beam and base

13

ROLER

Model file ROLER.SLDASM

Material AISI304

Restraints Fixed base

Fixed contact line

Loads none

Objective:

• Modal analysis

kL=2000N/m

m = 75.4kg

Contact line

14

ROLER

Θ

x

kL=2000N/m

m = 75.4kg

15

ROLER

Mode 1

Numerical result: 0.6Hz

Analytical result: 0.66Hz

Mode 2

Numerical result: 1544Hz

This is not a SDOF mode

STABILITY

16

STABILITY OF A SDOF SYSTEM

Non-oscillatory divergent motion is called DIVERGENT INSTABILITY

Oscillatory divergent motion is called FLUTTER

Over damped, critically damped, under damped motions are all well behaved. Amplitudes are finite, do not grow with time. If coefficients m, c, k are not positive, motion is not well behaved.

17

Divergent response

Flutter response

Stable response Marginally stable response Unstable response

STABILITY OF A SDOF SYSTEM

18

PENDULUM.SLDASM

Two linear springs500N/m each

m =1.0kg

Model file PENDULUM.sldasm

Material AISI 1020

Restraints Fixed base

Fixed hinge to arm

Loads 1. Gravity down

2. Gravity up

Objectives:

• Modal analysis

• Analysis of stability

STABILITY OF A SDOF SYSTEM

19

L1=0.1m

m =1.0kg

m = 1kgk = 500N/ml = 0.1mg = 9.81m/s²

Analytical solution FEA solution

Gravity down, k=500N/m Stable system

STABILITY OF A SDOF SYSTEM

20

m =1.0kg

Gravity down, k=196.2N/m Unstable system

Analytical solution FEA solution

L1=0.1m

STABILITY OF A SDOF SYSTEM

System becomes unstable when

21

L1=0.1m

m =1.0kg

Gravity up, no springs Stable system

Analytical solution FEA solution

STABILITY OF A SDOF SYSTEM

22