Semi-active implementation

of nonlinear damping for vibration isolation

By

Diala Uchenna

OutlineIntroduction.Aims and Objectives.Elements of a vibration isolation system.

Vibration isolation system with a linear damping characteristic .

Vibration isolation system with a Nonlinear damping characteristic.

Nonlinear (cubic) damping characteristic implementation with Magnetorheological damper (MR damper).

Results. Conclusion and Recommendation.

Vibration isolation deals with the control of unwanted vibrations to keep the adverse effects within acceptable limits (Z.Q. Lang et al., 2009).

Transmissibility is: a measure of the response of a vibration isolation system. also a measure it’s performance.

Viscous damping is incorporated to reduce vibration amplitude at resonance. However, if the effect of the viscous damper is linear, as

the damping level is increased to minimise the transmissibility in the resonant region, the transmissibility is increased in regions where isolation is desired.

Active vibration isolation system have been developed to resolve this problem but with limitations of cost and complexity.

Introduction

To study the effects of nonlinear viscous damping on a vibration isolation system.

To demonstrate it’s implementation using semi-active techniques with an MR damper. Nonlinear viscous damping measure is proposed

here using semi-active means to resolve the issue with active devices.

To design a controller to track the desired nonlinear viscous damping force.

Aims and Objectives

Vibration isolation system normally consists of a spring that offers stiffness to the system .a damping element (dashpot) to disperse input

energy.

Fig.1.0 : A mass-spring-damper system

Where, m = Mass of the objectk = Spring stiffnessc = Linear damper coefficient

Elements of a vibration isolation system

Vibration isolation system with a linear damping characteristic (Open loop)

The transmissibility is given as:

= nondimensional excitation freq ratio

= damping ratio

Fig.2.0: A mass-spring-damper system with linear damping characteristics

Vibration isolation system with a linear damping characteristic (Open loop) contd...

Transmissibility curves for freq ratio, r = 0:0.2:2 and damping ratio = 0.1, 0.2, 0.4 and 0.7 are shown in figure 3.0.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 240

45

50

55

60

65

70

75F

orce

tra

nsm

issi

bilit

y

freq ratio

E0.1__E0

E0.2__E0E0.4__E0

E0.7__E0

Fig. 3.0: Transmissibility curve for a vibration isolation system with linear damper

Region of desired isolation

Resonant region

Vibration isolation system with a nonlinear damping characteristic (Open loop)

The transmissibility is given as:

c1 = Linear damper coefficientc2 = Nonlinear damper coefficient

Fig.4.0: A mass-spring-damper system with nonlinear damping characteristics

Vibration isolation system with a nonlinear damping characteristic (Open loop) contd...Transmissibility curves for freq ratio, r = 0:0.2:2, zeta 1 = 0.1 and zeta

2 = 0, 0.2 and 0.4 are shown in figure 5.0.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 240

45

50

55

60

65

70

75

For

ce t

rans

mis

sibi

lity

freq ratio

E0.1__E0

E0.1__E0.2E0.1__E0.4

Fig. 5.0: Transmissibility curve for a vibration isolation system with nonlinear damper

Region of desired isolation

Resonant region

Vibration isolation system with a nonlinear damping characteristic (Open loop) contd...

Transmissibility curves for r = 0:0.2:2, zeta 1 = 0.2 and zeta 2 = 0, 0.2 and 0.4 are shown in figure 6.0.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 240

45

50

55

60

65

70F

orce

tra

nsm

issi

bilit

y

freq ratio

E0.2__E0

E0.2__E0.2E0.2__E0.4

Fig. 6.0: Transmissibility curve for a vibration isolation system with nonlinear damper

Region of desired isolation

Resonant region

Semi-active vibration isolation is realizable using mass control, stiffness control or damping control.

A spring in parallel with an adjustable damper (core of system) is used for this project.

Some controllable dampers include; • Magnetorheological dampers (MR dampers)• Electrorheological dampers (ER dampers)• Viscoelastic dampers etc.The MR damper was used in this project work.

Nonlinear (cubic) damping characteristic implementation with MR damper.

MR damper comprises a hydraulic cylinder containing ferromagnetic particles, of micron size, suspended in a fluid (often oil).

Exposure to electric or magnetic field, via a solenoid embedded inside, causes the MR material to modify from a free flowing viscous fluid to a semi-solid state in a few milli-seconds.

Fig.7.0: Schematic of an MR damper

Nonlinear damping characteristic implementation with MR damper contd...

-40 -30 -20 -10 0 10 20 30 40-3000

-2000

-1000

0

1000

2000

3000

Velocity (m/s)

mr d

ampe

r For

ce

1.5 A

1A

0A

0.25A

0.5A2A

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3000

-2000

-1000

0

1000

2000

3000

Displacement (m)

mr d

ampe

r for

ce (N

)

0A

0.25A

0.5A1A

1.5A

2A

Fig.8.0: Simulation results for MR damper force versus (a.) Velocity (m/s) (b) Displacement (m) plots

Nonlinear damping characteristic implementation with MR damper contd...

Nonlinear damping characteristic implementation with MR damper contd...(Closed loop with Proportional Integral, PI controller)

Fig.9.0: Schematic of a nonlinear damping characteristic implemented using the MR damper

Nonlinear damping characteristic implementation with MR damper contd...

Fig.10: Schematic of a nonlinear damping characteristic practically implemented using the MR damper.

Nonlinear damping characteristic implementation with MR damper contd...

Fig.11: SIMULINK model of the vibration isolation system with nonlinear damping implementation using the MR damper.

[dot_x1(t)]^3

u(t)

x1(t) Nonlinear cubic damper

force

MR Damper force

x' = Ax+Bu y = Cx+Du

State-SpaceScope4

Scope3

Scope2

Scope1

Vel. (m/s)

Current (A)Fmr (N)

MR Damper

200

Gain

Disturbance force

Direction of force

du/dt

Derivative Cubic

Cntrl input Cntrl signal

Controller Subsystem

Add-ve force

correction

Results

0 5 10 15 20 25-15

-10

-5

0

5

10

15

20

Freq(rad/sec)

Fou

t/F

in2

(dB

)

Force Transmissibility (Cubic Damper)

E3rms = 0

E3 = 0E3

rms = 0.2

E3 = 0.2

E3rms = 0.4

E3 = 0.4E3

rms = 0.7

E3 = 0.7

0 5 10 15 20 25-15

-10

-5

0

5

10

15

20

Freq(rad/sec)F

out/

Fin

2 (d

B)

Force Transmissibility (MR Damper)

E3rms = 0

E3 = 0E3

rms = 0.2

E3 = 0.2

E3rms = 0.4

E3 = 0.4E3

rms = 0.7

E3 = 0.7

Fig.12: Simulation results for transmissibility plots using desired Cubic damper model and MR damper model

Conclusion and recommendations

•The results show the beneficial effects of nonlinear viscous damping. •This implies same effect is achievable as an active device. •This would have significant implications for the engineering design of passive vibration isolators in a wide range of practical applications.•More investigations on the effect of more complex nonlinear viscous damping characteristics on vibration isolation could be carried out to improve on the results achieved in this study.

Thank you

Appendices