This document summarizes a student's research on modeling and simulating the dynamics of a negative stiffness vibration isolation mechanism. Key points: - Negative stiffness mechanisms aim to reduce stiffness without compromising load capacity, allowing for better vibration isolation at low frequencies. - The student models the mechanism as a nonlinear system and uses numerical integration and multi-harmonic balance methods to study its behavior under harmonic excitation. - Results show the system exhibits different periodic, quasi-periodic, and chaotic responses depending on excitation frequency and other parameters like mean load. - An averaging method is also used to analyze the mechanism's transmissibility and how parameters like geometry, excitation, and damping affect its isolation performance.

1. Dynamics of a Quasi Zero Stiffness Vibration Isolation
Mechanism
KIRAN MUKUND (CB.EN.P2EDN14006)
Guided by - DR. B. SANTHOSH
Department of Mechanical Engineering
Amrita School of Engineering
Amrita Vishwa Vidyapeetham
July 22, 2016
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2. INTRODUCTION
Vibration is undesirable in many domains
Vibration isolation is simply the process of isolating an object or a
space from the source of vibration
Passive isolation
Air isolator
Mechanical springs
Elastomer pads
Negative stiffness isolator
Active isolation
sensors and actuators that produce destructive interface with vibration
signals
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3. How passive isolation works
Contains mass, spring, damping element
0 2 4
−6
0
6
Frequency
Transmissibility
ζ=0
ζ=0.05
ζ=5
√
2
System attenuate effective isolation when ω =
√
2ωn
So isolation band width can be increased by reducing the natural
frequency of the system
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4. Reducing natural frequency
Natural frequency is defined by the mass and stiffness of the system
ωn = k
m
We can reduce natural frequency by reducing the stiffness
Reducing stiffness will leads to reduction in load bearing capacity of
the system
Here is the importance of Negative stiffness mechanism
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5. Negative stiffness mechanism (NSM)
Negative stiffness mechanism (NSM ) vibration isolation system -
passive approach for achieving low vibration environments and
isolation against low frequency vibrations
NSM reduces effective stiffness of the system without reducing the
weight bearing capacity and leads to High-Static Low-Dynamic
stiffness
Figure: Schematic representation of NSM
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6. OBJECTIVES
Modelling Negative stiffness mechanism as a fully nonlinear system
without approximations and understand the equilibrium states and its
stability characteristics
Investigate the dynamics of the system for external harmonic
excitation using Numerical Integration and Multi Harmonic Balance
Method
Investigate the effect of mean load on the dynamics of system
Study the isolation capabilities of the system
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7. Dynamic model of NSM
Equation of motion for forcing excitation
¨x +2ζ ˙x +x +rx 1−
1
√
α2 +x2
= σ +f cos(ωτ) (1)
Equation of motion for base excitation
¨z +2ζ ˙z +z +rz 1−
1
√
α2 +z2
= σ +Aω2
cos(ωτ) (2)
Kiran Mukund Dynamics of a QZS Vibration Isolation Mechanism 7/40

8. SQZS characteristics
For a Stable Quasi Zero Stiffness equilibrium, generalized second
order equation is
¨x +P(x) = 0
P(x0) = 0 P (x0) = 0
where P(x) = P1(x)−P2(x)
P1(x) = x +rx 1− 1√
α2+x2 , P2(x) = σ
Intersection points between curve P1(x) and P2(x) gives equilibrium
points
Stability of equilibrium points can be checked by P (x)
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9. 0 < α < r
1+r
0
0
x
P1(x)&P2(x)
P
1
(x)
σ
0
0
x 10
−3
x
P
′
(x)
0 < α < r
1+r
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10. α = r
1+r
0
0
x
P1(x)&P2(x)
P
1
(x)
0
0
x 10
−4
x
P
′
(x)
α = r
1+r
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11. α > r
1+r
0
0
x
P1(x)&P2(x)
P
1
(x)
σ
0
8
x 10
−4
x
P
′
(x)
α > r
1+r
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12. The potential function
V (x) =
x
0
rx 1−
1
√
α2 +x2
−2 0 2
0
2
x
V(x)
Double well potential
−2 0 2
0
2
x
V(x)
Neutral potential
−2 0 2
0
2
x
V(x)
Single well potential
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13. NUMERICAL SIMULATION
Bifurcation diagram with α = 0.54,ζ = 0.01,f = 0.01
−2 0 2
0
2
x
V(x)
Double well potential
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14. At ω = 0.07 period-2 solution exists.
−0.2 0 0.2
−0.05
0
0.05
displacement
P9/2
8.9 8.95
x 10
4
−0.2
0
0.2
displacement
Time
P
9/2
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15. At ω = 0.1 period-1 solution exists.
−0.2 −0.1 0 0.1 0.2
−0.04
−0.02
0
0.02
0.04
displacement
velocity P3/1
6.23 6.24 6.25 6.26
x 10
4
0
displacement
Time
P3/1
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16. At ω = 0.13 period-3 solution exists.
−0.2 −0.1 0 0.1 0.2
−0.06
0
0.060.06
displacement
velocity P7/3
4.8 4.81 4.82
x 10
4
−0.2
0
0.2
displacement
Time
P7/3
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17. At ω = 0.15 chaotic solution exists.
−0.2 0 0.2
−0.04
0
0.04
displacement
velocity
Chaos
4.2 4.24 4.28
x 10
4
−0.2
0
0.2
Time
Displacement
Chaos
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18. Multi Harmonic Balance method (MHBM)
To generate the steady state periodic solutions
Computationally more efficient than Numerical Integration and
Averaging method
The method possesses advantages in studying systems with strong
nonlinearities
Arc length continuation technique is used to trace unstable solution
branch
Floquet theory is used to study stability of solution
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19. Procedure of multi harmonic balance method
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20. Amplitude-Frequency response for zero mean load
0 0.2 0.4 0.6 0.8 1 1.2
0
0.4
0.8
1.2
ω
X
Stable solution
Unstable solution
Numerical Integration result
SN
Jump up
SN
SBB
Jump down
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21. Solutions at different frequencies
−0.2 0 0.2
−0.4
0
0.4
displacement
velocity
Figure: At ω = 0.08
−0.4 0 0.4
−0.4
0
0.4
displacement
velocity
Figure: At ω = 0.5
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22. EFFECT OF MEAN LOAD
Mean load σ is not zero in practical case
Here the effect of mean load in Amplitude-Frequency response is
studied
Equation of motion for mass excitation is
¨x +2ζ ˙x +x +rx 1−
1
√
α2 +x2
= σ +f cos(ωτ) (3)
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23. For σ = 0
−2 0 2
0
2
x
V(x)
Double well potential
At ω = 0.1 shows period-1 solution
−0.2 −0.1 0 0.1 0.2
−0.04
0
0.04
displacement
velocity
P3/1
6.046 6.047 6.048 6.049
x 10
5
−0.2
0
0.2
Time
displacement
P
3/1
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24. For σ = 0.01
−2 0 2
0
2
x
V(x)
Neutral potential
At ω = 0.1 shows chaotic solution
−0.1 0 0.1 0.2 0.3
−0.06
0
0.060.06
displacement
velocity
Chaotic
6.084 6.086 6.088
x 10
5
0
0.12
0.24
Time
displacement
Chaotic
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25. For σ = 0.05
−2 0 2
0
2
x
V(x)
Single well potential
At ω = 0.1 shows period-1 solution
0.23 0.24 0.25 0.26
−1
0
1
x 10
−3
displacement
velocity
P
1
6.04 6.045
x 10
5
0.24
0.26
Time
displacement
P
1
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26. For σ = 0.1
0 0.5 1 1.5 2
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
ω
X
At ω = 0.1 shows period-1 solution
0.31 0.32 0.33
−2
0
2
x 10
−3
displacement
velocity
P
1
6.06 6.07
x 10
5
0.32
0.33
Time
displacement
P1
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27. For σ = 0.5
0 0.5 1 1.5
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
ω
X
At ω = 0.1 shows period-1 solution
0.64 0.65
−7.8038
2.1962
x 10
−4
displacement
velocity
P
1
6.058 6.06
x 10
5
0.64
0.65
Time
displacement
P
1
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28. For σ = 1
0 0.5 1 1.5
0.7
0.8
0.9
1
1.1
1.2
ω
X
At ω = 0.1 shows quasi periodic solution
0.92 0.922 0.924 0.926 0.928 0.93 0.932 0.934
−3.6933
3.5067
x 10
−3
displacement
velocity
Quasi periodic
9.206 9.207 9.208
x 10
5
0.92
0.93
Time
displacement
Quasi periodic
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29. Amplitude-Frequency response for various mean loads
σ = 0
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
ω
X
σ = 0.01
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
ω
X
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30. σ = 0.05
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
ω
X
σ = 0.1
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
ω
X
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31. σ = 0.5
0 0.5 1 1.5
0
0.5
1
1.5
ω
X
σ = 1
0 0.5 1 1.5
0
0.5
1
1.5
ω
X
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32. VIBRATION ISOLATION STUDY ON SQZS
MECHANISM
Averaging method is used to study the effect of parameters on
transmissibility
Averaging approximate frequency - response relationship
2aζω2
+ −1(1+r)+aω2
+
ar
π
I(a)
2
−f 2
= 0 (4)
where
I(a) =
2π
0
cos2 ψ
α2 +a2 cos2 ψ
dψ =
4
a2
α2 +a2EllipticE
a2
α2 +a2
−
α2
α2 +a2
EllipticK
a2
α2 +a2
(5)
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33. Maximum force transmitted to the base
|Fmax | = (2aζω)2 +a2 (1+r)−
rα2
(α2 +a2)
3
2
2
(6)
where
T =
Fmax
f
(7)
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34. Effect of parameters on amplitude-frequency response
The effect of geometrical arrangement ratio α
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35. The effect of excitation magnitude f
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36. The effect of damping ratio ζ
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37. Transmissibility
0 0.4 0.8 1.2
0
1
2
ω
Transmissibility
α=0.54 ζ=0.01
α=0.54 ζ=0.03
α=0.54 ζ=0.05
α=0 ζ=0.01
α=0 ζ=0.03
α=0 ζ=0.05
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38. CONCLUSION
Investigated the dynamics of stable quasi zero stiffness vibration
isolation mechanism in time domain and frequency domain
The system exhibited periodic, quasi periodic and chaotic solutions
Saddle node, symmetry breaking and period doubling bifurcations are
identified
Introduction of mean load drastically changed the bifurcation
behaviour of the system
The isolation bandwidth increased compared to linear passive isolator
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39. PUBLICATIONS
Conference paper
Title : Dynamics of a stable-quasi-zero-stiffness isolator mechanism
using multi harmonic balance method
Status : Accepted in International Conference on Systems Energy and
Environment
Journal paper
Title : Effect of mean load on the dynamics of a quasi zero stiffness
isolator mechanism
Status : Will submit to Journal of Sound and Vibration
Kiran Mukund Dynamics of a QZS Vibration Isolation Mechanism 39/40

40. THANK YOU !!!
Kiran Mukund Dynamics of a QZS Vibration Isolation Mechanism 40/40