Transcript Introduction to Designing Elastomeric Vibration Isolators

Introduction to Designing
Elastomeric Vibration Isolators
Christopher Hopkins
OPTI 512
08 DEC 2009
Introduction
• Why elastomers?
• Key design parameters
– Loading
– Configuration
– Spring rates
• Design considerations
• Steps to designing a simple isolator
Elastomers
• An eastomer is any elastic polymer
– Silicone Rubber
– Butyl Rubber
– Fluorosilicone Rubber
• Material selection dependent on application
– Ulitimate Loading
– Sensitivity to Environment
– Internal Properties
Elastomeric Isolators
• Engineered properties can meet specific
applications
– Modulus of elasticity
– Internal dampening
• Homogeneous nature allows for compact
forms
• Easily manufactured
– Molded
– Formed in place
Key Isolator Design Parameters
• Configuration
• Loading
• Spring rate
– Shear, Bulk, and Young’s modulus
– Geometry
• Ultimate strength
• Internal dampening
• Maximum displacement
Simple Isolator Configurations
• Planer Sandwich Form
• Laminate
• Cylindrical
Spring Rates
• Isolator spring rate sets system resonant
frequency
– Ratio of resonant frequency to input frequency
plus dampening control amount of isolation
• For an elastomer, spring rate is determined by
– Shape factor
– Loading: shear, compression, tension
– Material Properties: bulk, shear, and young’s
modulus
Transmissibility
Transmissibility vs. Frequency Ratio
Transmissibility
10
η = 0.01
η = 0.05
1
0
0.5
1
1.5
2
2.5
3
η = 0.1
η = 0.5
0.1
fr/f
Shape Factor
• Ratio of load area to bulge area
• Easy to calculate for simple shapes simply loaded
– Planer sandwich forms are simple
– Tube form bearings are more difficult, but can be
approximated as a planer form
Shear Spring Rate
• Design isolator to attenuate in shear if possible
• Dependent on load area, thickness, and shear
modulus
• Shear modulus is linear up to 75%-100% strain
• Shear modulus for large shape factors is also
effected by high compressive loads
• When aspect ratio exceeds 0.25 a correction
factor is added to account for bending
Compression Spring Rate
• Designed properly, compression can
provide high stiffness
• Depends on load area, effective
compression modulus, and thickness
• Effective compression modulus
– Linear up to 30% strain
– Can be tricky to compute
Tension Spring Rate
• Try to avoid having elastomeric
isolators in tension
– Low modulus
– Can be damaged by relatively low loads
• Apply preload avoids this
– Easy to do for cylindrical isolators
– Must include correction to shape factor
Finding Modulus
• Many elastomers are listed with only with
– Durometer Shore hardness
– Ultimate strength (MPa or psi)
• Contacting manufacturer may be useful
• Perform tests
• Shear stress is 1/3 Young’s modulus as poisson’s ratio
approaches 0.5
• Use Gent’s relation between Shore A hardness and Young’s
modulus (if you gotta have it now)
Effective Compression Modulus, Ec
• Dependent on shape factor, young’s modulus and
bulk modulus
• Also know as the apparent compressive modulus
• As Poisson’s ratio approaches 0.5, Ec may be
separated into three zones depending on shape
factor
– For large shape factors: Ec ≈ bulk modulus
– For small shape factors: Ec ≈ young’s modulus
– Transisiton zone for intermediate shape factors
Computing Ec
• Gent provides a reference graph
• Hatheway found empirically that the
transition zone is (Ec/E)∙(t/D)1.583=0.3660
– Can calculate Ec or the simple case of a circular
load area of diameter D and thickness t
– Find the break points
• First break point: (t/D)1.583 = 0.366∙(E/EB)
• Second break point: (t/D)1.583 = 0.366
Compression Modulus vs. Shape
Factor [Gent]
Stiffening vs. Thickness Ratio [Hatheway]
Circular Cross Section, Eb=1.2 GPa, E=1.6 Mpa
1000
Stiffening Ratio Ec/E
100
10
1
0.0001
0.001
0.01
0.1
Thickness Ratio, t/D
1
10
100
Laminate Isolator
• Shear modulus is not effected by
shape factor (if aspect ratio is
<0.25)
• Effective compression modulus
strongly influence by shape factor
• Possible to design an isolator that is
very stiff in compression and
compliant in shear
Cylindrical Isolator
• Shear
– Axial loading
– Torsion
• Compression
– Radial loading
• Shape factor calculation for
compression
Design Considerations
•
•
•
•
•
•
What is being Isolated?
What are the inputs?
Are there static loads?
What are the environmental conditions?
What is the allowable system response?
What is the service life?
Example Design Process for a Simple
Isolator
•
•
•
•
•
Single excitation frequency
Circular cross section, planar geometery
All other components infinitely rigid
Low dampening
Attenuation provided in shear
Design Process
• Specifications
– Mass, input vibration, required attenuation, max
displacement
• Use transmissibility to determine resonance
frequency and spring rate
• Find isolator minimum area (A)
– Total number of isolators and max allowable stress
• Select modulus (G)
Design Process (con’t)
• Knowing area (A), modulus (G), and spring
rate (ks), calculate thickness (t)
– Calculate radius, verify aspect ratio < 0.25 to avoid
bending effect
• Find static deflection
– Is static plus dynamic deflection < max allowable
deflection?
• Find static shear strain
– Low strain reduces fatigue (<20%)
Conclusion
• Careful selection of parameters necessary to use
methods presented. Can get complicated quick
– Low strains
– Low loads
• Try to stay clear of the transition zone between
Young’s and the bulk modulus
• For multiple input frequencies, need to consider
if dampening (η) is necessary
• May need to include considerations other than
just isolation
– Stresses due to CTE mismatch
References
• P. M. Sheridan, F. O. James, and T. S. Miller, “Design of
components,” in Engineering with Rubber (A. N. Gent,
ed.), pp. 209{Munich:Hanser, 1992)
• A. E. Hatheway, “Designing Elastomeric Mirror
Mountings,”Proc. of SPIE Vol. 6665 (2007)
• Daniel Vukobratovich and Suzanne M. Vukobratovich
“Introduction to Opto‐mechanical design”
• A. N. Gent, “On The Relation Between Indentation
Hardness and Young’s Modulus,” IRI Trans. Vol. 34,
pg.46-57 (1958)
Questions?