Title: Headquartered in Upland, CA, the company Claremont BioSolutions LLC (CBS) offers a family of instruments to break up biological cells and spores in order to release their contents for analysis or purification. One of the instruments, the RapidLyser
1 RapidLyser Noise Reduction (AKA The noisy
vibrator problem)
Problem Presenter
Yousef Daneshbod, Department of Mathematics,
University of La Verne
Anna Belkine, Simon Frazer University, British
Columbia Chiaka Drakes, Simon Frazer University,
British Columbia Jose Pacheco, Cal-State
University, Long Beach Mark Morabito, University
of Massachusetts, Lowell
Team Members
Headquartered in Upland, CA, the company
Claremont BioSolutions LLC (CBS) offers a family
of instruments to break up biological cells and
spores in order to release their contents for
analysis or purification. One of the instruments,
the RapidLyser, has an oscillating arm that
moves a cartridge containing the liquid sample in
a packed bed of beads at a very high frequency.
The motion is similar to a metronome but at
much higher oscillation rates. In order to reduce
the noise produced during the operation of this
device, engineers at CBS are using a viscoelastic
material called Sorbothane to attach 4 circular
legs as bumpers at the corners of the
rectangular metal base that forms the bottom of
the instrument. Material properties of
Sorbothane are available at www.sorbothane.com.
In order to reduce manufacturing costs, it is
desired to use the minimal amount of Sorbothane
that still provides adequate noise reduction. The
participants at the math-in-industry workshop
could consider whether the use of 4 legs at the
corners is optimal or whether other arrangements
might work better. The number, shape, size,
thickness and placement of the bumpers can all be
varied. Parameters such as the mass of the
RapidLyser, the length of the oscillating arm,
the weight of the moving cartridge, the range of
oscillation frequencies, etc. will be provided at
the workshop. Also, although Sorbothane is the
material of choice, it would be nice to have a
model that is applicable to any viscoelastic
polymeric material.
2Outline
- Introduction
- Beam Model Rayleighs Principle
- Vibration Absorber (without damping)
- Vibration Absorber (with damping)
3Introduction
High speed oscillator And lysis cartridge
motor
Aluminum plate
Sorbothane dampers
Claremont Biosolutions
4Introduction
www.iqnewsnet.com
www.cntsa.com
Slider Crank Mechanism
5Introduction
- Originally the machine only had a thin metal
plate at the bottom (NOISIER!) - Possible Solutions
- Put it in a box
- Stiffening the structure
- Adding Damping (Crede,1951)
6Introduction
- What was done
- 2.54cm thick aluminum plate added to the bottom
- Four 2.54cm thick Sorbothane dampers included at
each corner - Why Sorbothane?
- Absorbs shocks efficiently
- Eliminates need for metal springs
- Has superior damping coefficient
(www.sorbothane.com)
7Introduction
- Limitations of Sorbothane
- Damping coefficient goes from 0.3 0.6 for given
excitation values (from 5 Hz 50 Hz) - RapidLyser oscillates at 250 Hz
- Important Note
- ½ wavelength gt thickness results in behaviour
like the SDOF system subjected to a harmonic
force. (Crede, 1951)
8The Beam Model
- Assume device can be modeled as a pinned (or
simply supported) beam
http//physics.uwstout.edu/StatStr/statics/Beams/b
dsn47.htm
9The Beam Model
- In turn, beam behaves like a simple oscillator
(Vibration Shock Isolation, Crede)
http//physics.uwstout.edu/StatStr/statics/Beams/b
dsn47.htm
http//upload.wikimedia.org/wikipedia/commons/arch
ive/9/9d/20070624031020!Simple_harmonic_oscillator
.gif
10Rayleigh Principle
Lord Rayleigh Theory of Sound (1877)
Pinned Beam
11Rayleigh Principle for Beam
Vibration and Shock Isolation,Crede,1951
Theory of Vibration with Applications,
Thomson,1972
12Simple Oscillator with Damper
13The Spring-Mass System
14Vibration Absorber
m2
15Force Equations (no damping)
From Newtons Second Law
m2
16Solve the system
From Newtons Second Law
m2
Assume both masses vibrate at same frequency.
17Solve the system
.Aaaand we lose time dependence!
Solve the algebraic equation for the amplitudes
of the two masses
Amplitude of the lower mass (our device)
Amplitude of the absorber
18What should the absorber be like?
How do we determine m2 and k2? The
natural frequency of the absorber system should
be the same as the frequency of the
forced vibrations.
19What should the absorber be like?
How do we determine m2 and k2? The
natural frequency of the absorber system should
be the same as the frequency of the
forced vibrations.
They also depend on the desired amplitude
of the absorber, X2 and the amplitude of
forced vibration, Fo
20What about k1?
Observe the denominator of the amplitude
equations
21Linear Second Order Non-Homogeneous System of
Equations
m2
Difficulties involved
22Transformation into 1st order ODEs
Any nth order Differential Equation CAN ALWAYS
be reduced into a system of n first order DEs
(Crede)
23Vectorizing the Problem
- Matrix A becomes With Forcing Term, g(t)
-
- Equation becomes
24Matrix A, with specific values
25Eigen Values(times 105) and Eigen Vectors
Solutions to the non-Homogeneous Equation
transient in nature
26non-Homogeneous Solutions
- Of the form
- solve numerically
- Difficulties
- Possible stiff solution
- Cannot solve analytically
- Time