RapidLyser™ Noise Reduction
(AKA “The noisy vibrator problem”)
Problem Presenter: Yousef Daneshbod, Department of Mathematics, University of La Verne
Team Members:
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
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.
Outline

Introduction

Beam Model & Rayleigh’s Principle

Vibration Absorber (without damping)

Vibration Absorber (with damping)
Introduction

The Problem
High speed oscillator
And lysis cartridge
motor
Aluminum
plate
Sorbothane
dampers
Claremont Biosolutions
Introduction
www.cntsa.com
www.iqnewsnet.com
Slider Crank Mechanism
Introduction

Originally the machine only had a thin metal
plate at the bottom (NOISIER!)

Possible Solutions:
1.
2.
3.
Put it in a box
Stiffening the structure
Adding Damping (Crede,1951)
Introduction

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)
Introduction

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 > thickness results in behaviour
like the SDOF system subjected to a harmonic
force. (Crede, 1951)
The Beam Model

Assume device can be modeled as a pinned
(or simply supported) beam
http://physics.uwstout.edu/StatStr/statics/Beams/bdsn47.htm
The Beam Model

In turn, beam behaves like a simple oscillator
(Vibration & Shock Isolation, Crede)
http://physics.uwstout.edu/StatStr/statics/Beams/bdsn47.htm
http://upload.wikimedia.org/wikipedia/commons/archive/9/9d/2007062403102
0!Simple_harmonic_oscillator.gif
Rayleigh Principle
R
1
 y 
R
Lord Rayleigh
Theory of Sound (1877)
y
x
Pinned Beam
Tmax
1 2
  y dm
2
U max
1
  Md 
2
Rayleigh Principle for Beam
Vibration and Shock Isolation,Crede,1951
Theory of Vibration with Applications, Thomson,1972
Simple Oscillator with Damper
F0 sin t
m1
k1
c
The Spring-Mass System
F0 sin t
m1
k1
c
Vibration Absorber
m2
F0 sin t
m1
k1
c
Force Equations (no damping)
From Newton’s Second Law:
m2
F0 sin t
m1
k1
c
Solve the system
From Newton’s Second Law:
m2
F0 sin t
m1
k1
c
Assume both masses vibrate
at same frequency.
Solve 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
What 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.
What 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
What about k1?
Observe the denominator of the amplitude equations…
Linear Second Order NonHomogeneous System of Equations
mx1  cx1  (k1  k2 ) x1  k2 x2  F0 sin t
m2
m2 x2  k2 x2  k2 x1  0
F0 sin t
m1
Difficulties involved:
cx1
k1
c
Transformation into 1st order ODE’s
Any nth order Differential Equation CAN
ALWAYS be reduced into a system of n first
order DE’s (Crede)

y
 y1 
y 
 2
 y3 
 
 y4 

 x1 
 x 
 1   y  y
1
2
 x2 
 
 x 2 
Vectorizing the Problem

Matrix A becomes
0

  (k1  k 2 )

m1

0

k2


m2

1
c
m1
0
0
Equation becomes
With Forcing Term, g(t)=
0
k2
m1
0
 k2
m2
0

0

1
0



y'  Ay  g (t )
0


 F0

sin

t
m

1


0




0
Matrix A, with specific values
Eigen Values(times 10^5) and Eigen
r t
r2t
r3 t
r4t
1
Vectors
yh  1e   2 e   3 e   4 e
Solutions to the non-Homogeneous Equation
*transient in nature
105 i
ri
i 1
i 1
i2
i3
i2
i3
i4
i4
i 1
non-Homogeneous Solutions




Of the form y  a sin t  b cost
p

*solve numerically

Difficulties
–
–
–
Possible stiff solution
Cannot solve
analytically
Time