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 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. 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 RapidLyser™ Noise Reduction (AKA “The noisy vibrator problem”) Team Members: Problem Presenter: Yousef Daneshbod, Department of Mathematics, University of La Verne

Outline Introduction Beam Model & Rayleigh’s Principle Vibration Absorber (without damping) Vibration Absorber (with damping)

Introduction The Problem motor Sorbothane dampers Aluminum plate High speed oscillator And lysis cartridge Claremont Biosolutions

Introduction Slider Crank Mechanism

Introduction Originally the machine only had a thin metal plate at the bottom (NOISIER!) Possible Solutions: 1. Put it in a box 2. Stiffening the structure 3. 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 (

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

The Beam Model In turn, beam behaves like a simple oscillator (Vibration & Shock Isolation, Crede) upload.wikimedia.org/wikipedia/commons/archive/9/9d/ !Simple_harmonic_oscillator.gif

Rayleigh Principle Lord Rayleigh Theory of Sound (1877) R Pinned Beam x y

Rayleigh Principle for Beam Vibration and Shock Isolation,Crede,1951 Theory of Vibration with Applications, Thomson,1972

Simple Oscillator with Damper m1 c k1

The Spring-Mass System m1 c k1

Vibration Absorber m1 c k1 m2

Force Equations (no damping) m2 m1 c k1 From Newton’s Second Law:

Solve the system m2 m1 c k1 From Newton’s Second Law: Assume both masses vibrate at same frequency.

Solve the system Solve the algebraic equation for the amplitudes of the two masses: ….Aaaand we lose time dependence! Amplitude of the lower mass (our device) Amplitude of the absorber

What should the absorber be like? How do we determine m 2 and k 2 ? 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 m 2 and k 2 ? 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, X 2 and the amplitude of forced vibration, F o

What about k 1 ? Observe the denominator of the amplitude equations…

Linear Second Order Non- Homogeneous System of Equations Difficulties involved: m1 c k1 m2

Transformation into 1 st order ODE’s Any nth order Differential Equation CAN ALWAYS be reduced into a system of n first order DE’s (Crede)

Vectorizing the Problem Matrix A becomes With Forcing Term, g(t)= Equation becomes

Matrix A, with specific values

Eigen Values(times 10^5) and Eigen Vectors Solutions to the non-Homogeneous Equation *transient in nature

non-Homogeneous Solutions Of the form *solve numerically Difficulties – Possible stiff solution – Cannot solve analytically – Time