Mechanical Engineering Department
Lowell, Massachusetts 01854
University of Massachusetts Lowell
978-934-4000
Cantilever Beam Experiment
Background
A disk drive manufacturer is redesigning several disk drive armature mechanisms. This is the result of
evaluation of the disk drive in “extreme” environments (drop loads). Typical disk drive armatures are
shown in Figure 1 along with snapshots of drop load using high-speed photography; these have revealed
significant armature/head displacements and accelerations resulting in undesirable separation of the
head mechanism from the armature.
Typical Armature Arrangement
Armature and Head Mechanism Separation - 130g/1msec Simulated Pulse
Figure 1 - Several Armatures and Snapshots using High-Speed Photography
As part of the redesign process, the armature’s dynamic characteristics must be identified. This is
needed to determine the natural frequency, damping and tip displacement/acceleration at the read/write
head of the armature for overall performance evaluations. While the first natural frequency is expected
to cause the largest displacements, the second natural frequency is also of concern due to the high
accelerations anticipated. A redesign of the armature along with both active and passive damping
treatments are anticipated to provide the best overall solution to the problem.
In order to evaluate this problem, several measurement scenarios are envisioned. Due to the high cost of
prototype development, a simple cantilever structure will be used to evaluate measurements using strain
gage, eddy current probe, accelerometer and LVDT measuring devices; three of these devices must be
used to assess the performance of the disk drive armature in the production vibration screening that will
be performed. Vibration screening tests will be routinely performed to confirm the adequacy of
production hardware due to vibration environments.
22.403 – Final Project - Cantilever Beam Experiment
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Rev 101806
Mechanical Engineering Department
Lowell, Massachusetts 01854
University of Massachusetts Lowell
978-934-4000
Assignment - Develop a mass, spring, dashpot model to characterize the cantilever beam
An analytical and experimental model needs to be developed for evaluation of the various designs.
Certain aspects of the model are to be developed using analytical approaches and others using
experimental approaches. However, where possible, both analytical and experimental approaches are
required together to assure that an accurate dynamic model is obtained. It is required that the tip
displacement of the cantilever be evaluated using both displacement and acceleration approaches using
three different measuring type devices. Correlation of the measured displacement and acceleration is
required to assure that an accurate dynamic model is obtained. The natural frequency and damping of
the system is to be evaluated. While the second natural frequency is of concern, the first steps in the
development of the analytical and experimental models will only be performed for the first natural
frequency of the system.
Theory
A mathematical model can be used to represent the characteristics of the system. Any vibration
textbook contains the material necessary; Reference 1 was used as the reference for the material
presented herein. However, all analytical models inherently have assumptions relating to the material
properties, elastic properties, boundary conditions and damping. As such, these analytical
representations must always be verified through the use of accurately measured dynamic data.
Th simplest model is the single degree of freedom lumped mass model defined by second order
differential equation with constant coefficients. This model is shown in Figure 2.
x(t)
f(t)
m
c
k
Figure 2 – Single Degree of Freedom Model
The equation of motion describing this is system can easily be shown to be
d2x
dx
m 2 + c + k x = f (t)
or
m&x& + c x& + k x = f ( t )
dt
dt
(1)
where m is the mass, c is the damping and k is the stiffness with the displacement, velocity and
acceleration and the forcing function as shown. Several pertinent relationships are shown in Reference
2. These include the natural frequency, damped natural frequency, damping factor, critical damping,
and techniques for estimating damping; other notes on signal processing and experimental modal
analysis are also contained in that document.
22.403 – Final Project - Cantilever Beam Experiment
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Rev 101806
Mechanical Engineering Department
Lowell, Massachusetts 01854
University of Massachusetts Lowell
978-934-4000
A continuous solution can be obtained for the cantilever beam or an analytical model can be developed
using the finite element technique in a software package such as MATLAB [3]. (Reference 4 contains
some brief notes concerning the finite element modeling process as well as some rudimentary MATLAB
script files for the generation of a simple cantilever beam model). Using some basic strength of
materials approximations along with the continuous beam vibration equation, an equivalent model can
be developed for analysis purposes.
The mode shapes for a continuous cantilever beam are given in [1] as
f n ( x ) = A n {(sin β n L − sinh β n L)(sin β n x − sinh β n x )
+ (cos β n L − cosh β n L)(cos β n x − cosh β n x )}
(2)
where n = 1,2,3,K ∞ and β n L = nπ
and the natural frequencies are given as
ω n = α 2n
EI
m L4
where α n = 1.875, 4.694, 7.855
(2)
The frequencies and mode shapes are shown in Figure 3.
Figure 3 – Frequencies and Mode Shapes for Cantilever Beam
22.403 – Final Project - Cantilever Beam Experiment
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Rev 101806
Mechanical Engineering Department
Lowell, Massachusetts 01854
University of Massachusetts Lowell
978-934-4000
From Strength of Materials, the deflection, x, at the tip of a cantilever beam is given by
x = P L3 / 3 E I
(3)
where
x
P
L
E
I
tip displacement
applied load (also referred to as F)
length of the cantilever beam
Young’s Modulus of Elasticity
bending moment of inertia
The deflection at the end of the cantilever beam can be expressed as
F=k x
(4)
and therefore, the stiffness of the cantilever beam can be expressed as
k = 3 E I / L3
(5)
Realizing that the natural frequency is
ωn =
k
m
(in rad/sec)
(6)
and the natural frequency for the first mode of the cantilever from the continuous solution is
ω n = (1.875) 2
EIg
w L4
(where w - weight and g – gravitational constant)
(7)
allows the effective mass at the tip of the cantilever beam to be determined. This approximation allows
the cantilever beam to be modeled as a single degree of freedom system since the mass and stiffness are
known.
22.403 – Final Project - Cantilever Beam Experiment
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Rev 101806
Mechanical Engineering Department
Lowell, Massachusetts 01854
University of Massachusetts Lowell
978-934-4000
Equipment
The Mechanical Engineering Laboratory has an assortment of equipment to evaluate this beam system.
Measurements on the beam must be made with a minimum of three different measuring devices
simultaneously to determine the beam tip displacement during vibration as well as the natural frequency,
damped natural frequency and damping in the system. Time domain and/or frequency domain
techniques can be employed in the solution of this problem.
Post Analysis and Report
The report should address (but is not limited to) the test setup, calibration, test procedure, digital data
acquisition system, underlying principle of operation for the transducers and acquisition system,
measurements made, numerical processing of the data, problems associated with the data collection
and/or reduction, digital signal processing considerations, accuracy, analog measuring devices used and
any other related information to substantiate the results presented.
The results and conclusions should address any problems that were observed. Recommendations to
improve the measurement system should be discussed. Recommendations for the production testing of
these disk drive armatures must be provided so that suitable equipment can be procured for the product
tests.
References
1) Dynamics of Structures, Humar, J.L., Prentice-Hall, 1990, ISBN 0-13-222068-7
2) Overview of Digital Signal Processing and Experimental Modal Analysis, Peter Avitabile,
excerpts of notes presented at IMAC 19 seminar on Modal Analysis, Orlando, Florida
3) MATLAB, Version 5.3, Math Works, Inc., Natick, Massachusetts
4) Some Brief Finite Element Modeling Notes (excerpted from Finite Element Modeling Notes)
and MATLAB scripts for Cantilever Beam Evaluation, Peter Avitabile, October 2001
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