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MECH ENG 3028 Dynamics & Control II – Vibrations

Assignment 3

To be submitted to the Dynamics & Control II submission box next to the School of

Mechanical Engineering office by 5pm on Monday 18th October 2010.

1) A light beam hanging vertically from a fixed ceiling has of modulus E is

loaded at equal intervals along the beam with lumped masses of 3m, 2.2m and

1.4m respectively. The masses are separated by a distance l/3, and the heavier

mass is closest to the ceiling (and a distance l/3 from the ceiling where the

beam is fixed)

a) Derive the equations of motion for the vertical displacement (i.e.

longitudinal vibration) of the masses in matrix form.

b) Write the characteristic equation for the symmetric eigenvalue

problem, and use Matlab to calculate the natural frequencies and the

mode shapes as a function of E, m and l without using Matlab's

symbolic calculation features. [Hint: this is possible by rearranging or

normalising the characteristic equation so that the transformed stiffness

matrix is only comprised of numbers.]

Show your final transformed stiffness matrix.

c) Plot the mode shapes and describe with diagrams the relative motion of

masses for each mode.

d) What are the modal frequencies if the beam is made of steel of total

length 1.5 m, and cross-sectional area 1 x 10-4 m

2? Also, m = 2.0 kg.

Note: m is not the mass of the beam but the scaling factor for the

loaded masses in the problem description.

You may assume that at static equilibrium the loaded masses have negligible

effect on the un-loaded length of the beam.

2) Suppose the system described in Q1 is released from a rest condition, where it

is initially under an additional under tension that displaces the bottom mass

downward whilst leaving the top two masses un-displaced relative to the

normal static equilibrium of the system. i.e. the bottom 3rd of the beam is

stretched slightly compared to its normal static length.

Determine the relative modal amplitudes of the normal modes of the 3-body

system for the ensuing motion.

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3) A motor is mounted on a platform that is observed to vibrate excessively at an

operating speed of 1500 rpm producing a 200 N excitation force.

a) Design a vibration absorber (un-damped) to add to the platform. Note

that in this case, the absorber mass will only be allowed to move a total

peak to peak displacement of 25 mm because of geometric and size

constraints.

b) Also calculate the range of motor speeds for which the vibration

absorber can be considered to be useful, if this is conservatively

defined to be 2<oF

Xk

Note: The motor operating speed is 1.2 times the measured natural

frequency of the motor/platform system, and the mass of the

motor/platform system is 20 kg.

c) What are the natural frequencies of the 2-body system produced by the

addition of the absorber?

4) A copper pipe is arranged as a fixed-fixed beam. The pipe has a length of 1.2

m, an outer diameter of 25 mm and a 2.5 mm wall thickness. Determine the

three lowest natural frequencies for longitudinal, torsional, and transverse

vibration of the beam. Hint: You will need to research values for the Young’s

modulus, density and Poisson ratio for copper.

5) Torsional modes of a beam of variable cross-section can be modeled by

considering the object to be constructed from a finite number of lumped

masses, connected by springs. This is modeling a continuous system in

torsional vibration with an n-dof system of lumped masses and springs in

linear 1-D motion, where n is the number of lumped masses chosen for the

model.

Consider a tapered, conical steel beam, clamped at its fixed end. The diameter

is 40 mm at the clamp, and it tapers down to 20mm diameter at its maximum

length of 160 mm.

• Assume the beam to be circular in cross-section at each position along

the beam

• Assume elastic deformation.

• Ignore gravity.

• Consider torsional vibrations only.

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Method

• Define n as a parameter.

• Determine the inertia and stiffness matrices for your model. Consider

your model springs stiffness’s and lumped masses very carefully so that

it best approximates the actual system. If n is large you will want to

automate the assignation of matrix elements (with “for” loops for

instance).

• Use appropriate MATLAB matrix operations to transform your problem

into an eigen-value problem, and then solve.

• When you are confident of your program, make n sufficiently large to

model a continuous system.

Issues you should comment on:

• How confident are you in your calculated frequencies? Is n large enough

to produce accurate vales.

• How confident are you with your model? What issues affect the validity

of the model?