Topic: Torsional/longitudinal/lateral Vibration of Rods

A Bio-MEMS device consists of a microscaled cantilevered rod that senses a biochemical reaction on the surface of the rod. In particular, the mass of the cantilever changes due to biochemical reaction products sticking to the surface and this in turn is detected via a change in the resonant frequency of the cantilever. The cantilever is made of silicon (Young's modulus, E = 150GPa, poissons ratio v=0.17, density ρ = 2330kg / m³) of diameter d = 5μm and length L=100μm.

a) Determine the first three natural frequencies for axial vibrations in MHz. Draw the corresponding mode shapes.

b) Determine the first three natural frequencies for torsional vibrations as a ratio of the fundamental axial natural frequency of vibration. Draw the corresponding mode shapes.

c) When in use, the Bio-MEMS device is found to pickup a layer of biomass that increases the mass per unit length of the beam by 5%. What % changes in fundamental axial, torsional and lateral frequencies result.

Topic: Transverse Vibration of Strings and Beams

On overhead catenary is used to transmit electrical power through a contact wire to trains as they move along tracks. A section of contact wire of length L and bending stiffness EI is modelled as being pinned between two masts. The tension of the contact wire is tuned so that the transverse string wave travels 4.5 times faster than the maximum speed of the train Vin„, to avoid undesirable standing wave interaction.

a) Determine the first three natural frequencies of transverse string vibration of the contact wire as a function of the length L and V_max. Ignore the bending stiffness of the wire and assume the contact wire tension is approximately constant throughout it's length.

b) Sketch the first three mode shapes of transverse string vibration.

c) Determine the first three natural frequencies of transverse beam vehicle excitation mast vibration of the contact wire as a function of the length L, mass m, and bending stiffness EL. Ignore the contact wire tension and model the boundary conditions as pinned.

d) Sketch the first three mode shapes of transverse beam vibration Tracks and explain any similarities or differences with the string modes identified in b).

e) Vehicle vibrations excite the contact wire at a moving contact point A as shown at an excitation frequency .0 of 20% of the fundamental beam natural frequency. Assuming transverse beam vibration, approximate the largest amplitude vibration deflection of the contact wire and the fractional position xi/L at which it occurs.

f) Explain if and how your answer in e) may vary if the excitation frequency Ω is 400% of the fundamental beam natural frequency.

g) Identify the conditions under which transverse string vibration assumptions are more appropriate as compared to transverse beam vibration assumptions.

Topic: Approximate Techniques - Rayleigh/DunkerleyMethods

For the overhead catenary problem in problem 2

a) The vehicle connection at A acts as an inertial mass and lowers the fundamental transverse natural frequency by a maximum of 20%, at an unknown position. Assuming beam behaviour, identify the position of the inertial mass where the maximum reduction occurs. Hence estimate the ratio of the inertial mass to the wire mass using Dunkerley's approximate method.

b) Repeat your estimate in a) using Rayleigh's method assuming an appropriate mode shape.

c) Hence determine the best estimate for the ratio of the inertial mass to the wire mass. Explain what effect the inertial mass has on all the other modes.