Reproducing the Velador Experiment
Lateral Beam Deflection
If the beam is allowed to rotate some angle θz about its central axis, a change in observed lateral deflection will occur as force components change in the instrumentation’s frame of reference. Elevational rotation can also add a small lateral deflection, although not as great as the same axial rotation. Additional error can be created if instrumentation is side-mounted on the support beam, further increasing its radius of axial rotation. Instrumentation should preferably be centrally mounted at the center of the beam’s end face to mimimize its axial radius and employ a beam path along the central axis of the support beam.
Lateral Force Due to Axial Rotation
Substitution of the Fx’ term into the formula for self-weight deflection yields an estimate for lateral deflection due to self-weight under axial rotation.
Dynamic lateral displacement greater than 61 μm is not possible due to axial rotation alone at less than some limiting angle θmax.
Table 3 - Limiting Angles of Rotation about Axis (Degrees)
Material | End Support | Center Support |
Yellow Pine 4x4” Wood Beam | 1.2 | 0.4 |
Yellow Pine 6”x6” Hollow Beam | 6.7 | 2.7 |
Yellow Pine 8”x8” Hollow Beam | 7.2 | 3.0 |
PVC 6” Pipe | 0.4 | 0.1 |
Carbon Steel 6” Pipe | 6.4 | 2.6 |
PVC 4” Pipe | 0.1 | 0.0 |
Carbon Steel 4” Pipe | 2.8 | 1.1 |
Actual angles of rotation should be limited to 25% or less of these values.
A greater limiting angle makes a beam less sensitive to self-weight displacement due to small axial rotations. 1”x6” & 2”x6” Yellow Pine Lumber and 6” Schedule 40 carbon steel pipe are superior to other selected materials in this regard. However, 6” schedule 40 carbon steel pipe is not an optimal construction material for the hobbyist. The weight of a 3m length of 6” schedule 40 carbon steel pipe exceeds the OSHA standard for medium lifting, and is not suitable for fine manipulation by hand. Its cost also typically exceeds US $100 per 3m section.
An effective compromise is the use of a hollow beam constructed of 1”x6”x10ft yellow pine lumber. The maximum self-weight deflection of this construction is comparable to that of 6” carbon steel pipe, and interior mounting of instrumentation is possible. The assembled weight of this support beam is comparable to that of an equivalent length of 4”x4” lumber, and (unlike a hollow beam assembled from 2”x6” lumber) its weight will meet the OSHA standard for medium lifting.
Schedule 40 PVC piping of either listed size is unacceptable as a support beam material.
Static self weight deflections are small enough to be neglected for all materials considered. However, dynamic self-weight deflections created by microscale variations in the mounting configuration during the course of azimuthal rotation remain an important source of measurement error. Simple supports in compression can introduce additional lateral deflection by introducing axial torque if the support’s contact surface dimensions vary by even a few dozen microns during azimuthal rotation. Minimizing these variations requires the use of as few supporting surfaces as possible with as small a contact area as possible, and care should be taken to choose a configuration with significant restoring forces to return the apparatus to the same vertical orientation each time it is rotated azimuthally. The support beam should preferably hang from a single pair of mounts on either side of its center of gravity. A triangular trapeze linking to a single suspension cable is desirable, although this will require an additional low tension end restraint on either the trapeze or the support beam itself to restrict rotation under any residual restoring torque. This end restraint may have an effect on vertical equilibrium, but the tolerance for elevational motion is much greater than for axial motion because elevational rotation introduces proportionally less lateral axial torque (in proportion to the ratio of width to length), and free vertical motion of the support beam end as great as 1 cm may be acceptable as long as there is no corresponding axial rotation.
Thermal Deformation
Flexure due to thermal expansion and contraction must also be considered.
Average coefficients of thermal expansion for the materials considered are:
Table 5
Material | αTE ( deg C) |
Carbon Steel Piping | 6.5·10-6 |
PVC Piping | 3.0·10-5 |
Yellow Pine | 1.1·10-5 |
Thermal expansion in yellow pine will be approximately double that evolved in carbon steel at the same temperature. PVC piping should not be considered because of its relatively large self-weight deflection.
Maximum possible lateral displacement of instrumentation due to thermal gradient across the beam is computed using equation 4.
Where l is the length of interest, depending on the gradient. (Note that the Equation 4 result is only ½ the maximum thermal deformation of the board. This value assumes centrally mounted instrumentation. Lateral deflection of side-mounted instrumentation will equal or exceed the thermal deformation of the board.) The largest lateral deflection will occur for the largest length of interest, which is the length of the support beam, L = 3.05 m. Maximum allowable thermal gradients ΔTmax can be computed:
Table 6
Material | αl2/2d (μm/K) | ΔTmax (deg C) |
6" Carbon Steel Piping | 180 | 0.33 |
6" PVC Piping | 829 | 0.07 |
Yellow Pine | 403 | 0.15 |
Actual thermal gradients should be limited to 25% or less of these values.
Yellow pine is an acceptable support beam material for lateral thermal gradients of less than 1 deg C/m. Carbon steel piping remains acceptable for lateral thermal gradients of up to 2 deg C/m. The limiting thermal gradient necessary for use of PVC pipe (0.4 deg C/m) is not attainable using the anticipated experimental design.
Average thermal conductivities of the materials under consideration are:
Table 7
Material | k (W/m) | R (m2/W) |
6" Carbon Steel Piping | 47 | 0.00031564 |
6" PVC Piping | 0.19 | 0.07807912 |
6" Yellow Pine | 0.15 | 0.38704762 |
Thermal resistance values are computed using equation 5.
In comparison to a 2.2cm thickness of wood, the insulation value of 0.7mm thick steel is negligible. This implies that the response of a 6” steel pipe support beam to any external thermal gradient will be up to 600 times more rapid than that of a 6” yellow pine beam. It should be possible to isolate thermal effects using large magnitude changes in the period of measurement cycles. (Thermal deformation is not instantaneous, and will respond to changes in the duration of measurement cycles.) However, the ratio of the period changes must be at least 600 times greater for a steel support beam to show the same difference in thermal flexure.
Yellow pine is a more effective insulator than carbon steel and equalizes to ambient temperature on a timescale which is more than two orders of magnitude greater with only double the flexure. This allows more detailed investigation of thermal effects using a support beam constructed of yellow pine.
Yellow pine appears to be the construction material of choice for this project. Oven-cured or well seasoned, non-pressure-treated wood should be employed, for its light weight. Care should be taken to prevent exposure to large amounts of moisture, and external paint may be desirable.
Because thermal deflection of the beam is time dependent, and it is desirable for this response to be as slow as possible in relation to the measurement period, additional insulation on both internal and external surfaces is recommended for further slow thermal responses.
Combined lateral deflections from all sources must not exceed 25% of that measured for the laser beam by Osadchey. This further reduces the allowed values and gradients to:
θALLOW = 1.7 degrees for end supports
= 0.3 degress For center supports
ΔTALLOW = 0.04 K (0.3 K/m)
Care must be taken to attain these allowed limits.
