In building construction, there is a reciprocal relationship between forces within and external to the structure that directly affect stability and longevity. Fundamentally, damped oscillations operate within a system where a resistance (oftentimes passive) is applied to the structure in direct conflict with the natural or imposed oscillation, thereby stopping the movement (Knight, 2007).
Forced oscillations also act upon a structure as harmonic resonance is neared. In such systems, there is a natural oscillating frequency (number of oscillations completed per second) which operates within the structure, and an external force termed the driving frequency which acts upon the system. Large differences between these two forces do not actively define the amplitude of the oscillations; however, when they are numerically equal to each other, harmonic resonance results, highlighting the maximum amplitude of the system (Knight, 2007). Damped systems utilise force to reduce the affect of oscillations on structural integrity, thereby preserving the lifecycle of the building.
To design and appropriately calculate the functions of a damped harmonic oscillator, the following components are needed where is equal to a constant and is equal to the structural mass.
This formula represents several functions of the structural stability including where in which a critically damped case arises and equilibrium is quickly established; where the system is over-damped and equilibrium is slowly reached; and where the system is under-damped and exhibiting transient behaviour (Elert, 2007).
Perhaps the most famous example of harmonic resonance leading to structural deformation, the Tacoma Narrows Suspension Bridge in the United States has been oft studied by scientists and scholars to determine the forced collapse. The product of a galloping oscillation, this structure was subjected to high wind forces (35mph) which excited the bridge’s transverse vibration mode, resulting in three hours of motion with an amplitude of 1.5 feet (Irvine, 1999). A supplemental increase in wind to 42 miles per hour caused dampening cables to snap, resulting in an unbalanced load condition that increased the amplitude to 28 feet, and ultimately ended in collapse (Irvine, 1999). More modern examples of structures functioning under the support of dampers include Victory Monument on Poklyonnaya Hill in Russia which features three dynamic oscillation dampers, fifteen flexural oscillation dampers, and one torsional oscillation damper (“High-Rise Constructions,” 2008). Representative of a system utilising both active and passive damping forces to reduce structural vibrations, this structure rises 141.8 m and retains a skeletal framework of steel latticework, requiring the enhanced oscillation damping.
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In high rise building construction, oscillations due to wind forces acting in both linear and non-linear capacities directly contribute to structural instability. Etkin and Hansen (1984) note that within such systems, artificial dampers limit the motions and resulting stresses and through a determination of the amplitude of response consistent with wind related variables, predictive measures can identify the maximum reaction. Similar forces directly contribute to structural destabilisation, undermining integrity through resonance and incumbent force variables. In 1994 a crane in Germany demonstrated the influence of galloping induced oscillations as it was destroyed by fatigue cracks in the tension bars during the bending mode at resting state (Hortmanns and Ruscheweyh, 1997). Resulting from wind based forces in a state of quasi-stationary vibrations, such structural conflicts oftentimes occur in large scale construction applications and cylindrical structures (Hortmanns and Ruscheweyh, 1997). Theoretical investigation of multiple vector damped linear systems has determined that resonant modes are free of coupling due to the damping forces, causing the system to behave as a sum of independent one-dimensional subsystems (Mathieu, 1965). The initial formulaic calculation for such a system is as follows:
When coupled with Raleigh’s assumption of proportion between stiffness and damping, this formula enables multiple variable calculations within a linear system that are mathematically independent of system counterparts (Mathieu, 1965). Effective in determining the relationship between frequency and harmonic balance at differing segments as well as varied temporal intervals, this equation is one more step towards comprehensive structural analysis. Considering that high rise structures demonstrate an incidence of wind-generated harmonic flux as well as torsional sinusoidal wave vibrations based on innate building oscillations, calculating amplitude at varied temporal positions offers long term predictions of stability and maximum collapse potential (Katagiri et al., 2001).
The preceding sections demonstrate the innate relationship between harmonic resonance and the forces which act both internally and externally to a structure. Ultimately, resistance is derived through damped systems or a forced resonance, thereby counteracting the effects of wind, torsional movement, and ground shifting. More modern investigation into differentials between linear and non-linear systems as exemplified by exploiting Raleigh’s formulae in Mathieu demonstrate that variables within the resonant system oftentimes operate individually, thereby prescribing unique forces without concurrent incidence within counterparts. Long term implications of such predictive mechanisms include dynamic damping integration which acts on a sector basis to minimise torsion.
References
Elert, G. (2007) The Chaos Hypertextbook. Glenn Ellert. Accessed on 29/11/08 From: http://hypertextbook.com/chaos/41.shtml.
Etkin, B; Hansen, J.S. (1984) “Effect of a Damper on the Wind-Induced Oscillations of a Tall Mast.” Journal of Wind Engineering and Industrial Aerodynamics, Vol. 17, pp. 11-29.
“High Rise Constructions.” (2008) Melnikov Institute. Accessed on 29/11/08 From: http://www.stako.ru/show_prj_list.php?&id=arch_high&lang=eng&data=arch_high&prn=yes.
Hortmanns, M; Ruscheweyh, H. (1997) “Development of a Method for Calculating Galloping Amplitudes Considering Nonlinear Aerodynamic Coefficients Measured with the Forced Oscillation Method.” Journal of Wind Engineering and Industrial Aerodynamics, Vol. 69, pp. 251-261.
Irvine, T. (1999) “The Tacoma Narrows Bridge Failure.” December. Accessed on 29/11/08 From: http://www.vibrationdata.com/Tacoma.htm.
Katagiri, J; Ohkuma, T; Marikawa, H. (2001) “Motion Induced Wind Forces Acting on Rectangular High-Rise Buildings with Side Ratio of 2.” Journal of Wind Engineering and Industrial Dynamics, Vol. 89, pp. 1421-1432.
Knight, R.D. (2007) Physics for Scientists and Engineers. New York: Pearson Education.
Mathieu, J.P. (1965) “On Damped Vibration Theory.” International Journal of Mechanical Science, Vol. 7, pp. 173-182.
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