Relationship between the reciprocity of transfer functions for mechanical vibration systems and optimal design formulas of dynamic vibration absorbers

Abstract

Transfer functions are often employed to evaluate the vibration characteristics of mechanical and electrical systems. Within a group of transfer functions, some particular ones have the property that swapping the coefficients of the two terms in the equation does not change the properties of the original function. Such functions are said to have a reciprocity property. Mechanical vibratory systems typically have three transfer functions; namely, compliance, mobility, and accelerance. For a single-degree-of-freedom system, the mobility transfer function has the reciprocity property; for a two-degree-of-freedom system, no transfer function has the reciprocity property; and for a three-degree-of-freedom system, one transfer function has the reciprocity property. This paper shows that the presence of transfer functions with the reciprocity property is extremely important for obtaining simple optimal design formulas for dynamic vibration absorbers.

Keywords

• dynamic vibration absorber
• damped primary system
• h∞ optimization
• h2 optimization
• formula manipulation
• exact algebraic solution
• mobility transfer function
• reciprocity