Mechanical impedance

Mechanical impedance is a measure of how much a structure resists motion when subjected to a harmonic force. It relates forces with velocities acting on a mechanical system. The mechanical impedance of a point on a structure is the ratio of the force applied at a point to the resulting velocity at that point.

The mechanical impedance is a function of the frequency of application of the force and can vary strongly as a function of the frequency itself. The lowest mechanical impedance of a system is found at resonance frequency, at this frequency the force to be applied to vibrate the system, at a given speed, will be the lowest.

Many physical systems that have in place a relationship between several physical quantities, can be described as a formed quantity that is in “input”, in a quantity that is in “output”. this relationship is called response function and is a propiety of the system. In reality the function can be of many properties of the system and therefore very complex. Often the goal is to determine this relationship starting from the most elementary components of the matter, to put it in simple words, dissecting the system. In many cases, however, it becomes more important, to analyze a response in a particular dynamic regime rather than its reaction to an arbitrary stimulus.

The impedance of a system is a property of the system, whose value becomes fundamental to describe its behavior under excitation. In the case of musical instruments, knowing the mechanical impedance of the materials it is made of is fundamental to predict the reactivity of the instrument, considering the one of the strings up to the one of the woods and alloys of which the hardware parts are made of. In essence, the kinetic energy produced by the motion of the strings of the guitar must be in a system that has an impedance value such that there is less dissipation as possible.

In an oscillating mechanical system to which a sinusoidal force \(F\) is applied, it is the complex ratio \(Z=F/v\), where \(v\) is the velocity. The mechanical impedance is considered, in general, only at the input of the system. In the case of a simple resonator with damping, it is:

\[Z=k’+i\left(\dfrac{k}{\omega}-m\omega\right)\]

where \(k’\) is the coefficient of proportionality relative to the friction force (ratio of friction force to velocity), \(k\) is the coefficient of proportionality relative to elasticity (ratio of force to elastic deformation), \(\omega =2\pi f\) where \(f\) is the frequency. The terms \(k’\) and \((k/\omega – m\omega)\) are respectively the real part R and the imaginary part X of the impedance; the latter cancels at the resonance frequency:

\[f=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\]

From the physical point of view, the impedance measures the resistance that the system opposes to an external action that tends to make it move with oscillatory motion and together the delay of its motion with respect to that external action; the phase shift φ is given by the relation tg φ = X/R. If it is zero R, the phase shift is ±90º; if it is zero X, the phase shift is zero, that is, the motion of the system is in phase with the external action.

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