Magnification function

In the steady state of an externally excited vibration system, the magnification function indicates the relationship between the input and output amplitudes as a function of the excitation frequency. The term magnification function is used in machine dynamics . The term resonance curve is used in electrical engineering and physics . The magnification function is also known as the amplitude- frequency response of the system. The amplitude response and the dimensionless magnification function can differ by a normalization factor. In order to express the magnification function independently of a special vibration system, the excitation frequency is also related to the undamped natural frequency. The magnification factor or amplification factor is the value of the magnification function at a given frequency.

overview

Example of force excitation

Mass, spring, damper system

Magnification function depending on the frequency ratio and the Lehr's damping

A linear damped oscillation system , such as the mass-spring-damper system shown opposite, can be excited to oscillate with a constant output amplitude by a periodic force acting on the mass . In this case, the amplitude of the periodically acting force represents the input variable, the oscillation amplitude of the mass is the output variable. The frequency-dependent ratio of the output variable to the input variable multiplied by a constant factor is the magnification function (see derivation of force excitation ): ${\displaystyle x}$ ${\displaystyle \alpha _{1}}$

${\displaystyle \alpha _{1}={\frac {1}{\sqrt {(1-\eta ^{2})^{2}+(2\,D\eta )^{2}}}}}$

It denotes:

• ${\displaystyle \eta }$the frequency ratio , ${\displaystyle \omega /\omega _{0}}$
• ${\displaystyle D}$the Lehr attenuation ,
• ${\displaystyle \omega }$the exciter circular frequency ,
• ${\displaystyle \omega _{0}}$ the undamped natural angular frequency of the oscillation system, also called the characteristic angular frequency.

At low excitation frequencies the enlargement function tends , at very high frequencies the enlargement function tends towards zero like . (See fig.) ${\displaystyle \eta \rightarrow 0}$ ${\displaystyle \alpha _{1}\rightarrow 1}$ ${\displaystyle \eta \gg 1}$ ${\displaystyle \alpha _{1}\approx \eta ^{-2}}$

With very little damping , the enlargement function converges towards the envelope: ${\displaystyle D\rightarrow 0}$

${\displaystyle \alpha _{1,max}={\frac {1}{\left|(1-\eta ^{2})\right|}}}$

resonance

For has the magnification function at a maximum ( amplitude resonance ) with the value ${\displaystyle D<{\sqrt {1/2}}}$ ${\displaystyle \alpha _{1}}$ ${\displaystyle \eta ={\sqrt {1-2\,D^{2}}}}$

${\displaystyle \alpha _{1,{\text{max}}}={\frac {1}{2D{\sqrt {1-D^{2}}}}}}$.

With low damping , the resonance amplitude is inversely proportional and the width of the resonance is directly proportional to . With strong damping , the amplitude maximum with the value 1 is fixed . ${\displaystyle D\ll 1}$ ${\displaystyle D}$ ${\displaystyle D\geq {\sqrt {1/2}}}$ ${\displaystyle \eta =0}$

With vanishing damping, theoretically infinitely large amplitudes occur in the case of resonance. Starting from the rest position, the amplitude builds up only linearly with time during the transient process, given the force amplitude and mass of the oscillation system ${\displaystyle {\hat {F}}}$ ${\displaystyle m}$

${\displaystyle {\hat {x}}={\frac {\hat {F}}{2\,m\,\omega _{0}}}\cdot t}$.

With, the steady oscillation lags the exciting force by exactly 1/4 period (phase response −90 °, also known as phase resonance ). Therefore the energy flow is always directed into the oscillation system, while otherwise it changes its sign twice per period because the phase difference is smaller than 90 ° at a lower frequency and greater than 90 ° (and up to 180 °) at a higher frequency. The energy content of the oscillation reaches its maximum. ${\displaystyle \omega =\omega _{0}}$ ${\displaystyle \omega =\omega _{0}}$

Suggestions

If the force amplitude is independent of the excitation frequency, the magnification function directly reproduces the amplitude response up to a constant factor. With centrifugal excitation, z. B. due to imbalance , the exciting force is quadratically dependent on the frequency. This is how the magnification function results

${\displaystyle \alpha _{3}=\eta ^{2}\alpha _{1}}$

Instead of a force excitation on the mass, the vibration system can also be excited via the spring / damper element. This type of excitation is also referred to as base point excitation or path excitation . This results in the magnification function (see vibration isolation , or derivation of path excitation ). An example is the quarter vehicle as the simplest model for the vibration behavior of a car. ${\displaystyle \alpha _{4}}$

Examples of the magnification functions for different types of excitation can be found in:

Derivation

Force excitation

The derivation of the magnification function in the case of force excitation is based on the differential equation for a forced oscillation. Find the amplitude x. With mass , spring constant and damping constant it follows: ${\displaystyle m}$ ${\displaystyle c}$ ${\displaystyle d}$

${\displaystyle m\;{\ddot {x}}+d\;{\dot {x}}+c\;x=F(t)}$.

By dividing with c:

${\displaystyle {\frac {m}{c}}\;{\ddot {x}}+{\frac {d}{c}}\;{\dot {x}}+x={\frac {1}{c}}\;F(t)}$.

one obtains the differential equation:

${\displaystyle {\frac {1}{\omega _{0}^{2}}}\;{\ddot {x}}+{\frac {2D}{\omega _{0}}}\;{\dot {x}}+x={\frac {1}{c}}\;F(t)}$.

Applying the Laplace transform one obtains the transfer function:

${\displaystyle {\frac {x(s)}{F(s)}}=G(s)={\frac {1}{c}}\;{\frac {1}{1+{\frac {2D}{\omega _{0}}}\;s+{\frac {1}{\omega _{0}^{2}}}\;s^{2}}}}$

With and you get the frequency response: ${\displaystyle s=j\omega \,}$ ${\displaystyle \eta =\omega /\omega _{0}\,}$

${\displaystyle G(\omega )={\frac {1}{c}}\;{\frac {1}{1-\eta ^{2}+j\cdot 2D\eta }}}$

The amplitude frequency response is obtained as the amount of the complex frequency response:

${\displaystyle \left|G(\omega )\right|={\frac {1}{c}}\;{\frac {1}{\sqrt {\left(1-\eta ^{2}\right)^{2}+\left(2D\eta \right)^{2}}}}={\frac {1}{c}}\;\alpha _{1}(\eta )}$

The expression is called the magnification function: ${\displaystyle \alpha _{1}(\eta )}$

${\displaystyle \alpha _{1}={\frac {1}{\sqrt {\left(1-\eta ^{2}\right)^{2}+\left(2D\eta \right)^{2}}}}}$

With a given force amplitude, the displacement amplitude is thus:

${\displaystyle {\hat {x}}(\eta )={\frac {\hat {F}}{c}}\;\alpha _{1}(\eta )}$

Path excitation

Magnification function for path excitation for different damping values

The oscillator is excited by the spring / damper element with z (t). This form of excitation is called base point excitation . Find the amplitude x. The differential equation is:

${\displaystyle m\;{\ddot {x}}+d\;{\dot {x}}+c\;x=d\;{\dot {z}}+c\;z}$.

By dividing with c:

${\displaystyle {\frac {m}{c}}\;{\ddot {x}}+{\frac {d}{c}}\;{\dot {x}}+x={\frac {d}{c}}\;{\dot {z}}+z}$.

one obtains the differential equation:

${\displaystyle {\frac {1}{\omega _{0}^{2}}}\;{\ddot {x}}+{\frac {2D}{\omega _{0}}}\;{\dot {x}}+x={\frac {2D}{\omega _{0}}}\;{\dot {z}}+z}$.

Applying the Laplace transform one obtains the transfer function:

${\displaystyle {\frac {x(s)}{z(s)}}=G(s)={\frac {1+{\frac {2D}{\omega _{0}}}\;s}{1+{\frac {2D}{\omega _{0}}}\;s+{\frac {1}{\omega _{0}^{2}}}\;s^{2}}}}$

With and you get the frequency response: ${\displaystyle s=j\omega \,}$ ${\displaystyle \eta =\omega /\omega _{0}\,}$

${\displaystyle G(\omega )={\frac {1+j\cdot 2D\eta }{1-\eta ^{2}+j\cdot 2D\eta }}}$

The magnification function results here directly as the amount of the complex frequency response:

${\displaystyle \alpha _{4}={\sqrt {\frac {1+\left(2D\eta \right)^{2}}{\left(1-\eta ^{2}\right)^{2}+\left(2D\eta \right)^{2}}}}}$

An isolating effect with regard to the excitation is only given from a frequency ratio of . At zero attenuation aim and against the same envelope. ${\displaystyle \eta >{\sqrt {2}}}$ ${\displaystyle \alpha _{4}}$ ${\displaystyle \alpha _{1}}$

Often it is not the amplitude of the oscillator that is of interest, but its acceleration. With

${\displaystyle {\frac {\hat {\ddot {x}}}{\hat {z}}}=\omega ^{2}\;{\frac {\hat {x}}{\hat {z}}}\;}$

one obtains the amplitude response:

${\displaystyle {\frac {\hat {\ddot {x}}}{\hat {z}}}=\omega _{0}^{2}\;\eta ^{2}\;{\sqrt {\frac {1+\left(2D\eta \right)^{2}}{\left(1-\eta ^{2}\right)^{2}+\left(2D\eta \right)^{2}}}}=\omega _{0}^{2}\;\eta ^{2}\;\alpha _{4}}$

The expression: is the dimensionless magnification function between the path excitation at the base point and the acceleration. ${\displaystyle \eta ^{2}\alpha _{4}}$

See also

• Resonance disaster
• Resonance frequency
• resonance
• Torsion pendulum

literature

• Hans Dresig , Franz Holzweißig : machine dynamics . Springer, Berlin 2011, ISBN 978-3-642-16009-7 , pp. 554 ( limited preview in Google Book search).

Individual evidence

1. ^ Gross, Hauger, Schröder, Wall: Technical Mechanics 3: Kinetics . Springer, ISBN 978-3-642-11264-5 .
2. ↑ Manfred Mitschke: Dynamics of motor vehicles. Volume B. Vibrations. 3. Edition. Springer-Verlag, 1997, ISBN 3-540-56162-5 .
3. ↑ K. Magnus, HH Müller: Fundamentals of technical mechanics . Teubner 1982, ISBN 3-519-02371-7 .
4. ↑ F. Švaříček: control technology. Lecture notes p. 9–12. (online) ( Memento from December 28, 2016 in the Internet Archive ) (PDF; 548 kB)
5. ^ Uwe Hollburg: Machine dynamics. 2nd Edition. ISBN 978-3-486-57898-0 . (on-line)
6. ↑ Wandinger: Elastodynamik 2. Lecture notes. (online) (PDF; 230 kB)