7.7.4 Stiffening of a Flexible String
A length of wire may be stabilized upside down by vertical HF excitation, due to an
increase in buckling load reflecting increased effective stiffness (Acheson and
Mullin 1998; Champneys and Fraser 2000; Fraser and Champneys 2002; Mullin
et al. 2003; Shishkina et al. 2008) – a phenomenon that Acheson (1997) has referred
to as ‘not quite the Indian rope trick’. The implication is that the stiffening effect is
to be taken into account under conditions of strong HF excitation, if predictions of
derived quantities such as natural frequencies, buckling loads, and equilibrium
positions should be meaningful.
Analytical predictions for the stiffening effect of HF excitation for a
clamped-free string has been derived (Thomsen 2003b), which compare well with
laboratory experiments for a wide range of conditions. We encountered a model for
this system in Sect. 7.6.3 (Fig. 7.8 and Eq. (7.123)), and derived analytical predictions for the stiffening effect for the actual case where resonance effect cannot be
ignored (GARP theory, Eq. (7.129)). The expressions for the change in effective
stiffness can be used to predict how much the string tip will lift in gravity, when the
string base is aligned and excited horizontally. This tip lift is a quantity that can
easily be observed and measured experimentally, and thus provides a means for
checking the quality of the theoretical predictions.
The laboratory experiment involved a clamped-free piano string, horizontally
aligned and subjected to longitudinal HF excitation at the clamped end. Gravity
causes the string to bend downwards, while the HF excitation generally causes it to
stiffen, and thus to straighten somewhat (Fig. 7.16). Measuring the lift of the string
tip (using an optical tracker) thus quantifies the stiffening effect.
Fig. 7.17 shows a set of acceleration responses, i.e. the tip lift as a function of
base acceleration, with the frequency kept constant at each of four values. As
appears the agreement between GARP theory and experiment is generally good.
7.8 Summing Up
Hopefully a beginning understanding of the possible effects that HF excitation may
have on mechanical systems, has by now emerged, as well as skills to analyze them
systematically. Recall from the introduction to Sect. 7.2 and Sect. 7.2.3 that the
MDSM is not the only analytical method available for this analysis; it just seems in
many cases to be more convenient than the alternatives. The examples given in this
chapter, though quite diverse, are not claimed to be inclusive or representative for
every relevant application. They are rather meant to be indicative of typical problems and phenomena, and to provide specific reference for the sections dealing with
generalized systems. If you want to expand on a beginning interest in this area,
some of the references cited at the end of Sect. 7.1 may be a place to start; in
particular Blekhman (2000) is recommended.
7.7 Specific Systems and Results – Some Examples
441
A length of wire may be stabilized upside down by vertical HF excitation, due to an
increase in buckling load reflecting increased effective stiffness (Acheson and
Mullin 1998; Champneys and Fraser 2000; Fraser and Champneys 2002; Mullin
et al. 2003; Shishkina et al. 2008) – a phenomenon that Acheson (1997) has referred
to as ‘not quite the Indian rope trick’. The implication is that the stiffening effect is
to be taken into account under conditions of strong HF excitation, if predictions of
derived quantities such as natural frequencies, buckling loads, and equilibrium
positions should be meaningful.
Analytical predictions for the stiffening effect of HF excitation for a
clamped-free string has been derived (Thomsen 2003b), which compare well with
laboratory experiments for a wide range of conditions. We encountered a model for
this system in Sect. 7.6.3 (Fig. 7.8 and Eq. (7.123)), and derived analytical predictions for the stiffening effect for the actual case where resonance effect cannot be
ignored (GARP theory, Eq. (7.129)). The expressions for the change in effective
stiffness can be used to predict how much the string tip will lift in gravity, when the
string base is aligned and excited horizontally. This tip lift is a quantity that can
easily be observed and measured experimentally, and thus provides a means for
checking the quality of the theoretical predictions.
The laboratory experiment involved a clamped-free piano string, horizontally
aligned and subjected to longitudinal HF excitation at the clamped end. Gravity
causes the string to bend downwards, while the HF excitation generally causes it to
stiffen, and thus to straighten somewhat (Fig. 7.16). Measuring the lift of the string
tip (using an optical tracker) thus quantifies the stiffening effect.
Fig. 7.17 shows a set of acceleration responses, i.e. the tip lift as a function of
base acceleration, with the frequency kept constant at each of four values. As
appears the agreement between GARP theory and experiment is generally good.
7.8 Summing Up
Hopefully a beginning understanding of the possible effects that HF excitation may
have on mechanical systems, has by now emerged, as well as skills to analyze them
systematically. Recall from the introduction to Sect. 7.2 and Sect. 7.2.3 that the
MDSM is not the only analytical method available for this analysis; it just seems in
many cases to be more convenient than the alternatives. The examples given in this
chapter, though quite diverse, are not claimed to be inclusive or representative for
every relevant application. They are rather meant to be indicative of typical problems and phenomena, and to provide specific reference for the sections dealing with
generalized systems. If you want to expand on a beginning interest in this area,
some of the references cited at the end of Sect. 7.1 may be a place to start; in
particular Blekhman (2000) is recommended.
7.7 Specific Systems and Results – Some Examples
441