Modal interactions can be especially pronounced when two or more of the linear
natural frequencies of a system are commensurate or near-commensurate, that is,
when the natural frequencies are related by integers or near-integers – i.e.
x 2 % 2x 1 , x 2 % 3x 1 , x 3 % x 2 ± x 1 , x 3 % 2x 2 ± x 1 , etc. Depending on the order
of the nonlinearity, such frequency relationships can cause the corresponding
modes to be strongly coupled, and internal resonance is said to exist. For example,
if the system has quadratic nonlinearities, then internal resonances may arise when
x i % 2x j or x i % x j ± x k , whereas for cubic nonlinearities the possible conditions
are x i % 3x j , x i % 2x j ± x k , or x i % x j ± x k ± x l . Internal resonance is
responsible for several interesting phenomena, especially when combined with
external resonance. Nayfeh and co-workers at the Virginia Polytechnic Institute and
State University have produced an impressive number of significant contributions
within this area (e.g., Nayfeh and Balachandran 1989; Nayfeh 1989; Balachandran
and Nayfeh 1991; and in particular the monograph Nayfeh 1997 and references
cited there). See also Cartmell’s textbook (1990), and the surveys on autoparametric
resonance by Verhulst (1996b) and Tondl et al. (2000).
Sometimes some of the system frequencies ore stiffnesses are not fixed but may
vary in time, e.g. with the state of the system. Then some special and sometimes
useful variants of nonlinear action can take place, in particular with weakly coupled
nonlinear systems, such as, e.g., energy pumping/targeted energy transfer
(Gendelman et al. 2001; Vakakis et al. 2001; Jian et al.), and resonance/resonant
capture (Quinn et al. 1995; Vakakis et al. 2001; Jian et al. 2003).
Section 4.2 discusses the autoparametric vibration absorber – a device purposefully designed to operate at conditions of combined internal and external resonance. Section 4.3 deals with the nonlinear dynamics of the non-shallow arch, an
example of a structure for which internal resonance is more or less unavoidable.
Brief mentions of other internal resonant structures are given in Sect. 4.4. In
Sect. 4.5 we turn to the follower-loaded double pendulum, which besides displaying
rich dynamics allows us to demonstrate how to employ multiple scales perturbation
for systems written in first-order matrix form. Finally, in Sects. 4.6–4.8, we consider
nonlinear interactions associated with vibration induced sliding of mass.
4.2 The Autoparametric Vibration Absorber
4.2.1 The System
Consider the system in Fig. 4.1, having two degrees of freedom x(t) and h(t).
Essentially this is the vibration absorber suggested by Haxton and Barr (1972) (see
also Cartmell and Lawson (1994)), which for some applications can be a suitable
alternative to the traditional tuned mass damper (Hunt 1979; den Hartog 1985;
Korenev and Reznikov 1993; Krenk and Høgsberg 2014). The spring-supported
mass m 1 , which is externally excited by a force F(t), represents the system to be
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4 Nonlinear Multiple-DOF Systems: Local Analysis
natural frequencies of a system are commensurate or near-commensurate, that is,
when the natural frequencies are related by integers or near-integers – i.e.
x 2 % 2x 1 , x 2 % 3x 1 , x 3 % x 2 ± x 1 , x 3 % 2x 2 ± x 1 , etc. Depending on the order
of the nonlinearity, such frequency relationships can cause the corresponding
modes to be strongly coupled, and internal resonance is said to exist. For example,
if the system has quadratic nonlinearities, then internal resonances may arise when
x i % 2x j or x i % x j ± x k , whereas for cubic nonlinearities the possible conditions
are x i % 3x j , x i % 2x j ± x k , or x i % x j ± x k ± x l . Internal resonance is
responsible for several interesting phenomena, especially when combined with
external resonance. Nayfeh and co-workers at the Virginia Polytechnic Institute and
State University have produced an impressive number of significant contributions
within this area (e.g., Nayfeh and Balachandran 1989; Nayfeh 1989; Balachandran
and Nayfeh 1991; and in particular the monograph Nayfeh 1997 and references
cited there). See also Cartmell’s textbook (1990), and the surveys on autoparametric
resonance by Verhulst (1996b) and Tondl et al. (2000).
Sometimes some of the system frequencies ore stiffnesses are not fixed but may
vary in time, e.g. with the state of the system. Then some special and sometimes
useful variants of nonlinear action can take place, in particular with weakly coupled
nonlinear systems, such as, e.g., energy pumping/targeted energy transfer
(Gendelman et al. 2001; Vakakis et al. 2001; Jian et al.), and resonance/resonant
capture (Quinn et al. 1995; Vakakis et al. 2001; Jian et al. 2003).
Section 4.2 discusses the autoparametric vibration absorber – a device purposefully designed to operate at conditions of combined internal and external resonance. Section 4.3 deals with the nonlinear dynamics of the non-shallow arch, an
example of a structure for which internal resonance is more or less unavoidable.
Brief mentions of other internal resonant structures are given in Sect. 4.4. In
Sect. 4.5 we turn to the follower-loaded double pendulum, which besides displaying
rich dynamics allows us to demonstrate how to employ multiple scales perturbation
for systems written in first-order matrix form. Finally, in Sects. 4.6–4.8, we consider
nonlinear interactions associated with vibration induced sliding of mass.
4.2 The Autoparametric Vibration Absorber
4.2.1 The System
Consider the system in Fig. 4.1, having two degrees of freedom x(t) and h(t).
Essentially this is the vibration absorber suggested by Haxton and Barr (1972) (see
also Cartmell and Lawson (1994)), which for some applications can be a suitable
alternative to the traditional tuned mass damper (Hunt 1979; den Hartog 1985;
Korenev and Reznikov 1993; Krenk and Høgsberg 2014). The spring-supported
mass m 1 , which is externally excited by a force F(t), represents the system to be
210
4 Nonlinear Multiple-DOF Systems: Local Analysis