Shock amplification, curve veering and the role of damping

Highlights

• High acceleration amplification in coupled systems can result from a veering pair of modes with a high mass ratio.

• A two DOF model reveals the main parameter dependence.

• High amplification can be mitigated using non-proportional damping.

Abstract

The circumstances are investigated under which high peak acceleration can occur in the internal parts of a system when subjected to impulsive driving on the outside. Previous work using a coupled beam model has highlighted the importance of veering pairs of modes. Such a veering pair can be approximated by a lumped system with two degrees of freedom. The worst case of acceleration amplification is shown to occur when the two oscillators are tuned to the same frequency, and for this case closed-form expressions are derived to show the parameter dependence of the acceleration ratio on the mass ratio and coupling strength. Sensitivity analysis of the eigenvalues and eigenvectors indicates that mass ratio is the most sensitive parameter for altering the veering behaviour in an undamped system. Non-proportional damping is also shown to have a strong influence on the veering behaviour. The study gives design guidelines to allow permissible acceleration levels to be achieved by the choice of the effective mass and damping of the indirectly driven subsystem relative to the directly driven subsystem.

Introduction

There are many situations in which an object with fragile internal components may be subjected to external impacts. It is important to be able to predict the shock level on these internal components, and to design the whole system to limit this to an acceptable level. A typical example would be the design of packaging systems for the transportation of fragile art objects to cope with, for example, airport freight handling facilities. Any such system can be broadly divided into two coupled subsystems: the external case which is directly subjected to impact, and the internal system which is indirectly excited through the coupling. Their response could explain the performance of many physical systems. An important engineering application is in the transmission of vibration between coupled structures. An issue of general interest is whether under some unfavourable circumstances the peak internal acceleration may be as great as or even greater than that directly imposed on the external case. This is the phenomenon of shock amplification.

This problem represents a particular small corner of the study of vibration in coupled systems, which of course has a huge literature. Methods like Statistical Energy Analysis (see for example [1]) explicitly involve vibration analysis in terms of coupled subsystems. Numerical approaches such as the Finite Element method may use substructuring as a way to improve efficiency. However, the particular problem treated here does not seem to have received a definitive treatment in the literature, despite its relevance in a number of industrial settings.

In an earlier study [2], an idealised system consisting of two coupled beams was analysed both experimentally and theoretically to explore the phenomenon of shock amplification. It was shown that amplification can indeed occur, even within the limits of linear vibration theory and when the coupling between the two subsystems is weak. The analysis carried out in that paper [2] indicated that the worst amplification was generally associated with one or more pairs of modes exhibiting “veering” behaviour: when a mode of one beam falls close in frequency to a mode of the other beam, and these couple to produce a pair of modes each showing significant modal amplitudes on both beams.

The phenomenon of veering has also generated its own significant literature. Although related phenomena were already well known in atomic physics, where the effect is usually called “level repulsion”, curve veering was first identified within the structural dynamics context in the 1970s, for a rectangular membrane problem [3]. Initially there were concerns about whether it was an artefact of mathematical approximations used to estimate frequencies. However studies of the vibration of orthotropic rectangular plates [4] and sagged cables [5] confirmed that veering was a physical phenomenon. Triantafyllou noted that when the ends of a sagged cable are at the same level, the eigenvalues exhibit mode cross-over but when the cable is inclined (ends at different levels), mode crossing never occurs. The existence of curve veering in continuous models was verified by Perkins and Mote [6].

The pattern by which mode shapes change during a veering event relates to a wider phenomenon of “mode localisation” [7], in which the modal or forced response within a coupled system is sometimes found to be restricted to one region rather than distributed throughout the system. The localisation phenomenon for vibration in mechanical systems has received considerable attention in recent years, especially in relation to periodic structures [8], [9], [10], [11] made up of assemblies of nominally identical substructures: see for example the review by Bendiksen [12].

It is now clearly understood that veering is the norm, not the exception: equal natural frequencies, implying the crossing of eigenvalue locus curves, only occurs under special circumstances. Balmes [13] stated that three types of conservative structures allow modal crossing:

• Symmetric structures have multiple modes that can be characterised using the algebraic properties of the group of symmetry—in particular, cyclically symmetric structures.

• Multidimensional substructures for which motion in different orientations uncouple, such as a beam having a bending and a torsional mode at the same frequency.

• Structures with fully uncoupled substructures.

In the context of the coupled-beam problem studied earlier [2], a veering pattern was found to be a prerequisite for good excitation on the directly driven beam simultaneously with high response on the indirectly driven beam. The specific aspects of veering relevant to the shock amplification problem do not appear to have been studied in the existing literature, and these aspects are investigated here using the simplest possible theoretical model, given by a two-degree-of-freedom system. This system allows closed-form solutions that reveal important aspects of the parameter dependence on the shock amplification effect.

The simplified spring–mass system shown in Fig. 1 will be used. In the first instance the system will be undamped, but the effect of damping will be investigated later. This system is an idealisation for the coupling of any pair of modes from two coupled subsystems as described above. The mass and stiffness matrices of the system are given by

$$\mathbf{M}=\left[\begin{array}{cc}\hfill m_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_2\hfill \end{array}\right],\mathbf{K}=\left[\begin{array}{cc}\hfill k_1+k_c\hfill & \hfill -k_c\hfill \\ \hfill -k_c\hfill & \hfill k_2+k_c\hfill \end{array}\right].$$

The case of most interest will involve weak coupling, represented in the model by small values of $k_c$ relative to $k_1$ and $k_2$. When the two subsystems, represented by the two masses here, are strongly coupled it would not be surprising if external impacts resulted in high internal acceleration. The design of any protection and cushioning system tends to incorporate weak coupling, precisely with the intention of isolating the internal system from external excitation. The aim here is to explore when this isolation fails. The simplified system is familiar from analysis of tuned-mass dampers (see e.g. [14]), but the question of interest is different here: not to achieve low motion on the directly driven mass, but how to get high acceleration on the indirectly driven one. The present problem also differs from the tuned-mass damper in terms of the relative masses of the two systems: a damper will normally have a much lower mass than the main structure, but there is no reason to expect this to be the case for a fragile object being transported in, and protected by, an outer case. A wide range of mass ratios could be of interest, and this study will not emphasise the low-mass case particularly.

The veering phenomenon is readily illustrated with this system. For simplicity the case $m_1=m_2$ is chosen for this initial plot. Fig. 2 shows how the two natural frequencies vary when $k_2$ is varied over a range encompassing the value of $k_1$, all other parameters being kept fixed. In the absence of coupling, i.e. with $k_c=0$, the result is simply two lines that cross, but with non-zero coupling “veering” or “avoided crossing” occurs. At both edges of the diagram the mode shapes are more or less the same as in the uncoupled case, but when $k_1=k_2$ the character of the modes changes. Considerations of symmetry in this case dictate that the modes must take the form ${[11]}^{\mathrm{T}}$ and ${[1-1]}^{\mathrm{T}}$. Instead of localised modes, where the energy is largely confined to one oscillator or the other, the modes become global, with equal amplitudes on each mass. To satisfy orthogonality there must be a sign reversal between the two modes.