William J. O’Connor Ming Zhu
Mechanical Engineering, Room 213 Engineering Building,
UCD Belfield, Dublin 4, Ireland.
Tel: +353 1 7161887 e-mail: william.oconnor@ucd.ie
This paper shows how pure travelling waves in cascaded, lumped, uniform, mass-spring systems can be defined, established, and maintained, by controlling two boundary actuators, one at each end. In most cases the control system for each actuator requires identifying and measuring the notional component waves propagating in opposite directions at the actuator-system interfaces. These measured component waves are then used to form the control inputs to the actuators. The paper also shows how the boundaries can be actively controlled to establish and maintain standing waves of arbitrary standing wave ratio, including those corresponding to the classical modes of vibration of such systems with textbook boundary conditions. These vibration modes are achieved and maintained by controlled reflection of the pure travelling wave components. The proposed control systems are also robust to system disturbances: they
react to overcome external disturbances quickly and so to re-establish the desired steady motion.
Keywords: travelling waves; standing waves; boundary control; lumped systems; mass-spring systems; standing wave ratio
This paper considers the creation and maintaining of wave-like motion in systems consisting of strings of lumped masses and springs, connected in series, driven by two actuators, one at each end, such as in Fig.1.
A recent paper [1] shows that the minimum number of actuators to achieve pure travelling waves in a uniform, distributed, rectilinear system is two, one at each boundary. This is an intuitive result. Figure 1 shows the corresponding lumped system. It is assumed that two boundary actuators are necessary and sufficient also in the lumped case. The problem then is how to control these two actuators first to establish, and then to maintain, any desired, physically possible, travelling wave, or standing wave
(vibration mode), or standing wave ratio, within the lumped, cascaded system. x
0 x
1 x
2 x n x n+1 m
1 m
2
2 m n
Actuator 1 Actuator 2
Fig.1 A lumped mass-spring string controlled by two, position-controlled, boundary actuators

Regarding the actuation, note that if, instead of having only two, boundary actuators, each mass has more direct, controlled actuation, then arbitrary, wave-like motion along the system can be achieved by suitably moving each discrete mass to create the desired overall wave effect, whether propagating or standing. In this case any dynamic coupling between the masses becomes secondary, indeed dispensable.
The control problem is then primarily a kinematic one, with each actuator moving locally to create any desired overall wavelength, wave frequency, wave speed and waveform along the system.
But, in this work, the only controlled actuation is at the boundaries. The springs in the cascade provide the coupling of motion between the masses. The actuator control systems must then understand, respect and exploit these mass-spring dynamics and the way motion propagates through them.
At first sight the problem seems simple enough. For example, suppose it is desired to set up an harmonic travelling wave, of a specified frequency, propagating along the cascade of Fig.1, beginning at one actuator and ending at the other. Then, at steady state, clearly the two actuators must move harmonically, with a common amplitude and frequency, and a time delay (or phase lag) between them corresponding to the wave travel time along the system. That much is clear and apparently simple.
But a number of challenges immediately arise. First of all, what is the appropriate phase lag between the two actuators? In other words, what is the wave travel time at the specified frequency? For such a lumped system the wave speed (assuming it can be clearly defined) is certainly frequency dependent. The system is also inherently dispersive, even assuming ideal behaviour and perfectly known parameters, so in general waves will distort as they travel, more obviously in the case of transients.
Secondly, how does one reach steady state? If the system is starting from any initial state, whether known
(at rest, say) or unknown, even if the correct, steady-state, phase delay is known beforehand, simply setting the two actuators to move with the correct phase delay will not achieve steady motion. Transients will certainly arise and they can be of large amplitude and quite anarchic. If damping is negligible the
“transients” (so-called) could endure indefinitely, spoiling the desired travelling waves. If damping is significant, the transients might (one hopes) die out, but now the travelling wave will also be attenuated as it propagates. So in addition to the phase differences, the amplitudes of the two actuators should no longer be equal if a pure, attenuated, travelling wave is to be created, but it is not clear what these amplitude and phase differences should be. Thus, with or without damping, there are unresolved practical and theoretical difficulties.
If setting up a desired travelling wave is challenging, it proves no easier to establish standing waves: in other words, waves of equal magnitude and frequency propagating in opposite directions, which when superposed create standing waves, typically with nodes and antinodes at fixed points along the system. A more general problem is how to set up a specified intermediate situation where there is a “standing wave ratio”, characterised by amplitude variations whose envelope forms a fixed spatial pattern. This situation can be interpreted as a travelling wave component combined with a standing wave component, arising from wave reflection and absorption conditions at the actuators, or as the resultant of two waves of equal frequency but unequal magnitude travelling in opposite directions in the system, producing the standing pattern.
The standard, textbook approach to modelling cascaded lumped systems is modal analysis. In modal analysis, boundary conditions are specified, whether dynamic (also called “natural”), kinematic
(“geometric”), or mixed. Resonance frequencies and mode shapes are then determined by assuming

synchronous, steady-state motion. But what if, instead of doing a theoretical exercise, one is attempting to control a physical system such as Fig.1, and it is desired to get the actuators to reproduce such specific vibratory responses? This is now a practical control question. The actuators can be taken to correspond to the system boundaries of modal analysis. But how should their controllers work to establish a desired mode (starting from rest, for example), and then how should they maintain it (despite inevitable minor disturbances)? The actuators are required to get the system to a steady synchronous motion, which includes having the actuators reproduce the specified boundary conditions (which, paradoxically, might include fixed boundaries). But before reaching and maintaining the specified steady state, the actuators will need to pass through a very specific, controlled, start-up transient, and it is not obvious how to specify this. Again this apparently simple, practical problem proves quite challenging and raises non-trivial questions.
A further question considered in this paper is how to simulate more general boundary conditions, where one of the actuators can simulate the interaction of the given cascaded system with another dynamic system with its own dynamic response (other than simple reflection or absorption). This is a generalization of the “natural” boundary conditions of modal analysis.
The present paper presents novel solutions to all of these challenges. For simplicity the following assumptions will be made. a) Only uniform systems (with all masses and all springs of equal value) will be considered. b) The actuators will be assumed to be ideal, with output motion equal to the requested input. c) Damping, both internal (between masses) and external (masses to ground) will be assumed to be negligible.
None of these assumptions is essential. The work can be extended to non-uniform systems, systems with real actuators, and systems with damping. These assumptions are chosen simply to reduce the variety of issues to be treated and so limit the length of the paper. The other cases will be considered separately.
Before presenting the method, it is worth briefly considering if solutions to these challenges already exist, and why one might be interested.
As far as the authors are aware, unlike in distributed systems such as transmission lines, strings, rods, beams and plates, the problems described above have not been solved to date explicitly for lumped systems. Similar problems have been discussed in the literature in various contexts. Pure travelling waves have been applied to create propulsion (e.g. in an ultrasonic motor), squeeze-film levitation, transportation devices and snake-like motion in robotics. Gabai and Bucher [1] have done research in actuation, sensing and tuning travelling waves in finite systems. They developed wave identification methods and discuss active boundary conditions that can form travelling waves. Loh and Ro [2] built a transport system by creating flexural ultrasonic progressive waves in an elastic beam. They investigated the relationships between transport speed, transport direction and excitation frequency. Chen, Wang, Ma
& Li [3] analysed kinematic problems of a snake-robot’s locomotion. Boundary control has also been used for active vibration damping of strings, beams and plates. See for example [4-5].
However, the travelling waves discussed in these and similar papers differ from the present work, either because they arise in distributed (rather than lumped) systems, or because they are produced by multiple

actuators, often with as many actuators as degrees of freedom. The need for wave analysis and control of under-actuated, lumped systems, considered in this paper, can arise in various situations where the physical nature of the system is such that it is appropriately modelled by lumped elements, or where a lumped configuration is used as a lower order model of a distributed system. Examples of inherently lumped systems include many robots, articulated space structures, and certain swimming and transport devices.
In the classical discussion of motion waves, a key feature is that there is a finite delay before motion in one part of a system affects another part. The motion analysis can then be purely local (that is, uncoupled from the total system). It is characterised by PDEs, derived by analyzing a vanishingly small system element, whose inherent dynamics are independent of the rest of the system and its boundaries. By contrast, in the lumped cascaded system, analysis should be global, in that the motion and inherent dynamics of every component is instantaneously linked to that of all the others in the system, with no transport delay. So at first sight, waves in lumped systems are not possible, at least not waves as classically defined. In lumped systems the motion of components is described by ODEs, for which the concept of a transfer function can be useful. The coupling and immediate interaction among all system components can be described by transfer functions relating the motion of any part of the system or boundary to that of any other part. There is no finite delay, or associated propagation speed, both of which characterise wave behaviour.
Thus some clarification is required to justify speaking about waves in lumped systems, even before describing how to achieve them in practice. The topic of defining and measuring such waves has been considered in some detail in [6].
For present purposes the relevant considerations are now briefly reviewed.
X
0 k m k m
X
1
X
2
…
X
0
+
-
A
0
A
1
G G
A
2
…
+ + +
B
0 G
B
1
G
B
2
… k m k m
X n-1
X n
G
G
A n-1
G
A n
+ +
G
B n-1
B
G n
Fig.2. A uniform system (upper part) and its wave model (below), with X i
=A i
+B i
, i=0, 1, …n.
Figure 2 shows a lumped system with one actuator. Below it in the figure is the corresponding wave model, consisting of a looped arrangement of connected transfer functions and a summing junction. For the uniform case considered in this paper, the transfer functions, G, are identical, and are given by
G(s) = 1 +
1
2
(s/ω n
) 2
− (s/ω n
)√1 + (s/2ω n
) 2 (1) where ω n
= √k/m .

These have been called “wave transfer functions” (WTFs), and have interesting properties, including the absence of finite poles or zeros. They describe the transfer function between the motions of two adjacent masses within a uniform system extending to infinity. It can be shown that the sum of the motions in the upper and lower branches in the wave model of Fig.2 reproduces exactly the motion of the dynamic system above it, in response to any actuator input, x
0
(t), or X
0
(s) in the s-domain. (Capitals indicate
Laplace Transforms of the corresponding time-domain variables.) Thus Fig.2 provides one way to resolve the lumped system’s motion, X i
, into (notional) outgoing waves, A i
, propagating from the actuator to the n th mass, and returning waves, B i
, propagating from the tip mass back to the actuator, with X i
= A i
+ B i
.
The sign convention adopted here is as follows. The reference direction for motion of masses is positive to the right. The leftwards-propagating, returning waves, B i
, are positive when they contribute positive
(rightwards) motion to the masses. This implies that whereas positive outgoing (rightwards) waves are associated with compression of the springs, the leftwards waves (when positive) cause extension of the springs.
In the example of Fig.2, in the wave model the actuator is shown as a summing junction, where returning waves B
0
are added to any external incoming waves from the actuator motion X
0
, with X
0
= A
0
+ B
0
. The other end is shown as “free”, with the outgoing wave, A n
, becoming a returning wave, B n
, having passed through another WTF G. If, instead of being free, the right hand end has another actuator, for example, then this arrangement will be replaced by a second summing junction, as seen below in Fig.3.
For practical implementation of schemes involving G(s), some method is needed to model the corresponding behaviour in the time domain. The inverse Laplace transform of G(s) involves Bessel functions of time, which are challenging to work with in practical controllers. Instead G(s) is modelled by carefully designed rational polynomial transfer functions which are easily implemented. The accuracy of these approximations increases with polynomial order. Even a second order approximation is fine for some purposes, but, as will be seen, higher order systems are needed in the present work. It can be shown that successively better approximations, G
0
, G
1
, G
2
, … G i
, can be achieved from the previous approximation using the recursive formula [7]
G i
(s) =
1
2
2 i−1
(s))
1−G i−1
⁄
G
) 2
(2) beginning with the second order approximation
G
0
(s) =
ω
2
G s 2 +ω
G s+ω 2
G
.
For example, the next higher approximation is the 5 th order transfer function
G
1
(s) = s 5
ω
2
G
+2√2ω
G s
3
+2√2ω
3
G s
2 s 4 +8ω
2
G s 3
+6ω
+8√2ω
3
G
4
G s+4√2ω s 2 +12ω
4
G
5
G s+4√2ω
5
G
(3)
(4) where ω
G
= √2k/m .
If it is accepted that a wave model, such as Fig.2, provides a working definition of waves in a cascaded lumped system, the next question is how to measure these component waves, assumed to be present in any motion of the system. It transpires that two measured values, processed through two WTFs, can resolve the motion in one part of the system, X i
, into the counter-propagating wave components, A i
and B i
,

with X i
=A i
+ B i
. Such an arrangement is shown in the lower parts of Fig.3, where the motions of the two actuators are each resolved into two component motions. In other words, X
0
, the motion of Actuator 1, is resolved into A
0
and B
0
, while X n+1
(Actuator 2) is resolved into A n+1
and B n+1
. The arrows in Fig.3 indicate the direction of information flow and are not reference directions for displacements.
R=A
0 +
+
X
0
Actuator 1 mass 1 mass 2 mass n
Actuator 2 X n+1
+
+
L
X
0
+ A
0
-
B
0
G
G
A
1
X
1
+
-
B
1
X n
+
-
A
B n n
G
G
X n+1
A n+1
-
+
B n+1
C a k
Fig. 3 Resolving actuator motions into counter-propagating wave components, A & B.
To determine the components A
0
and B
0
of X
0
, the two measurands are the actuator’s own position, X
0
, and that of the first mass, X
1
. (The second measurement could also be the force at the actuator output [8,
9], but here only displacement measurements will be used.) These two measured displacements are processed through the two WTFs as shown, where the lines labelled A
0
and B
0
give the desired components of X
0
. The WTFs provide memories of the dynamic history in the system close to the actuator, essential for resolving these motions into wave components. It can be seen by inspection that, for example, X
0
= A
0
+ B
0
, X
1
=A
1
+ B
1
, and that A
1
=G.A
0
and that B
0
=G.B
1
, all of which are as required by the wave theory and wave model of the system.
In Fig. 3, the input to the actuator, X
0
, is set to be the sum of two components, an arbitrary external input
R plus the measured return wave B
0
. This arrangement is a way to ensure that the actuator absorbs any returning waves arriving back to the actuator, measured as B
0
. With an ideal actuator, because X
0
=A
0
+B
0
, it follows that A
0
= R. That is to say, the rightward-propagating wave, which is launched into the system at
Actuator 1, is R.
Similarly, to determine A n+1
and B n+1
, measurements are taken of the positions of the last mass, X n
, and that of the right hand actuator, X n+1
. Again, using two WTFs, the rightwards going component A n+1
and the leftward component B n+1
of the actuator’s motion X n+1
are determined.
Also shown in Fig.3 is that the right hand actuator, like the left hand one, is set as the sum of two inputs.
These are an arbitrary external input, L, (which will not be used in this work) and C a
times the rightwardstravelling wave component in the actuator, A n+1
. By changing C a
, different right hand boundary conditions can be set up, as will be seen below.
Because ideal actuators are assumed, for simplicity, the output of each actuator is taken to be equal to the input, so in Fig.3 the labels X
0
and X n+1
appear twice. With real actuators, their outputs will in general be different from their inputs, so different variable names would be required. Real actuators will have

their own subcontrollers, attempting to move them to their requested, input positions, subject to the actuators’ inherent dynamics and external load from the mass-spring string. The real response will approach the ideal if the actuators are sufficiently rapid and powerful, with a tight local control loop. For measurement purposes, however, there is no need to assume such ideal behaviour because the wave measurements described above use the actually achieved actuator outputs, X
0
and X n+1
, rather than their requested inputs. The measured wave components are therefore those actually achieved, propagating in the system.
As analysed above, motion in lumped system can be interpreted as the superposition of two travelling waves travelling in opposite directions. One wave is caused by the input to the system at Actuator 1 at the left; the other is caused by reflection of the boundary at Actuator 2. To achieve a pure travelling wave, that is, to eliminate any reflection at the right hand end, Actuator 2 needs to respond to motion arriving there in a precisely defined way, to absorb it from the system. Under the theory behind this work, this precisely defined way is as follows: the actuator needs to move as if it were the next mass in a notional extension of the mass-spring system, to infinity, to the right (from which wave motion would never return). This is arguably the equivalent, in a lumped system, of the concept of a matched boundary impedance as used in distributed systems (especially in electrical transmission line theory).
The time-domain implementation of this criterion as a practical control system is shown in Fig. 3, provided the gain C a
is set to 1 (and L is set to zero, i.e. no input to Actuator 2 from an external signal). This arrangement attempts to make the motion of Actuator 2, X n+1
, in response to the motion of the n th mass,
X n
, to correspond to the WTF G, which is that of an infinite, uniform system, as required. If everything is perfectly implemented, including ideal actuator behaviour and perfect modelling of the WTFs G, then there will be no returning wave, and B n+1
= 0 = B n
. To the extent that conditions are not ideal, or undergoing transient behaviour, there will be a finite returning wave, B n
, present in X n
, which will then propagate back towards Actuator 1.
Because of this, the control system for Actuator 1 needs to be able to detect any such returning waves and move to absorb them actively. In this way one avoids creating recirculating waves, trapped in the system due to repeated reflections at each end. Active absorption is therefore required at both ends, certainly to cope with initial transients before the travelling wave becomes established, but also to cope with subsequent disturbances and non-ideal behaviour which will inevitably arise in a physical implementation.

1.5
0
-0.5
1
0.5
X0
X1
X2
X3
X4
-1
-1.5
0 10 20 30
Time (s)
40 50 60
Fig.4. Response of a 3-mass system to a unit sinusoidal input, starting from rest, set up to achieve a pure travelling wave (Fig.3, with Ca=1, L=0, n=3). X0 and X4 are the two actuator motions.
These ideas were extensively tested in computer simulation. As one example, Fig.4 shows the time response of a three-mass system (n=3), as in Fig.3, with C a
=1, L=0. The masses are 1kg, the springs have stiffness k = 1 N/m. The driver is a unit sinusoidal signal of frequency 0.45 rad/s, which is entered at R=A
0
.
(The frequency is arbitrary.) As can be seen, the system starts from rest and quickly (within one cycle) achieves and maintains the desired uniform travelling wave pattern. In this case, the WTFs G were modelled using an 11th order approximation. If after the travelling wave is established the system is disturbed by an external impulsive force on one of the masses, the control system responds quickly, to absorb the extraneous motion and to recover the pure travelling wave.
Comment 1. This paper speaks about “pure” travelling waves. What exactly this means when the medium is not uniform is challenging to define. But at least in uniform, undamped, lumped systems, it is intuitively attractive to assume that a pure, uni-directional, travelling wave will feature perfectly uniform motion at steady state. In other words, the amplitude (and frequency) of the motion of all the masses (and of the actuators) will be exactly equal, with equal phase lag between each successive mass. This is what this control arrangement establishes and works to maintain.
Comment 2. The wave analysis and modelling of lumped systems, on which this work is based, involves decisions which are arbitrary to an extent [6]. The waves are notional, and there are many ways to define valid wave models, and so many different ways to define the waves and the wave components. All such wave models can claim to be equally valid to the extent that the superposition of their components

correctly reproduces the lumped system dynamics. In previous work it was proposed to remove this ambiguity, by invoking the idea of a system extending uniformly to infinity, to define one-way waves. It is therefore reassuring that this decision leads to waves in uniform systems which correspond to the intuitive notion of pure travelling waves. The present results therefore retrospectively strengthen the arguments for previous choices.
Comment 3. In testing the control arrangement, it was found that the modelling of the 4 WTFs G of Fig.3 needed to be of relatively high order. In previous work on wave-based, rest-to-rest, position control with a single actuator, it was found that the modelling of G was not critical [10]. Very good responses were achieved with low-order models of G (e.g. 2 nd order). In the present work, such low order G models did not achieve an exactly uniform travelling wave. Rather small standing fluctuations were observed, which can be interpreted as a standing wave, even if they are very small. (In terms of the discussion below, the standing wave ratio is close to unity). To obtain the results in Fig.4, the 4 WTFs were modelled by 11th order transfer functions.
Comment 4. One conclusion of the present work therefore is that there are more stringent requirements for the creation of pure travelling waves (and so the avoiding of circulating waves) than for position control with active vibration damping. A second conclusion is that, to create pure travelling waves, corresponding to wave propagation in an apparently infinite system, it seems to be necessary first to achieve a good model of the response of an apparently infinite system, which is not a trivial matter.
Comment 5. The only sensing required for this control system is at, or close to, the boundary actuators.
Comment 6. Reference [1] begins with modal analysis and works towards travelling waves, whereas this paper could be said to go in the opposite direction. It begins with travelling waves and, in the following, uses these to understand, establish and control vibration modes. (In this it is similar to reference [11] which uses travelling wave concepts to achieve modal analysis.)
Having achieved pure, single-direction, travelling waves, characterised by perfectly uniform motion, it then becomes possible to create systems with a controlled amount of reflection at the actuators, thereby creating standing waves of any desired kind, as the outgoing and returning wave become superposed. In telecommunications, the transmission efficiency of a channel is often measured by a standing wave ratio
(SWR), which is related to the extent to which waves, travelling from a source at one end to a load at the other end, are reflected at the load end. The SWR is the ratio between the maximum and minimum values within the standing wave envelope. The present work makes it possible to extend this idea to lumped systems, both theoretically and practically, implemented as a controlled system to achieve any desired
SWR.
Consider Fig. 3, again with the no external, active input at the Actuator 2, that is L=0. The amount of reflection will be controlled by adjusting the value of the gain, C a
, which determines the proportion of the arriving wave, A n+1
, which is re-inserted into the system, as B n+1
, to travel leftwards. This in turn will determine a reflection coefficient, C r
defined as
C r
=
B n+1
A n+1
=
X n+1
−A n+1
A n+1
=
C a
A n+1
−A n+1
A n+1
= C a
− 1 (5)

The standing wave ratio will then correspond to
SWR =
|A n+1
|+|B n+1
|
|A n+1
|−|B n+1
|
=
1+|C r
|
1−|C r
|
=
1+|C a
1−|C a
−1|
−1| where │·│ indicates maximum positive values, or sinusoidal amplitudes. Three limit cases will be considered first.
(6) o C r
= –1 (C a
= 0): With C a
set to zero, the right hand actuator becomes fixed (X n+1
= 0) regardless of the motion in the mass-spring string. This corresponds to a fixed boundary in vibrating systems. The reflected wave is equal in magnitude to the incident, but inverted, so B n+1
= – A n+1
. o C r
= 0 (C a
= 1): If C a
is set to 1, in response to the motion of the n th mass, the actuator will move as the (n+1) th mass of a mass spring chain that extends to infinity (X n+1
=A n+1
). The actuator will therefore move to absorb rightwards-travelling waves from the n th mass, there will be no reflected wave, and a pure travelling wave is achieved, as shown before in Fig. 4. o C r
= 1 (C a
= 2): If C a is set to 2, a maximum positive reflection will occur (B n+1
= A n+1
). The reflected wave will have the same amplitude as the forward wave, but now with the same sign. This case corresponds to a free boundary, with X n+1
=2A n+1
. Due to a subtle effect, however, the free-boundary it will then be modelling will not be that of a uniform mass-spring string. To make the actuator behave like the last mass in a uniform free-boundary system of n+1 masses, C a
should be set to (1+G) rather than 2. To see why, consider the free end of Fig.2 and its wave model, where the last mass has motion (1+G) times the arriving wave.
Thus, as C a
varies between 0 and 2, C r
varies between -1 and +1. At intermediate values, Actuator 2 will move to simulate other dynamic boundary conditions, where there is partial absorption and partial reflection of arriving waves, with or without inversion (change of sign), thus creating a standing wave with an intermediate SWR given by Eq.(6).
The ideas above have been widely tested on systems of different sizes and component values. Sample simulation results for different boundary conditions are now considered, using a sinusoidal reference input, R, at Actuator 1, and different values of C a
at Actuator 2.
The case of C a
=1, producing a pure travelling wave, has been described already, with results shown in
Fig.4. The input frequency is arbitrary, and for each frequency the control system quickly finds the appropriate phase differences (corresponding to uniform wave travel times). The higher the frequency the slower the wave speed and the shorter the wavelength. There is an upper frequency limit for travelling waves, not due to the control system, but due to the inherent dynamics of the lumped system itself, which acts as a low pass filter. This occurs at

= 2
 n
, = 2√(k/m), when there is a cut-off effect, with
180° phase shift between successive masses, indicating opposing oscillatory behaviour, which inhibits propagation of energy, momentum, or waves. The wavelength then corresponds to two mass-spring lengths. At and above this frequency the waves leaving Actuator 1 become evanescent rather than propagating.

2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
X0
X1
X2
X3
X4
-2.5
0 10 20 30
Time (s)
40 50 60
Fig.5. Response of 3-mass system with C a
=0, fixed right hand boundary, driven at the first natural frequency of a fixed-fixed system.
When C a
is set to 0, the right hand boundary is fixed, so clearly no travelling wave is then possible. (Indeed, for this particular case no control is needed at Actuator 2: it simply remains stationary.) Figure 5 shows a response from rest, with displacements of the two actuators (X
0
and X
4
) and three masses (X
1
, X
2
and X
3
).
The input frequency in Fig.5 was set to a precise value, ω source
= √0.5858 ∗ k/m = 0.7654
rad/s, which is the resonance frequency of the lowest mode of the fixed-fixed, 3-mass, vibrating system. Under the control arrangement of Fig.3, it can be seen that the system quickly establishes the expected first mode of vibration. Thus the left hand actuator, X
0
, first establishes the mode and then comes to rest, as it should for the fixed-fixed mode. All the while the input signal, R, at the resonant frequency continues, but it is cancelled by the addition of return wave B
0
of equal magnitude but with 180° phase shift. The correct mode shape can be seen in the relative amplitudes of the mass motions, with X
0
and X
4
stationary, X
1
=X
3
, and X
2
having maximum displacement.
If the system is driven at one of the natural frequencies corresponding to the free-fixed system, then the amplitude at actuator 1 will quickly settle at twice the input amplitude, with the rest of the system adopting the correct mode shape. By setting C a
=2 (or better, C a
= 1 + G) a free boundary condition is specified at the right hand actuator, and fixed-free and free-free modes can easily be established. In this way, the system can be made to reproduce all the modes corresponding to each set of boundary conditions (fixed-fixed, fixed-free, free-fixed and free-free).

Note that the modes are achieved very rapidly from any initial conditions, stationary or moving.
Furthermore, if subsequently the system is disturbed, the control systems quickly react to re-establish the modes. For example, when Actuator 1 is simulating a fixed boundary, if the system is disturbed, it momentarily begins to oscillate with just the right amplitude to recover the correct model behaviour, before returning to rest.
These modes of vibration can be thought of as standing waves [11], with nodes and antinodes. Fixed boundaries will define nodes, free boundaries antinodes. For given boundary conditions successively higher modes will each add a further node to the response. From a wave perspective, these are the result of counter-travelling waves of successively shorter wavelengths.
If the frequencies corresponding to the desired modes are not known, or the system parameters are poorly defined, the modal frequencies and shapes can be found experimentally, by varying the input frequency until the desired boundary conditions are observed at the actuators. The experimental search is helped by observing the measured actuator wave components, which in turn suggest how to fine-tune the frequency to achieve a desired node or antinode at an actuator.
Achieving pure modal behaviour is just one possibility. By setting C a
to intermediate values, one can also investigate situations where there is partial transmission and partial reflection at the right hand boundary.
The intention may be to set up a partial standing wave in the system, or to simulate how the mass-spring system will react when connected to another system of different dynamics.
1.5
1
0.5
0
-0.5
X3
X4
X5
X6
X7
X8
X0
X1
X2
X9
X10
X11
-1
-1.5
0 10 20 30 time(sec)
40 50 60
Fig.6. Response from rest for a 10-mass system with a target SWR =1.5.

Figure 6 shows a case where a specified target SWR of 1.5 is quickly established in a 10-mass system, starting from rest, by setting C a
=0.8. Figure 7 shows the envelope of the response, from which the SWR can be confirmed to be 1.2/0.8, which is indeed the target value. The standing wave pattern is most clearly seen when the system is long and the frequency is well below cut-off. But the wave analysis and interpretation still apply perfectly even when the system is very short or the frequency high. The modal responses above can be considered as standing waves of infinite SWR where the minimum value is zero. magnitude envelope
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0 1 2 3 4 5 6
Location
7 8 9 10 11
Fig.7. The envelope of the standing waves of Fig.6, verifying that SWR = max/min =1.5
Instead of establishing a target SWR it may be desired to simulate the system’s interaction with another dynamic system, or load, at the right hand end. In this case the force at the right hand boundary can be measured, and then Actuator 2 can be moved to reproduce the resulting motion of the boundary of the attached system in response to this force. The attached system may be active or passive. In terms of waves, this means that A n+1
will be the motion transmitted to the attached system, while the reflected wave, B n+1
can be considered as a combination of a partial reflection of A n+1
plus any new wave arriving into the string from the attached system. If the attached system is active, this new wave may carry new energy and momentum into the system. If passive, any entering energy or momentum will have been absorbed earlier and temporarily stored in the passive system.
The above is achieved in the time domain, as required for implementation in the control systems. Looked at in the frequency domain, the resulting returning wave, B n+1
, will generally have a different magnitude and phase with respect to the arriving wave, A n+1
. The A n+1 and B n+1
components will be perfectly in phase only in the case of a purely viscous load, giving a load impedance which is real. In this case the system response can be obtained by setting C a
to be the ratio between them. This is what was done to create
Figures 6 & 7, so these can also be considered as responses to a viscous load. The more general case of an

arbitrary load can also be considered as either adding another input at L in Fig.3, or giving a timedependant characteristic to C a
, or both. But in practice it will generally be simpler to drive the actuator, as described in the second sentence of the previous paragraph, determining X n+1
directly, and then the component waves will emerge automatically.
The SWR is often seen as a measure of power transmission from the source (Actuator 1) to the load
(Actuator 2). A pattern such as in Fig.7 can be understood as a combination of a travelling wave, transmitting energy through the system, and a standing wave, holding energy within the system. If there is a node anywhere (such as in the modal responses above) there is no transmitted energy (SWR=∞). The other extreme, with maximum energy transfer, is when Actuator 2 produces a pure travelling wave
(SWR=1), as discussed above, with no standing component. In the frequency domain, this corresponds to a perfect impedance matching between the load and the system dynamics. Note that this perfect matching will not be produced by any real load impedance, no matter what value is chosen, but by the impedance which would be presented by an infinite mass-spring string.
This paper has shown how pure travelling waves in cascaded, lumped, uniform, mass-spring systems can be a) defined, b) established, and c) maintained, by suitably controlling two boundary actuators, one at each end. These are not trivial challenges, despite first appearances. In most cases, the control system for each actuator requires identifying and measuring the notional component waves propagating in opposite directions at the actuator-system interfaces. These component waves are then used to make up the inputs to the actuator sub-controllers.
The paper has also shown how the boundaries can be actively controlled to establish and maintain standing waves of arbitrary standing wave ratio, including those corresponding to the classical modes of vibration of such systems with standard boundary conditions. These are achieved by controlled reflection of the pure travelling waves. The proposed control systems rapidly achieve the desired steady-state responses from arbitrary initial conditions. They are robust to system disturbances, reacting to overcome external disturbances quickly and so to re-establish the desired steady motion.
This work, believed to be novel, has focussed on uniform, undamped systems. Future papers will deal with more general cases, including non-uniform systems under boundary control. Simply defining travelling waves then becomes challenging. If a definition is agreed, there is a further challenge in designing control systems to establish and maintain them, and to control boundary reflections.
Considerable progress, however, has now been made on all these issues and they will be the topic of another paper.
The authors gratefully acknowledge support for this work from the China-Ireland Scholarship Fund.

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