Research Article Mathematical Modeling of a Transient ...eprints.uanl.mx/15222/1/316.pdf · Research Article Mathematical Modeling of a Transient Vibration Control Strategy Using

Research ArticleMathematical Modeling of a Transient Vibration ControlStrategy Using a Switchable Mass Stiffness Compound System

Diego Francisco Ledezma-Ramirez,1 Neil Ferguson,2 and Adriana Salas Zamarripa1

1 Universidad Autonoma de Nuevo Leon, Facultad de Ingenierıa Mecanica y Electrica. Avenida Universidad s/n,66451 San Nicolas de los Garza, NL, Mexico

2 Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK

Correspondence should be addressed to Diego Francisco Ledezma-Ramirez; [email protected]

Received 1 November 2013; Accepted 10 March 2014; Published 31 March 2014

Academic Editor: Jeong-Hoi Koo

Copyright © 2014 Diego Francisco Ledezma-Ramirez et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

A theoretical control strategy for residual vibration control resulting from a shock pulse is studied. The semiactive control strategyis applied in a piecewise linear compound model and involves an on-off logic to connect and disconnect a secondary mass stiffnesssystem from the primary isolation device, with the aim of providing high energy dissipation for lightly damped systems. Thecompound model is characterized by an energy dissipation mechanism due to the inelastic collision between the two masses andthen viscous damping is introduced and its effects are analyzed.The objective of the simulations is to evaluate the transient vibrationresponse in comparison to the results for a passive viscously damped single degree-of-freedom system considered as the benchmarkor reference case. Similarly the decay in the compound system is associated with an equivalent decay rate or logarithmic decrementfor direct comparison. It is found how the compound system provides improved isolation compared to the passive system, and thedamping mechanisms are explained.

1. Introduction

Mechanical shock is a common problem characterized by asuddenly applied excitation in a short period of time. Usuallyit involves very large forces and displacements which couldlead to damage to sensitive equipment, human discomfort,and other effects [1]. Thus the effective isolation of shockgenerated vibration is a very importantmatter in engineering.Shock isolation is normally achieved through energy storageby elastic foundations but optimum isolation is compromiseddue to the high energy levels requiring large deformations ofthe isolatorwhere normally space is a constraint. Additionallythe isolation system must be able to dissipate the storedenergy quickly once the shock has finished in order tominimize residual vibrations.

The classical approach to shock isolation is based ona single degree-of-freedom system with linear stiffness andviscous damping elements. Many shock scenarios can beanalyzed considering this method to select proper isolators.

Most of the literature related to shock isolation dates from1950 to 1960 when authors like Ayre [2], Snowdon [3], andEshleman and Rao [4] studied this phenomenon and settledthe fundamental theory of shock analysis and isolation.However, linear passive elements are limited. For instance,there is the compromise aforementioned between isolationperformance and space limitations. In order to improve shockisolation, the use of variable or switchable rate elements hasbeen considered. Optimal shock isolation has been consid-ered by Balandin et al. [5] where a performance index anda design constraint are used to design an isolator, obtainingtime optimal functions for the isolator. The concept of anearly warning or preacting isolator has also been studied,showing a substantial performance increase over typicalisolators [6].Waters et al. devised a dual rate damping strategywhere the damping was reduced to a lower value whilst ashock input is applied [7]. Ledezma et al. applied severalwell-known variable damping skyhook strategies to a singledegree-of-freedom system subjected to pulse excitations [8].

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 565181, 10 pageshttp://dx.doi.org/10.1155/2014/565181

2 Shock and Vibration

k − Δk

m

�

Δk

Figure 1: Single degree-of-freedom system with on-off switchablestiffness.

Ledezma has also recently presented a switchable stiffnessstrategy in two stages, namely, the control during a shockand the later stiffness switching to reduce residual vibrations.This study demonstrated theoretically and experimentallythat reducing the stiffness for the duration of a shock reducesthe peak response of the system [9, 10].

The approach presented in this work follows a semiactivetheoretical strategy based on a piecewise linear compoundmodel, comprised of a main system, that is, the mass to beisolated supported by elastic elements, and a secondary massspring model which can be attached to or disconnected fromthe main system. By the use of such model it is expectedto gain improved shock isolation and quick dissipation ofresidual vibrations. The hypothesis is that by transferringpart of the energy of the main system to the secondarysystem it can be dissipated faster as the secondary systemoscillates at a higher frequency once it is disconnected.The analysis presented here studies the energy dissipationmechanism for residual vibrations only, when the compoundsystem is subjected to an initial velocity, that is, a very shortpulse, focusing on the times when the secondary system isdisconnected, and then connected by an inelastic impact tothemainmass, thus dissipating energy. It is found how energydissipation can bemaximised if proper values of themass andstiffness ratios are chosen.

2. A Semiactive Control Strategy for ResidualVibration Control

A control strategy presented by Onoda et al. [11] involvesthe semiactive switching of the stiffness during the residualperiod, that is, when a certain shock pulse ends and thesystem undergoes free vibration. The objective is to quicklydissipate the energy stored by the elastic element during theshockwithout adding an external dampingmechanism. Con-sidering a single degree-of-freedom system with a switchablestiffness elementΔ𝑘 subjected to an initial velocity impulse asgiven in Figure 1, the control law is given by

𝑘effective = {𝑘 ]] ≥ 0

𝑘 − Δ𝑘 ]] < 0} , (1)

where ] is the velocity of the mass.

The residual control strategy has to ensure that theamplitude of vibration decreases every cycle. The stiffnessshould be maximum and equal to 𝑘 when the product ]]is positive and minimum and equal to 𝑘 − Δ𝑘 when ]]is negative. When the displacement response satisfies thecondition ]] ≥ 0 the displacement ] and the velocity ]have the same sign. As a result the secondary spring, Δ𝑘,is disconnected when the absolute value of the displacementof the mass is a maximum. It is connected again when theabsolute value of the velocity is amaximum, when the systempasses through its equilibrium position.The phase plane plotpresented in Figure 2(a) shows how the switching occurseffectively dissipating the energy at every stiffness reductionpoint. Figure 2(b) depicts a time history corresponding to thisexample.

A comprehensive study of this strategy is also presentedin [9], concluding that a greater stiffness reduction leads togreater rate of reduction of the residual vibrations. The studypresented by Ledezma-Ramirez et al. [9, 10] assumed that themodel involved massless elastic elements. The incorporationof mass with the secondary elastic element represents afurther step in the modeling of the system.

3. Compound Model for ResidualVibration Suppression

The concept introduced here considers a secondary spring-mass system that is allowed to connect and disconnect froma main system following a switching logic as depicted inFigure 3. Figure 3(a) shows the rigidly connected model withthe response quantity ](𝑡) whilst Figure 3(b) shows the dis-connected systemswith response quantities ]

1(𝑡) and ]

2(𝑡) for

the primary and secondary systems, respectively. Althoughthe system might look like a two degree of freedom model,the secondary and main systems are not coupled thus theyrepresent two separate SDOF systems when disconnected.

The secondary spring-mass system is allowed to connectand disconnect from the main mass following the controllaw as given by (1). Considering that the secondary mass-spring system will oscillate independently during the offpart of the control law (low stiffness stage), it is importantto ensure that the secondary mass is exactly at the staticequilibrium position at the moment of stiffness recovery,when the secondary stiffness Δ𝑘 reconnects to the primarymass. If this is achieved, both the main and the secondarysystem will coincide at the correct time as given by thecontrol law.This also requires that the primary and secondarysystems oscillate in such a way that they do not collide duringthe time they are disconnected.

One of the principal characteristics of the simple modelconsidered in previous studies [9, 10] is the immediateenergy loss at the time at which the secondary stiffness isdisconnected, which are the stiffness reduction points shownin Figures 2(a) and 2(b).However, the new approach shown inFigure 3 considers no energy loss during the disconnection.The total energy is the sum of the energy in the main and thesecondary system, the latter has a certain amount of potential

Shock and Vibration 3

k − Δk

k − Δk

A

E

FD

k

k

k

C

B�max

�/�max

(a)

A E

F

D

C

B

�max

t/Tm

�/�max

(b)

Figure 2: Free vibration of the on-off switchable system illustrating the effects of the stiffness change (— high stiffness; - - - low stiffness).(a) Phase plane plot, (b) time history of the displacement response. Response quantities have been normalized to their respective maximumvalues, and time is normalized considering the mean period 𝑇

𝑚= (𝑇on + 𝑇off )/2.

k − Δk

Δk

m − Δm

𝜉(t)

�(t)

Δm

(a)

k − Δk

Δk

m − Δm

𝜉(t)

�1(t)�2(t)

Δm

(b)

Figure 3: Compound model comprising two single degree-of-freedom models that can oscillate together (a) or independently (b).

energy when it is disconnected, and no external or internalform of damping is considered.

For this model the energy dissipation mechanism isattributed solely to the subsequent connection and the impactbetween the main mass𝑚−Δ𝑚 and the secondary mass Δ𝑚.When the impacting masses stick together after the impactthen the collision is said to be perfectly inelastic. In this case,the ratio between the velocities of separation and approachof the two masses involved in the impact, also called thecoefficient of restitution [12], is zero. The masses will thenhave the same velocity immediately after the impact, which is

according to the conservation of momentum principle givenby

]0=(𝑚 − Δ𝑚) ]

1+ Δ𝑚]

2

𝑚, (2)

where ]1and ]

2are the velocities of the two masses imme-

diately before the impact and ]0is the common velocity of

the masses once they are moving together immediately afterthe impact.This condition on the collisionmight be achievedpractically if a rapid clamping mechanism is used to attachthe secondary mass to the primary mass.


t/TmA E

F

D

C

B

t0

�/�max

Figure 4: Response of the main mass 𝑚 − Δ𝑚 before the stiffnessreduction (–) and after the reduction (--).The dotted line representsthe response of the secondary system when disconnected. Responseis normalized to the maximum value, and time is normalizedconsidering the mean period 𝑇

𝑚= (𝑇on + 𝑇off )/2.

When the systems are attached, the equation of motion isas given by equation

𝑚] + 𝑘effective] = 0, (3)

where 𝑚 is the total mass and 𝑘effective is the effective totalstiffness in the system. Following the control logic describedby (1), when the displacement of the mass is maximum thestiffness is switched to its low value. At this point, the primaryand secondary systems will oscillate independently and theirequations of motion are

(𝑚 − Δ𝑚) ]1+ (𝑘 − Δ𝑘) ]

1= 0,

Δ𝑚]2+ Δ𝑘]

2= 0.

(4)

The point at which the spring is disconnected is shown inFigure 4 as point B at time 𝑡 = 𝑡

0.

Figure 4 also shows a general time-displacement responsefor the main mass 𝑚 − Δ𝑚 until the stiffness recovery point.The system oscillates with a new natural frequency resultingfrom the effective stiffness andmass change until point C.Thesecond mass is allowed to oscillate independently in a waythat they do not collide or come together until a specifiedpoint in the cycle of vibration, which is marked as C inFigure 4 of the primarymass𝑚−Δ𝑚.This process will repeatagain from point D where the systems will be disconnectedand eventually recombined at point E.

4. Energy Dissipation in the Compound Model

To obtain an expression for the energy dissipated during theimpact, it is necessary to consider the exact values of velocityfor each mass at the moment of contact. Considering thecontrol law, the main mass𝑚 − Δ𝑚 is assumed to be passingthrough the static equilibrium position at the moment ofcontact or recovery, and the solution for the displacement of

each mass can be found provided that the initial conditionsare those from the stiffness reduction point B in Figure 4, thatis, maximum displacement and zero velocity:

]1= ]max cos𝜔𝑝 (𝑡 − 𝑡

0) ,

]2= ]max cos𝜔𝑠 (𝑡 − 𝑡

0) .

(5)

The corresponding velocities are given by

]1= −𝜔𝑝]max sin𝜔𝑝 (𝑡 − 𝑡

0) ,

]2= −𝜔𝑠]max sin𝜔𝑠 (𝑡 − 𝑡

0) .

(6)

The natural frequencies for the disconnected single degree-of-freedom systems 𝜔

𝑝and 𝜔

𝑠(primary and secondary

systems) are defined as

𝜔𝑝= √

𝑘 − Δ𝑘

𝑚 − Δ𝑚,

𝜔𝑠= √

Δ𝑘

Δ𝑚.

(7)

The time 𝑡0is the time it takes for both masses to oscillate for

the first quarter of the cycle and it can be expressed as

𝑡0=

𝜋

2𝜔𝑛

, (8)

where 𝜔𝑛is the natural frequency of the compound system

when the masses are connected and is given by 𝜔𝑛= √𝑘/𝑚.

At the moment of contact between the masses (marked asC in Figure 4), 𝑡 − 𝑡

0= 𝜋/2𝜔

𝑝and the corresponding

displacements and velocities of each mass can now berewritten as

]1= 0, (9)

]2= ]max cos(

𝜋𝜔𝑠

2𝜔𝑝

) , (10)

]1= −𝜔𝑝]max, (11)

]2= −𝜔𝑠]max sin(

𝜋𝜔𝑠

2𝜔𝑝

) . (12)

As stated by the control law the secondary mass needsto be at its static equilibrium point since its velocity is amaximum and this will increase the energy dissipation. Itcan be seen from (10) that this condition is possible onlywhen the frequency ratio 𝜔

𝑠/𝜔𝑝takes odd integer values;

that is, 𝜔𝑠/𝜔𝑝

= 1, 3, 5, 7, . . .. The subsequent results arecalculated using this condition. Thus, it is useful at this pointto introduce the frequency ratio Ω = 𝜔

𝑠/𝜔𝑝, the frequency

ratio for the disconnected systems. It is now possible tocalculate the energy lost during each impact. The kineticenergy after the impact is given by

𝑇0=1

2𝑚]20, (13)


where ]0is the common velocity after the masses collide as

given by (2).The initial total potential energy of the system is(1/2)𝑘]2max = (1/2)𝑚]2max, where ]max is the maximum peakdisplacement of the main system just before the secondarysystem is disconnected, and there is no energy lost until thepoint of zero displacement for the primary mass 𝑚 − Δ𝑚

when the impact occurs. At this point, the energy dissipatedduring the impact is

𝐸𝑑=1

2𝑘]2max −

1

2𝑚]20. (14)

Hence the corresponding percentage of energy dissipatedcan be expressed as a percentage of the energy in the systembefore the impact:

%𝐸𝑑= (1 −

𝑚]20

𝑘]2max) × 100. (15)

Combining (2), (14), and (15) the common velocity afterthe impact can be written as

]0=

]max𝑚

× [(𝑚 − Δ𝑚)𝜔𝑝+ Δ𝑚𝜔

𝑠sin(

𝜋𝜔𝑠

2𝜔𝑝

)] .

(16)

Equations (15) and (16) can be combined to give thepercentage of energy dissipated as

%𝐸𝑑= [1 −

1

𝑘𝑚

×[(𝑚 − Δ𝑚)𝜔𝑝+ Δ𝑚𝜔

𝑝sin(

𝜋𝜔𝑠

2𝜔𝑝

)]

2

] × 100.

(17)

Equation (17) can bewritten in a nondimensional formbyusing the parameters 𝜎 = Δ𝑘/𝑘, the stiffness reduction ratio,and the frequency ratio Ω = 𝜔

𝑠/𝜔𝑝, the frequency ratio, to

give

%𝐸𝑑= [1 −

𝜎

1 + Ω2 ((1/𝜎) − 1)

× [(1

𝜎− 1)Ω + sin(𝜋

2Ω)]

2

] × 100.

(18)

The mass ratio 𝜇 = Δ𝑚/𝑚 as defined previously is usedagain. To guarantee that the masses coincide at the staticequilibriumdisplacement position at the time required,𝜇 and𝜎 must have values that satisfy Ω = 𝜔

𝑠/𝜔𝑝= 1, 3, 5, . . ., that

is, odd integers. However, in the case of 𝜇 = 𝜎, which givesΩ = 1, the amount of energy dissipated is zero.This is becausethe velocity of the secondary mass is equal in magnitude andphase to the velocity of the main mass when they collide; thatis, the relative approach velocity is zero. There is no changein the velocity of the masses before and after the impact, sothere is no energy lost during the contact.

Ener

gy d

issip

ated

(%)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

Ω

Figure 5: Percentage of energy dissipated at the first reconnectionas a function of the stiffness reduction ratio 𝜎 for different values ofthe secondary to primary systems frequency ratio Ω (—Ω = 3; - - -Ω = 7; ⋅ ⋅ ⋅ Ω = 11; —Ω = 5; − − Ω = 9; - - - Ω = 13).

In order to assure maximum energy dissipation duringthe impact the common velocity of the masses after thecontact must be as small as possible; that is, the kineticenergy is minimized. Since the velocity of the mass 𝑚 − Δ𝑚

is always a maximum just before the impact, the velocityof Δ𝑚 should preferably be a maximum and have oppositesign at the point of impact. This is not necessarily true forall the odd integer values of Ω as the energy dissipated isconsiderably higher when Ω = 3, 7, 11, . . .. On the otherhand, when Ω = 5, 9, 13, . . . the energy dissipated is smaller.It is possible to maximize the energy dissipation when Ω =

3, 7, 11, . . . because the velocity of the masses is out of phaseat the moment of impact. However, when Ω = 5, 9, 13, . . .

the velocities, although different inmagnitude, have the samephase; therefore the energy dissipation is considerably lesscompared with the previous case.This fact can be seen clearlyin Figure 5, which shows the energy dissipation as a functionof the stiffness ratio, for the values ofΩmentioned above.

Although no damping is included in the modeling of thesystem, it is useful to obtain an equivalent viscous dampingratio in order to quantify the energy dissipated in the modeland compare with the well-knownMKC system. A plot of theequivalent viscous damping ratio as a function of the stiffnessreduction factor 𝜎 is presented for several values of Ω inFigure 6. Although the decrement of peak amplitudes is notlogarithmic, it can be considered as such for simplicity andcomparison to the viscously damped system. As a result, (18)can be used to calculate the amplitude ratio of consecutivepeaks and then the logarithmic decrement as 𝛿 = ln(]

1/]2)

and hence the equivalent damping ratio is estimated usingthe well-known equation 𝜁 = 𝛿/√4𝜋2 + 𝛿2; this plot furtherconfirms the energy dissipation behaviour. The equivalentviscous damping ratio was obtained numerically by obtainingthe peak displacements for the impacting model, using afourth order Runge-Kutta routine, and then considering the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜁 eq

𝜎

Figure 6: Equivalent damping ratio 𝜁eq as a function of the stiffnessreduction ratio 𝜎 for different values of the secondary to primarysystem frequency ratioΩ (—Ω = 3; - - -Ω = 7; ⋅ ⋅ ⋅ Ω = 11; —Ω = 5;− − Ω = 9; - - - Ω = 13).

decay rate. The calculations were made for Ω = 3, 5, 7, 9, 11,and 13 as a function of the stiffness reduction factor.

When Ω = 3, 7, 11, . . . there is an optimum combinationof 𝜇 and 𝜎 for each value ofΩ, which dissipates all the energyduring the first impact. As a result both masses return to restimmediately after the impact. This condition can be statedmathematically considering the total momentum after theimpact is zero, as follows:

(𝑚 − Δ𝑚) 𝑥max𝜔𝑝 − Δ𝑚𝑥max𝜔𝑠 = 0. (19)

Noting thatΩ = 𝜔𝑠/𝜔𝑝, (19) can be written as

𝑚 − Δ𝑚

Δ𝑚= Ω. (20)

Thus, for Ω = 3, 7, 11, . . . the relationship between theoptimummass ratio and the frequency ratio is given by

𝜇 =1

Ω + 1. (21)

Using (7) the frequency ratio can also be expressed asΩ =

√(1 − 𝜇)𝜎/(1 − 𝜎)𝜇. As a result, the relationship between thestiffness ratio and the frequency ratio can be calculated as

𝜎 =Ω

Ω + 1. (22)

The values of 𝜇 and 𝜎 calculated using (21) and (22)represent the peaks observed in Figure 6 for Ω = 3, 7, 11, . . ..These peaks tend to indicate an equivalent damping ratioof 1, which is true considering the definition of criticaldamping as the systemno longer possesses oscillation.Hence,it is possible to maximize the energy dissipation for thesesituations. This can be further validated by obtaining thederivative of (18), which can be used to obtain the maxi-mum value of energy dissipation for particular values of Ω.

0 3 7 11 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ω

𝜇,𝜎

Figure 7: Values of the stiffness reduction ratio 𝜎 and mass ratio𝜇 corresponding to different values of secondary to primary systemfrequency ratio Ω which give maximum energy dissipation in theimpacting model (— 𝜎; - - - 𝜇).

However, asΩ increases, the stiffness reduction required alsoincreases, which could be difficult to achieve in practice. Thecorresponding values of𝜇 and𝜎 for discrete values ofΩwhichgive maximum energy dissipation are shown in Figure 7,where it is clearly seen that higher stiffness reduction isneeded asΩ increases.

In general the stiffness and mass ratios can be relatedusing the following equation:

𝜇 =1

1 + Ω2 ((1/𝜎) − 1). (23)

As an example, consider the plot shown in Figure 8(a)which depicts a time history showing the displacement,velocity, and acceleration responses for the impacting model(bold line) where 𝜎 = 0.5 and 𝜇 = 0.1. The sudden changesin velocity are due to the impact between the masses afterevery half cycle. This combination will give Ω = 3 but is notoptimized for the case of maximum energy dissipation. Theoptimum situation when the energy dissipated is maximized,that is, 𝜇 = 0.25 and 𝜎 = 0.75, is presented in Figure 8(b).

Figure 9(a) shows the secondary system response (dashedline) during the times it remains disconnected and oscillatesindependently, showing that both masses coincide at thesame point during the impact. Figure 9(b) shows the kinetic,potential, and total energy for the system.This plot shows thesumof the energies in the primary and the secondary systems.Both Figures 9(a) and 9(b) are for values of 𝜎 = 0.5 and𝜇 = 0.1. The results corresponding to the optimum situationoccurringwhen𝜇 = 0.25 and𝜎 = 0.75 are shown in Figure 10,where it is clearly seen that the masses are at rest immediatelyafter the first impact.


0 0.5 1 1.5 2 2.5 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/Tm

(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

t/Tm

(b)

Figure 8: Time response for the impacting model. The time isnormalised with respect to the mean period 𝑇

𝑚. (a) 𝜎 = 0.5 and

𝜇 = 0.1, (b) 𝜎 = 0.75 and 𝜇 = 0.25. The frequency ratio betweensecondary and primary system is Ω = 3 (— displacement; - - - -velocity; ⋅ ⋅ ⋅ acceleration).

5. Comparison with the MasslessSecondary System

In order to establish a relationship between this impactingmodel and the basicmodel considered in [7] one can considerthe behaviour of the impacting model when the secondarymass tends to zero. The maximum energy in the basicswitchable stiffness model can be written in terms of thepotential energy as (1/2)𝑘]2max. The percentage of energydissipated after the first stiffness reduction (half a cycle) canbe expressed as

%𝐸𝑑=(1/2) 𝑘]2max − (1/2) 𝑘]2min

(1/2) 𝑘]2max× 100, (24)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

t/Tm

� 1/�

max

,�2/�

max

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ener

gy/in

itial

ener

gy

0 0.5 1 1.5 2 2.5 3

t/Tm

(b)

Figure 9: (a) Displacement response for both the main (—) and thesecondary system (- - -). (b) Energy levels in the system (— totalenergy; - - - kinetic energy; ⋅ ⋅ ⋅ potential energy). Time is normalisedwith respect to the mean period 𝑇

𝑚. For this example the stiffness

reduction ratio is 𝜎 = 0.5 and the mass ratio 𝜇 = 0.1, giving asecondary to primary system frequency ratio Ω = 3.

where ]min is the subsequent negative peak displacement(marked as D in Figure 2) and it is related to ]max by ]min =

]max√1 − 𝜎. It is important to note that there are two stiffnessreductions for the basic model, as there are in general apartfrom the optimum cases two impacts in the impacting modeleach cycle. Hence, (24) reduces to

%𝐸𝑑= 𝜎 × 100. (25)

Equation (25), which gives the energy dissipation as aresult of the first stiffness reduction in the cycle, coincideswith the energy dissipation in the impacting model duringthe first impact, assuming a very small fixed value of thesecondary mass; that is, 𝜇 ≈ 0. This can be easily shown if


0 0.5 1 1.5 2 2.5 3

t/Tm

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

� 1/�

max

,�2/�

max

(a)

0 0.5 1 1.5 2 2.5 3

t/Tm

0

0.2

0.4

0.6

0.8

1

Ener

gy/in

itial

ener

gy

(b)

Figure 10: (a) Displacement response for both the main (—) andthe secondary system (- - -). (b) Energy levels in the system. Timeis normalised with respect to the mean period 𝑇

𝑚. For this example

the stiffness reduction ratio is 𝜎 = 0.75 and the mass ratio 𝜇 = 0.25,giving a secondary to primary system frequency ratio Ω = 3 andmaximum energy dissipation (— total energy; - - - kinetic energy;⋅ ⋅ ⋅ potential energy).

(18) is expressed in terms of the mass ratio 𝜇 and the stiffnessratio reduction 𝜎 giving

%𝐸𝑑

=[[

[

1 − [

[

√1 − 𝜎√1 − 𝜇 + √𝜇𝜎 sin(𝜋

2

√(1 − 𝜇) 𝜎

(1 − 𝜎) 𝜇)]

]

2

]]

]

× 100.

(26)

By setting 𝜇 = 0 in (26) and simplifying the resultingexpression equals (25) thus showing how the compoundmodel reduces to the simple model when the secondary massis negligible.

6. Effect of Damping and Equivalent ViscousDamping Ratio

The objective of the switchable stiffness strategy is to reduceor minimise the residual vibration in lightly damped systemsafter a shock has been applied to a system. So far, theimpacting model has been analyzed, without taking intoaccount any damping. In this section, a brief investigation isconducted as to whether the impact strategy would have anybenefit if applied to a system that already has some dampingpresent. This is important because all real systems inherentlyhave some form of damping.

Figure 11 represents a viscously damped single degree-of-freedom system with two parallel springs one of whichcan be disconnected using the control law given by (1).When disconnected there are two independent mass-spring-damper systems. When both systems are attached an equiva-lent damping constant is calculated from the constants 𝑐

𝑝and

𝑐𝑠corresponding to the damping constants of the primary

and secondary systems, respectively. Additionally, viscousdamping ratios for the primary and secondary systems areintroduced as 𝜁

𝑝and 𝜁𝑠, respectively.

The displacement response for the primary and sec-ondary masses from the maximum displacement point (i.e.,point B in Figure 4, but now considering damping) is given,respectively, by

]1= 0, (27)

]2= ]max 𝑒

−𝜁𝑠𝜔𝑠(𝑡−𝑡0) cos(𝜔

𝑠√1 − 𝜁2

𝑠(𝑡 − 𝑡0)) . (28)

The time 𝑡0is given by (8) as in the undamped case.

The percentage of the energy dissipated can be obtainedcalculating the common velocity of the masses ]

0after they

impact and oscillate together, as expressed by (2). Using thederivatives of (27) and (28) and then combining them into(2) give the common velocity after the impact. As a resultthe percentage of energy dissipated during the impact in thedamped impacting model can be expressed as

%𝐸𝑑

= [1 −𝜎

1 + Ω2

𝑑((1/𝜎) − 1)

× [𝑒−𝜋𝜁𝑝/2√1−𝜁

2

𝑠 (1

𝜎− 1)Ω

𝑑√1 − 𝜁2

𝑝+ 𝑒𝜋𝜁𝑠Ω𝑑/2√1−𝜁

2

𝑝

× [𝜁𝑠cos(𝜋

2Ω𝑑) + √1 − 𝜁2

𝑠sin(𝜋

2Ω𝑑)]]

2

] × 100,

(29)

where the frequency ratioΩ has been replaced by the dampedfrequency ratio, since the model is now damped. This ratio isdefined as

Ω𝑑=

𝜔𝑠√1 − 𝜁2

𝑠

𝜔𝑝√1 − 𝜁2

𝑝

. (30)


c1

c2

k − Δk

m − Δm

Δm

Δk

�1

(a)

c1

c2

k − Δk

m − Δm

Δm

Δk

�1�2

(b)

Figure 11: On-off stiffness model with viscous dampingmodel considering a secondary spring withmass: (a) the systems are rigidly attached;(b) the secondary mass is disconnected and oscillates independently of the main mass.

As described in the previous section, one wants thefrequency ratio to have values Ω

𝑑= 3, 7, 11, . . . in order

to maximize energy dissipation and ensure the systems canrecombine at the required times.Thus, the physical propertiesof the system must be tuned to keep the required frequencyratio.

In order to evaluate the performance of this system akey parameter to compare is the equivalent damping ratioof the system. It is difficult to obtain an analytical expressionfor the equivalent damping, since the effective damping willcomprise the effect of the viscous damping present in thesystem and the energy dissipated by the stiffness reduction.Additionally, the damping ratio of the systemwill change overtime as a result of the stiffness and mass variations. How-ever, it is relatively straightforward to estimate the effectivedamping by using (26) to calculate the energy dissipationand then find the consecutive peaks used to obtain theequivalent logarithmic decrement. For the numerical resultsin this section, the damping constants 𝑐

𝑝and 𝑐𝑠are selected

so that the damping ratio for both systems is the same; thatis, 𝜁1= 𝜁2. The effective damping ratio is shown in Figure 12

for several values of the initial damping ratio in the system forthe on period, as a function of the stiffness reduction factor,consideringΩ

𝑑= 3.

This condition will shift the optimum values of 𝜇 and𝜎, but the frequency ratio Ω

𝑑will remain the same. The

equivalent damping ratio will be enhanced as the viscousdamping in the system increases. However, the main conclu-sion from this figure is that there is no significant change ifthe system is lightly damped, for instance, when the fractionof critical damping is less than 5%. There is a limit on howmuch equivalent damping can be obtained depending onthe amount of physical damping present in the system. Itis important to remember that the strategy is suitable forlow damping systems, where the addition of any other formof damping is not straightforward. Otherwise, if the systemis already highly damped it can be more convenient froma practical point of view not to use a semiactive strategy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜎

𝜁 eq

Figure 12: Equivalent damping ratio 𝜁eq for the impacting systemconsidering viscous damping in both primary and secondary sys-tems, considering the same damping ratio in both systems, so 𝜁 =

𝜁1= 𝜁2. The frequency ratio is Ω

𝑑= 3 (—𝜁 = 0; —𝜁 = 0.01; − − 𝜁 =

0.1; - - -𝜁 = 0.3).

but simply to add another form of passive damping. Finally,Figure 13 compares two situations; the first one depicted bythe continuous line represents the equivalent damping ratiowhen Ω

𝑑= 3 and a damping ratio of 0.01. On the other

hand, the dotted line shows the equivalent damping for thebasic model as explained in [9], considering a damping ratioof 0.01. As a result, it can be seen that the impacting modelpresents a practical limiting value of stiffness reductionwherethe energy dissipation is optimized, rather than the physicallyunrealisable value of 100% stiffness reduction of the basicmodel when energy dissipation is maximized. Moreover, itappears that the impacting model at low values of stiffnessreduction always exceeds the performance of the basic on-off stiffness model in terms of energy dissipation. If the


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜎

𝜁 eq

Figure 13: Equivalent damping ratio comparison between animpacting model, considering a frequency ratio of Ω = 3 (—), andthe samemodel when the secondarymass approaches zero (⋅ ⋅ ⋅ ).Theviscous damping ratio considered for both models is the same andis equal to 1%.

parameters of the model are adequately chosen, the energylost by the inelastic impacts is higher than the energy lost inthe simple model by the disconnecting spring.

7. Conclusions

An alternative modelling approach has been proposed, inorder to investigate the energy dissipation in a switchablestiffness system and to provide a valid mathematical andphysical model. This approach involved a secondary springwith a smallmass and considers the impact between thismassand the main mass at the moment of stiffness recovery. Theenergy is solely dissipated due to this inelastic impact.There isa trade-off between the mass and stiffness ratios; high energydissipation can be achieved for certain combinations of theseparameters which are physically allowed. It was found that asthe secondary mass is reduced to zero, this system effectivelyreduces to the energy dissipation characteristics for the basicon-off model.

Furthermore, the inclusion of viscous damping wasstudied in both the basic and the impacting systems. Themain conclusion obtained being that when the system islightly damped the performance is not affected. However,highly damped systems experience a drawback in vibrationsuppression, and the effective damping ratio could becomelower than that for the viscous damping ratio present for thepassive linear original system. This supports the hypothesisthat the strategy is only suitable and beneficial to improveresidual vibration control in lightly damped systems.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the Mexican Councilfor Science and Technology CONACyT for the financialsupport and the Universidad Autonoma de Nuevo Leon,UANL.

References

[1] C. M. Harris and C. E. Crede, Shock and Vibration Handbook,McGraw-Hill, New York, NY, USA, 1996.

[2] R. S. Ayre, Engineering Vibrations, McGraw-Hill, NewYork, NY,USA, 1958.

[3] J. C. Snowdon, Vibration and Shock in Damped MechanicalSystems, Wiley and Sons, New York, NY, USA, 1968.

[4] R. L. Eshleman and P. Rao, “Response of mechanical shockisolation elements to high rate input loading,” Shock andVibration Bulletin, Shock and Vibration Information Center, vol.40, no. 5, pp. 217–234, 1969.

[5] D. V. Balandin, N. N. Bolotnik, and W. D. Pilkey, “Review:optimal shock and vibration isolation,” Shock andVibration, vol.5, no. 2, pp. 73–87, 1998.

[6] D. V. Balandin, N. N. Bolotnik, and W. D. Pilkey, “Pre-actingcontrol for shock and impact isolation systems,” Shock andVibration, vol. 12, no. 1, pp. 49–65, 2005.

[7] T. P. Waters, Y. Hyun, and M. J. Brennan, “The effect of dual-rate suspension damping on vehicle response to transient roadinputs,” Journal of Vibration and Acoustics, Transactions of theASME, vol. 131, no. 1, 2009.

[8] D. F. Ledezma, N. S. Ferguson, and M. J. Brenan, “Shock per-formance of different semiactive damping strategies,” Journal ofApplied Research andTechnology, vol. 8, no. 2, pp. 249–259, 2010.

[9] D. F. Ledezma-Ramirez, N. S. Ferguson, and M. J. Brennan,“Shock isolation using an isolator with switchable stiffness,”Journal of Sound andVibration, vol. 330, no. 5, pp. 868–882, 2011.

[10] D. F. Ledezma-Ramirez, N. S. Ferguson, andM. J. Brennan, “Anexperimental switchable stiffness device for shock isolation,”Journal of Sound and Vibration, vol. 331, no. 23, pp. 4987–5001,2012.

[11] J. Onoda, T. Endo, H. Tamaoki, and N. Watanabe, “Vibrationsuppression by variable-stiffness members,” AIAA journal, vol.29, no. 6, pp. 977–983, 1991.

[12] D. K. Anand and P. F. Cunniff, Engineering Mechanics: Dynam-ics, Houghton Mifflin Company, Boston, Mass, USA, 1973.

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