Displacement transmissibility of a Coulomb friction ...energy dissipation, friction may also be introduced in mechanical and civil structures to serve purposes such as isolation and

Nonlinear Dyn (2019) 98:2595–2612https://doi.org/10.1007/s11071-019-04983-x

ORIGINAL PAPER

Displacement transmissibility of a Coulomb frictionoscillator subject to joined base-wall motion

Luca Marino · Alice Cicirello · David A. Hills

Received: 14 December 2018 / Accepted: 25 April 2019 / Published online: 10 May 2019© The Author(s) 2019

Abstract This study investigates the displacementtransmissibility of single-degree-of-freedom systemswith a Coulomb friction contact between a mass anda fixed or oscillating wall. While forced vibration andbase motion problems have been extensively investi-gated, little work has been conducted on the joinedbase-wall problem. Based on the work of Den Hartog(Trans Am Soc Mech Eng 53:107–115, 1930), analyti-cal expressions of the displacement transmissibility arederived and validated numerically. The mass absolutemotion was analysed in the joined base-wall motioncase with a new technique, with results such as: (1) thedevelopment of a method for motion regime determi-nation; (2) the existence of an inversion point in trans-missibility curves, after which friction damping ampli-fies the mass response; (3) the gradual disappearing ofthe resonant peak when the ratio between friction andelastic forces is increased. Moreover, numerical anal-ysis provides further insight into the frequency regionwheremass sticking occurs in the basemotion problem.

L. Marino (B) · A. Cicirello · D. A. HillsDepartment of Engineering Science, University of Oxford,Parks Road, Oxford OX1 3PJ, UKe-mail: [email protected]

A. Cicirelloe-mail: [email protected]

D. A. Hillse-mail: [email protected]

Keywords Coulomb damping · Friction · Displace-ment transmissibility · Base motion · Joined base-wallmotion

1 Introduction

Developing a fundamental understanding of the roleplayed by friction damping in structural dynamics isnowadays an important challenge for exploring suitablestructural designs. Many engineering systems, suchas aerospace and automotive vehicles, turbo-machinesand buildings, are in fact characterised by the pres-ence of frictional joints. As an important source ofenergy dissipation, friction may also be introduced inmechanical and civil structures to serve purposes suchas isolation and vibration control. At an early stage ofa design, the performance of a frictional contact underdynamic load is usually investigated by consideringa single-degree-of-freedom (1-DoF) system subject toharmonic base excitation. This contribution is focusedon the investigation of themain features of the dynamicresponse of such a model, both to base motion and tojoined base-wall motion. The latter is of interest formany industrial applications, such as the dovetail rootof a turbine blade; however, its response features areyet mostly unexplored.

The dynamic behaviour of a 1-DoF system underbase motion is usually investigated with the displace-ment transmissibility, which is the ratio between theamplitudes of mass and base motions. This metric can

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2596 L. Marino et al.

Fig. 1 1-DoFmass-spring-viscous dampersystem under harmonic basemotion (a) and itstheoretical displacementtransmissibility evaluated atdifferent damping ratios (b) m

x(t)

y(t)k c

(a)

0 1 2 3 4 5

Frequency ratio

0

1

2

3

4

5

6

Dis

plac

emen

t tra

nsm

issi

bilit

y

.51

2

.25

.1

= 0

(b)

be augmented by including the evaluation of phase shiftbetween input and output displacements. In the viscous1-DoF system shown in Fig. 1a, the base excitation istransmitted to the mass through a spring and a viscousdashpot in parallel. The response can be obtained asa superposition of the analytical forced responses tothe elastic and the viscous forces [2]. The displace-ment transmissibility for different values of dampingratio is portrayed in Fig. 1b as a function of the ratiobetween forcing and natural frequencies. It is possibleto observe that as the damping increases, the resonantpeak is shifted to the left and for a frequency ratio below√2 the transmissibility is reduced; above this value,

there is an inversion of the curves and so the dampingamplifies the transmissibility. This behaviour needs tobe accounted during the design stage.

In a 1-DoF system with a Coulomb friction contact,the problem is complicated by the nonlinearity of theequations which can lead to a periodic sticking duringthe mass vibration, defined as stick-slip behaviour (seee.g. [3,4]). A general solution to the governing equationcannot been found in a closed form, andnumerical tech-niques are usually employed [2]. Nonetheless, the basemotion problem can be investigated as an equivalentforced vibration problem [1] that can be solved withalternative analytical approaches.These approaches arebriefly reviewed in what follows. In 1930, Den Hartogproposed in [1] a method for the analysis of forcedvibration with Coulomb friction; the author suggesteda local solution for the steady sliding motion, allow-ing the description of quantities such as the amplitudeof the mass motion or the phase shift between input

load and response, and a condition for locating thetransition between stick-slip and continuous motion.Only after many years, the same problem has beenapproached with different techniques, such as numeri-cal integration in time domain [5], phase-plane meth-ods [6], incremental harmonic balance [7], equivalentlinearisation method [8] and others. In 2001, Hong andLiu [9] proposed a new approach aiming to improveDen Hartog’s solution. This includes the calculation ofthe conditions which lead to a purely sliding motionand a new estimation of the maximum velocity of theslidingmass and its time lag. Hong and Liu have shownthat the same expressions as Den Hartog’s for the max-imum displacement and its phase lag are recovered.Moreover, their analytical results were validated witha numerical solution [10].

The base motion problem has been addressed as aforced vibration problem in references [1,6,10,11]. Inparticular, Den Hartog [1] proposed an extension of histheory to the classical base motion problem and also tothe case where base and wall are jointed so that the wallis oscillating with the same harmonic motion imposedon the base. However, for the base motion case, someimplications were not sufficiently discussed, includingdetails about the stick-slip occurrence at high frequencyratios and the existence of an upper limit of the ratiobetween friction and spring force amplitudes allowinga continuous motion.

For the joined base-wall motion case, a series ofresults based on Den Hartog’s theory are representedfor the first time. However, such results are referredexclusively to the relative mass-wall motion. The pre-

123

Displacement transmissibility of a Coulomb friction 2597

sented analysis goes much further, discussing the con-ditions for mass sticking and reaching a formulationholding for mass absolute motion, which allows theevaluation of both the displacement transmissibilityand the phase shift between excitation and response.

A numerical approach based on the resolution of thegoverning equationswas introduced in order to validatethe analytical transmissibility curves. Furthermore, thenumerical evidence allows the assessment of the ana-lytical forecast about the transition between continuousand stick-slipmotion. Being not based onDenHartog’sassumptions, this technique is able to extend the trans-missibility curves also to the stick-slip region. Particu-lar attention is paid to the minimum number of param-eters necessary to describe the problem both in an ana-lytical and in a numerical approach, leading to a moresuitable formulation of the equations.

The paper is organised as follows: Sect. 2 is ded-icated to detailed review of Den Hartog’s theory onforced vibration; in Sect. 3, the analytical formula-tion of the base motion problem is discussed; Sect. 4presents the results regarding the joined base-wallmotion case; the numerical approach and the valida-tion are introduced in Sect. 5.

2 Review of Den Hartog’s theory on forcedvibrations

In reference [1],DenHartog introduced an approach forthe analysis of the forced vibration of a 1-DoF mass-spring system under harmonic load in the presence of aCoulomb friction contact between mass and a ground-fixed wall. The main outcomes from his work are thepossibility to determine analytically the conditions forwhich themass is sticking or slipping during the steady-state motion and the amplitude of mass displacementwhen such motion is continuous. The governing equa-tions, as well as the main results, are discussed in thissection.

Let us consider a 1-DoF system composed by amassm connected to ground through a springwhose stiffnessis k (Fig. 2); be such mass rubbing against a ground-fixed wall in the presence of a compression load N , insuch a way as to form a contact characterised by a fric-tion coefficient μ. A monoharmonic load, with ampli-tude F0 and frequency ωb, is assumed to be applieddirectly to the mass. The general equation of motion ofthe system represented in Fig. 2 is:

m

x(t)

y(t) kµ

N

Fig. 2 1-DoF mass-spring system with a friction contact underharmonic load

mx + kx + μNsgn(x) = F0 cosωbt (1)

Suppose that the steady-state condition has beenreached, and the mass is vibrating with a steady andcontinuous periodic motion; no assumptions are madehere regarding the harmonicity of this response. Massmotion is supposed to exhibit the same period as theforce, so it is possible to consider a time interval[0, 2π/ωb] between a generic couple of subsequentmaxima (Fig. 3) as completely representative of thesteady-state motion. Further, it is reasonable to assumethat a phase shift will be present in general between theapplied force and the mass displacement. Due to thesymmetric loading, the motion is assumed as symmet-ric with respect to the origin of the x-axis and the min-imum condition will therefore occur when t = π/ωb.Being the motion assumed as continuous, x < 0 in allthe internal points of the interval [0, π/ωb], so Eq. (1)will be reduced to:

x + ω2n

(x − μN

k

)= F0

mcos(ωbt + φ) (2)

Equation (2) is linear and its boundary conditionscan be written in terms of displacement and velocity atboth the ends of the half period:

⎧⎪⎨⎪⎩x(0) = x0, x(0) = 0

x

(π

ωb

)= −x0, x

(π

ωb

)= 0

(3a)

(3b)

The maximum absolute displacement x0 and the phaseangle φ are both unknown. It must be underlined thatsuch a phase angle is intended by Den Hartog asreferred to the maxima of force and motion and thephase angle between their zeros will be different ingeneral.

123


0 /2 b / b 3 /2 b 2 / bTime

-F0/k

-x0

0

x0

F0/kD

ispl

acem

ent

Mass responseBase motion

Fig. 3 Steady-state continuous motion of a 1-DoF system withCoulomb friction

Introducing the frequency ratio:

r = ωb

ωn

as the ratio between the driving and the natural frequen-cies of the excited system, the solution of Eq. (2) canbe expressed as:

x(t) = An cosωnt + Bn sinωnt

+ F0k

1

1 − r2cos(ωbt + φ) + μN

k(4)

where the constants An and Bn can be removed byimposing the conditions (3a):

x(t) = x0 cosωnt + μN

k(1 − cosωnt)

+ F0k

1

1 − r2

[cosφ(cosωbt − cosωnt)

+ sin φ(r sinωnt − sinωbt)

](5)

The imposition of the conditions (3b) allows thefinding of the unknown values of x0 and φ; substitutingthem in Eq. (5) and in its derivative:

{A cosφ + B sin φ + C = 0

P cosφ + Q sin φ + R = 0

(6a)

(6b)

where:

A = − F0k

1

1 − r2

(1 + cos

π

r

)

B = F0k

1

1 − r2sin

π

r

C = x0

(1 + cos

π

r

)+ μN

k

(1 − cos

π

r

)

P = F0ωn

k

1

1 − r2sin

π

r

Q = F0ωn

k

1

1 − r2

(1 + cos

π

r

)

R = −(x0 − μN

k

)ωn sin

π

r

(7a)

(7b)

(7c)

(7d)

(7e)

(7f)

Hence:

cosφ = BR − CQ

AQ − PBsin φ = CP − AR

AQ − PB(8)

Introducing the damping function U (r) and theresponse function V (r):

U (r) = sin π/r

r(1 + cosπ/r)V (r) = 1

1 − r2(9)

it is possible to write Eq. (8) as:

cosφ = kx0F0

1

Vsin φ = −μN

F0

U

V(10)

Substituting the results (10) in Eq. (5) and applyingthe condition (3b), it is finally possible to determinethe amplitude of the mass motion:

x0 = F0k

√V 2 −

(μN

F0

)2

U 2 (11)

Equation (11) indicates as the ratio between the ampli-tude of mass vibration and its static displacement F0/kis not directly dependent on the applied force, but it israther related to the ratio between friction and externalforce amplitudes; Den Hartog’s curves for such ampli-tude ratio are plotted in Fig. 4 at different values of theforce ratio.

A fundamental issue of this approach is understand-ing when the solution (11) is valid. As the assumptionof x < 0 was posed for all the internal points of theconsidered time interval, the solution x(t)must clearly

123


0 0.5 1 1.5 2Frequency ratio

0

1

2

3

4

5A

mpl

itude

ratio

Boundary

.6

.4.6

.2.8

F /F0 = 0

.2

.4

0

Fig. 4 Dimensionless amplitudes of vibration in a 1-DoF systemwith Coulomb friction

respect this condition. Applying it to the derivative ofEq. (5), the condition obtained is:

x0 >μN

k

S

r2(12)

where:

S(r) = max0≤t≤ π

ωb

[r sinωnt +Ur2(cosωbt − cosωnt)

sinωbt

]

From (11) and (12):

x0 >F0k

√√√√√√V 2

1 +(Ur2

S

)2 (13)

so there is a unique limit condition for the validity ofEq. (11), independently of the ratio between frictionand external force. The meaning of this condition (dot-ted in Fig. 4) is that, if x = 0 in any of the internal pointsof the time interval, a stop condition is occurring and themotion is no longer continuous. A particular behaviouris further highlighted in Fig. 4: only the curve corre-sponding to μN/F0 = 0.8 shows a finite peak value atresonance, which is also shifted to the left (r < 1). DenHartog suggested that the lower value of force ratio forwhich a finite resonance is observed is:

μN

F0= π

4(14)

The numerical simulation of the mass motion led inSect. 5 allows to observe that this phenomenon is dueto stick-slip occurrence at r = 1. This means that Eq.(14) can be obtained by evaluating the force ratio atwhich the displacement transmissibility and boundary

0 0.5 1 1.5 21/r

0

30

60

90

120

150

180

Pha

se a

ngle

(d

eg)

BoundaryF /Fk = 0

0.2

.4

.6

.4

.2

.6

.8

Fig. 5 Phase-angle diagram for a 1-DoF system with Coulombfriction

curves intersect for r → 1. It can be shown that S = 1at resonance; from Eqs. (11) and (13):

μN

F0= lim

r→1

(− V

U

)

= limr→1

r(1 − cosπ/r)

sin π/r(r2 − 1)= π

4(15)

Den Hartog derives another limit condition for thenon-stopmotion fromEqs. (11) and (12); it can bewrit-ten in terms of the ratio between friction and externalforces:

μN

F0<

√√√√√√V 2

(S

r2

)2

+U 2

(16)

Finally, a limit condition can be obtained also onthe phase angle by using Eqs. (10), (11) and (16). Theresult from Den Hartog’s theory is plotted in Fig. 5.

3 Vibration transmission in base motion with fixedwall case

Den Hartog affirms that the approach introduced inSect. 2 can be applied to the absolute motion of themass excited from a harmonic base motion; the rub-bing wall is supposed to be fixed. In this section, themain results deriving from this statement will be pre-sented and discussed in terms of their impact on vibra-tion transmission. Furthermore, a better understandingabout the mass motion regime at high frequencies iscarried out.

Consider the 1-DoF system introduced in Fig. 2; bethemass excited from themonoharmonic motion of the

123


m

x(t)

y(t) kµ

N

Fig. 6 1-DoF mass-spring system with friction contact underharmonic base excitation

base, rather than from an external force (Fig. 6). Theequation of motion of such system can be written as:

mx + k(x − y) + μNsgn(x) = 0 (17)

so, if the base motion is:

y(t) = Yb cosωbt (18)

Eq. (17) becomes:

mx + kx + μNsgn(x) = kYb cosωbt (19)

Comparing Eqs. (1) and (19), it is plain how the ana-lytical results will be exactly the same, as observedin base motion problems for undamped systems [2],if F0 = kYb, completely agreeing with Den Hartog’sobservation. It is worth noting as this position is in noway introducing alterations in the response able to alterits accordance with Den Hartog’s assumptions. There-fore, the main results are obtained applying this trivialsubstitution and reported in Eqs. (20–23). The phaseangle between force and displacement will be:

cosφ = x0Yb

1

Vsin φ = −μN

kYb

U

V(20)

and themaximumamplitude ofmass displacementwillbe:

x0 = Yb

√V 2 −

(μN

kYb

)2

U 2 (21)

The displacement transmissibility is defined as theratio between the maximum response magnitude andinput displacement magnitude at the input frequency[2]; it is usually plotted as a function of the frequencyratio. Therefore, here it is given exactly by the ratiox0/Yb (Fig. 7):

0 1 2 3 4 5Frequency ratio

0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y Boundary

.8

F /Fk = 0

.6

.4.2

.6

.4

0

.2

Fig. 7 Theoretical displacement transmissibility in a 1-DoF sys-tem under base motion with ground-fixed wall

|X ||Y | =

√V 2 −

(μN

kY

)2

U 2 (22)

The limit conditions for the non-stop motion haveexactly the same formulation introduced in Sect. 2:

|X ||Y | >

√√√√√√V 2

1 +(Ur2

S

)2

μN

kY<

√√√√√√V 2

(S

r2

)2

+U 2

(23)

The most important implications of the presentedresults are:

– the possibility of a priori evaluation of the ampli-tude of the mass motion, if continuous;

– the possibility of forecasting the presence of masssticking along the steady-state motion;

– being U , V and S functions of r , the above resultsdepend only on two parameters, i.e. the frequencyratio itself and the force ratio between friction andspring force amplitudes:Fμ

Fk= μN

kYThis last property is particularly important for itsimpact on the design of friction dampers as it impliesthat the same result, in terms of steady-statemotion, canbe achieved with different set of parameters, as long asfrequency and force ratios are unaltered; this propertywill be discussed extensively in Sect. 5.

The mass will indefinitely remain still if:

kY ≤ μN (24)

123


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Frequency ratio

0

0.2

0.4

0.6

0.8

1Fo

rce

ratio

Fig. 8 Transition force ratio value as a function of frequencyratio

In this case, i.e. if the force ratio is at least equal to1, the elastic force is never enough to overcome thefriction force. When the condition (24) is not verified,the mass may exhibit both a continuous or a stick-slipmotion but never be stationary.

In Fig. 7, it is not clear if the transmissibility curvesalways intersect the boundary line at high frequencies.A better way to analyse this problem is shown in Fig. 8,where the force ratio corresponding to the transitionbetween continuous and stick-slip motion, described incondition (23b), is plotted as a function of the frequencyratio. It is possible to observe as the limit force ratiocurve tends to an asymptotic value when r → +∞;this value is evaluated as:

limr→+∞

√√√√√V 2(

S

r2

)+U 2

= 2√4 + π2

∼= 0.5370 (25)

It is therefore possible to affirm that regime transitionat high frequencies will occur at a certain point only ifthe force ratio is higher than the above value.

Another important result which can be deductedfrom Fig. 8 is the existence of a maximum value ofthe transition force ratio, lower than 1, above whichthe continuous motion is not possible. Such value hasbeen found to be:

max

√√√√√V 2(

S

r2

)+U 2

∼= 0.8265 (26)

and it is obtained at r ∼= 0.8490. This is the last valueof frequency ratio for which it is possible to observe acontinuous motion while increasing the force ratio.

m

x(t)

y(t) kµ

N

Fig. 9 1-DoF mass-spring system with friction contact underharmonic joined base-wall excitation

4 Vibration transmission in joined base-wallmotion case

Another extension of Den Hartog’s theory is to vibra-tion transmission when base and wall are movingtogether. Among his final annotations, Den Hartog [1]affirms:

If the rubbing wall be tied to the upper end of thespring and this end with the wall be subjected tothe motion y(t) = Yb cosωbt , the above solution(Sect. 2) holds for the relative motion betweenmass and wall, if only F0/k be replaced by Ybr2.

This section discusses the implications of such state-ment in terms of the relative motion between mass andwall, presenting the results deriving from the suggestedposition. Amore complete approach will be introducedwith the goal of describing the mass absolute motionand, consequently, the actual displacement transmissi-bility and phase shift between excitation and response.The results show interesting similarities with viscouslydamped systems and also a more complex scenario interms of possible motion regimes, due to the introduc-tion of the frequency ratio in the above position.

Let us consider a 1-DoF system whose only differ-ence with the system in Fig. 6 is that base and wall arejointed, therefore also the wall is subject to harmonicmotion (Fig. 9). The equation of motion of such systemdiffers from Eq. (17) only in terms of friction force: asthe wall is moving, the relative velocity between themass and the wall must be taken into account:

mx + k(x − y) + μNsgn(x − y) = 0 (27)

123


Introducing the new variable z = x−y and substitutingthe expression of y(t), Eq. (27) becomes:

mz + kz + μNsgn(z) = mω2bYb cosωbt (28)

whence:

mz + kz + μNsgn(z) = kYbr2 cosωbt (29)

This equation is written in a coordinate system integralwith base and wall in the form a forced vibration. Theapparent external force acting on the mass is:

F(t) = kYbr2 cosωbt (30)

so, in agreement withDenHartog’s statement, the solu-tion holds for the relative mass-wall motion if the rela-tion:

F0 = kYbr2 (31)

is verified. Applying this condition to the solution (5),it yields:

z(t) = z0 cosωnt + μN

k(1 − cosωnt)

+ Ybr2

1 − r2

[cosφ(cosωbt − cosωnt)

+ sin φ(r sinωnt − sinωbt)

](32)

where the values of φ and z0 are obtained from theboundary conditions (3a–3b), expressed in terms of thevariable z. So:

cosφ = z0Yb

1

Vr2sin φ = − μN

kYbr2U

V(33)

and:

z0 = Ybr2

√V 2 − 1

r4

(μN

kYb

)2

U 2 (34)

The dimensionless amplitude of the relative motionbetweenmass andwall is then represented as a functionof the frequency ratio in Fig. 10; the limit of validityof Den Hartog’s theory for such motion can be derivedfrom the condition z < 0, exactly as in Sect. 2, or moresimply imposing the position (31) in Eq. (13):

|Z ||Y | > r2

√√√√√√V 2

1 +(Ur2

S

)2 (35)


0

1

2

3

4

5

|Z|/|

Y|

Boundary

1.4.8

.6.4

.2

F /Fk = 0

1.01.2

Fig. 10 Dimensionless amplitude of the relativemotion betweenmass and wall

0 0.5 1 1.5 21/r

0

20

40

60

80

100

120

140

160

180

Pha

se a

ngle

zy

(deg

)

Boundary

F /Fk = 0

.2

.4

.4

.6

.6.8

1.01.2

1.4

.20

Fig. 11 Phase angle between excitation and relative mass-walldisplacement

Observing Fig. 10, it is also clear as there willbe no intersections between the amplitude curves andthe boundary described by Eq. (35) at high frequen-cies, where Den Hartog’s assumptions will be there-fore always verified. Figure 11 shows the phase anglebetween the excitation y(t) and the relative mass-wallmotion z(t) as evaluated from Eq. (33). From now on,such angle will be indicated as φyz in order not to con-fuse it with the phase shift between base and massmotion.

If the relativemass-wallmotion can be easily studiedas an extension of Den Hartog’s theory, this is not truefor themass absolutemotion. Even if a relation betweenz0 and x0 can be found as:

x0 = z0 + y0 = z0 + Yb cosφyz

= z0 + z0Vr2

= z0r2 + 1 − r2

r2= z0

r2(36)

123


0 ( - yz)/ b /2 b ( - xz)/ b / b

Time

-z0

-Yb

-|X|-x

0

0

x0

|X|

Yb

z0

Dis

plac

emen

t

x(t)y(t)z(t)

Fig. 12 Steady-state continuous motion of a 1-DoF systemunder joined base-wall motion

it appears evident from Fig. 12 that x0 is different fromthe maximum absolute value |X | assumed by x(t) inthe time interval [0, π/ωb]. In order to evaluate thedisplacement transmissibility, it is necessary to knowthe amplitude of the motion x(t), i.e.:

|X | = maxt∈[0, π

ωb]|x(t)| (37)

where:

x(t) = y(t) + z(t) (38)

By using Eqs. (18) and (32), x(t) can be written as:

x(t) = x0r2 cosωnt + μN

k(1 − cosωnt)

+ Yb cos(ωt + φyz) + Ybr2

1 − r2[cosφyz(cosωbt − cosωnt)

+ sin φyz(r sinωnt − sinωbt)

](39)

The approach developed for the evaluation of thedisplacement transmissibility is based on a maximumproblem rather than on a specific analytical expression.For this reason, it is not possible to impose a limit con-dition for the continuous motion in terms of this quan-tity, as done in Eqs. (13) and (23a). Nevertheless, it ispossible to write it in terms of force ratio fromEq. (16):

Fμ

Fk< r2

√√√√√√V 2

(S

r2

)2

+U 2

(40)


0

1

2

3

4

5

Forc

e ra

tio

Continuousmotion

Relativestick-slipmotion

Sticking

Fig. 13 Graphical identification of mass motion regime on fre-quency ratio–force ratio plane

As confirmed numerically in Sect. 5, if the force ratio islarger than such limit value, a relative stick-slip motionwill occur betweenmass and wall. An important reflec-tion must be made also on the condition for mass com-plete sticking: if Eq. (24) states that in the fixed wallcase this is achieved for a unitary force ratio, inde-pendently of the frequency ratio, in the joined base-wall motion case mass motion is not possible when theamplitude of the apparent external force acting on themass itself does not overcome the friction force ampli-tude:

kYbr2 ≤ μN (41)

whence:

r ≤√

Fμ

Fk(42)

The relation (42) has two important implications:

– mass motion is possible also at force ratio valueslarger than 1, if the frequency ratio is high enough;

– mass may stick completely at low frequencies evenif the force ratio is minor than 1.

Equations (40) and (42) result in three mass behaviourswhich can be identified for specific values of frequencyand force ratios also via graphic approach as proposedin Fig. 13. The above conditions divide the frequencyratio–force ratio plane into three regions correspondingto as many motion regimes: it is particularly interest-ing to observe as there is no overlap between the twocurves. Consequently, whatever the force ratio, there isalways a range of frequencies for which it is possibleto observe stick-slip motion; nevertheless, such rangeis notably narrower when the force ratio is around 0.5.

123



0

1

2

3

4

5D

ispl

acem

ent t

rans

mis

sibi

lity

.2.4

.8

1.0

.6

1.21.4

F /Fk = 0

(a)

1.4 1.45 1.5 1.55 1.6Frequency ratio

0.7

0.75

0.8

0.85

0.9

0.95

1

Dis

plac

emen

t tra

nsm

issi

bilit

y

F /Fk = 0

.6.4

.2

.8

1.2

1.0

1.4

1.41.21.0

.6 .8.4

0.2

(b)

Fig. 14 Theoretical displacement transmissibility in a 1-DoF system under joined base-wall motion (a) and detail of the inversion (b)

The displacement transmissibility |X |/|Y | has beenevaluated in the continuous motion region by deter-mining numerically the maximum value of x(t) asdescribed in Eqs. (37–39) and the resulting curves areportrayed in Fig. 14a at different force ratios. The limitcondition, represented by Eq. (40), can be imposed dur-ing the evaluation of each transmissibility curve, insuch a way to plot the value only when the motionis continuous. Particularly, the dotted line has beenobtained reiterating the procedure for all the force ratiosin the interval [0,1.5] and provides a graphical repre-sentation of the transition from stick-slip to continuousmotion.

Furthermore, Fig. 14a shows as the condition (14) isvalid also for the joined base-wall motion: all the res-onant peaks obtained for high force ratios have a finitevalue and increasing such ratio it is notable as theyget more and more smoothed and shifted at higher fre-quencies. When r � 1, transmissibility tends to zero,so the mass will have more and more narrow oscilla-tions independently of wall motion till getting asymp-totically stationary.

One of the most interesting elements highlighted inFig. 14a is the presence of an inversion of the curves(in details in Fig. 14b), a phenomenon observable alsoin viscous systems subject to base motion: the vibra-tion transmission becomeswider by effect of the damp-ing. It is worth underlining the physical similitudebetween the motion here analysed and the well-knownbasemotion problem applied to viscous 1-DoF systems

(Fig. 1): in both these cases, the motion is transmit-ted not only through the spring but also through thedamper (whatever dashpot or friction contact). Despiteno resemblance being found between the analyticalmodels employed for viscous and dry Coulomb damp-ing, these similarities offer some interesting elementsfor a comparison:

– whereas the inversion always occurs at rinv = √2

in the presence of viscous damping, here the phe-nomenon appears to occur at very slightly higherfrequencies and in a very narrow region 1.4 <

rinv < 1.55 rather than in a single point;– transmissibility value at the inversionwithCoulombdamping is not equal to 1 as in the viscous case buta bit lower.

It is opportune to point out that, although possible,transmissibility curves have not been plotted at forceratios higher than 1.4; the motivation of this choiceresides in the impossibility to catch the inversion orany other phenomena of interest at larger values of suchratio, due to stick-slip occurrence (see Fig. 13).

A last challenge for achieving a full description ofmass response to joined base-wall motion is the deter-mination of the steady-state phase angle φxy betweenthe excitation and the response. Whereas the phaseangle φyz has already been obtained analytically fromEq. (33) and represented in Fig. 11, φxy must clearly beobtained numerically, as it is referred to x(t) and y(t)maxima.

123


0 0.5 1 1.5 21/r

0

10

20

30

40

50

60

70

80

90P

hase

ang

le

xz (d

eg)

F /Fk = 0

.4

.6

.4.6

.2

0

.8

.2

1.0

1.21.4

Fig. 15 Phase angle between relative mass-wall displacementand absolute mass motion

0 0.5 1 1.5 21/r

0

20

40

60

80

100

120

140

160

180

Pha

se a

ngle

xy

(deg

)

0

.6

.4

.2

F /Fk = 0

.2

.8.6

.4

1.01.2

1.4

Fig. 16 Phase angle between relative excitation and absolutemass motion

The first step is the numerical evaluation of the timeinstant txz ∈ [0, π/ωb]where the maximum of |x(t)| isreached; this allows the calculation of the phase angleφxz between absolute and relative mass motions:

⎧⎪⎨⎪⎩

φxz = 2π − ωbtxz, if x(txz) ≥ 0

φxz = π − ωbtxz, if x(txz) < 0

(43)

In Fig. 15, it is possible to see that φxz , just as φyz ,always belongs to the interval [0, 180◦] (see Fig. 11):this means that both x(t) and y(t) peaks anticipate z(t)(which is the origin of the time coordinate system) andtheir observed absolute maximum value will be nega-tive as a consequence. Also, φyz is always larger thanφxz and so (as expected) the negative peak of x(t) willalways follow the negative peak of y(t). Therefore,Fig. 12 is well representative of the problem in terms

of peaks position and it is completely fair to evaluateφxy as:

φxy = (π − φxz) − (π − φyz) = φyz − φxz (44)

The results are finally reported in Fig. 16.

5 Numerical validation and extension to stick-slipmotion regime

A numerical model in MATLAB [12] has beenemployed for the validation of the theory presented inthe previous sections. The code is based on the numeri-cal resolution of the differential equations of motion ofthe 1-DoF systems introduced above; beyond the vali-dation goal, this approach is able to provide results alsooutside the region of validity of Den Hartog’s theory,for the stick-slip motion, as no assumptions were madehere regarding mass motion continuity.

5.1 Base motion with fixed wall case

The base motion problem introduced in Sect. 3 is gov-erned by Eq. (19). Den Hartog’s approach reveals thedependency of the motion on two parameters, the fre-quency ratio and the force ratio: it is possible to demon-strate how this property can be extended also to the gen-eral solution, including transients and the possibility ofa stick-slip behaviour.

The equation of motion presents seven variables:t , m, k, μ, N , Yb and ωb. Only three dimensions arerequired to describe such quantities: [kg], [m] and [s].According to Buckingham’s theorem [13], it should bepossible to identify at least four dimensionless groupsinEq. (19). Therefore, introducing, for instance, dimen-sionless time and position as:

τ = ωnt x = x

Yb(45)

it would be expected to obtain a dimensionless equa-tion where only two other dimensionless parametersare present. Introducing the statement (45) in Eq. (19),it is possible to write:

mω2nYb

d2 x

dτ 2+ kYbx + μNsgn

(ωnYb

dx

dτ

)

= kYb cos rτ (46)

123


0 1 2 3 4 50

1

2

3

4

5D

ispl

acem

ent t

rans

mis

sibi

lity

Frequency ratio

F /Fk = 0

.2.4

.6

.8

.6

.4

.2

0

(a)

0 0.2 0.4 0.6 0.8Frequency ratio

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1.7

Dis

plac

emen

t tra

nsm

issi

bilit

y

00.20.40.60.8

Force ratio

(b)

Fig. 17 Numerical evaluation of displacement transmissibility in a mass-spring-fixed wall system (a) and detail of the curves outsidethe boundaries of the analytical forecast (b)

Dividing by kYb and removing any positive constantsfrom the argument of the sgn function:

mω2n

kx ′′ + x + μN

kYbsgn(x ′) = cos rτ (47)

therefore:

x ′′ + x + Fμ

Fksgn(x ′) = cos rτ (48)

Equation (48) is a dimensionless form of the governingequation of the problem, and it shows clearly the solu-tion dependency on the above-mentioned ratios only, aswell as on the dimensionless time. This result impliesthat it is possible to approach the transmissibility anal-ysis for a stick-slip motion numerically still referringonly to r and Fμ/Fk , without the necessity to know allthe other mentioned parameters.

In order to solve Eq. (48) in MATLAB with theode45 function, it is required to write it as a system offirst order differential equations [12]. This is achieveddefining x1 = x , so that:⎧⎨⎩x1 = x2

x2 = cos rτ − x1 − Fμ

Fksgn(x2)

(49)

The transmission parameters must be investigatedonly in the steady-state condition, so the first part ofthe response, which is dominated by the initial condi-tions, can be disregarded. As for the time interval, itis fundamental to impose a fixed time step during the

simulation in order to obtain correct information in fre-quency domain; recalling that the sampling frequencyis:

fs = 1

dt(50)

it will be necessary to keep it at least two times higherthan the maximum frequency considered in the prob-lem [14]. Introducing the dimensionless time, it is pos-sible to refer directly to the dimensionless frequencyf = f/ fn ; this quantity is exactly the frequency ratior when f = fb. Substituting τ and f in Eq. (50), thedimensionless sampling frequency is evaluated as:

fs = 2π

dτ(51)

The length of the time interval will affect the resolutionof the frequency spectrum, and anyway, it has to be suchnot to lose information on the steady-state behaviour.

Analysing the frequency spectra of x(τ ) and y(τ ),it will be possible to obtain the displacement transmis-sibility as:

|X (r)||Y (r)|and to plot the response in all the range of interest inorder to observe if other contributes are present at dif-ferent frequencies. The code can, at last, be easily reit-erated for all the values of frequency ratio in the chosenrange 0 ≤ r ≤ 5.

The overall displacement transmissibility is por-trayed in Fig. 17a at different force ratios, while

123


Fig. 18 Comparisonbetween analytical andnumerical displacementtransmissibility in amass-spring-fixed wallsystem


0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y

NumericalAnalyticalBoundary

(a) Force ratio = 0.2


0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y


(b) Force ratio = 0.4


0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

blity


(c) Force ratio = 0.6


0

0.5

1

1.5

2

Dis

plac

emen

t tra

nsm

issi

bilit

y


(d) Force ratio = 0.8

Fig. 17b shows in detail the evolution of the transmis-sibility curves at low frequency ratios, where Den Har-tog’s theory cannot be applied. An excellent agreementbetween theoretical and numerical transmissibilities isachieved in Fig. 18. Nevertheless, some minor discrep-ancies can be observed at the transition from stick-slipto continuous regime.

In Fig. 17a, b it is possible to observe a series ofproperties across the different curves, also outside theregion of validity of Den Hartog’s theory, related to thedifferent motion regimes represented in Fig. 19.

– Thequasi-staticmotion is characterisedbyvery lowvalues of transmissibility, inversely proportionallyto force ratio. This behaviour occurs during themul-tiple stops regime (Fig. 19a), where friction effectis very strong.

– In the two stops regime (Fig. 19b), the transmissi-bility is increased by friction, except in the case of

very high force ratio (Fμ/Fk = 0.8). Further, thiseffect appears to end always at r ∼= 0.6, where aninversion of the curves with Fμ/Fk < π/4 can beobserved.

– Even if Den Hartog’s forecast on regime transitionis accurate in terms of r value, displacement trans-missibility is higher than expected around the tran-sition frequency and there is no real intersectionbetween the numerical curves and the theoreticalboundary condition.

– At higher driving frequencies, when the regime isplainly continuous (Fig. 19c), the theoretical resultsare extremely accurate.

– When the driving and the natural frequencies arevery close, the beating phenomena will occur(Fig. 19d).

– While the resonant peak is infinite for Fμ/Fk <

π/4 and the response never reaches a steady-state

123


Fig. 19 Examples of massresponse in different motionregimes

0 50 100 150 200 250 300 350 400-2

0

2

(a) Multiple stops stick-slip motion (r = 0.1, Fµ/Fk = 0.2).

0 20 40 60 80 100 120 140 160 180 200-2

0

2

(b) Two stops stick-slip motion (r = 0.35, Fµ/Fk = 0.2).

0 20 40 60 80 100 120 140 160 180 200-2

0

2

(c) Continuous motion ( r = 1.5, Fµ/Fk = 0.2).

0 200 400 600 800 1000 1200 1400 1600 1800 2000-10

0

10

(d) Beating phenomenon (r = 0.9, Fµ/Fk = 0.2).

0 10 20 30 40 50 60 70 80 90 100-0.5

0

0.5

(e) Stick-slip occurency at r = 1, Fµ/Fk = 0.8.

condition, Fig. 19e shows how for high force ratiosa stick-slip behaviour occurs also when the systemis driven at resonance and a steady-state conditionis reached after a few cycles. In this case, the peakvalue is finite and it appears at frequency ratioslower than 1, where the motion is still continuousand has a larger amplitude (Fig. 18d).

– At very high values of r stick-slip will occur againif the force ratio is high enough, as stated in Eq.(25).

5.2 Joined base-wall motion case

The approach introduced in Sect. 5.1 holds also forthe joined base-wall motion problem. The governing

123



0

1

2

3

4

5D

ispl

acem

ent t

rans

mis

sibi

lity

F /Fk = 0

.2.4

.6.8

1.41.2

1.0

Fig. 20 Numerical evaluation of displacement transmissibilityin a mass-spring-moving wall system

equation for the relative mass-wall motion (29) can beexpressed in the same form of Eq. (48) introducingthe dimensionless time presented in Eq. (45) and thedimensionless relative position:

z = z

Yb(52)

Therefore, the final expression is:

z′′ + z + Fμ

Fksgn(z′) = r2 cos rτ (53)

while mass absolute motion can be obtained from:

x = z + cos rτ (54)

By imposing z1 = z, the state-space representation ofEq. (53) is written as:⎧⎨⎩z1 = z2

z2 = r2 cos rτ − z1 − Fμ

Fksgn(z2)

(55)

Thenumerical transmissibilities, portrayed inFig. 20as functions of the frequency ratio, offer a very goodagreement with the curves obtained in Sect. 4, as shownin Fig. 21. It is possible to appreciate also the agree-ment with the forecast on motion regime introduced inFig. 13.

– At the beginning of each curve, where the mass issupposed to remain still on the wall, the transmis-sibility is always equal to 1, as there is no relativemotion respect to wall (see Eq. 42).

– When the motion starts, a very narrow decrease oftransmissibility can be observed unless the forceratio is very low as in Fig. 21. This means that theoccurrence of the relative stick-slipmotion betweenmass andwall, shown inFig. 22, implies a reductionin amplitude in the mass absolute motion.

An interesting result can be observed in Fig. 21e, f.When the force ratio is very large, the resonant peakgradually decreases until it disappears. This type ofbehaviour is quite similar to the one observed for over-damped viscous systems. In this case, the peak dis-appears because, increasing the force ratio, completesticking and stick-slip regimes occur at higher andhigher frequency ratios until damping completely theoriginal peak.

6 Concluding remarks

A new approach has been developed for the evaluationof themain properties of a 1-DoF systemwithCoulombfriction under harmonic (i) base excitationwith ground-fixed wall; (ii) joined base-wall excitation.

DenHartog’s theory on forced vibrations in the pres-ence of Coulomb damping [1] was extended success-fully to base motion with fixed wall case, allowing abetter understanding of the problem and highlightingits impact also in friction dampers design; a few innova-tive results were presented in terms of stick-slip occur-rence at high frequencies and maximum value of forceratio allowing a continuous mass motion at least at onevalue of frequency ratio.

Joined base-wall motion problem was dealt withextensively, extending Den Hartog’s theory to rela-tive mass-wall motion and introducing a new analysisregarding mass absolute motion. The key results were:(i) the development of new understanding about themotion regimes and how they are supposed to occurdepending on frequency and force ratios; (ii) the evalu-ation of the displacement transmissibility and a qualita-tive comparison with its evolution in viscous systems,particularly focused on the existence in both the casesof a inversion of the curves at high frequencies, wheredamping amplifies the transmission; (iii) a method forthe evaluation of the phase shift between excitation andresponse.

The numerical resolution of the differential govern-ing equations of these problems allowed: (i) the com-parison between analytical and numerical displacementtransmissibilities, which showed an excellent agree-ment between the two approaches; (ii) the validationof the theoretical boundary identifying stick-slip andcontinuous motion regions in the frequency ratio–forceratio plane; (iii) an interesting insight into transmis-sibility evolution in the stick-slip region. The devel-

123


Fig. 21 Comparisonbetween analytical andnumerical displacementtransmissibility in amass-spring-moving wallsystem. Dark grey, lightgrey and white regionsrepresent, respectively, theanalytical forecast forsticking, stick-slip andsliding motion

0 0.5 1 1.5 2

Frequency ratio

0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(a) Force ratio = 0.4

0 0.5 1 1.5 2

Frequency ratio

0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(b) Force ratio = 0.6

0 0.5 1 1.5 2

Frequency ratio

0

1

2

3

4

5

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(c) Force ratio = 0.8

0 0.5 1 1.5 2

Frequency ratio

0

0.5

1

1.5

2

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(d) Force ratio = 1


0

0.5

1

1.5

2

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(e) Force ratio = 1.2

0 0.5 1 1.5 2

Frequency ratio

0

0.5

1

1.5

2

Dis

plac

emen

t tra

nsm

issi

bilit

y

AnalyticalNumerical

(f) Force ratio = 1.4

123


Fig. 22 Numericalevidence of relativestick-slip motion betweenmass and wall (r = 0.95,Fμ/Fk = 0.8)

0 10 20 30 40 50 60 70 80 90 100

Dimensionless time

-1.5

-1

-0.5

0

0.5

1

1.5

Dim

ensi

onle

ss d

ispl

acem

ent

Relative mass-wall motion Joined base-wall motion

opment of an alternative formulation of the differen-tial equations highlighted how mass motion can bedescribed through only two parameters, the frequencyratio and the force ratio, independently of DenHartog’sassumption. This means that also the stick-slip motioncan be analysed relying only on these parameters anda complete description of the transmissibility can beachieved integrating the theoretical and the numericalapproaches. Finally, the numerical curves allowed toobserve the gradual disappearing of the resonant peakwhen the force ratio increases over π/4. The main goalof the numerical approach was to provide a validationfor the proposed analytical results. Further validationof the presented results can be achieved with an exper-imental investigation; however, this is not the focus ofthis paper.

Overall, the presented results give information thatcan support the design of friction joints for several engi-neering structures. Experimental testing validating theresults presented in this paper will be discussed in aseparate publication.

Acknowledgements Luca Marino thanks the EPSRC andRolls-Royce for an industrial CASE postgraduate scholarship.Alice Cicirello gratefully acknowledges the financial supportprovided by Balliol College for a Career Development Fellow-ship. David Hills and Alice Cicirello thanks Rolls-Royce plc andtheEPSRC for the support under the Prosperity PartnershipGrant“Cornerstone: Mechanical Engineering Science to Enable AeroPropulsionFutures”,GrantRef:EP/R004951/1.The experimentswere performed in the Dynamics, Vibration and Uncertainty Lab(University of Oxford), which has been established thanks to theJohn Fell Fund (163/029).

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrest-ricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

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