1979 hundal - response of a base excited system with coulomb and viscous friction

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~our rt al of Sound and Vi brat i on 1979) 64 3), 371-378

RESPONSE OF A BASE EXCITED SYSTEM WITH COULOMB

AND VISCOUS FRICTION

M. S. HUNDAL

Department of

Mechanical

Engineeri ng,

The University of Vermont, Burlington, Vermont 0540 5. U.S.A.

Received 12 October 1978, and in revisedform 10

January 1979)

The response of a single degree of freedom springgmass system with viscous and

Coulomb friction, with harmonic base excitation, is determined. Closed form analytical

solutions of the equation of motion are found for two cases: (a) continuous motion of the

mass and (b) motion of mass w ith two stops per cycle. Results are presented in non-

dimensional form as magnification factors versus frequency ratios as functions of viscous

and Coulom b friction parameters.

1. INTRODUCTION

Den Hartog [l] presented

a closed form analytical solution for the response of a

harmonically excited system with Coulom b and viscous friction. His system, shown in

Figure l(a), consists of a force excited m ass with friction forces acting b etween it and the

ground. He showed both theoretically and by experiment that, depending on system

parame ters, the mass may move continuously or it may come to stop during parts of each

cycle.

Harmon i c

m o t i o n

Coo l omb

f r i c t i on

b )

Figure 1. (a) System analyzed by Den Hartog [l]: (b) system analyzed by Levitan [2].

371

0022460X/79/1 10371+08 02.00:0

V 1979 Academic Press Inc. (London) Limited



372

M S HUNDAI

Levitan [23 analyzed the motion of a system with harmonic displacement of the base,

shown in Figure l(b). The friction forces in his model act between the base and the mass.

He used a Fourier series approximation for the Coulomb friction force.

This paper presents a closed form solution for the response of the system shown in

Figure 2. It is a base excited system in which the viscous friction force acts between base

and the mass, whe reas the Coulomb friction oppo ses the mass motion relative to the

ground. The model also represents a system in which an active device is mounted on the

mass. The device would act as a vibration controller, applying a constant force to the

mass in a direction opposite to its velocity. Curves of magnification factor \;ersus

frequency ratio are given, showing the effects of the two friction param eters. It is shown

that the mass motion may be continuous or one with stops during each cycle, depending

on system parameters.

Y

X’

k

m

c

Coulomb

I

frlctlon

, 1,1, II // ,/ ,/ l’,,

,,x, /‘-/ii-7

Figure 2. System with Coulomb friction between mass and ground, with harmonic excitation of the base

2. EQUATION OF MOTION

The equation of motion for the system is

md2.x’ldt2 + cd.x’/dt + kx’ + F’ = cd~‘/dt + 12~

(1)

where ) =

Ycos wt+p)

is the base motion and F’ is the force of Coulomb friction (see

Figure 2). The other symbols in equation (1) have their customary meanings (a list of

symbols is given in the Appendix). When the mass is in motion F’ = F sign(dx’/dt). When

the mass is at rest and the applied force is insufficient to overcom e the Coulomb friction,

the magnitude of the Coulomb force is then given by

F’ = - kx’ + L’

dJt/dt +

ky.

2)

Equation (1) is now put in a non-dimensional form by defining the following dimension-

less variables and parameters: time T =

w,t:

displacement x = x’/Y: Coulomb force

X; = F’lk Y; frequency ratio r = w/w,.

Here o, =

k/m)‘12.

Also z =

c/2 km)’

damping ratio. Equation (1) now becom es

,? + 22i + x + X; = -2~ sin(rT+p) + cos rT+p),

whe re dots represent differentiation with respect to

T.

2 is the

3)

3. CONTINUOUS MOTION OF THE MASS

The system is assumed to have reached steady state, and it is further assumed that the

mass is in continuous motion between its extrem e positions x = X, and x = -x,.



COULOMB AND VISCOUS FRICTION

373

x, = X6&:’

Y is thus the magnification factor. For the half cycle of mass motion when

.i- < 0, equation (3) becom es

jl + 2z.t + s = X, - 2~ sin(rl”+p) +

cos rT+p),

(4)

where X, = F/kY. The boundary conditions on equation (4) are

s = x,,, and

q-0

. at T = 0,

5)

s = -X0.

and .q = 0, at T = n r.

(6)

Such a motion of the mass and that of the base a re shown in Figure 3, which also clarifies

the role of the phase angle p. Equation (4) can now be solved for the steady state response.

The general solution is

x(T) = e

“(C, sin q7+C , cos

qT) + Xf + X,

sin(rT+r)

(7)

where

q =

(1 -z2)r12. The last term in equation (7) represents the steady state harm onic

response of the system in the absence of Coulomb friction. Thus

X, = ([l +(2zr)2],‘[(1 -12)2 +(2z,.)x])1,‘Z

(8)

and th e phase angles r and p are related by

X,[-2zrsine+(l-r’)cosr] = -sinp + 2zrcos p,

X,[(I-r2)sine+2zrcose] = cosp-2zrsinp.

(9)

Applying the boundary conditions (5) yields

C, = X, - X,- - X, sin e,

C, = zC,j’q - X, r cos e)/q.

10,

11)

For the sake of brevity the symbols

S, = sin(qn/r),

C, = cos(qxjr),

E = eern*

are defined, and also

A

11 = zS,E/q+l +C,E)X,, A,2 = rS,EX,/q,

A

2 1 = z2/q + q) S, EX,,

A,, = (zS,

E/q - 1 - C, E) rX,,

B, = x, + Xf + X,-Xf) C,+zS,jq)E,

B, = X, - Xf) z2/q + q) S, E. (12)

Figure 3. Harmonic base motion and continuous motion of the mass



374 M S HUNDAI

Upon substituting the values of C, and C, in equation (7) and applying the boundary

conditions (6), the following two equations are obtained:

A,, sine + A,,cose = B,,

A,,

sine +

A,, cos e = B,.

(13)

Equations (13) can now be solved for sine and cos e. With

M

and N defined as functions

of system parameters by

M =

(l-2zS,Ejq-E’)/(l +C,E+E*),

N = 2S,E/rq(l+2C,E+E2),

the solution of equations (13) is

sin e = (X, + MX f)iX,, cos e = NX ,/X,,

which yields an explicit expression for X,:

X, = -

MX , + X; - N’X;)?

(14)

(15)

16)

4. LIMITS OF CONTINUOUS MOTION

The solution of equation (4) given in section 3 is valid only when Z? < 0 for 0 < T < n/r.

This condition is used to determine the maximum values of z and X, for which a

continuous motion is possible. Upon differentiating equation (7) and substituting from

equations (lo), (11) and (15), the required condition is found to be

i = e-‘TIXf(l +M)(sin

qT)/q + zrXf

N(sin

qT)/q - rXf N cos qT]

+ rX, N cos rT - r X, +X, M) sin rT < 0,

for 0 <

T < n/r.

(17)

5. MOTION OF MASS WITH STOPS

When the magnitude of the Coulomb friction force is sufficiently high, the mass, once it

come s to a stop at either end of its stroke, will not begin to move until the applied force is

larger than the Coulomb force. Let the mass be in motion for 0 d T d T, and to rest for

T, < T < n/r

during the half cycle considered above, as illustrated in Figure 4. Thus

during 0 <

T < T,

equation (4) and its solution, equation (7), are valid, with the boundary

conditions

x = x,, and i = 0, at

T = 0,

18)

x = -x,,

and k-0, at

T=T,.

(19)

During the interval T, < T Q rc/r, from equation (3),

x - X> =

-2zr

sin(rT+p) +

cos rT+p),

which is also valid at

T = 0.

At this instant, as motion is about to ensue, X; = X,, and

therefore

X, -X, = -2zrsinp + cosp.

20)

There is, thus, an additional unknown T,, and an additional equation, (20).

Applying the boundary conditions (18) yields the same values of C, and C, as given in

equations (10) and (11). Nex t, by applying the boundary conditions (19), using equations




375

t

Figure 4. Harmonic base motion and mass motion with one stop per half cycle.

(20) and (9) to eliminate X, and then p, respectively, two equations in sin e and cos e are

obtained. Aga in, for brevity, the symbols S, = sinrT,, C, =

cosrT ,, S,, =

sinqT,,

C’,, =

cosqT,

and

E, = e-

zTo

re defined, and further,

D,, = 1 -

r2 + C, - r2E,(CqO +zSq0/q) ,

D , , = 2zr + S, + 2zrE,C, , + r(2z2-1)E0S4,, /q,

D 2, = -S, + r , ,/q, D22 = co - J (C,, +zS,o/d.

(21)

The equations in sin e and cos e are then

D,,sine +

D , ,cose = -2X , /X, , D , ,

sine +

D ,,cose = 0.

(22)

Equations (22) can be solved numerically for the unknow ns T, and e. The correspond-

ing values of p and X , can then be found from equations (9) and (20) respectively.

6. RESULTS

Plots of the magnification factor X, are shown in Figure 5 for various values of

dam ping ratio and dimensionless Coulom b friction force. The broken lines represent the

boundaries of the regions in which continuous motion of the mass and motion with stops

exist. These broken lines are obtained by applying the condition given by equation (17).

The portions of the curves lying above the broken lines correspond to continuous

motion, and are defined by equation (16). Magnification factors for the case of mass

motion with two steps per cycle are given by the curves lying below the broken lines in

Figure 5. These portions of the curves are obtained as described in section 5.

It is evident that for z = 0.1, see Figure 5(a), the mass motion is nearly zero for X, = 1.

It can be shown that for X, > 1, for a range of frequency ratios, the mass motion will be

zero. The condition for no response of the mass is that the limiting C oulomb friction force

be greater than the applied force: i.e., the sum of spring and viscous friction forces. From

equation (4), this condition is

X, > [1+(2zr)2]1/2,

or

r < ( XT -

1)1/2/(2z).

(23)

At high frequency ratios the magnification factor approaches constant values. As

r -+ co

equation (16) becomes X, = X,-X ,, since

E -+ 0 . A4 +

1 and N -+ 0.



3 7 6

M. S. HLJNDAL.

2

I I

e)

I

Xf

Figure 5. Magnification factor DS.frequency ratio. Portions of curves above broken lines represent continuous

motion of mass, below broken lines one stop per half cycle. (a) z = 0.1; (b) z = 0.2; (c) z = 0.3: (d) z = 0.4: (e)

z = 0.5.

7. CONCLUSION

Closed form solutions for the response of the system shown in Figure 2 with harmonic

base excitation have been obtained. Conditions governing the transition from a con-

tinuous motion of the mass to one with two stops during each cycle have been

determined. These two types of motion are illustrated in Figure 6.

In the absence of viscous friction the system is mathematically identical to that of Den

Hartog [l] under the same restriction. Furthermore, as pointed out by Den Harto g, mass

motion with more than two stops per cycle is also possible. The condition for this type of

motion, which occurs at low frequency ratios, is that 1 > 0 for 0 < T < To he expression

for the velocity being equation (17).




77

Cb)

A

P-

x(T)

I-

2L

Figure 6. M ass motion us. base motion. z = 0.3: r = 1. (a) X, = 0.2: (b) X, = 0.8

REFEREN ES

1. J . P. DEN HARTOG 1931 Transactions of the American Society ofMecha nical Engineers 53, 107- I 15.

Forced vibrations with combined Coulom b and viscous friction.

2. E. S. LEVITA N 1960 Journal of the Acoustical Society ofAmerica 32, 1265-126 9. Forced oscillation

of a spring-mass system having combined Coulomb and viscous damping.



378

M S HIJNDAL

APPENDIX: NOTATION

A, coefficients in equations (13) defined by equations (12)

Bi right-hand side terms in equations (13) defined by equations

(12)

Ci unknown coefficients in equation (7)

co cos rT,

C, cos(&r)

C,, cos qT,

c coefficient of viscous friction

D, coefficients in equations (22), defined by equations (21)

E ,-Zn r

EO

;

F

k

M

P

4

SO

s

@J

T

’

Y

function of system parameters, defined by equations (14)

phase angle of base displacement; see Figure 3

dimensionless natural frequency w ith viscous friction, = ( 1 z’)l/’

frequency ratio, = w/w ,

sin

rTo

sin(yn/r)

sin

qTo

dimensionless time, = w, t

dimensionless time, in the case of motion with stops, at which mass c omes to a stop

time

limiting value of dimensionless Coulomb friction force , =

F/kY

dimensionless Coulomb friction force, =

F’/k Y

magnification factor with Coulomb and viscous friction, = x,,,jY

magnification factor with viscous friction only

dimensionless displacement, = x’/Y

displacement

Z

w

amplitude of base displacement

base displacement

damping ratio,

=c/2 km)“’

base excitation frequency

w, undamped natural frequency

e-zT,,

phase angle associated w ith steady state harmonic response; see equation (7)

limiting value of Coulomb friction fo rce

Coulomb friction force, <F

spring stiffness

function of system parameters, defined by equations (14)

mass

1979 hundal - response of a base excited system with coulomb and viscous friction

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