2. Benchmark solution#

The reference solution, which is analytical, is calculated as follows.

Either

the x-axis of the mass in the fixed coordinate system and

the abscissa of the vibrating support in this same frame of reference.

Initially, it is assumed that the mass is adherent to its support. It then remains so for some time after the initial moment

. It therefore undergoes the acceleration imposed by the rigid support, i.e.

. The tangential force exerted by the mass on the support is then

(zero at the initial moment, which justifies the initial hypothesis that initially, the mass is adherent to its support). The mass remains adherent as long as

. Si

, the mass therefore remains adherent to its support indefinitely, and its movement is exactly the same as this one. By introducing the dimensionless coefficient

, the condition of permanent adherence is written

. The acceleration curve of the mass, as of the support, then has the following appearance as a function of time:

As for speed, it looks as follows (the only primitive with zero mean):

, there is a smaller time

Such as

. This smallest time is necessarily such that

, which makes it possible to remove the absolute value in the previous expression, and to obtain an explicit expression

. In particular,

After this moment, the mass slides to the left with respect to the support, so it verifies the dynamic equation

, or

. Its speed therefore increases linearly with time, starting at

Of the negative value

(in fact,

Movement for

, « stick-slip » regime, succession of grip and slipping

Necessarily, for a certain amount of time

satisfying

, the speed of the mass becomes equal to the speed of the support again. At this moment, the movement becomes adherent again if and only if the acceleration experienced by the mass at the beginning of the adhesion is less than in absolute value than

. The translation of this condition is examined below. To begin with, we express the value of

The weather

Satisfy the equation

, or

This equation, transcendent, allows the determination of

depending on

and

, or finally, taking into account the expression of

, the determination of

depending on the physical parameters of the system

and

. If the acceleration of support in

is less in absolute value than

, the movement then remains adherent until a moment

for which the acceleration of the support and the mass reach the value

, an instant that, for reasons of clear symmetries on the graphs above, satisfies exactly

. The mass then begins a sliding phase until an instant.

, after which the movement is repeated periodically.

It is understood that for sufficiently small values of

, the movement will not be able to become a member over time

, because the acceleration of the mass would exceed the threshold

. So there is a critical value.

Such as for

, the movement of the mass passes without an adherence phase from a sliding to a sliding in the opposite direction. A reflection on the continuity of the mass velocity response function with respect to the parameter

Show that for

, the subsequent movement is always slippery (« slip-slip » regime, with alternating meanings). For

, the movement periodically alternates phases of grip and sliding.

The critical value

Admits a simple analytic expression. In fact, for

, the moments

and

are confused. So

And the equation

becomes

. By squaring, we get

, or

For

, the movement is only asymptotically periodic. The suite

Moments of change in the direction of sliding check

when

tends to infinity. The figure below shows the typical shape (broken line) of the mass’s speed in the slip-slip situation.

Movement for

: « slip-slip » regime, no adherence

Let’s summarize the findings:

We have the dimensionless coefficient

and its critical value* such as

If	the established regime is of the « stick-slip » type: alternation of adhesion and sliding phases;





If ,	the established regime is of the « slip-slip » type: alternating permanent sliding;







If ,	the established regime is of the « stick » type: permanent adhesion with the base.

In the following analytical calculation/Aster comparison results, the amplitude choices

are such that these three situations are visited. In fact, we take \(m\mathrm{=}1\mathit{kg}\), \(g\mathrm{=}10m\mathrm{/}{s}^{2}\), \(\mu \mathrm{=}0.1\),

\({a}_{0}\mathrm{=}15m\mathrm{/}{s}^{2}\), \({a}_{0}\mathrm{=}1.5m\mathrm{/}{s}^{2}\), \({a}_{0}\mathrm{=}1.01m\mathrm{/}{s}^{2}\), and \({a}_{0}\mathrm{=}0.99m\mathrm{/}{s}^{2}\).

The wear power is physically zero during the adhesion phases.

In Code_Aster, with the dyna_ VIBRA operator used here, adhesion is not detected because the integration of the movement is done by regularizing the law of friction. Compliance with the zero result of the wear power during adhesion phases required the introduction of a criterion on the sliding speed, so that below a certain value, it must be considered zero, and the adherent movement. You can consult the reference documentation Wear Calculation Operator/Archard Model [R7.04.10].

During the sliding phases, the wear power follows the law

, where

is the relative sliding speed of the mass on the support. In the stick-slip regime situation, in which the movement becomes strictly periodic after a finite amount of time, the wear energy during a half-period is exactly

The transcendent formulation of

apparently does not make it possible to simplify the expression of this energy of wear and tear. The average wear power

Is simply wear energy

above divided by the half-period of the response

In the case of a movement that is always slippery (

), the integration interval to take is of the form

with

big enough, so that

is close enough to the limit value

. We can avoid the numerical calculation by recurrence of this sequence, knowing that the mean of the asymptotic speed is zero. In fact, the following

Has a finite limit

. The properties satisfied by

are illustrated in the following figure:

The line segment has the equation

And for

, the speed

Or take the opposite value

, which gives the equation

either

;

whose solution is

Note that we find although for

, the acceleration of the support calculated at time

Give the limit value

. Indeed

In the case of the always sliding movement, the wear energy during an asymptotic period is exactly given by the formula

which can be explained according to the previous calculation, by taking

and

, which gives

either

The average wear power (over a period) asymptotic is then

The following Maple program allows the calculation of the exact wear power in a specified time interval, as well as the plot of the graph showing the convergence of the mass speed function to a periodic limit function, for any value of physical and excitation parameters such as the regime is of the slip-slip type (\(\eta \mathrm{\le }{\eta }^{\text{*}}\)), and the exact value of the average wear power over a period (the only useful value for what we are interested in) in the case of stick-slip.

# This program does the calculation, on the transitory part

# of the start of the signal, of the exact wear power,

# up to a time specified at the start of the program.

Digits: = 20:

pi: = evalf (Pi):

T: = 1: # period of support movement

omega: = 2*pi/T:

rpm: = 4:

tmax: = 12: # duration of the transient in question

ncycle: = floor (tmax/T) +2: # calculation iteration number of ti [i] and tf [i]

Nmax: = 100*ncycle: # to replace the sin function with a broken line

m: = 1:

g: = 10:

Mu: = 0.1:

a0: = 1.5:

eta: =mu*g/a0:

omega: = 2*pi/T:

etarstar: = 2/sqrt (pi^2+4):

ti [1] := 1/omega*arcsin (eta):

dX: = t -> -a0/omega*cos (omega*t):

dxminus [0] := dX (t):

lines x: = [ti[1], dX (ti [1])]:

Wear: = 0: # there is no wear during the adhesion phase [0, ti[1]]

# Note that ti [i+1] is necessarily in the interval [i*T-T/4,i*T+T/2]

# and that tf [i] is necessarily in the interval [i*T-3*T/4, i*T].

# These two intervals overlap, but we still have tf [i] <ti [i+1].

If eta<etaetoile then # slip-slip regime

For I from 1 to ncycle do

dxplus [i] := mu*g* (t-ti [i]) + subs (t=ti [i], dxminus [i-1]):

f [i] := fsolve (x (t) =dxplus [i], t= (i*t-3*t/4).. (I*t)):

linex: = linexx, [tf [i], dX (tf [i])]:

tinf: = max (ti [i], tmin):

tsup: = min (tf [i], tmax):

If tinf<tsup then

Wear: = Ease + int (m*g* (dX (t) -dxplus [i]), t=tinf.. tsup):

End:

dxminus [i] := -mu*g* (t-tf [i]) + subs (t=tf [i], dxplus [i]):

ti [i+1] := fsolve (dX (t) =dxminus [i], t= (i*t-T/4).. (t/2+I*t)):

linexx: = linexx, [ti[i+1], dX (ti [i+1])]:

tinf: = max (tf [i], tmin):

tsup: = min (ti [i+1], tmax):

If tinf<tsup then

Ease: = Ease + int (m*g* (dxminus [i] -Dx (t)), t=tinf.. tsup):

End:

ID:

# curveX: = plot ([eq ([J*tmax/max, x (J*tmax/nmax)], j=0.Nmax)]):

# curves: = plot ([lines]):

# with (plots):

# display ([curveX, curvex]);

theta: = arccos (pi*eta/2) /omega:

dxinfinity: = t -> mu*g* (t-theta) +dX (theta):

Vginfini: = xinfinity - xX:

Eumoyana: = - int (M*G*vginInfinity (t), t=theta.. (theta+pi/omega)):

Eumoyanaana: = m*g*a0/omega^2*sqrt (4-eta^2*pi^2):

Pumoyana: = 2*Eumoyana/T:

Pumoyanaana: = 2*Eumoyanaana/T:

Powder: = Wear/ (tmax-tmin);

elif (eta>etaetoile and eta<1) then # stick-slip regime

lines x: = [ti[1], dX (ti [1])]:

dxplus [1] := mu*g* (t-ti [1]) + subs (t=ti [1], dxminus [0])):

f [1] := fsolve (dX (t) =dxplus [1], t= (T-3*T/4).. T):

dxplus: = unapply (dxplus [1], t):

Vg: = dxplus - dX:

Eu: = -int (M*g*vg (t), t=ti [1].. tf [1]):

Pusuremoy: = 2*EU/T;

else # permanent adherence regime

Eu: = 0;

End:

The Aster solution considered is the calculation of the average wear power during a transitory phase ranging from \(4\) to \(\mathrm{11,99}\mathit{secondes}\) (from \(8\pi \mathrm{/}\omega\) to \(24\pi \mathrm{/}\omega\)). The wear energy during this transitory period differs somewhat from the average wear energy (asymptotic) over this period (both in a stick-slip and slip-slip situation). It is therefore appropriate, in order to compare it precisely with the Aster results, to make an exact calculation of this energy in the time interval \(\mathrm{[}\mathrm{4s},\mathrm{11,99}s\mathrm{]}\).

For \({a}_{0}\mathrm{=}\mathrm{15m}\mathrm{/}{s}^{2}\), the asymptotic mean wear power is \(\mathrm{15,1146144886}\mathit{Watt}\) while the average wear power over the time interval \(\mathrm{[}\mathrm{4s},\mathrm{11,99}s\mathrm{]}\) is \(\mathrm{15,257521794}\mathit{Watt}\). It is this last value that constitutes the reference result.

Note:

As an average power calculation, the wear power calculated over an interval does not have to increase with the duration of the interval. If you add to the interval a period of time over which there is adhesion, the average wear power will be lower.

2.1. Benchmark results#

Acceleration value \(\mathit{max.}\mathit{a0}\) ( \({\mathit{ms}}^{\mathrm{-}2}\) )	Average wear power value On the range \(\mathrm{[}\mathrm{4s},\mathrm{11,99}s\mathrm{]}\) , in Watt
15 (slip-slip)	15,26709959
1.5 (stick-slip)	0.40906245
1.01 (stick-slip)	2.261641E-4
0.99 (stick)	0

2.2. Uncertainty about the solution#

Quasi-analytical solution (presence of transcendent equations solved numerically with arbitrary precision).

2.3. Bibliographical references#

1. WESTERMO, F. UDWADIA: Periodic Response of a sliding oscillator system to harmonic excitation. Earthquake Engineering and Structural Dynamics Vol 14 135-146 (1983)
Code_Aster documentation [R7.04.10]