Dynamic analysis of the continuous-time damped- oscillator with dry friction

As it was shown above, the desired zero position is always unstable, and it was also discussed that this type of instability can be modeled as a continuous-time spring force and a negative damping. Therefore, the following continuous-time model is considered for the analysis of the dynamics of the sampled data sliding-oscillator defined in Sec. 3.3

m¨x(t)−bx(t) +˙ kx(t) =







−C⁺, x(t)˙ >0 C⁻, x(t)˙ <0

. (4.7)

In this equation,xrepresents the generalized coordinate corresponding to the modeled degree- of-freedom, m is the generalized mass that takes its meaning based on the definition of x, parameter b is the damping factor, and k denotes the stiffness. In addition, C⁺ and C⁻

are the magnitudes of the generalized dry friction force associated with the positive- and negative velocities, respectively. Note, due to the negative sign in term −bx(t), the damping˙ coefficientb is necessarily positive in the numerical calculations. The simplest and most well- known system model which can be characterized by this generalized model is the one degree- of-freedom sliding-oscillator shown in Fig. 4.1, wherexwould correspond to the displacement of the center of mass of the body, and m is the mass of the body.

C⁺ -C⁻ f_fr C^±

k

b m

0 x

˙ x

Figure 4.1: Introductory model (left) and asymmetric Coulomb friction model (right)

At zero velocity, the friction force can take values in the interval [−C⁻, C⁺]. IfC⁺ =C⁻ then the Coulomb friction model is symmetric, otherwise, it is asymmetric. Since a certain asymmetry of the friction torques was observed in the experiments presented in Ch. 2, in the followings the asymmetric case is analyzed. Reducing the number of free parameters, Eq. (4.7) becomes

¨

x(t)−2ζω_nx(t) +˙ ω²_nx(t) =







−f₀⁺ω_n², x(t)˙ >0 f₀⁻ω_n², x(t)˙ <0

, (4.8)

where ω_n=p

k/mis the undamped natural angular frequency, ζ=b/(2mω_n) is the damping ratio which here corresponds to the considered negative damping, and f₀^± =C^±/k.

4.1.1 Motion induced by initial condition

The solution of Eq. (4.8) is derived for the case when the damping ratio is in the range 0< ζ <1 with initial conditions

x(0) =x₀ >0 and x(0) =˙ v₀ = 0. (4.9) In this case, the first half-period of the motion takes place with negative velocity, therefore the solution of Eq. (4.8) until the first velocity reversal can be obtained as

x⁻(t) = e^ζωnt(c₁cos(ω_dt) +c₂sin(ω_dt)) +f₀⁻, t ∈[0, T /2] (4.10) where the damped natural angular frequency is ω_d =ω_np

1−ζ² and the time period of oscillation is T = 2π/ω_d. The parameters c₁ and c₂ in the first period of oscillation are obtained from the initial conditions in Eq. (4.9) as

c1 =x0−f₀⁻ and c2 = ζω_n(f₀⁻−x₀)

ω_d . (4.11)

Substituting c₁ and c₂ into Eq. (4.10) and applying some trigonometric manipulation, the solution for the first period of oscillation, is derived in the form

x⁻(t) = C₀

p1−ζ² e^ζωntcos(ω_dt+ arcsinζ) +f₀⁻, t ∈[0, T /2] , (4.12) where C₀ = x₀ −f₀⁻. When the direction of motion of the block changes, and the velocity becomes positive, the solution for the following half-period of oscillation becomes

x⁺(t) = C₁

p1−ζ² e^ζωntcos(ω_dt+ arcsinζ)−f₀⁺, t ∈[T /2, T] , (4.13) where

C₁ =C₀−(f₀⁻+f₀⁺)e⁻

√ζπ

1−ζ2. (4.14)

Similarly to the method presented above, the constants C_n can be determined for the n-th half-period of the motion as

C_n =C₀−(f₀⁻+f₀⁺)

n

X

k=1

e⁻

√kζπ

1−ζ2, n = 1,2, . . . . (4.15) The motion will stop at the end of the n-th half-period, i.e., at the so-called sticking region, when

−f₀⁺≤x n^T₂

≤f₀⁻. (4.16)

In case the Coulomb friction model is symmetric, then f₀⁻=f₀⁺=f₀ and

C_n=x₀−f₀−2f₀

n

X

k=1

e⁻

√kζπ

1−ζ2, n= 1,2, . . . (4.17)

x

˙ x

S₁

S₂

Σ Σb

-f₀⁺ f₀⁻

x1 x0

Φ₁ Φ2

Figure 4.2: Phase portrait of the piecewise smooth system

4.1.2 Piecewise smooth analysis

The system of damped oscillator with dry friction is a type of piecewise smooth dynamical system [62], because the applied Coulomb friction law has discontinuity at zero velocity. By transforming the equation of motion in Eq. (4.8) into the system of first-order differential equations, the system of equations which describes the dynamics is

˙ y=







Ay+a⁻, if c^Ty<0 Ay−a⁺, if c^Ty>0

, (4.18)

where

y(t) =



 x(t)

˙ x(t)



, A=





0 1

−ω_n² 2ζω_n



, a^±=



 0 f₀^±ω_n²



 and c=



 0 1



. (4.19)

The state space R² is split into two subsets S₁ and S₂ by the switching surface Σ (see in Fig. 4.2), which is defined by the switching function

h(y) = c^Ty= ˙x= 0. (4.20)

The subspaces S₁,S₂ and the switching surface Σ are characterized by

S₁ ={y∈R² :c^Ty<0}, Σ ={y∈R² :c^Ty= 0}, S₂ ={y∈R² :c^Ty>0}. (4.21)

In the subspaces S₁ and S₂, the two vector fields are

F₁(y) =Ay+a⁻ and F₂(y) =Ay−a⁺. (4.22) These vector fields are defined by differential equation Eq. (4.18); therefore the trajectories are continuous, but non-smooth, because F₁(y) 6= F₂(y) on the switching surface Σ which results in a so-called Filippov system [62].

In Filippov systems that part of the switching surface where the trajectories are attracted by both sides is called the sliding region. This can be determined by

Σ =b {y∈Σ :n·F₁(y)>0 ∩ n·F₂(y)<0}, (4.23) where n = ∇h(y) = c^T. Substituting Eq. (4.19) back into the Eq. (4.22), the location of sliding region can be determined as −f₀⁺ ≤x≤f₀⁻, which is identical to Σ (see also inb Fig. 4.2).

The vector field in the switching surface Σ is not defined, but the so-called sliding vector field F_S can be determined, for example, by means of Filippov’s convex method [62]

F_S(y) = (1−γ(y))F₁(y) +γ(y)F₂(y), (4.24) where

γ(y) = nF₁(y)

n(F₁(y)−F₂(y)). (4.25)

The given vector field FS(y) is interpreted only in the sliding region Σ instead of the wholeb switching surface Σ. Substituting Eq. (4.22) into the Eqs. (4.24)-(4.25), F_S(y) and γ(y) become

FS(y) =Ay+a⁻−γ(y)(a⁻+a⁺), (4.26) where

γ(y) = c^T(Ay+a⁻)

c^T(a⁻+a⁺). (4.27)

Therefore F_S(y) =0, so in the sliding region, the motion will stop. This result corresponds to Eq. (4.16).

In case when the Coulomb friction model is symmetric (a⁺=a⁻=a), F_S(y) and γ(y)

simplifies to

F_S(y) =Ay+ (1−2γ(y))a, (4.28)

where

γ(y) = c^T(Ay+a)

2c^Ta . (4.29)

Finally, based on these two expressions the sliding vector fieldF_S is also zero, i.e.,F_S(y) = 0.

4.1.3 Limit cycles and vibrations

The main goal of this section is the derivation of the condition for critical initial position at zero initial velocity, when unstable limit cycle can be arisen. A damped oscillator with dry friction is a piecewise linear system, therefore a limit cycle may exist [72]. If there is a limit cycle, the trajectory closes itself after one switch. If the motion starts with the initial condition y₀ = [x₀ 0]^T with x₀>0, then the trajectoryΦ₁(t) can be expressed as

Φ₁(t) = e^At y₀+A⁻¹a⁻

−A⁻¹a⁻, c^Ty<0. (4.30) The result in Eq. (4.30) can also be determined based on x⁻(t) of Eq. (4.12). Substituting the necessary elements of Eq. (4.19) into Eq. (4.30), the trajectory Φ1(t) becomes

Φ₁(t) = e^ζωnt ωd





(x₀−f₀⁻)(ω_dc_ω−ζω_ns_ω) (f₀⁻−x₀)ω²_ns_ω



+



 f₀⁻

0



 (4.31)

with c_ω = cos(ω_dt), and s_ω = sin(ω_dt). Note that the matrix exponential exp(At) can be analytically determined, for example, by the method of spectral decomposition [71], which results in the form

e^At= e^ζωnt ω_d





ωdcω−ζωnsω sω

−ω²_nsω ωdcω+ζωnsω



 (4.32)

After time t₁, the trajectoryΦ₁(t) reaches the switching surface at point y₁ =Φ₁(t₁) = [x₁ 0]^T. From this point the motion continues with positive velocity, which trajectory can also be determined by the solution x⁺(t) of Eq. (4.13), or by

Φ₂(t) = e^At y₁−A⁻¹a⁺

+A⁻¹a⁺, c^Ty>0. (4.33)

Substituting the necessary elements of Eq. (4.19) into Eq. (4.33), and using the same notations as above, trajectory Φ₂(t) is given by

Φ₂(t) = e^ζωnt ω_d





(x₁+f₀⁺)(ω_dc_ω−ζω_ns_ω)

−(x₁+f₀⁺)ω²_ns_ω



−



 f₀⁺

0



 . (4.34)

Finally, after time t₂, the trajectory closes itself at point y₂ = Φ₂(t₁ +t₂) = [x₀ 0]^T. From Eqs. (4.31) and (4.34), parameters t₁,t₂, x₀ and x₁ can be determined analytically, so t₁ =t₂ =π/ω_d and

x_0,cr=x₀ = f₀⁻e

√ζπ 1−ζ2

+f₀⁺ e

√ζπ

1−ζ2 −1

or x_1,cr =x₁ =−f₀⁺e

√ζπ 1−ζ2

+f₀⁻ e

√ζπ

1−ζ2 −1

. (4.35)

It follows that periodic motion takes place, if the motion is initiated from the critical initial positions, x0,cr and x1,cr , with zero initial velocity. In case the Coulomb friction model is symmetric (f₀⁻=f₀⁺=f₀), x₁=−x₀, where

x_0,cr=x₀ =f₀e

√ζπ

1−ζ2 + 1 e

√ζπ

1−ζ2 −1

=f₀coth π 2

ζ p1−ζ²

!

. (4.36)

The local stability behavior of the limit cycle can be decided based on the Jacobian of the corresponding Poincar´e map. According to [72], this is given byP=P₂P₁ with

P₁ =

I−(Ay₁+a⁺)c^T c^T(Ay₁+a⁺)

e^At1, and P₂ =

I− (Ay₀−a⁻)c^T c^T(Ay₀−a⁻)

e^At2. (4.37) where I is the identity matrix. Substituting t₁=t₂=π/ω_d and the elements of Eq. (4.19) into Eq. (4.37), the matrices P₁ and P₂ become

P₁ =P₂ = e

√ζπ 1−ζ2





−1 0

0 0



. (4.38)

This is a non-trivial result which shows that the local stability behavior is independent of the Coulomb friction, and the Jacobian P is the same for the asymmetric and symmetric

models. The characteristic polynomial of matrix P is z

z−e

√2ζπ 1−ζ2

= 0, (4.39)

and the characteristic roots are

z₁ = 0 and z₂ = e

√2ζπ

1−ζ2. (4.40)

The limit cycle is stable if the magnitude of the eigenvalues of Pare less then one [64]. Thus, abs(z₁) = 0 is always less then one and

abs(z₂)<1 ⇔ 2ζπ

p1−ζ² <0 ⇔ ζ <0. (4.41) As in this problem 0< ζ < 1, the limit cycle is locally unstable regardless of the symmetric or asymmetric Coulomb friction models. The magnitude and symmetry properties of friction, however, affect the size of the unstable limit cycle.

4.1.4 Relationship between the initial conditions

The main goal of the previous section is the condition for critical initial position x_0,cr at zero initial velocity, v₀ = 0, when the unstable limit cycle arises as the function of the system parameters. In order to generalize, the relationship is looked for between the cases when the motion is initiated from arbitrary position with arbitrary velocity in the phase space. First, the cases are examined when x(0) = x₀ = 0 and ˙x(0) = v₀ > 0 or ˙x(0) = v₀ < 0. In this case, after time t_peak, vibration peaks x_peak occurs when ˙x(t_peak) = 0.

The general solution of Eq. (4.8) is derived for the case when the damping ratio is in the range 0< ζ <1 as

x(t) = e^ζωnt(c₁cos(ω_dt) +c₂sin(ω_dt)) +







−f₀⁺, x(t)˙ >0 f₀⁻, x(t)˙ <0

, (4.42)

Table 4.1: Relationship between the initial conditions and the critical initial positions v₀ >0 v₀ = 0 v₀ <0

x₀ >0 x^∩_peak+x₀ =x_0,cr x₀ =x_0,cr x^∪_peak−x₀ =x_1,cr x₀ = 0 x^∩_peak =x_0,cr - x^∪_peak =x_1,cr x0 <0 x^∩_peak−x0 =x0,cr x0 =x1,cr x^∪_peak+x0 =x1,cr

In Eq. (4.42), the parameters c₁ and c₂ can be obtained using the initial conditions

c₁ =







f₀⁺, v₀ >0

−f₀⁻, v₀ <0

and c₂ = v₀

ω_d − ζω_n

ω_d c₁. (4.43)

By solving the equation ˙x(t_peak) = ˙x⁺(t_peak) = ˙x⁻(t_peak) = 0 for t_peak, the time, when the first vibration peak occurs, is determined as follows

t_peak= 1

ω_darctan

ω_dc₂+ζω_nc₁ ω_dc₁ −ζω_nc₂

. (4.44)

Substituting back the parametersc₁ andc₂ of Eq. (4.43) into Eq. (4.44), the time instants when local maxima x^∩_peak and local minima x^∪_peak occur can be given as follows

t^∩_peak = 1 ω_datan

ω_dv₀

−f₀⁺ω_n²−ζω_nv₀

and t^∪_peak = 1 ω_datan

ω_dv₀ f₀⁺ω_n²−ζω_nv₀

. (4.45) With these the first local minimax^∪_peak or local maximax^∩_peakcan be determined and using the critical initial positions in Eq. (4.35), the condition for the existence of a limit cycle is x^∩_peak =x_0,cr orx^∪_peak =x_1,cr.

Furthermore, when both initial conditions are non-zero, using the method written above, the critical initial conditions can be determined within a simple extra step. First, based on only the initial velocity, the first peak is determined. If the sum of the magnitude of this peak and the given initial position is equal to the critical initial position, then there will be a limit cycle. These cases are summarized in Tab. 4.1

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