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Abstract

In this article, the dynamical behavior of a single degree-of-freedom impact oscillator with impulse excitation is studied, where the mass impacts at one stop and is shocked with impulse excitation at the other stop. The existing and stability conditions for periodic motion of the oscillator are established. The effects of system parameters on dynamical response are discussed under different initial velocities. It is found that smaller shock gap than impact gap could make the periodic motion more stable. The decrease in natural frequency would consume less impact energy, make the vibration frequency smaller, and reduce the vibration efficiency. Finally, the dynamical properties are further analyzed under a special case, that is, the shock gap approaches zero. It could be seen that the larger shock coefficient and impact restitution coefficient would make vibration period smaller. Based on the stability condition, there are an upper limit for the product of shock coefficient and impact restitution coefficient, so that a lower limit of corresponding vibration period exists.

Keywords

Impact oscillator impulse excitation periodic motion stability analysis

Introduction

There are often low-speed reciprocating impacts between some mechanical components, which may lead to fatigue damage of mechanical parts or lower the system efficiency.¹ Therefore, it is necessary to study the relationship between working performance and dynamical properties of the impact system.

Numerical simulation method has been widely used in the impact vibration system due to its complicated nonlinearities. Wen et al.² introduced an impact damper system in which the discontinuous jumping to quasi-periodic attractors from non-impact grazing periodic orbits were revealed, and the origin of the quasi-periodic attractor and coexistence of different solutions were analyzed. Li et al.³ detected the existence of asymmetric sub-harmonic orbits outside the homoclinic orbits of an inverted pendulum impacting on rigid wall under external periodic excitation. Li et al.⁴ extended the well-known Melnikov method for smooth system to a class of piecewise smooth planar system, used the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds, and studied the chaotic dynamics in the piecewise smooth system numerically. Chu et al.⁵ investigated the impact vibration characteristics of a shrouded blade with asymmetric gaps under wake flow excitation. U Andreaus and MD Angelis⁶ researched nonlinear dynamic response of a base-excited single degree-of-freedom (SDOF) oscillator with double-side constraints. Dimentberg and Iourtchenko⁷ considered an SDOF oscillator with one-side rigid barrier at its equilibrium position and found that the excitation bandwidth ratio was a key governing parameter. Dynamic response of an SDOF oscillator excited by a base acceleration and constrained by two unilateral constraints was also studied.⁸

All the above analyses were primarily based on numerical method, but analytical method was also used in some impact systems, especially in the majority with analysis of unilateral and bilateral symmetric impact. Shaw and Holmes⁹ studied an SDOF piecewise linear oscillator, found the coexistence of sub-harmonic and chaotic motions, and analyzed different bifurcation phenomena in the oscillator. Xu et al.¹⁰ analytically obtained homoclinic orbits and Melnikov function for a Duffing vibro-impact oscillator. Pavlovskaia and Wiercigroch¹¹ presented a semi-analytical method which could be developed to compute periodic solutions for a new model of an impact oscillator with a drift. A linear oscillator with one or two stops and rigid block under periodic excitation was considered,¹² and the stability condition and bifurcation mechanism were investigated. A sinusoidally driven oscillator with linear stiffness and impacts at rigid stops was studied by S Foale,¹³ and the analytical results revealed the transition mechanism of grazing bifurcation and codimension-two bifurcation. Hogan¹⁴ provided an impact model where a rigid block lied on a horizontal surface between two symmetrically placed walls and found the symmetry-breaking and period-doubling bifurcations. Periodic motions of the impact oscillator were also studied by analytical methods, and the stability condition of different periodic motions was investigated.¹⁵ U Andreaus et al.¹⁶ considered a soft impact bilinear oscillator and obtained bifurcation diagrams, Lyapunov coefficients, return maps, and phase portraits of the response. Peterka¹⁷ also studied the dynamics of oscillators with soft impacts by bifurcation diagrams, time series, phase trajectories, and Poincare maps of periodic and chaotic impact motions. JP Chávez et al.¹⁸ presented a bifurcation analysis of two periodically forced Duffing oscillators coupled via soft impact. Li et al.¹⁹ computed the spectra of Lyapunov exponents for periodic motions and chaos and calculated the largest Lyapunov exponents in a large parameter range in a two-degree-of-freedom vibro-impact system. The calculation of the spectrum of Lyapunov exponents for impact-vibrating systems with rigid constraints was given in detail by Jin et al.²⁰ Yue and Xie²¹ developed a method of calculating the Lyapunov exponents in vibro-impact systems with symmetric rigid constraints. Moreover, other scholars^22–41 studied different impact oscillators and found some typical different phenomena.

It is important to analyze the dynamics of the impact oscillator in detail, so as to enlarge the profitable effects including optimum parameter design and high reliability, and minimize the adverse effects such as high noise levels, fatigue damage, and unstability. In this article, a simple but typical impact oscillator is established at first, where the oscillator impacts at one stop and is shocked with impulse excitation at the other stop instantaneously in section “Mathematical model.” This oscillator could model many mechanical systems, such as drilling jumbo, vibrating hammer, and pile driver. The existence condition for periodic motion and its properties are analyzed in sections “Dynamical analysis” and “Properties of periodic-1 motion,” respectively. Then, section “Parameter analysis” presents the detailed analysis of system parameters on dynamical response. Finally, the obtained conclusions are summarized.

Mathematical model

The considered SDOF mechanical model is shown in Figure 1. In the model, the mass m is connected to supporting face through a linear spring with stiffness coefficient as k. When its displacement is $λ δ$ and supposing the velocity before shock is v₀, it instantaneously suffers from a shock which we consider an impulse exciting force at Stop B and moves reversely at an initial velocity as −e₂v₀. When its displacement is −δ, the mass m will impact with rigid Stop A, change its movement direction again, and move at a new initial velocity. Due to the effects of impact and shock (impulse excitation), the linear vibration oscillator becomes a nonlinear vibration system.

Figure 1.

An SDOF impact oscillator with impulse excitation.

The differential motion equation of the oscillator is

{\begin{matrix} m \overset{\cdot\cdot}{x} + kx = 0 \\ {\overset{\cdot}{x}}_{A}^{+} = - e_{1} {\overset{\cdot}{x}}_{A}^{-}, x = - δ, 0 < e_{1} < 1 \\ {\overset{\cdot}{x}}_{B}^{+} = - e_{2} {\overset{\cdot}{x}}_{B}^{-}, x = λ δ, e_{2} > 1 \\ x (0) = λ δ, \overset{\cdot}{x} (0) = v_{0} \end{matrix}

(1)

where m, k, e₁, e₂, $δ$ , and $λ δ$ are mass, linear stiffness, impact restitution coefficient at Stop A, shock coefficient at Stop B, impact gap, and shock gap, respectively. The superscripts − and + in equation (1) denote the instantaneous state before and after impact (shock), respectively. It should be remembered that the initial velocity $v_{0}$ in this article is the velocity before shock with Stop B. All the units of the physical quantities are the international system of units. When the oscillator is shocked or impulsively excited at Stop B, the system energy will be increased so that the restitution coefficient in this case is greater than 1. On the contrary, the general impact happens at Stop A and its restitution coefficient is less than 1.

Dynamical analysis

Period-1 motion

When $- δ < x < λ δ$ holds consistently, there is no impact and shock. This is the simplest case and the oscillator vibrates linearly. We will not consider it in the following parts of this article and focus on impact (shock) vibration.

Obviously, equation (1) has the solution as

x (t) = c_{1} \cos ω t + c_{2} \sin ω t

(2)

where $ω$ is the natural frequency. Then, we can analyze its motion piecewisely.

The motion from Stop B to Stop A is considered first. The initial conditions are $x (0) = λ δ$ , $\overset{\cdot}{x} (0)^{-} = v_{0}$ , and $\overset{\cdot}{x} (0)^{+} = - e_{2} v_{0}$ . Based on equation (2), equation (3) could be obtained

c_{1} = λ δ, c_{2} = - e_{2} v_{0} / ω

(3)

Substituting equation (3) into equation (2), it will yield

x (t) = λ δ \cos ω t - (e_{2} v_{0} / ω) \sin ω t

(4a)

\overset{\cdot}{x} (t) = - λ δ ω \sin ω t - e_{2} v_{0} \cos ω t

(4b)

Supposing $T_{1}$ is time from Stop B to Stop A, one could obtain

\overset{\cdot}{x} (T_{1})^{-} = - v_{1}, x (T_{1}) = - δ

Substituting these two conditions into equation (4), it yields

\sin ω T_{1} = \frac{ω δ (e_{1} v_{0} + λ v_{1})}{δ^{2} ω^{2} λ^{2} + e_{2}^{2} v_{0}^{2}}

(5)

\cos ω T_{1} = \frac{- δ^{2} ω^{2} λ + e_{2} v_{0} v_{1}}{δ^{2} ω^{2} λ^{2} + e_{2}^{2} v_{0}^{2}}

(6)

Combining equations (5) and (6), one could get

v_{1} = \sqrt{- δ^{2} ω^{2} + δ^{2} ω^{2} λ^{2} + e_{2}^{2} v_{0}^{2}}

(7)

It could be found that different $λ$ and initial velocity $v_{0}$ will determine whether $v_{1}$ is real. When λ > 1, $v_{1}$ is always real. If $λ < 1$ , the initial velocity should meet

v_{0} > \frac{δ ω \sqrt{1 - λ^{2}}}{e_{2}}

(8)

so that $v_{1}$ is real and the mass could arrive at Stop A. On the contrary, imaginary $v_{1}$ means the mass could not arrive at Stop A and no impact exists.

2. Then, the motion from Stop A to Stop B is studied. The initial conditions are $x (T_{1}) = - δ$ and $\overset{\cdot}{x} (T_{1})^{+} = e_{1} v_{1}$ , and it yields

x (t) = - δ \cos ω t + (e_{1} v_{1} / ω) \sin ω t

(9)

\overset{\cdot}{x} (t) = δ ω \sin ω t + e_{1} v_{1} \cos ω t

(10)

In consideration of periodic impact vibration, the boundary conditions $x (T_{2}) = λ δ$ and $\overset{\cdot}{x} (T_{2})^{-} = v_{0}$ must be held within a shock and impact period, where $T_{2}$ is time from Stop A to Stop B. Substituting those conditions into equations (9) and (10), one can get

x (T_{2}) = - δ \cos ω T_{2} + (e_{1} v_{1} / ω) \sin ω T_{2}

(11)

\overset{\cdot}{x} (T_{2})^{-} = δ ω \sin ω T_{2} + e_{1} v_{1} \cos ω T_{2}

(12)

As the similar procedure, one can obtain

\overset{\cdot}{x} (T_{2})^{-} = v_{0} = \sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{2}^{2} v_{1}^{2}}

(13)

Substituting equation (7) into equation (13), it will yield

v_{0} = \sqrt{\frac{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2})}{1 - e_{1}^{2} e_{2}^{2}}}

(14)

According to equations (8) and (14), the existing conditions for period-1 motion of impact vibration system could be established as follows:

When λ = 1, that is, shock gap is the same as impact gap, the condition is $e_{1} e_{2} = 1$ .

If λ > 1, that is, shock gap is larger than impact gap, the condition should be $e_{1}^{2} e_{2}^{2} > 1$ .

When λ < 1, the conditions are $e_{1}^{2} e_{2}^{2} < 1$ and $v_{0} > (δ ω \sqrt{1 - λ^{2}}) / e_{2}$ .

Figures 2 –4 show the phase trajectories for λ = 1, λ > 1, and λ < 1, respectively. Obviously, they agree well with the above analysis.

Figure 2.

The phase plane of (x, v) when λ = 1.

Figure 3.

The phase plane of (x, v) when λ > 1.

Figure 4.

The phase plane of (x, v) when λ < 1.

Other possible periodic motion

The period-1 impact motion is elucidated in the above analysis. In this subsection, other possible periodic motions, that is, period-n motions, will be investigated. Here, n denotes the number of shocks and impacts existing simultaneously in a vibration period.

Supposing the mass departures from Stop B with the initial velocity $- e_{2} v_{B (0)}^{-}$ and it returns to Stop B after a period, one could establish the Poincare map as

v_{B (1)}^{-} = \sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2} {v_{B (0)}^{-}}^{2}}

(15)

Similarly, the oscillator return to the stop B after two shocks, one could obtain

v_{B (2)}^{-} = \sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2} {v_{B (1)}^{-}}^{2}}

(16)

Under the condition of period-2 motion, that is, $v_{B (2)}^{-} = v_{B (0)}^{-}$ , one could easily get

v_{B (2)}^{-} = v_{B (1)}^{-} = v_{B (0)}^{-}

(17)

It is obvious that the solution of period-2 motion is the same as that of period-1 motion. That means there is no single period-2 motion in the oscillator. Using the mathematical induction, period-n motion can also be derived as

v_{B (n)}^{-} = v_{B (n - 1)}^{-} = \dots = v_{B (1)}^{-} = v_{B (0)}^{-}

Accordingly, only period-1 motion shown in section “Period-1 motion” exists in the oscillator if impact and shock exist simultaneously.

Properties of periodic-1 motion

Poincare map and stability analysis

Based on the mathematical model, one could establish the iterative relationship between the velocity of n + 1 impact and that of n impact

v_{B (n + 1)}^{-} = \sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2} v^{2}}

(18)

Equation (18) could be rewritten as

v_{B (n + 1)}^{-} = f (v_{B (n)}^{-})

(19)

In order to determine the stability of equilibrium in equation (19), that is, period-1 motion, one should apply Taylor expansion to equation (19) at the equilibrium and obtain

Δ v_{B (n + 1)}^{-} = f' (v_{B (n)}^{-}) Δ v_{B (n)}^{-} + o (Δ v_{B (n)}^{-})

By analyzing the values of $f^{'} (v_{B (n)}^{-})$ , one can obtain the stability condition. From equations (18) and (19), it yields

f' (v_{B (n)}^{-}) = \frac{e_{1}^{2} e_{2}^{2} v_{B (n)}^{-}}{\sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2} v^{2}}}

(20)

Substituting $v_{B (n + 1)}^{-} = v_{B (n)}^{-} = v_{0}$ into equation (20), one can get the explicit form for stability condition of period-1 motion as

f' (v_{B (n)}^{-}) = e_{1}^{2} e_{2}^{2}

Obviously, there will be three cases for the stability of period-1 motion:

When λ < 1, one could obtain $e_{1}^{2} e_{2}^{2} < 1$ . That means $f' (v_{B (n)}^{-}) < 1$ and will make period-1 motion asymptotically stable.

The case λ > 1 will make $e_{1}^{2} e_{2}^{2} > 1$ and $f' (v_{B (n)}^{-}) > 1$ . Accordingly, period-1 motion will be unstable in this case.

The critical case $λ = 1$ will make period-1 motion stable, but not asymptotically stable.

Possibility of multiple periodic motion

Between two consecutive shocks, the periodic motion of the system may exist in one of the following forms: (1) multiple impacts, (2) without impact, and (3) only one impact. Case 3 has been investigated in section “Dynamical analysis.” Then, we will study the possibility for other two possible periodic motions within a shock period.

Case 1 may occur only when λ > 1. If the oscillator cannot come back to Stop B after an impact within an excitation period, the impact will continue and the system energy will become less and less gradually. That means the oscillator will never arrive at stop B again, so that there is no periodic impact motion in this case.

Case 2 may occur only when λ < 1. Here, the oscillator is unable to reach Stop A initially within an excitation period, so that the impulsive excitation will continue constantly. If the system energy accumulates to a certain level, impact will appear. Then, the oscillator will vibrate as Case 1 or Case 3. Because Case 1 does not exist, it can only be in the form of Case 3.

Therefore, there is only one stationary period-1 motion within an excitation period.

Parameter analysis

Initial velocity $v_{0}$

The effect of initial velocity $v_{0}$ on dynamical response is analyzed as follows. If initial velocity is small enough and λ < 1, the early motion will have only shock and without impact.

Assuming the actual initial velocity is $v_{0}^{'}$ and impact will occur after n times of shocks, the motion velocity in Stop B is

{\overset{\cdot}{x}}_{B (n)}^{+} = - e_{2}^{n} v_{0}^{'}

(21)

Thus, the velocity at Stop A is

{\overset{\cdot}{x}}_{A (n)}^{-} = \sqrt{- δ^{2} ω^{2} + δ^{2} ω^{2} λ^{2} + e_{2}^{2} {\overset{\cdot}{x}}_{B (n)}^{-}^{2}}

(22)

The increment of system energy after n times of shocks in a cycle is

Δ J_{(n)} = \frac{1}{2} m {\overset{\cdot}{x}}_{B (n)}^{- 2} (e_{2}^{2} - 1) - \frac{1}{2} m {\overset{\cdot}{x}}_{A (n)}^{- 2} (1 - e_{1}^{2})

(23)

that is

Δ J_{(n)} = \frac{1}{2} m ({\overset{\cdot}{x}}_{B (n)}^{-}^{2} - v_{0}^{2}) (e_{1}^{2} e_{2}^{2} - 1)

(24)

From equation (24), it is easy to find that the system energy will be increased when λ < 1 and $e_{1}^{2} e_{2}^{2} < 1$ , while the system energy is decreased if λ > 1 and $e_{1}^{2} e_{2}^{2} > 1$ . If $Δ J_{(n)} > 0$ , the system energy will be increased. On the contrary, λ > 1 and $e_{1}^{2} e_{2}^{2} > 1$ will make $Δ J_{(n)} < 0$ , so that system energy will be dissipated. In order to analyze energy variation during a cycle, the next velocity at Stop B is

{\overset{\cdot}{x}}_{B (n + 1)}^{-} = \sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2 n} v_{0}^{' 2}}

(25)

Comparing ${\overset{\cdot}{x}}_{B (n + 1)}^{-}$ and $v_{0}$ , ${\overset{\cdot}{x}}_{B (n)}^{- 2} = e_{2}^{2 n - 2} {v'_{0}}^{2} < v_{0}^{2}$ , equation (26) could be found

\frac{v_{B (n + 1)}^{-}}{v_{0}} < \frac{\sqrt{δ^{2} ω^{2} (1 - e_{1}^{2}) (1 - λ^{2}) + e_{1}^{2} e_{2}^{2} v_{0}^{' 2}}}{v_{0}}

(26)

Based on comparison of equations (14) and (26), it could be found $v_{B (n + 1)}^{-}$ is smaller than $v_{0}$ . Accordingly, one could find the following conclusions based on equation (23):

When λ < 1 and actual initial velocity is smaller $(v_{0}^{'} < v_{0})$ , system energy will be increased after many impacts and shocks. Therefore, velocity at Stop B before shock constantly increases and will approach $v_{0}$ . If initial velocity is larger $(v_{0}^{'} > v_{0})$ , system energy still dissipates after many impacts and shocks, so that the velocity at Stop B before shock will approach $v_{0}$ . The results of numerical simulation in Figures 5 –10 prove the correctness of the analysis, and $v_{0}$ is the condition for asymptotical stable periodic motion of this system.

When λ > 1 and initial velocity is smaller $(v_{0}^{'} < v_{0})$ , vibration gradually attenuates and will become free vibration without shock and impact in the end, which is shown in Figures 11 –13. If initial velocity is larger $(v_{0}^{'} > v_{0})$ , system energy constantly increases and vibration velocity will become larger and larger. Therefore, the system is not stable, which is shown in Figures 14 and 15. This is also in accordance with the stability analysis in section “Poincare map and stability analysis.”

Figure 5.

The time history of the velocity when λ < 1 and $v_{0}^{'} > v_{0}$ .

Figure 6.

The phase plane when λ < 1 and $v_{0}^{'} > v_{0}$ .

Figure 7.

The time history of the energy when λ < 1 and $v_{0}^{'} > v_{0}$ .

Figure 8.

The time history of the velocity when λ < 1 and $v_{0}^{'} < v_{0}$ .

Figure 9.

The phase plane when λ < 1 and $v_{0}^{'} < v_{0}$ .

Figure 10.

The time history of the energy when λ < 1 and $v_{0}^{'} < v_{0}$ .

Figure 11.

The time history of the velocity when λ > 1 and $v_{0}^{'} < v_{0}$ .

Figure 12.

The phase plane when λ > 1 and $v_{0}^{'} < v_{0}$ .

Figure 13.

The time history of the energy when λ > 1 and $v_{0}^{'} < v_{0}$ .

Figure 14.

The time history of the velocity when λ > 1 and $v_{0}^{'} > v_{0}$ .

Figure 15.

The time history of the energy when λ > 1 and $v_{0}^{'} > v_{0}$ .

Furthermore, periodic motion can be obtained only when initial velocity is exactly equal to the condition of periodic solution in theory if λ > 1. In the actual situation, such periodic motion is difficult to be obtained, and the system is quite sensitive to small variations in initial conditions and system parameters. If the initial velocity changes slightly during actual implementation, the system responses will be free oscillation or unstable motion. Therefore, parameter condition λ > 1 should not be taken into account during actual system design.

Shock energy

According to energy equation and initial conditions, that is, $x (0)^{-} = v_{0}$ and $\overset{\cdot}{x} (0)^{+} = - e_{2} v_{0}$ , energy variation of one shock is

J = \frac{1}{2} {mv}_{0}^{2} (e_{2}^{2} - 1)

It could be found that shock energy of periodic motion depends on initial velocity, and it will become larger with the increase in initial velocity or e₂. Our purpose is to analyze the shock energy of the periodic motion. Therefore, substituting the condition of periodic solution into the equation of shock energy, one could obtain

J = \frac{1}{2} m \frac{δ^{2} ω^{2} (1 - e_{1}^{2}) (e_{2}^{2} - 1) (1 - λ^{2})}{1 - e_{1}^{2} e_{2}^{2}} = \frac{1}{2} k \frac{δ^{2} (1 - e_{1}^{2}) (e_{2}^{2} - 1) (1 - λ^{2})}{1 - e_{1}^{2} e_{2}^{2}}

According to the above analysis, the condition for periodic motion is $e_{1} = e_{2} = 1$ in the case of λ = 1. It will not appear in the actual situation and thus is not taken into account. When λ > 1, shock energy will become larger with the increase in λ. If λ < 1, maximum shock energy occurs when λ = 0, and shock energy reduces with the increase in λ.

It can also be seen that shock energy is influenced by k, e₁, and $δ$ , and it will be larger with the increase in k, e₁, and $δ$ . Because $δ$ is always small and e₁ depends on the actual impact objects, shock energy can be reduced by reducing k.

Vibration frequency

For simplicity, the vibration frequency from Stop B to Stop A is denoted by $Ω_{1}$ and the vibration frequency from Stop A to Stop B is denoted by $Ω_{2}$ , respectively:

According to the boundary condition $x (0) = λ δ$ and $x (π / Ω_{1}) = - δ$ , one could get the displacement and velocity from Stop B to Stop A

\begin{matrix} x (t) = \frac{- δ}{\sin \frac{ω}{Ω_{1}} π} [λ \sin (ω t - \frac{ω}{Ω_{1}} π) + \sin (ω t)], \\ \overset{\cdot}{x} (t) = \frac{- δ ω}{\sin \frac{ω}{Ω_{1}} π} [λ \cos (ω t - \frac{π ω}{Ω_{1}}) + \cos (ω t)] \end{matrix}

(27)

The initial velocity can be expressed as

{\overset{\cdot}{x}}_{B}^{+} = \overset{\cdot}{x} (0) = \frac{- δ ω}{\sin \frac{ω}{Ω_{1}} π} (λ \cos \frac{ω}{Ω_{1}} π + 1)

When λ > 1, the existing condition for periodic motion is ${\overset{\cdot}{x}}_{B (1)}^{+} < 0$ . One could obtain

0 < \frac{ω}{Ω_{1}} < \arccos (- \frac{1}{λ}) / π

(28)

If λ < 1, the existing conditions for periodic motion should be ${\overset{\cdot}{x}}_{B}^{+} < 0$ and ${\overset{\cdot}{x}}_{A}^{-} = \overset{\cdot}{x} (ω / Ω_{1}) π < 0$ . Accordingly, equation (29) could be found

0 < \frac{ω}{Ω_{1}} < \arccos (- λ) / π

(29)

2. Based on the boundary conditions $x (π / Ω_{1}) = - δ$ and $x ((π / Ω_{1}) + (π / Ω_{2})) = λ δ$ , one could obtain the displacement and velocity from Stop A to Stop B

\begin{matrix} x (t) = \frac{δ}{\sin \frac{ω}{Ω_{2}} π} [\sin (ω t - \frac{ω}{Ω_{1}} π - \frac{ω}{Ω_{2}} π) + λ \sin (ω t - \frac{ω}{Ω_{1}} π)], \\ \overset{\cdot}{x} (t) = \frac{δ ω}{\sin \frac{ω}{Ω_{2}} π} [\cos (ω t - \frac{ω}{Ω_{1}} π - \frac{ω}{Ω_{2}} π) + λ \cos (ω t - \frac{ω}{Ω_{1}} π)] \end{matrix}

(30)

The initial velocity of Stop A to Stop B can be expressed as

\overset{\cdot}{x} (\frac{π}{Ω_{1}}) = \frac{δ ω}{\sin \frac{ω}{Ω_{2}} π} (\cos \frac{ω}{Ω_{2}} π + λ)

When λ > 1, the existing conditions for periodic motion are ${\overset{\cdot}{x}}_{A}^{+} = \overset{\cdot}{x} (π / Ω_{1}) > 0$ and ${\overset{\cdot}{x}}_{B}^{-} = \overset{\cdot}{x} ((π / Ω_{1}) + (π / Ω_{2})) > 0$ . It yields

0 < \frac{ω}{Ω_{2}} < \arccos (- \frac{1}{λ}) / π

(31)

If λ < 1, the existing condition for periodic motion should be ${\overset{\cdot}{x}}_{A}^{+} = \overset{\cdot}{x} (π / Ω_{1}) > 0$ . It will yield

0 < \frac{ω}{Ω_{2}} < \arccos (- λ) / π

(32)

From the above analysis, one could find the following results:

When λ > 1, vibration frequencies $Ω_{1}$ and $Ω_{2}$ decrease with increase in $λ$ and increase with the enlargement of natural frequency $ω$ .

If λ < 1, vibration frequencies $Ω_{1}$ and $Ω_{2}$ will increase with the enlargement of $λ$ or $ω$ .

Therefore, small $ω$ and λ < 1 will reduce shock energy and vibration frequencies will decrease accordingly. This will reduce work efficiency and should be avoided.

In the following part, the special case, that is, λ = 0 is considered.

According to equation (27), one could obtain

{\overset{\cdot}{x}}_{B}^{+} = \overset{\cdot}{x} (0) = \frac{- δ ω}{\sin \frac{ω}{Ω_{1}} π} = - e_{2} {\overset{\cdot}{x}}_{B}^{-} = - e_{2} v_{0}

Vibration period from Stop B to Stop A could be obtained

T_{1} = \frac{1}{ω} \arcsin \frac{δ ω}{e_{2} v_{0}}

(33)

According to equation (30), it yields

\overset{\cdot}{x} (\frac{π}{Ω_{1}} + \frac{π}{Ω_{2}}) = \frac{ω δ}{\sin \frac{ω}{Ω_{2}} π} = v_{0}

Vibration period from Stop A to Stop B is

T_{2} = \frac{1}{ω} \arcsin \frac{δ ω}{v_{0}}

(34)

According to equations (33) and (34), vibration period can be written as

T = \frac{1}{ω} \arcsin \frac{δ ω}{e_{2} v_{0}} + \frac{1}{ω} \arcsin \frac{δ ω}{v_{0}}

(35)

Substituting $v_{0} = \sqrt{\frac{δ^{2} ω^{2} (1 - e_{1}^{2})}{1 - e_{1}^{2} e_{2}^{2}}}$ into equation (35), one could get

T = \frac{1}{ω} \arcsin \frac{1}{e_{2}} \sqrt{\frac{1 - e_{1}^{2} e_{2}^{2}}{1 - e_{1}^{2}}} + \frac{1}{ω} \arcsin \sqrt{\frac{1 - e_{1}^{2} e_{2}^{2}}{1 - e_{1}^{2}}}

(36)

λ = 0 is a special case of λ < 1, and it satisfies the relationship between vibration period and natural frequency analyzed as above. According to equation (32), the relationship between $e_{1}$ , $e_{2}$ , and vibration frequency is quite clear. Vibration period reduces and vibration frequencies increase with the increase in $e_{1} / e_{2}$ . Therefore, $e_{1} / e_{2}$ shall be as large as possible. However, when λ < 1, periodic motion should meet the condition $e_{1}^{2} e_{2}^{2} < 1$ . Therefore, there is an upper limit of the product of $e_{1}$ and $e_{2}$ and lower limit of vibration period if λ = 0.

Conclusion

In this article, the existing and stability conditions of period-1 motion in an SDOF impact oscillator with impulse excitation are established. The possibilities of other periodic motions are also studied, and it is found that only period-1 motion exists for this oscillator. The effects of system parameters under different initial velocities are discussed in detail. Some important properties, such as shock energy and vibration frequency, are also analyzed. The results are important to design or control this kind of impact systems.

Footnotes

Academic Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to the support by National Natural Science Foundation of China (No. 11372198), the Cultivation plan for Innovation team and leading talent in Colleges and universities of Hebei Province (LJRC018), the Program for advanced talent in the universities of Hebei Province (GCC2014053), and the Program for advanced talent in Hebei Province (A201401001).

References

Jin

. Impact and vibration control. Beijing: Science Press, 2005.

Wen

Yin

. Analysis of grazing bifurcation from periodic motion to quasi-periodic motion in impact-damper systems. Chaos Soliton Fract 2016; 83: 112–118.

Zhang

. Asymmetric type II periodic motions for nonlinear impact oscillators. Nonlinear Anal: Theor 2008; 68: 2681–2696.

Zhang

Hao

. Melnikov-type method for a class of discontinuous planar systems and applications. Int J Bifurcat Chaos 2014; 24: 1450022.1–1450022.10.

Chu

Cao

Sun

. Impact vibration characteristics of a shrouded blade with asymmetric gaps under wake flow excitations. Nonlinear Dynam 2013; 72: 539–554.

Andreaus

Angelis

. Nonlinear dynamic response of a base-excited SDOF oscillator with double-side unilateral constraints. Nonlinear Dynam 2016; 84: 1447–1467.

Dimentberg

Iourtchenko

. Stochastic and/or chaotic response of a vibration system to imperfectly periodic sinusoidal excitation. Int J Bifurcat Chaos 2005; 15: 2057–2061.

De Souza

Caldas

Viana

. Basins of attraction changes by amplitude constraining of oscillators with limited power supply. Chaos Soliton Fract 2005; 26: 1211–1220.

Shaw

Holmes

. A periodically forced piecewise linear oscillator. J Sound Vib 1983; 90: 129–155.

10.

Feng

Rong

. Melnikov’s method for a general nonlinear vibro-impact oscillator. Nonlinear Anal: Theor 2009; 71: 418–426.

11.

Pavlovskaia

Wiercigroch

. Periodic solution finder for an impact oscillator with a drift. J Sound Vib 2003; 267: 893–911.

12.

Ivanov

. Bifurcations in impact systems. Chaos Soliton Fract 1996; 7: 1615–1634.

13.

Foale

. Analytical determination of bifurcations in an impact oscillator. Philos T Roy Soc A 1994; 347: 353–364.

14.

Hogan

. Rigid block dynamics confined between side-walls. Philos T Roy Soc A 1994; 347: 411–419.

15.

Aidanpää

Gupta

. Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. J Sound Vib 1993; 165: 305–327.

16.

Andreaus

Placidi

Rega

. Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator. Commun Nonlinear Sci 2010; 15: 2603–2616.

17.

Peterka

. Behaviour of impact oscillator with soft and preloaded stop. Chaos Soliton Fract 2003; 18: 79–88.

18.

Chávez

Brzeski

Perlikowski

. Bifurcation analysis of non-linear oscillators interacting via soft impacts. Int J Nonlin Mech 2017; 92: 76–83.

19.

Chen

Wei

. The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system. Int J Nonlin Mech 2011; 46: 197–203.

20.

Jin

Twizell

. A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. J Sound Vib 2006; 298: 1019–1033.

21.

Yue

Xie

. Lyapunov exponents and coexistence of attractors in vibro-impact systems with symmetric two-sided rigid constraints. Phys Lett A 2009; 373: 2041–2046.

22.

Quinn

Bairavarasu

. Near-simultaneous impacts. Int J Impact Eng 2006; 32: 889–904.

23.

Tai

Liu

. Stability and steady-state response analysis of a single rub-impact rotor system. Arch Appl Mech 2015; 85: 133–148.

24.

Wiercigroch

Sin

VTW

. Experimental study of a symmetrical piecewise base-excited oscillator. J Appl Mech: T ASME 1998; 65: 657–663.

25.

Bapat

Popplewell

Mclachlan

. Stable periodic motions of an impact-pair. J Sound Vib 1983; 87: 19–40.

26.

Karpenko

Wiercigroch

Pavlovskaia

. Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring. Int J Mech Sci 2002; 44: 475–488.

27.

Jin

. Periodic vibro-impacts and their stability of a dual component system. Acta Mech Sinica 1997; 13: 366–376.

28.

Yang

. Optimization of impact position and simulation of small vibratory pile driver. J Hydraul Eng 1999; 30: 72–76.

29.

Luo

. Hopf-flip bifurcations of vibratory systems with impacts. Nonlinear Anal: Real 2006; 7: 1029–1041.

30.

Knudsen

Massih

. Dynamic stability of weakly damped oscillators with elastic impacts and wear. J Sound Vib 2003; 263: 175–204.

31.

Luo

ACJ

. Period-doubling induced chaotic motion in the LR model of a horizontal impact oscillator. Chaos Soliton Fract 2004; 19: 823–839.

32.

Luo

ACJ

George

. Analytical dynamics of a mass–damper–spring constrained system. In: Tenreiro Machado

Baleanu

Luo

ACJ

(eds) Discontinuity and complexity in nonlinear physical systems (Nonlinear systems and complexity). New York: Springer, 2014, pp.323–343.

33.

Czolczynski

. On the existence of a stable periodic solution of an impacting oscillator with damping. Chaos Soliton Fract 2004; 19: 1291–1311.

34.

Bishop

Wagg

. Use of control to maintain period-1 motions during wind-up or wind-down operations of an impacting driven beam. Chaos Soliton Fract 1998; 9: 261–269.

35.

Luo

Zhang

. Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops. Nonlinear Anal: Real 2007; 8: 1272–1292.

36.

Newman

Makarenkov

. Resonance oscillations in a mass-spring impact oscillator. Nonlinear Dynam 2015; 79: 111–118.

37.

Jović

Raičević

. Vibro-impact forced oscillations of a heavy mass particle along a rough circle excited by a single-frequency force. Acta Mech 2012; 223: 1153–1168.

38.

Luo

. Dynamic analysis and suppressing chaotic impacts of a two-degree-of-freedom oscillator with a clearance. Nonlinear Anal: Real 2009; 10: 756–778.

39.

Budd

Dux

Cliffe

. The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators. J Sound Vib 1995; 184: 475–502.

40.

Pavlovskaia

Wiercigroch

. Computation of periodic solutions for piecewise oscillators. Int J Mech Sci 2002; 44: 475–488.

41.

Xie

. The mathematical model for the impact hammer and global bifurcations. Acta Mech Sinica 1997; 29: 456–463.

Dynamical analysis of a single degree-of-freedom impact oscillator with impulse excitation

Abstract

Keywords

Introduction

Mathematical model

Dynamical analysis

Period-1 motion

Other possible periodic motion

Properties of periodic-1 motion

Poincare map and stability analysis

Possibility of multiple periodic motion

Parameter analysis

Initial velocity v 0

Shock energy

Vibration frequency

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Initial velocity $v_{0}$