Impact-meshing vibration characteristics of high-speed gear systems with Backlash

Establishment of mathematical model

A typical four-inertia unit comprises a drive, gear pair and load connected by flexible shafts. When the gear teeth are in contact, the bodies oscillate as a four-inertia system with a mean rotation, as shown in Figure 1. In either case, owing to the relative velocity between the gears, the impact occurs on the forward sides of the gear teeth when contact occurs.

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Figure 1.

Gear torsional system.

A mathematical model of the geared torsional system is shown in Figure 2, wherein the gear pair is represented by an impact pair. To simplify the model, the torques were represented by horizontal forces, and the angular rotations relative to the mean rotation were represented as horizontal displacements. The mesh sliding friction was ignored, and only the torsional vibration of the gear transmission system was considered.

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Figure 2.

Model of a geared torsional system using an impact pair.

In this model, xⁱ (i = 1, 2, 3, 4) represents the torsional angular displacement of the gear teeth, input shaft and output shaft. The moments of inertia of the gear teeth, input shaft and output shaft are denoted as m_i (i = 1, 2, 3, 4). The backlash of the gear is 2b. The constants k₃ and k₄ represent the torsional stiffnesses of the input and output shafts, respectively. The equivalent viscous damping coefficients of the input shaft, output shaft and gear teeth are c₃, c₄ and c, respectively. F₁ and F₂ represent the load constant, while F₃ and F₄ represent the load fluctuation components. The function k (t) is the time-varying meshing stiffnesses of the gears. The law of the fluctuation of the speed of the active shaft is expanded into a Fourier series, and the magnitude of the excitation amplitude is an important factor affecting the flapping performance of the system.

Owing to the existence of a backlash, many interesting phenomena can be observed in the gear transmission system. Three motion domains were considered to explain the gear teeth movement based on the backlash. The vibrator can vibrate when both vibrators (m₁ and m₂) are not in contact. This state is defined as j = 2. To identify the regions of the right- and left-side impact motions, j = 1 and 3, respectively. According to the geometric conditions, the motion between both gears can be classified into three cases as expressed as follows.

{ C a s e I : z = x 1 − x 2 > b , j = 1 C a s e II : − b < z = x 1 − x 2 < b , j = 2 C a s e III : z = x 1 − x 2 < − b j = 3

(1)

Based on the gear meshing state, the equation of motion for this rotating system can be obtained using Newton’s laws as expressed below.

{ m 1 x ·· j 1 + c 3 ( x · j 1 − x · j 3 ) + k 3 ( x j 1 − x j 3 ) + c h j + k ( t ) f j = 0 m 2 x ·· j 2 + c 4 ( x · j 2 − x · j 4 ) + k 4 ( x j 2 − x j 4 ) − c h j − k ( t ) f j = 0 m 3 x ·· j 3 + c 3 ( x · j 3 − x · j 1 ) + k 3 ( x j 3 − x j 1 ) = F 1 + F 3 cos ( ω 1 t ) m 4 x ·· j 4 + c 4 ( x · j 4 − x · j 2 ) + k 4 ( x j 4 − x j 2 ) = − ( F 2 + F 4 cos ( ω 2 t ) )

(2)

Here, ‘•’ and ‘··’ are the first and the second derivatives, respectively, with respect to time t. The relative penetration depth f j is expressed as

f j = { x j 1 − x j 2 − b , j = 1 0 , j = 2 x j 1 − x j 2 + b , j = 3

(3)

The relative penetration velocity h j is expressed as

h j = { x · j 1 − x · j 2 , j = 1 , 3 0 , j = 2

(4)

This represents the torque transmitted by the gear teeth, which has a non-zero value only when the gear teeth are in contact. The time-varying meshing stiffness k(t) is expressed as

k ( t ) = k 0 + k w cos ( ω m t ) .

(5)

Here, k₀ is the mean mesh stiffness and k_w the variation amplitude of mesh stiffness. For the convenience of analysis, the differential equation (equation (2)) must be dimensionless. Introducing the nominal dimension b c (rad) and nominal time scale τ = ω mn t , where ω mn = 100 (s⁻¹) is the natural frequency of gear 1, the dimensionless displacement, velocity and acceleration can be expressed as ( i = 1 , 2 , 3 , 4 ; j = 1 , 2 , 3 ).

The dimensionless gear backlash is b ¯ = b b c , and the dimensionless equation of the gear motion can be expressed as

{ x ¯ ·· j 1 + 2 ζ 1 ( x ¯ · j 1 − x ¯ · j 3 ) + η 1 ( x ¯ j 1 − x ¯ j 3 ) + 2 ζ 5 h ¯ j + k 1 ( τ ) f ¯ = 0 x ¯ ·· j 2 + 2 ζ 2 ( x ¯ j 2 − x ¯ · j 4 ) + η 2 ( x ¯ j 2 − x ¯ j 4 ) − 2 ζ 6 h ¯ j − k 2 ( τ ) f ¯ j = 0 x ¯ ·· j 3 − 2 ζ 3 ( x ¯ · j 1 − x ¯ · j 3 ) − η 3 ( x ¯ j 1 − x ¯ j 3 ) = F ¯ 1 + F ¯ 3 ( τ ) x ¯ ·· j 4 − 2 ζ 4 ( x ¯ · j 2 − x ¯ · j 4 ) − η 4 ( x ¯ j 2 − x ¯ j 4 ) = − ( F ¯ 2 + F ¯ 4 ( τ ) ) .

(6)

The relative penetration velocities h j can be expressed as

h ¯ j = { x ¯ · j 1 − x ¯ · j 2 , j = 1 , 3 0 , j = 2 .

(7)

The dimensionless non-linear displacement functions for the backlash are expressed as

f ¯ j = { ( x ¯ j 1 − x ¯ j 2 − b ¯ ) , j = 1 0 , j = 2 ( x ¯ j 1 − x ¯ j 2 + b ¯ ) , j = 3 .

(8)

The dimensionless time-varying meshing stiffnesses are expressed as

{ k 1 ( τ ) = η 5 + η 6 cos ( ω m ω mn τ ) k 2 ( τ ) = η 7 + η 8 cos ( ω m ω mn τ ) .

(9)

Further,

ζ 1 = c 3 / 2 m 1 ω mn , ζ 2 = c 4 / 2 m 2 ω mn , ζ 3 = c 3 / 2 m 3 ω mn ζ 4 = c 4 / 2 m 4 ω mn , ζ 5 = c / 2 m 1 ω mn , ζ 6 = c / 2 m 2 ω mn η 1 = k 3 / k 0 , η 2 = k 4 m 1 / k 0 m 2 , η 3 = η 4 = k 3 m 1 / k 0 m 3 η 5 = 1 , η 6 = k w / k 0 , η 7 = m 1 / m 2 , η 8 = k w m 1 / k 0 m 2 F ¯ 1 = F 1 m 1 m 3 k 0 b c , F ¯ 2 = F 2 m 1 m 4 k 0 b c , F ¯ 3 ( τ ) = F 3 m 1 m 3 k 0 b c cos ( ω ¯ 1 τ ) , F ¯ 4 ( τ ) = F 4 m 1 m 4 k 0 b c cos ( ω ¯ 2 τ )

where ω ¯ 1 = ω 1 / ω mn , ω ¯ 2 = ω 2 / ω mn and ω ¯ 1 and ω ¯ 2 are the dimensionless excitation frequencies.

Boundary and analytic conditions

Because the discontinuous boundary in the absolute coordinate system changes with time, the analytical conditions for the motion transformation in a multi-constrained four-degrees-of-freedom impact vibration system cannot easily be obtained. For simplicity, a relative coordinate system was introduced. The relative displacement, velocity and acceleration of block 1 as regards block 2 are defined as z = x 1 − x 2 , z · = x · 1 − x · 2 and z ·· = x ·· 1 − x ·· 2 , respectively.

In a relative coordinate system, the relative motion regions of the object blocks m₁ and m₂ are expressed as

Ω 1 = { ( z ¯ 1 , z ¯ · 1 ) | z ¯ 1 ∈ ( b ¯ , + ∞ ) } Ω 2 = { ( z ¯ 2 , z ¯ · 2 ) | z ¯ 2 ∈ ( − b ¯ , b ¯ ) } Ω 3 = { ( z ¯ 3 , z ¯ · 3 ) | z ¯ 3 ∈ ( − ∞ , − b ¯ ) } } .

(10)

When both gear teeth enter the meshing domain from the free-motion domain, an energy transition occurs, that is, the impact loses energy. When the gear teeth are separated, the energy does not change, but the meshing force disappears. For an intuitive analysis, we further classified the boundary into impact and meshing boundaries. The relative impact boundary is expressed as follows.

{ Bd Ω 2 ∞ R + = { ( z ¯ , z ¯ · ) | φ 2 ∞ R + ≡ z ¯ = b ¯ , z ¯ · > 0 } Bd Ω 2 ∞ R − = { ( z ¯ , z ¯ · ) | φ 2 ∞ R − ≡ z ¯ = b ¯ , z ¯ · < 0 } Bd Ω 2 ∞ L + = { ( z ¯ , z ¯ · ) | φ 2 ∞ L + ≡ z ¯ = − b ¯ , z ¯ · > 0 } Bd Ω 2 ∞ L − = { ( z ¯ , z ¯ · ) | φ 2 ∞ L − ≡ z ¯ = − b ¯ , z ¯ · < 0 }

(11)

The corresponding relative boundary is expressed as

{ Bd Ω 21 = Bd Ω 12 = { z ¯ | φ 21 ≡ z ¯ = b ¯ , z ¯ · = 0 } Bd Ω 23 = Bd Ω 32 = { z ¯ | φ 23 ≡ z ¯ = − b ¯ , z ¯ · = 0 }

(12)

where the subscript ‘21’ denotes the boundary between Domains 2 and 1 and subscript ‘23’ the boundary between Domains 2 and 3. Based on the properties of the sets, the global set Ω can be expressed as

Ω = Ω 1 ∪ Ω 2 ∪ Ω 3 ∪ Bd Ω 2 ∞ R ∪ Bd Ω 2 ∞ L ∪ Bd Ω 21 ∪ Bd Ω 23

(13)

where Bd Ω 2 ∞ R = Bd Ω 2 ∞ R + ∪ B d Ω 2 ∞ R − and Bd Ω 2 ∞ L = Bd Ω 2 ∞ L + ∪ B d Ω 2 ∞ L − . The relative motion domains of the block and their corresponding boundaries are shown in Figure 3.

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Figure 3.

Motion boundary and motion domain.

The possible motion and boundary of the gear is shown in Figure 4.

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Figure 4.

Possible motions and boundaries of the oscillators.

In the relative coordinate system, the following vectors are introduced for the relative motions of m₁ and m₂:

{ Λ j = ( z j , z · j ) T = ( z j , v j ) T g j = ( z · j , v · j ) T = ( v j , g j ) T , j = 1 , 2 , 3

(14)

where v j = x · j 1 − x · j 2 and g j = x ·· j 1 − x ·· j 2 , (j = 1, 2, 3). j = 1, 2 and 3 represent three different relative motion domains.

For the motion in meshing domains 1 and 3, the relative forces per unit mass of m₁ and m₂ are expressed as follows:

g j = g ( z j , x ¯ j 2 , x ¯ · j 2 , x ¯ j 3 , x ¯ · j 3 , x ¯ j 4 , x ¯ · j 4 , τ ) = ( 2 ζ 2 ( x ¯ · j 2 − x ¯ · j 4 ) + η 2 ( x ¯ j 2 − x ¯ j 4 ) − 2 ζ 6 h ¯ j − k 2 ( τ ) f ¯ j ) − ( 2 ζ 1 ( z · j + x ¯ · j 2 − x ¯ · j 3 ) + η 1 ( z j + x ¯ j 2 − x ¯ j 3 ) + 2 ζ 5 h ¯ j + k 1 ( τ ) f ¯ j )

(15)

The normal vector of the relative boundary is expressed as

n ∂ Ω α β = ∇ φ α β = ( ∂ φ α β ∂ z , ∂ φ α β ∂ v ) T ,

(16)

where ∇ = ( ∂ ∂ z , ∂ ∂ v ) T is the Hamiltonian.

From equations (10), (11) and (15), the normal vector of the discontinuous boundary of the system is obtained as follows:

{ n ∂ Ω 2 ∞ R = n ∂ Ω 2 ∞ L = ( 1 , 0 ) T n ∂ Ω 21 = n ∂ Ω 23 = ( 0 , 1 ) T

(17)

Combined with the theory of flow switchability in discontinuous dynamical systems in Luo and O’connor, ⁶ a G-function is used to analyse the singularity of the discontinuous dynamical system boundaries. In a relative coordinate system, the zero- and first-order G-functions on the discontinuous boundaries can be simplified as follows:

G α ( z α , x ¯ α 2 , x ¯ · α 2 , x ¯ α 3 , x ¯ · α 3 , x ¯ α 4 , x ¯ · α 4 , τ m ± ) = n Bd Ω α β T · [ g α ( z α , , x ¯ α 2 , x ¯ · α 2 , x ¯ α 3 , x ¯ · α 3 , x ¯ α 4 , x ¯ · α 4 , τ m ± ) − g α β ( z α , x ¯ α 2 , x ¯ · α 2 , x ¯ α 3 , x ¯ · α 3 , x ¯ α 4 , x ¯ · α 4 , τ m ± ) ]

(18)

where g α represents a state vector in the domain, g α β the state vector on the boundary Bd Ω α β , τ m the moment when the domain flow changes, and τ m − and τ m + the moments before and after the stream transition occurs, respectively.

The onset conditions for the impacting chatter in Domain 2 are v 2 > 0 on Bd Ω 2 ∞ R + or v 2 < 0 on Bd Ω 2 ∞ L − .

Conditions under which tooth meshing in domain 2 are

or g 1 > 0 g 2 > 0 } on Bd Ω 21 g 2 > 0 g 3 > 0 } on Bd Ω 23 }

(19)

The condition of tooth meshing vanishing is

v 1 < 0 on Bd Ω 2 ∞ R − or g 1 > 0 g 2 > 0 } on Bd Ω 21 or v 1 > 0 on Bd Ω 2 ∞ L + or g 2 < 0 g 3 < 0 } on Bd Ω 23 }

(20)

Mapping structure

Because the discontinuous boundary in an absolute coordinate system changes with time, the analytical conditions for the motion transformation cannot easily be obtained. Because the impact time is extremely short (usually a few thousandths or tens of thousands of seconds), the restitution coefficient e of an impact is adopted to account for the energy loss owing to the impact.

By defining impact planes according to boundaries,

{ ∑ 2 ∞ R + = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = b ¯ , z ¯ · k > 0 ] ∑ 2 ∞ R − = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = b ¯ , z ¯ · k < 0 ] ∑ 2 ∞ L + = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = − b ¯ , z ¯ · k > 0 ] ∑ 2 ∞ L − = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = − b ¯ , z ¯ · k < 0 ] ,

(21)

The meshing switching planes for sticks are defined as

{ ∑ 21 = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = b ¯ , z ¯ · k = 0 ] ∑ 23 = [ ( t k , z ¯ · , x k ( 2 ) , x · k ( 2 ) ) | z ¯ k = − b ¯ , z ¯ · k = 0 ] .

(22)

To characterise the complex dynamics in the meshing process of the gear teeth, we introduced impact mapping. When the gear teeth collide, it increases the speed, and the impact transmission law is used to describe the motion continuity of the gear transmission system. In this study, the motion process of the gear was classified into impact and meshing phases, and the restitution coefficient, e, was used to describe the energy loss during the gear impact.

The velocity of any object block after impact can be expressed as

x · 2 1 + = I 1 ( 1 ) x · 2 1 − + I 2 ( 1 ) x · 2 2 −

(23)

Where ‘−’ represents before the impact, ‘+’ after the impact and i = 1, 2 denote Vibrators 1 and 2, respectively.

I 1 ( 1 ) = m 1 − m 2 e m 1 + m 2 , I 2 ( 1 ) = ( 1 + e ) m 2 m 1 + m 2 I 1 ( 2 ) = m 2 − m 1 e m 1 + m 2 , I 2 ( 2 ) = ( 1 + e ) m 1 m 1 + m 2 }

(24)

According to the gear motion state, the gear tooth meshing process can be sectioned into multistage chattering and meshing. The mapping structure was sectioned into free and meshing mappings. The free mapping represents the motion state when the main and driven gear teeth are not in contact.

In the relative frame, the mappings are also expressed as

P 1 : ∑ 21 → { ∑ 2 ∞ R − , ∑ 21 } P 2 : { ∑ 2 ∞ R − , ∑ 21 } → { ∑ 2 ∞ R + , ∑ 21 } P 3 : { ∑ 2 ∞ R − , ∑ 21 } → { ∑ 2 ∞ L − , ∑ 23 } P 4 : ∑ 23 → { ∑ 23 , ∑ 2 ∞ L + } P 5 : { ∑ 2 ∞ L + , ∑ 23 } → { ∑ 2 ∞ L − , ∑ 23 } P 6 : { ∑ 2 ∞ L + , ∑ 23 } → { ∑ 2 ∞ R + , ∑ 21 } }

(25)

P1 and P4 refer to the meshing map of the movement in the meshing domain ( α = 1 , 3 ), and ‘ A → B ’ represents the mapping relationship from set A to B. Figure 5 shows several possible scenarios for the above-mentioned mapping.

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Figure 5.

Schematic of the decomposition of the mapping structure.

To simplify the equation, impact maps are implicit in the mapping structure.

{ t k + 1 = t k x k + 1 ( 1 ) = x k ( 1 ) x k + 1 ( 2 ) = x k ( 2 ) x · k + 1 ( 1 ) = I 1 1 x · k ( 1 ) + I 2 1 x · k ( 2 ) x · k + 1 ( 2 ) = I 1 2 x · k ( 2 ) + I 2 2 x · k ( 1 )

(26)

The subscript k represents the pre-impact state and k + 1 the post-impact state. To study the meshing motion of the vibration-impact model of the gear transmission system, the basic mapping notation proposed in Sakaridis et al. ¹⁹ was used to describe the mapping motion.

P n k ⋯ n 2 n 1 = P n k ° ⋯ ° P n 2 ° P n 1 , n k ∈ { 1 , 2 , … , 6 }

(27)

Numerical simulation of impact and meshing

For the first time, we refined the gear meshing process into impact and meshing. The impact and meshing phenomena of the gears during the starting process were the most typical, capable of elucidating the kinematic mechanisms of the gear impact and meshing. The system analysis diagrams are presented when both the input and output torques are F1 = F3 = 400 N·m. As shown in Figure 6(a), when the rotational speed of the input shaft is 1000 r/min, the gear teeth enter a long-term stable meshing state after multiple impact-meshing cycles under the action of the driving force and load during the start-up process. As shown in Figure 6(b), the gear teeth have three disengagement phenomena, and each disengagement produces multiple impacts. During the process of gear teeth from impact to meshing, the driving gear teeth experienced seven impacts with a very short period before entering the first meshing. The system enters a stable meshing state after a period of time. This corresponds with the working state of the actual gear system.

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Figure 6.

Operation analysis diagram when the input shaft rotates at 1000 r/min: (a) initial impact and meshing displacement diagrams and (b) drawing of the partial enlargement.

When the active wheel gives the driven wheel a speed, the driven wheel accelerates instantaneously and it is separated from the active wheel at this time, and it is difficult to continue to maintain the fit, thus forming an impact-meshing cycle process.

Figure 7 indicates that the active gear teeth may enter a stable meshing state after multiple collision impacts during the initial meshing process, or they may be in the collision-mesh-re-collision-re-meshing cycle process for a long time.

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Figure 7.

Vibration response of oscillator 1 at different input speeds: (a) Initial shock and meshing displacement plot at 17,550 r/min and (b) Initial shock and meshing displacement plot at 19,000 r/min.

Dynamic response of the system at different speeds

To investigate the response of the gear system under a stable operation, we define the gear system as a steady state after 10,000 iterations, and obtain the phase diagram and spectrogram of the system.

As shown in Figure 8, when n = 1000 r/min, the gear system moves in one cycle after stable operation. When n = 17,550 r/min, the number of disengagement increases, and the gear teeth move in multiple cycles under stable operation. When n = 19,000 r/min, the gear teeth cannot enter the stable meshing state, and the system enters the chaotic motion state.

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Figure 8.

Phase and Poincaré diagram of the gear system at different input speeds: (a) n = 1000 r/min, (b) n = 17,550 r/min and (c) n = 19,000 r/min.

To portray the cause of the long-term disengagement, the FFT plot of the gear system at different speeds is solved, as shown in Figure 9. The FFT diagram of the system is obtained by considering the position of the gear teeth at the centre of the backlash as the initial position and the gear system as a stable state after 10,000 iterations.

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Figure 9.

Spectrogram of the gear system at different speeds: (a) n = 1000 r/min, (b) n = 17,550 r/min and (c) n = 19,000 r/min.

As shown in Figure 9, when n = 1000 r/min, the p–1 periodic solution is dominated by the fundamental vibration with a similar frequency to the meshing frequency of the gear teeth, accompanied by higher harmonics. When n = 17,550 r/min, which does not include shock, the second harmonics and their integral multiorder harmonics appear in the p–n periodic solution of the system, while the fundamental frequency vibration amplitude is still the maximum. When n = 19,000 r/min, there is also a peak in the fundamental frequency component of the chaotic solution with shock; however, the low-frequency component is remarkably abundant. The impact of the gear teeth caused generating complex instability factors in the system and rendered the dynamic behaviour of the flow on the transition boundary relatively more complex and diverse. The disengagement behaviour produced a rough and uneven peak signal in the frequency domain, as shown in Figure 9(c). The rotation and meshing frequencies of the shaft are the main frequency components of the vibration signal of a gear pair in the gear transmission system.

To study the influence of the shaft speed in the high-speed gear transmission system on the gear impact-meshing phenomenon, we drew a waterfall diagram of the gear system.

The waterfall diagram presented in Figure 10 shows that a large number of modulation frequencies occur when the rotational speed lies between 25,000 r/min and 35,000 r/min. The modulation frequency disappears when the speed rises above 35,000 r/min. We observed a noticeable impact-meshing cycle phenomenon of the gear teeth in the resonant frequency band of the gear system. A large number of modulation frequencies are observed because the gear system cannot enter a stable meshing state in this speed range, resulting in a long-term collision-meshing cycle. The resonance caused by the joint action of the rotational frequency and meshing frequency is the primary cause of the long-term impact meshing of the gear transmission system. Concurrently, a distinct frequency band containing the gear meshing frequency is observed, and it can be concluded that the meshing frequency of the gear system is the primary cause of the vibration.

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Figure 10.

Waterfall diagram of a gear system controlled by the input speed.

Influence of driving load variation on impact-meshing characteristics of gear system

The external excitation of the gear system is an important factor affecting the stability of the gear system. In this study, numerical simulation is used to analyse the shock response of the gear system under different load parameters. Here, the input shaft speed is n = 7200 r/min, load fluctuation amplitude F₃ = F₄ = 4 N·m and the constant component of load F₂ = 400 N·m. The relative displacements of the gear teeth in the starting process under a stable load and different initial driving torques are shown in Figure 11.

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Figure 11.

Impact and meshing of the gear teeth under different driving torques.

As shown in Figure 11, the relative penetration depth of the teeth can be observed to be the largest in the initial meshing state, which gradually decreased until stable meshing. With the increase in the initial drive torque, the time required for the gear to enter the stable meshing state is shorter. With the increase in the constant component F1 of the driving load, the relative meshing depth between the gear teeth increased, but the meshing impact time of the gear system is reduced correspondingly. However, the number of shock-meshing is not necessarily related to the increase in the constant load. When starting, the driving force was slightly larger than the load, which is beneficial in terms facilitating the rapid entry of the gears into stable meshing.

Select the input shaft speed n = 7200 r/min, F₄ = 4 N·m and F₁ = F₂ = 400 N·m. The relative displacements of the gear teeth in the starting process under stable load and different amplitudes of driving torque fluctuation are shown in Figure 12.

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Figure 12.

Impact and meshing of the gear teeth under different driving torque fluctuation amplitudes.

The driving load fluctuation component F3 has a minor influence on the impact-meshing characteristics of the system, while the driving load fluctuation component F3 mainly affects the periodic response characteristics of the gear system under the stable running state.