Mechanism of vibration suppression induced by roller pitch diameter modification in planetary roller screw mechanism

Abstract

To elucidate the intrinsic mechanism of vibration suppression induced by roller pitch diameter modification in planetary roller screw mechanism (PRSM), a coupled dynamic model incorporating modification effects is developed. The influence of different modification magnitudes on the vibration response of key components is systematically investigated. First, based on helical surface contact theory, a thread contact model considering roller modification is established. The clearance distribution induced by the modification is introduced into the load distribution analysis as the initial condition. Subsequently, Hertzian contact theory is employed to obtain the non-uniform load distribution. A lubricated sliding friction model for the roller–screw interface is further established and incorporated into the dynamic equations as an excitation source. By analyzing the spatiotemporal evolution of friction forces under different modification levels, the vibration suppression mechanism is clarified. Finally, the proposed model is validated through model degeneration analysis and experimental comparisons, confirming its reliability and accuracy. However, the synergistic vibration suppression effect of gear pair modification and thread pair modification was not considered in the present study. Future work will focus on establishing a coupled modification model of the thread pair and gear pair to further reveal the vibration suppression mechanism and vibration characteristics of PRSM.

Keywords

vibration suppression contact characteristics modification planetary roller screw mechanism dynamic

1. Introduction

The planetary roller screw mechanism (PRSM) structure is shown in Figure 1, it has been widely applied in aerospace, high-end CNC equipment, and electromechanical actuation systems due to its superior load-carrying capacity, superior transmission accuracy, and suitability for high-speed and heavy-load operating conditions (Chen, 2019; Zhang et al., 2024).

Figure 1.

Schematic representation of the PRSM structure.

With the increasing demand for high-speed, high-precision, and low-noise transmission systems, vibration and noise in PRSM have become key factors limiting their performance and engineering applications. Existing studies indicate that the primary excitation sources originate from contact nonlinearity and friction within internal transmission pairs, especially the sliding friction in the roller–screw thread pair (Fu et al., 2018). However, the vibration generation mechanism caused by the coupling of friction and contact nonlinearity, as well as corresponding vibration suppression methods, has not been systematically studied. Therefore, it is necessary to investigate the friction-induced vibration characteristics and suppression mechanisms of PRSM to support the optimal design and low-vibration operation of high-performance systems. In this study, only the thread pair modification is considered, as it is the primary load-bearing and sliding contact interface governing the dominant friction-induced excitation. The gear pair is assumed unmodified due to its relatively small contact force, and thus limited influence on the overall vibration response. This simplification enables the model to focus on the dominant vibration mechanism while maintaining physical relevance.

To address vibration issues in PRSM, previous studies have mainly focused on dynamic modeling, load distribution, and lubrication characteristics. In dynamic modeling, Wu et al. developed a torsional model based on contact behavior and performed modal analysis (Wu et al., 2020), and later improved it by incorporating deformation effects (Wu et al., 2024). Mo et al. established a dynamic model considering time-varying gear meshing stiffness and investigated system stability under different operating conditions (Mo et al., 2024). Xu et al. developed a load distribution model including gear meshing excitation and found that time-varying meshing stiffness induces periodic variations in thread load distribution (Xu et al., 2026).

Regarding load distribution of thread pair, Zhang et al. was first introduced in load distribution models under different installation configurations, although the load sharing among rollers was not considered (Zhang et al., 2016). To address this limitation, Xing et al. introduced a multi-roller load-sharing model for PRSM (Xing et al., 2024a). In addition, various load equalization methods have been studied. Yao et al. presented an optimization approach to realize the most uniform load distribution through lead design (Yao et al., 2022). Fu et al. found that load-sharing characteristics are exhibits higher sensitivity to the positional relationship between the screw and nut, while being less sensitive to roller position errors (Fu et al., 2017).

Regarding friction and lubrication, Xing et al. developed a kinematic model of PRSM to analyze the effects of structural and operating parameters on sliding velocity and direction at the roller–screw and roller–nut interfaces (Xing et al., 2023). Xie et al. established a mixed-lubrication model and systematically studied the effects of rotational speed, load, and surface roughness on lubrication performance (Xie et al., 2019). They further developed a Lagrangian-based friction–dynamic model to investigate the influence of operating conditions and creep parameters on friction torque, transmission efficiency, and relative sliding velocity (Xie et al., 2023). In addition, Xing et al. derived entrainment velocity, oil film thickness, and pressure distribution considering eccentricity error and analyzed their effects on lubrication characteristics (Xing et al., 2024b).

Significant progress has been made in understanding the effects of friction and lubrication on the vibration behavior of transmission systems from tribological and dynamic perspectives. Liu et al. investigated the dynamic behavior of rolling bearings based on elastohydrodynamic lubrication (EHL) theory and Hertzian contact mechanics (Liu et al., 2025). Zhang et al. analyzed the vibration characteristics of rotor–bearing systems under high-temperature conditions by combining tribology and dynamics (Zhang et al., 2022). Li et al. proposed a method for calculating time-varying friction torque in ball bearings (Li et al., 2026), while Sulollari et al. established the relationship between friction and vibration using a mass–spring–damper model (Sulollari et al., 2024). In addition, Kumar et al. developed a control strategy to suppress stick–slip vibration in single-degree-of-freedom systems (Kumar and Malas, 2024). Khaldoon F. developed gear dynamic models showing that tooth surface friction significantly enhances vibration response in the off-line-of-action direction (Khaldoon et al., 2015), and further demonstrated that considering elastohydrodynamic lubrication, changes in lubricant viscosity significantly alter the vibration response of the gear system by affecting the friction coefficient (Khaldoon et al., 2022).

Existing studies often treat contact geometry, load distribution, and friction behavior independently, lacking a coupled framework along “structural modification–contact state–lubrication friction–dynamic response.” Roller pitch diameter modification introduces axial contact clearance, regulating load distribution and improving friction conditions; however, its influence on friction excitation and vibration response remains unclear. This study develops coupled contact and load distribution models incorporating modification, and integrates lubricated sliding friction into the dynamic equations for unified modeling. Based on this framework, the effects of modification magnitude on contact state, friction evolution, and vibration response are systematically analyzed, revealing the underlying vibration suppression mechanism.

2. Contact characteristics and clearance of thread pairs

The load distribution of PRSM thread pairs shows that the roller–screw contact force decreases with increasing thread index, while the roller–nut contact force increases. Due to axial reciprocating motion, contact forces are highest at both ends of the roller threads. Therefore, reducing end contact forces is crucial for improving load capacity, reducing friction, and suppressing vibration.

During the machining of the threads located at both ends of the roller, the pitch diameter of the threads is reduced by increasing the feed rate. This machining approach is referred to as roller pitch diameter modification. After modification, the contact relationship between the roller and the screw/nut threads is illustrated in Figure 2.

Figure 2.

Contact relationship after roller pitch diameter modification.

Blue solid lines: screw and nut profiles; gray solid line: unmodified roller profile; orange dashed line: modified roller profile; green dashed line: mean diameter of the modified roller. Yellow ellipses: contact areas of the unmodified roller ( $o_{SR i}$ , $o_{NR i}$ ) and green ellipses: contact areas of the modified roller ( $o_{SR i}^{P m}$ , $o_{NR i}^{P m}$ ). $F_{SR i}$ , $F_{NR i}$ and $F_{SR i}^{P m}$ , $F_{NR i}^{P m}$ denote the corresponding normal contact forces at the screw–roller and nut–roller interfaces.

As shown in Figure 2, the axial clearance between the modified roller end threads and the screw/nut threads increases. This enlarged axial clearance delays the engagement of the roller end threads, thereby reducing the corresponding contact forces.

A coordinate system $w_{R} o_{R} u_{R}$ is established based on the roller cross-section, and the modified tooth profile of the roller is shown in Figure 3. In the figure, $P_{RU}$ and $P_{RB}$ denote the upper and lower arc profiles of the unmodified roller, while $P_{RU}^{'}$ and $P_{RB}^{'}$ represent those after modification. $o_{TR}$ and $r_{TR}$ are the center and radius of the roller’s arc, and $c_{R}$ is the half tooth thickness of the roller.

Figure 3.

Profile of the roller with modified threads.

According to Figure 3, when the roller pitch diameter is reduced by $P m_{i}$ , the axial clearance between the roller and the screw or nut increases by $g_{XR i}^{P m}$ . The contact point between the roller and the screw/nut shifts along the direction of increasing axial clearance, that is, toward the crest of the modified roller thread. At this stage, the angle between the tangent at the contact point and the $u_{R}$ -axis is defined as the modified contact angle $β_{XR i}^{P m}$ .

The coordinates $(u_{TR}, w_{TR})$ of the center $o_{TR}$ of the modified roller arc $P_{RU}^{'}$ are expressed as

{\begin{cases} u_{TR} = - r_{TR} \sin β_{R} \\ w_{TR} = - r_{TR} \cos β_{R} + c_{R} \end{cases}

(1)

Where

β_{R}

is the thread flank angle.

The equation of the modified roller arc $P_{RU}^{'}$ is given by

{(u_{R} + r_{TR} \sin β_{R} + P_{m i})}^{2} + {(w_{R} + r_{TR} \cos β_{R} - c_{R})}^{2} = r_{TR}^{2}

(2)

Where

P_{m i}

denotes the modification amount of the roller pitch diameter.

When $u_{R} = 0$ , the coordinate of the contact point $o_{XR i}^{P m}$ along the $w_{R}$ -axis can be expressed as

w_{R} = \sqrt{r_{TR}^{2} - {(r_{TR} \sin β_{R} + P_{m i})}^{2}} - r_{TR} \cos β_{R} + c_{R}

(3)

The increment in axial clearance after modification is twice the distance between $o_{XR i} o_{XR i}^{P m}$ . Since the coordinate of $o_{XR i}$ along the $w_{R}$ -axis is $c_{R}$ , the additional axial clearance between the roller and the screw or nut threads after modification can be expressed as

g_{XR i}^{P m} = 2 (r_{TR} \cos β_{R} - \sqrt{{r_{TR}}^{2} - {(r_{TR} \sin β_{R} + P_{m i})}^{2}})

(4)

Where the subscripts S and N denote the screw and nut, respectively.

By differentiating equation (2), we obtain

\frac{d w_{R}}{d u_{R}} = - \frac{u_{R} + P_{m i} + r_{TR} \sin β_{R}}{w_{R} + r_{TR} \cos β_{R} - c_{R}}

(5)

Substituting the coordinates of point $o_{XR i}^{P m}$ into equation (5), the contact angle between the roller and the screw or nut threads can be expressed as

β_{XR i}^{P m} = \arctan \frac{r_{TR} \sin β_{R} + P_{m i}}{\sqrt{r_{TR}^{2} - {(r_{TR} \sin β_{R} + P_{m i})}^{2}}}

(6)

The helix angle $λ_{X R i}^{P m}$ at the contact point $o_{XR i}^{P m}$ between the modified roller and the screw or nut is given by

λ_{XR i}^{P m} = \arctan \frac{L_{R}}{π d_{i}^{P m}}

(7)

Where

d_{i}^{P m} = d_{R} - 2 P_{M i}

is the modified pitch diameter of the thread.

According to existing literature (Fu et al., 2016), the clearances between the unmodified roller–screw and roller–nut thread pairs can be calculated using Equations (8)–(11)

g_{SB - RU} = \frac{P}{2} - (c_{S} + c_{R}) + r_{TR} \cos β_{R} + (r_{SR} - r_{S}) \tan β_{S} - \sqrt{{r_{TR}}^{2} - {(r_{RS} - r_{R} + r_{TR} \sin β_{R})}^{2}} - \frac{ϕ_{SR} L_{S} + ϕ_{RS} L_{R}}{2 π}

(8)

g_{SU - RB} = \frac{P}{2} - (c_{S} + c_{R}) + r_{TR} \cos β_{R} + (r_{SR} - r_{S}) \tan β_{S} - \sqrt{{r_{TR}}^{2} - {(r_{RS} - r_{R} + r_{TR} \sin β_{R})}^{2}} - \frac{ϕ_{SR} L_{S} + ϕ_{RS} L_{R}}{2 π}

(9)

g_{SR} = g_{SB - RU} + g_{SU - RB}

(10)

g_{NR} = g_{NU - RB} + g_{NB - RU} = P - 2 (c_{N} + c_{R})

(11)

Where P is the pitch,

c

is the half tooth thickness,

r

is the radius, and

L

is the lead. The subscripts R, S, and N represent the roller, screw, and nut, respectively.

r_{SR}

and

r_{RS}

denote the meshing radii of the screw and roller helical surfaces, while

ϕ_{SR}

and

ϕ_{SR}

represent their respective meshing offset angles.

After modification, the clearance between the roller–screw and roller–nut thread pairs is equal to the sum of the original clearance and the additional clearance induced by modification, as expressed in equation (12)

g_{XR i} = g_{XR i}^{P m} + g_{XR}

(12)

3. Load distribution and friction force

3.1. Load distribution

To evaluate the friction force after roller pitch diameter modification, the contact force distribution must first be determined. In this study, modification at both roller ends increases axial clearance, leading to delayed engagement. The introduced clearance is treated as a compensation for elastic deformation and incorporated into the load distribution model. Based on this, the modified load distribution is calculated using Hertzian contact theory.

The elastic deformation of a thread tooth is equal to the global axial deformation of the system minus the thread clearance after roller pitch diameter modification.

The relationship between thread contact deformation and thread clearance is expressed as

δ_{XR i} = δ_{XR} - g_{XR i}

(13)

Where

δ_{X R}

denotes the overall axial deformation of the roller and screw or roller and nut.

The Hertzian contact force of the PRSM thread pair can be derived based on Hertzian contact theory. Therefore, the contact force of the modified roller and screw or roller and nut thread pair is given by:

F_{XR i}^{P m} = C_{XR i} {(δ_{XR} - g_{XR i})}^{3 / 2}

(14)

Where

C_{X R i}

denotes the damping coefficient of the thread contact between the roller and the screw or nut.

According to the force analysis of the PRSM, the axial contact force of each thread tooth after roller pitch diameter modification can be expressed as

F_{XR a i}^{P m} = \frac{C_{XR i} {(δ_{XR} - g_{XR i})}^{3 / 2}}{\cos λ_{XR i}^{P m} \cos β_{XR i}^{P m}}

(15)

Thus, the contact force of the roller and screw or roller and nut thread pair after modification is given in equation (16)

F_{XR}^{P m} (δ_{XR}) = \sum_{i = 1}^{n} \frac{C_{XR i} {(δ_{XR} - g_{XR i})}^{3 / 2}}{\cos λ_{XR i}^{P m} \cos β_{XR i}^{P m}}

(16)

Before solving the thread tooth contact forces, a contact detection step is performed for each thread tooth to ensure that all teeth participating in engagement undergo elastic deformation, that is,

δ_{XR i} > 0

(17)

An initial value of the global deformation of the roller and screw or roller and nut system is set as the sum of the minimum thread clearance and the convergence tolerance, as expressed in equation (18)

δ_{XR 0} = \min (g_{XR i}) + ε

(18)

A nonlinear equation is then established based on the contact force formulation, as shown in equation (19)

f (δ_{XR}^{k}) = \sum_{i = 1}^{n} \frac{C_{XR i} {(δ - g_{XR i})}^{3 / 2}}{\cos λ_{XR i}^{P m} \cos β_{XR i}^{P m}} - F_{a}

(19)

The derivative of the contact force function at the contact point between the roller and the screw/nut is obtained as

f^{'} (δ_{XR}^{k}) = \sum_{i \in C (δ_{XR})} \frac{\frac{3}{2} C_{XR i} {(δ_{XR} - g_{XR i})}^{1 / 2}}{\cos λ_{XR i}^{P m} \cos β_{XR i}^{P m}}

(20)

The Newton–Raphson iteration scheme is applied

s^{k} = - \frac{f (δ_{XR}^{k})}{f^{'} (δ_{XR}^{k})}

(21)

δ_{XR}^{k + 1} = δ_{XR}^{k} + η^{k} s^{k}

(22)

Where

η^{(k)} \in (0, 1]

represents the iteration update term.

Finally, a convergence criterion is defined for the iterative solution

| δ_{XR}^{k + 1} - δ_{XR}^{k} | < ε

(23)

Based on the load non-uniformity coefficient for PRSM thread pairs reported in Zhang et al. (2016), the contact forces of the R–S and R–N thread are expressed as

F_{XR i}^{P m} = ξ_{XR i} F_{XR i}^{P m}

(24)

Where

ξ_{XR i}

denotes the load non-uniformity coefficient of the thread.

Accordingly, the load distributions of the roller–screw and roller–nut thread pairs considering load equalization can be obtained from equation (24).

3.2. Friction force

Due to relative sliding between the roller and the screw and pure rolling between the roller and the nut, the roller–screw interface exhibits combined rolling and sliding friction. Since rolling friction is negligible compared to sliding friction, this study neglects rolling friction and roller–nut friction, considering only the sliding friction at the roller–screw interface. The sliding friction force is expressed as

f_{SR i}^{P m} = μ_{SR i}^{P m} \sum_{i = 1}^{n} F_{SR i}^{P m}

(25)

Where

μ_{SR i}^{P m}

denotes the sliding friction coefficient of the roller–screw pair after roller pitch diameter modification, and

F_{SR i}^{P m}

represents the contact force of the roller–screw thread pair after modification

F_{SR i}^{P m} = F_{SR i}^{P m} [\begin{array}{c} \cos ϕ_{RS i}^{P m} \tan β_{RS i}^{P m} - \sin ϕ_{RS i}^{P m} \tan λ_{RS i}^{P m} \\ - \sin ϕ_{RS i}^{P m} \tan β_{RS i}^{P m} - \cos ϕ_{RS i}^{P m} \tan λ_{RS i}^{P m} \\ - 1 \end{array}] / \sqrt{1 + \tan^{2} λ_{RS i}^{P m} + \tan {β^{2}}_{RS i}^{P m}}

(26)

F_{SR i}^{P m} = F_{SR i z}^{P m} \sqrt{1 + \tan^{2} λ_{RS i}^{P m} + \tan {β^{2}}_{RS i}^{P m}}

(27)

In addition,

ϕ_{RS i}^{P m}

β_{RS i}^{P m}

, and

λ_{RS i}^{P m}

denote the meshing offset angle, contact angle, and helix angle at the roller–screw contact point on the modified helical surface, respectively.

F_{SR i z}^{P m}

represents the axial contact force of the modified roller–screw thread pair. Based on the above expression, the contact force of the roller–screw pair after modification can be obtained.

The sliding contact in PRSM roller–screw threads is analogous to bearing contacts, and the friction coefficient can be described using a typical Stribeck curve (Wen and Huang, 2018). As shown in Figure 4, the bearing friction coefficient varies with the bearing characteristic number, corresponding to boundary, mixed, and hydrodynamic lubrication regimes.

Figure 4.

Stribeck curve of the bearing.

Based on the friction coefficient curves under different lubrication regimes shown in Figure 4, the friction coefficient function is obtained through curve fitting, as expressed in Equations (28) and (29)

μ (I_{c}) = 10^{y (x)}, x = \log_{10} (I_{c})

(28)

y (x) = {\begin{cases} - 0.058 x - 1.18 \\ 4.2 x^{2} + 68.5 x + 277 \\ 0.62 x + 3.25 \end{cases} \begin{array}{l} , \\ , \\ , \end{array} \begin{array}{l} I_{c} < 6 \times 10^{- 9} \\ 6 \times 10^{- 9} \leq I_{c} \leq 1.2 \times 10^{- 8} \\ I_{c} > 1.2 \times 10^{- 8} \end{array}

(29)

Where

Ι_{c}

is the bearing characteristic number, which can be calculated using equation (30)

Ι_{c} = \frac{η U}{p}

(30)

Where

η

is the grease viscosity,

U

is the grease film entrainment velocity, and

p

is the grease film pressure.

The sliding velocity at the contact interface of the screw–roller pair thread is given by

v_{RS i}^{s l i d e} = v_{SR i}^{P m} - v_{RS i}^{P m}

(31)

Where

v_{SR i}^{P m}

and

v_{RS i}^{P m}

denote the velocities of the screw helical surface and roller helical surface at the roller–screw contact point after pitch diameter modification, respectively

v_{SR i}^{P m} = w_{S} [\begin{array}{l} - r_{SR i}^{P m} \sin ϕ_{SR i}^{P m} \\ r_{SR i}^{P m} \cos ϕ_{SR i}^{P m} \\ 0 \end{array}]

(32)

v_{RS i}^{P m} = [\begin{array}{l} - r_{SR i}^{P m} \sin ϕ_{SR i}^{P m} w_{P} + n_{S} r_{RS i}^{P m} \sin ϕ_{RS i}^{P m} w_{P} \\ r_{SR i}^{P m} \cos ϕ_{SR i}^{P m} w_{P} + n_{S} r_{RS i}^{P m} \cos ϕ_{RS i}^{P m} w_{P} \\ - (w_{S} L_{S}) / (2 π) \end{array}]

(33)

Specifically, $r_{SR i}^{P m}$ and $ϕ_{SR i}^{P m}$ represent the engagement radius and engagement offset angle of the roller–screw contact point on the modified roller helical surface, while $r_{RS i}^{P m}$ and $ϕ_{RS i}^{P m}$ denote the corresponding meshing radius and meshing offset angle on the screw helical surface. $w_{S}$ and $w_{P}$ denote the screw angular velocity and the roller revolution angular velocity, respectively.

Based on the flow separation method for solving non-Newtonian lubrication problems, the Reynolds equation for the Ostwald non-Newtonian fluid model can be derived as

\frac{\partial}{\partial x} [ρ {h_{SR i}}^{2} {(\frac{h_{SR i}}{ϕ} \frac{\partial p_{SR i}}{\partial x})}^{\frac{1}{n}}] + \frac{\partial}{\partial y} [ρ {h_{SR i}}^{2} {(\frac{h_{SR i}}{ϕ} \frac{\partial p_{SR i}}{\partial y})}^{\frac{1}{n}}] = \frac{2^{1 + \frac{1}{n}} (2 n + 1)}{n} [\frac{\partial (U_{SR i x} ρ h_{SR i})}{\partial x} + \frac{\partial (U_{SR i y} ρ h_{SR i})}{\partial y}]

(34)

Where

p_{SR i}

and

h_{SR i}

represent the lubricating grease film pressure and film thickness in the i-th thread contact interface region, respectively;

ϕ

ρ

and

n

denote the plastic viscosity, density, and flow behavior index of the lubricant.

U_{SR i x}

and

U_{SR i y}

denote the entrainment velocities in the

X_{SR}

-and

Y_{SR}

-directions of the thread contact region, respectively.

Based on the relative sliding velocity between the roller and screw, the entrainment velocity at contact point is formulated as

U_{SR i} = \frac{v_{SR i}^{P m} + v_{RS i}^{P m}}{2}

(35)

When the PRSM is loaded, elastic deformation and contact stress occur at the roller–screw interface, and the contact region forms an elliptical area. The contact stress at an arbitrary point in the elliptical contact region of the i-th thread tooth is formulated as

p_{SR i}^{P m} = p_{H i} \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}}

(36)

Where a and b are the semi-major axis and semi-minor axis of the elliptical contact area, respectively, and

p_{H i}

is the maximum pressure at the center of the thread contact ellipse

p_{H i} = \frac{3 F_{SR i}^{P m}}{2 π a b}

(37)

By substituting the entrainment velocity $U_{SR i}$ , contact pressure $p_{SR i}^{P m}$ , and lubricant viscosity into Equations (28) and (29), the friction coefficient of the roller–screw thread pair can be obtained. Finally, by substituting the friction coefficient and the contact force obtained from equation (26) into equation (25), the sliding friction force between the roller and the screw can be determined.

4. Effect of profile modification on the vibration characteristics of PRSM

4.1. Dynamic model

Friction in the roller–screw contact interface is recognized as the dominant source of excitation governing the steady-state vibration of the PRSM system, with sliding friction being the primary contributor. In this study, only the roller–screw interface is considered for modification analysis, while the gear pair is assumed to remain unmodified, since its contribution to vibration excitation is relatively minor compared with the sliding contact behavior of the thread interface.

Based on this assumption, the contact and sliding characteristics under roller pitch diameter modification are first analyzed, and a corresponding sliding friction model is established. The friction-induced excitation is then incorporated into the dynamic equations to develop a lumped-parameter dynamic model of the PRSM considering modification effects. The model is formulated using the lumped mass method combined with force equilibrium analysis and is used to investigate the vibration responses of key components, providing a theoretical basis for structural optimization and vibration reduction.

The governing dynamic equation of the PRSM is given in equation (38)

M x^{″} + (C_{b} + C_{m} + 2 ω_{c} G) x^{'} + (K_{b} + K_{m}^{P m} - ω_{c}^{2} K_{Ω}) x = T + f_{SR}^{P m}

(38)

Where

x

denotes the displacement vector of each component, M is the mass matrix,

C_{b}

is the support damping matrix,

C_{m}

is the mesh damping matrix, and

G

is the gyroscopic matrix.

K_{b}

represents the support stiffness matrix,

K_{m}^{P m}

denotes the contact stiffness matrix considering roller pitch diameter modification, and

K_{Ω}

is the radial stiffness matrix. T is introduced to represent the generalized external excitation acting on the PRSM system,

f_{SR}^{P m}

denotes the sliding friction force between the roller and the screw after modification.

The governing equation of motion for the screw is expressed in equation (39), the internal ring gear in equation (40), the nut in equation (41), the roller in equation (42), and the carrier in equation (43), respectively

{\begin{cases} m_{S} (x_{S}^{″} - 2 ω_{C} y_{S}^{'} - {ω_{C}}^{2} x_{S}) + \sum_{i = 1}^{N} c_{SR i} {δ^{'}}_{SR i}^{P m} \cos φ_{S i} \cos α_{S} + \sum_{i = 1}^{N} k_{SR i}^{P m} δ_{SR i}^{P m} \cos φ_{S i} \cos α_{S} + c_{S} x_{S}^{'} + k_{S} x_{S} = - f_{SR x}^{P m} \\ m_{S} (y_{S}^{″} + 2 ω_{C} x_{S}^{'} - {ω_{C}}^{2} y_{S}) + \sum_{i = 1}^{N} c_{SR i} {δ^{'}}_{SR i}^{P m} \sin φ_{S i} \cos α_{S} + \sum_{i = 1}^{N} k_{SR i}^{P m} δ_{SR i}^{P m} \sin φ_{S i} \cos α_{S} + c_{S} y_{S}^{'} + k_{S} y_{S} = 0 \\ \frac{I_{S}}{{r_{SP}}^{2}} u_{S}^{″} + \sum_{i = 1}^{N} c_{SR i} {δ^{'}}_{SR i}^{P m} \cos α_{S} + \sum_{i = 1}^{N} k_{SR i}^{P m} δ_{SR i}^{P m} \cos α_{S} + \frac{c_{S t}}{r_{SP}^{2}} u_{S}^{'} + \frac{k_{S t}}{r_{SP}^{2}} u_{S} = \frac{T_{i n}}{r_{SP}} \end{cases}

(39)

{\begin{cases} m_{r} (x_{r}^{″} - 2 ω_{c} y_{r}^{'} - {ω_{c}}^{2} x_{r}) + \sum_{i = 1}^{N} c_{rR i} δ_{rR i}^{'} \sin φ_{r i} + \sum_{i = 1}^{N} k_{rR i} δ_{rR i} \sin φ_{r i} + c_{r} x_{r}^{'} + k_{r} x_{r} = 0 \\ m_{r} (y_{r}^{″} + 2 ω_{c} x_{r}^{'} - {ω_{c}}^{2} y_{r}) + \sum_{i = 1}^{N} c_{rR i} δ_{rR i}^{'} \cos φ_{r i} + \sum_{i = 1}^{N} k_{rR i} δ_{rR i} \cos φ_{r i} + c_{r} y_{r}^{'} + k_{r} y_{r} = 0 \\ \frac{I_{r}}{r_{r}^{2}} u_{r}^{″} + \sum_{i = 1}^{N} c_{rR i} δ_{rR i}^{'} + \sum_{i = 1}^{N} k_{rR i} δ_{rR i} + \frac{c_{r t}}{r_{r}^{2}} u_{r}^{'} + \frac{k_{r t}}{r_{r}^{2}} u_{r} = 0 \end{cases}

(40)

{\begin{cases} m_{N} (x_{N}^{″} - 2 ω_{c} y_{N}^{'} - {ω_{c}}^{2} x_{N}) + \sum_{i = 1}^{N} c_{NR i} {δ^{'}}_{NR i}^{P m} \cos φ_{N i} \cos α_{N} + \sum_{i = 1}^{N} k_{NR i}^{P m} δ_{NR i}^{P m} \cos φ_{N i} \cos α_{N} + c_{N} x_{N}^{'} + k_{N} x_{N} = 0 \\ m_{N} (y_{N}^{″} + 2 ω_{c} x_{N}^{'} - {ω_{c}}^{2} y_{N}) + \sum_{i = 1}^{N} c_{NR i} {δ^{'}}_{NR i}^{P m} \sin φ_{N i} \cos α_{N} + \sum_{i = 1}^{N} k_{NR i}^{P m} δ_{NR i}^{P m} \sin φ_{N i} \cos α_{N} + c_{N} y_{N}^{'} + k_{N} y_{N} = 0 \\ \frac{I_{N}}{r_{N}^{2}} u_{N}^{″} + \sum_{i = 1}^{N} c_{NR i} {δ^{'}}_{NR i}^{P m} \cos α_{N} + \sum_{i = 1}^{N} k_{NR i}^{P m} δ_{NR i}^{P m} \cos α_{N} + \frac{c_{N t}}{r_{N}^{2}} u_{N}^{'} + \frac{k_{N t}}{r_{N}^{2}} u_{N} = 0 \end{cases}

(41)

{\begin{cases} \frac{I_{R i}}{r_{R}^{2}} u_{R i}^{″} - c_{SR} {δ^{'}}_{SR i}^{P m} \cos α_{S} - k_{SR i}^{P m} δ_{SR i}^{P m} \cos α_{SR} - c_{NR} {δ^{'}}_{NR i}^{P m} \cos α_{N} \\ - k_{NR i}^{P m} δ_{NR i}^{P m} \cos α_{N} - c_{rR} δ_{rR i}^{'} - k_{rR} δ_{rR i} + \frac{c_{R t}}{r_{R}^{2}} u_{R i}^{'} + \frac{k_{R t}}{r_{R}^{2}} u_{R i} = f_{SR y}^{P m} \end{cases} \begin{array}{l} , & i = 1 \sim N \end{array}

(42)

\frac{I_{c}}{r_{c}^{2}} u_{c}^{″} + \frac{c_{c t}}{r_{c}^{2}} u_{c}^{'} + \frac{k_{c t}}{r_{c}^{2}} u_{c} = 0

(43)

In the above equations,

m

represents the mass, and

I

denotes the equivalent rotational inertia, respectively.

k

and

c

are the support stiffness and damping, while

k_{t}

and

c_{t}

represent the thread stiffness and damping. N is the number of rollers.

α

is the angle between the normal contact force and the axial section, and

φ

is the phase angle of the roller.

T_{i n}

denotes the input torque of the screw, and

r_{SP}

represents the contact radius of the screw. The subscripts S, R, N, r, and C denote the screw, roller, nut, internal ring gear, and carrier, respectively. The subscripts x and y represent radial and tangential directions, respectively, and u denotes the torsional direction.

The formulations of the displacement vectors, damping matrices, time-dependent meshing stiffness of the gear pair, gyroscopic matrix, and radial stiffness matrix in the PRSM dynamic model can be found in Xu et al. (2026) and are not repeated here for brevity.

4.2. Numerical example

The roller pitch diameter modification is mainly applied to the threads at both ends of the roller, while the threads in the middle remain unmodified.

The roller threads are divided into five segments, and the modification amount for each segment is defined as

P m_{i} = {\begin{cases} P m_{\max} & , & | i - \frac{z + 1}{2} | \geq \frac{z}{3} \\ P m_{\max} \frac{| i - \frac{z + 1}{2} | - \frac{z}{5}}{\frac{z}{3} - \frac{z}{5}} & , & \frac{z}{5} < | i - \frac{z + 1}{2} | < \frac{z}{3} \\ 0 & , & | i - \frac{z + 1}{2} | \leq \frac{z}{5} \end{cases}

(44)

Where

z

is the total number of roller threads, and

P m_{\max}

is the maximum modification amount of the roller pitch diameter. By varying

z

, the spatial extent of the modified threads along the axial direction can be adjusted, while

P m_{\max}

controls the magnitude of the modification at the thread level. Together, these two parameters enable flexible control of both the range and magnitude of the thread modification.

The PRSM parameters used in this study are listed in Table 1.

Table 1.

PRSM parameters.

Parameters part	Pitch diameter (mm)	Pitch (mm)	Number of starts	Half thread thickness (mm)	Thread flank angle (°)
Screw	18	1	5	0.227	45
Roller	6	1	1	0.242	45
Nut	30	1	5	0.249	45

The maximum modification amount $P m_{\max}$ is set to 0.001, 0.005, 0.01, 0.02, and 0.03, respectively. Substituting the parameters in Table 1 into Equations (4)–(12), the thread clearances are obtained, as observed in Figure 5.

Figure 5.

Effect of roller pitch diameter modification on thread pair clearance. (a) Clearance of roller–screw thread pair (b) Clearance of roller–nut thread pair.

Figure 5 shows the influence of roller pitch diameter modification on the thread clearance. It can be observed that the clearance distribution is consistent with the modification distribution: the clearance is largest at both ends of the roller threads, while the middle threads maintain the same clearance as the unmodified case. This distribution delays the contact of the end threads, thereby reducing the contact force in the thread pairs.

Since this study focuses on the vibration suppression effect of roller pitch diameter modification, the non-uniform load distribution among rollers is neglected, and the nut load is assumed to be evenly distributed. A nut load of 50 kN and 10 rollers are considered. By substituting the modified clearance, contact angle, and helix angle into Equations (13)–(24), the load distributions of the thread pairs under different modification amounts are obtained, as shown in Figure 6.

Figure 6.

Effect of roller pitch diameter modification on thread pair load distribution. (a) Load distribution of roller–screw thread pair (b) Load distribution of roller–nut thread pair.

According to Figure 6, roller pitch diameter modification effectively reduces the contact force at the end threads of both the roller–screw and roller–nut pairs, with a more pronounced reduction observed on the roller–screw side.

As the modification increases, the maximum contact force at the middle threads of the roller–screw pair exceeds the unmodified case at 0.01 mm, while a similar trend occurs for the roller–nut pair at 0.006 mm. However, since the roller–screw contact force is significantly higher, it is taken as the governing criterion. Therefore, the maximum allowable modification is selected as 0.01 mm.

After obtaining the contact angles and load distributions under roller pitch diameter modification and the grease viscosity $η$ is set to 200 Pa·s, the friction coefficient of the roller–screw thread pair under different modification amounts is calculated using Equations (25)–(37), as shown in Figure 7.

Figure 7.

Effect of roller pitch diameter modification on the friction coefficient of the roller–screw thread pair.

As shown in Figure 7, the friction coefficient of the roller–screw thread pair decreases after roller pitch diameter modification. This is because the contact point shifts toward the thread crest, increasing contact stress and lubricant film pressure, while the sliding velocity between the roller and screw is reduced. The combined effect leads to a lower bearing characteristic number and reduced friction coefficient, indicating that the PRSM operates in the hydrodynamic lubrication regime under the present conditions.

By incorporating the sliding friction force of the roller–screw thread pair into the PRSM dynamic model, the vibration accelerations of various components after roller pitch diameter modification are obtained, as shown in Figures 8 –11.

Figure 8.

Effect of roller pitch diameter modification on the vibration acceleration of screw. (a) Screw x vibration acceleration (b) Screw y vibration acceleration (c) Screw u vibration acceleration.

Figure 9.

Effect of roller pitch diameter modification on the vibration acceleration of nut. (a) Nut x vibration acceleration (b) Nut y vibration acceleration (c) Nut u vibration acceleration.

Figure 10.

Effect of roller pitch diameter modification on the vibration acceleration of roller. (a) Roller x vibration acceleration (b) Roller y vibration acceleration (c) Roller u vibration acceleration.

Figure 11.

Effect of roller pitch diameter modification on the vibration acceleration of ring gear. (a) Internal ring gear x vibration acceleration (b) Internal ring gear y vibration acceleration (c) Internal ring gear u vibration acceleration.

As shown in Figures 8 –11, due to the combined excitation of thread contact and gear meshing, the roller exhibits the highest vibration acceleration. In addition, in this rotational transmission system, the torsional vibration is significantly greater than the tangential and radial components. After applying roller pitch diameter modification, the vibration amplitudes of all components are reduced, among which the reductions for the screw (tangential and radial directions) and the roller are the most significant. This is mainly attributed to the increase in thread clearance and the reduction in contact deformation after modification, which decrease the contact force and consequently reduce the sliding friction coefficient and friction force. When the modification amount is 0.01 mm, the tangential acceleration of the roller decreases from 15.7 m/s² to 8.01 m/s², corresponding to a reduction of 48.98%, indicating that the proposed method achieves an effective vibration suppression effect.

4.3. Model validation

Since accurate reference results for load distribution of modified threads are not available, a model degradation approach is adopted by setting the roller pitch diameter modification to zero and comparing the results with those reported by Zhang et al. (2016). Under identical structural parameters and operating conditions, the load distributions of the unmodified roller–screw and roller–nut pairs are calculated, as shown in Figure 12.

Figure 12.

Comparison of load distribution calculation results.

As shown in Figure 12, the calculated load distributions agree well with published results in both trend and magnitude, with maximum relative errors of 2.45% for the roller–screw pair and 1.27% for the roller–nut pair. The slightly larger deviation in the roller–screw case is mainly due to the inclusion of the contact misalignment angle, which was neglected in the reference, as well as differences in stiffness modeling. Previous studies indicate that Hertzian contact stiffness accounts for about 97% of the total thread pair stiffness in PRSM; thus, only Hertzian contact stiffness is considered in this study, while bulk stiffness is neglected.

To further validate the proposed dynamic model considering friction, experimental measurements of the nut vibration acceleration are conducted. Due to the high cost and time required for roller modification, a degradation experimental validation is performed. The measured vibration acceleration of the unmodified nut is evaluated against the theoretical results to verify the accuracy of the model. The experimental setup for measuring vibration acceleration is shown in Figure 13.

Figure 13.

Experimental setup for vibration acceleration of the PRSM.

As shown in Figure 13, the test rig consists of a motor, loading system, transmission system, a well-lubricated test specimen, and vibration acceleration sensors. The accelerometer is mounted on the surface of the nut to measure its radial vibration acceleration.

To ensure repeatability, the vibration acceleration of the nut is measured twice. The screw is operated at a rotational speed of 500 r/min, and the sampling frequency of the acceleration sensor is 200 Hz. The two sets of measured radial vibration acceleration results are shown in Figure 14(a). The corresponding theoretical results are presented in Figure 14(b).

Figure 14.

Nut radial vibration acceleration. (a) Experimental results (b) Calculated results.

As shown in Figure 14(a), a sudden increase in vibration acceleration occurs at startup due to the thread clearance. During the steady-state stage (0.1 s–0.4 s), the root mean square (RMS) value of the measured nut vibration acceleration is 0.331 m/s². The theoretically predicted RMS value is 0.308 m/s², resulting in a relative error of 6.95%. The proposed model is validated by the good agreement between experimental and theoretical results.

5. Conclusions

A coupled dynamic model of the PRSM considering roller pitch diameter modification, contact mechanics, and lubrication–friction effects is developed to reveal the vibration suppression mechanism. The results show that pitch diameter modification improves load distribution and reduces friction-induced excitation, leading to significant vibration attenuation. The main conclusions are summarized as follows.

(1) The modification changes the contact geometry, shifts the contact region toward the thread crest, and redistributes load by reducing edge loading and enhancing central load capacity.

(2) The coupled contact–lubrication–friction effect reduces the friction coefficient and sliding friction forces, thereby weakening friction-induced excitation.

(3) The optimal modification magnitude is approximately 0.01 mm, achieving a maximum vibration reduction of 48.98%, while the numerical–experimental deviation is within 6.95%.

(4) The proposed model provides guidance for vibration-oriented design of PRSM, and future work will consider thermal–wear effects and system-level coupling under more complex conditions.

Footnotes

ORCID iD

Qianjin Xu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is supported by the National Key R&D Program of China (Grant No. 2024YFB4708700).

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.

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