Dynamic characteristics analysis is very important for the design and application of compliant mechanisms, especially for dynamic and control performance in high-speed applications. Although pseudo-rigid-body (PRB) models have been extensively studied for kinetostatic analysis, their accuracy for dynamic analysis is relatively less evaluated. In this paper, we first evaluate the accuracy of the PRB model by comparing against the continuum model using dynamic simulations. We then investigate the effect of mass distribution on dynamics of PRB model for compliant parallel-guided mechanisms. We show that when the beam mass is larger than 10% of the motion stage, the error is significant. We then propose a new PRB model with a corrected mass distribution coefficient which significantly reduces the error of the PRB model. And the dynamic responses are also analyzed according to the corrected mass distribution coefficient. At last, a compliant double parallel-guiding mechanism is used as a case study for validation of the new PRB model for dynamics of compliant mechanisms.

## Introduction

Comparing with traditional rigid-body mechanisms, compliant mechanisms for flexures transfer motion, energy, and force depending upon the deflection of their flexible structure. This characteristics offer compliant mechanisms prominent advantages, such as simple/compact structure, high precision, low friction, reduced assembly, and manufacturability [1]. Compliant mechanisms are widely used in high-precision instrumentation, harsh environments, aerospace, and micro-electromechanical systems [27]. In recent year, many researchers studied the configuration design and optimization, and kinetostatic (kinematic and static) analysis of compliant mechanisms [812]. The finite element analysis and the pseudo-rigid-body (PRB) model [13,14] are the two most often used methods for kinetostatic analysis of compliant mechanisms with larger deflections. Recently, the PRB 3R model proposed in Ref. [15] improved the accuracy of the conventional PRB 1R model for a cantilever beam with a general tip load.

However, research on the dynamic analysis of complaint mechanism is relatively less explored [16]. However, dynamic analysis is very important for compliant mechanisms, especially when they are in high speed tasks. Other than the finite element analysis method, the assumed mode method, the Nodal coordinate method, have been proposed for dynamic behavior analysis of compliant beams. Lobontiu et al. extensively studied the flexure hinges [17] and established the dynamic model [18] including the inertia and damping properties of flexure hinges. Rösner et al. [19] presented a reduced-order model by using Krylov subspace reduction for system matrices. Based on a PRB model, Lyon et al. [20,21] and Boyle et al. [22] discussed the first-order frequency of compliant parallel-guiding mechanisms and developed a dynamic model for a compliant constant-force compression mechanism. Pei et al. [23] proposed a new method to establish PRB models for beam-based compliant mechanism, where the behavior of parallel linear spring stage beams are imitated by a rigid bar with two pin-joints. Li and Kota [24] studied dynamics analysis of compliant mechanisms including natural frequencies, modes, dynamic response, and frequency characteristics, by using the finite element method. Zhao et al. [25] developed a dynamic model based on Lagrange equation for a compliant linear-motion mechanism, then investigated the effect of material properties, shape parameters, and geometric parameters on natural frequency for providing quantitative insight to designers. Yu et al. [26] developed a dynamic model of compliant mechanisms based on traditional 1R PRB model. Li et al. [27] studied the dynamic analysis for the curved-type compliant mechanism based on a multirevolute joints PRB model established based on stiffness and curvature characteristics of the mechanism.

Given these foundational work, the accuracy of using PRB models to dynamic analysis of compliant mechanisms is not systematically evaluated. In this work, we will study how the mass distribution in PRB models affects the accuracy of dynamic analysis. Because the vibration of compliant mechanism is nonlinear large deformation in low frequency, the first-order natural frequency in main motion direction is considered as the inherent quality for the research of the mechanism dynamic response based on the PRB model and the continuum model. Based on this study, we further propose to a new mass distribution coefficient to correct the PRB model for a more accurate prediction of first-order natural frequency.

The organization of the paper is described in the following: First, the effect of the mass distribution on PRB model by comparing first-order natural frequency against the continuum model is studied, and the error distribution is also discussed in Sec. 2. Second, a nondimensional model for first-order natural frequency is presented to eliminate the influence of structural parameters. The relationship between mass distribution coefficient and mass ratio is developed in Sec. 3. The compliant double parallel-guiding mechanism is taken as a case study for validation of the mass distribution coefficient in Sec. 4. Finally, the conclusion is presented in Sec. 5.

## Motivation: The Effect of Mass Distribution on the Accuracy of Pseudo-Rigid-Body Model in Dynamic Analysis

### The Accuracy Assessment of Pseudo-Rigid-Body Model for Dynamic Analysis.

It has been extensively studied that PRB model for cantilever beams works very well for kinetostatic analysis [28]. And a dynamic modeling of a two-dimensional compliant link has been studied for safety evaluation in human–robot interactions [29]. However, their accuracy in dynamic analysis is much less assessed. Here, let us take the compliant parallel guide mechanism as an example. We compare the dynamic simulation of this mechanism using the PRB model and the continuum model. The PRB model is simulated in MSC Software's adams program, and the simulation of the continuum model is conducted in abaqus fea program. Suppose the beams have a length L and a uniform cross section with the second moment of area I. The elastic modulus of the material is E.

Figure 1(a) shows the main parameters of the compliant parallel guide mechanism, on which a horizontal force $F$ is exerted on the motion stage during dynamic analysis. The main structural parameters of the fully compliant planar stage are given in Table 1. As shown in Fig. 1(b), the beams and the motion stage in the compliant parallel guide mechanism are modeled by using Beam 21 elements in abaqus. The PRB model shown in Fig. 1(c) (implemented in adams software) is equivalent to a parallel four-bar linkage with four torsion springs, i.e., each beam is composed of three rigid links connected by two pin joints with torsion springs. The location of the two torsion springs is determined by the characteristic radius factor $γ=0.85$ [1]. To simplify the analysis, the two torsion springs are symmetrically distributed on th3e beam. And each torsion spring has a characteristic stiffness factor as $KΘ=2.67$. Note this means that the true torsion spring constant is $KΘ(EI/L)$.

Then, the first-order natural frequency of these two models will be obtained from dynamic simulations. Let $mo$ be the mass of motion stage, $mb$ be the mass of body links, and $χ=mo/mb$ be their ratio. Let us define the horizontal direction as the x direction. When the same horizontal force $F$ is applied to the stage in the two models, the relationship between the mass ratio of stage and body links and the first-order natural frequency according to the PRB model and continuum model are plotted as the continuous line in Fig. 2.

As shown in Fig. 2, the first-order natural frequency of continuum model approaches to a constant value when the mass ratio increases to a certain value. To validate the simulation, let $Kx$ be the total stiffness in the x direction, and $dx$ be the displacement in x direction. Based on the small deflection Euler beam theory, $Kx$ can be obtained as
$Kx=Fdx=24EIL3$
(1)
According to the structural parameters, the total stiffness can be given as $Kx$. When the mass of motion stage is far greater than the mass of the body links, the mass of beams can be neglected, and the frequency based on the equation of frequency is obtained as
$f=12πkm$
(2)

Thus, the frequency of x direction can be calculated as $f=83.163$ Hz, which corresponds to the case when the beam mass can be neglected. When the mass ratio $χ>10$, comparing the calculated frequency from Eq. (2) with the abaqus simulation result using the continuum model, the deviation is less than 0.151 Hz, i.e., an error of 0.18%.

Also from Fig. 2, we can see that the frequency by the continuum model is very close to that of the PRB model when the mass ratio is very large, e.g., $χ>10$. However, when the mass ratio is small, the frequency values from these two models are clearly different. More specially, the frequency value based on the PRB model is less than that obtained from the continuum model, especially when the mass of the stage is far less than that of the beams. The result shows that when the mass ratio is small, the error of the PRB model will be large if we employ the kinematic characteristic radius factor $γ$ for mass distribution in the dynamic modeling. In other words, the effect of mass distribution on the dynamic characteristic analysis of PRB model cannot be neglected for this case.

### The Error Distribution Analysis.

Let $fPRB$ be the actual frequency of the PRB model, and $fc$ be the frequency of the continuum model, thus, the error can be described as
$e=fPRB−fcfc=Δffc%$.

The relationship between e and $χ$ can be plotted in Fig. 3. As we can see, the error value is obviously larger when the mass of stage is far less than that of beams. The error value is quickly reduced with the increase of the mass ratio. When the mass ratio value is more than 8, the error approaches to zero. Therefore, when the mass ratio value is $χ ∈[0 10]$, it is essential to adjust the mass distribution in the PRB model. The problem can be solved through a proposed correction coefficient of mass distribution, which is discussed in Sec. 3.

## Method: The Mass Distribution Coefficient

### The Nondimensional Model for First-Order Natural Frequency.

It is well known that the first-order natural frequency of a compliant mechanism is a function of structural and material property parameters, including mass, length, second moment of area, and elastic modulus of the material. To derive a more general PRB model for dynamic analysis, a nondimensional model is preferred in order to eliminate the effect of structural parameters. Based on Eqs. (1) and (2), the frequency can be approximated as
$f=12π24EImL3$
(3)
Let $f¯$ be the nondimensional form of frequency. $f¯$ can be calculated as
$f¯=fmL3EI$
(4)

where, m in Eqs. (3) and (4) is the total mass of the mechanism.

Let $f¯PRB$ be the frequency nondimensional form of PRB model, and $f¯c$ be the frequency nondimensional form of the continuum model. When the mass ratio $χ$ is greater than 8, the frequency of continuum model $f¯c$ is approximating the calculation value, that is fc = 83.16 Hz.

In the rest of the paper, we are going to study the nondimensional frequency $f¯PRB$ of the PRB model and $f¯c$ of the continuum model. Also the error is also redefined as
$e¯=f¯PRB−f¯cf¯c=Δf¯f¯c%$.

### The Mass Distribution Coefficient η.

Let $η$ be the mass distribution coefficient and the structural parameters be constant. The mass of the PRB link is $ηmb$ ($η∈[0 1]$), and the mass of stage is obtained as $mo′=m0+(1−η)mb$. These parameters are implemented in the multibody dynamic simulation program adams. By equalling the first-order natural frequency of the PRB model with that of the continuum model under same simulation conditions, we can obtain the mass distribution coefficient $η$ for a given mass ratio $χ$, as plotted dots in Fig. 4. As we can see, $η$ approaches to zero when the mass ratio $χ$ increases to a certain value, but changes dramatically when the mass ratio is small.

Since χ is less than 10 for most practical applications, we zoom in the plot and show it in Fig. 5. As we can see, the correction mass distribution coefficient gradually decreases with the increase of mass ratio. When the mass of stage is far less than that of the beams, the value of correction mass distribution coefficient approaches 1. When the mass of stage increases with the mass ratio closing to 10, the coefficient value is close to 0.4. The relationship between $η$ and $χ$ can be fit by using three piecewise quadratic functions as

$η={0.073χ2−0.198χ+1.005 0.1<χ≤0.80.012χ2−0.129χ+0.986 0.8<χ≤3 0.003χ2−0.078χ+0.911 3<χ≤10$
(5)

Then, the mass distribution coefficient η is applied to dynamic simulation of the PRB model in adams. Meanwhile, we keep the kinematic parameters of PRB model the same, i.e., the characteristic radius factor γ = 0.85 and spring constant Kθ = 2.76EI/L. See the PRB model with all parameters for dynamic analysis in Fig. 6.

The frequency comparison among the PRB model with and without the correction mass coefficient and the continuum model in abaqus is shown in Fig. 7. Using the continuum model as the base line, the corresponding errors of the PRB models are also shown in Fig. 7. As we can see, the frequencies of the PRB model with mass correction coefficient match very well with the continuum model, with the error being less than 2%. Therefore, the mass distribution coefficient is verified to be effective for the dynamics analysis of the PRB model.

### Validation of the Modified Pseudo-Rigid-Body Dynamics Response According to Step Excitation.

The dynamic response of the modified PRB according to step excitation is also analyzed in this section. Suppose the step excitation F = 2000 N($t∈[00.05]$ s) and the simulation duration is $t∈[00.1]$ s. The parameters of the modified PRB model are fixed as the above analysis. The according dynamic responses such as displacement, velocity, acceleration are illustrated as the following.

The dynamic responses comparison between the PRB model with and without the application of correction mass coefficient and continuum model for large deformation analysis are shown in Figs. 810. As we can see, displacement deformation range as $d∈[041] mm$, which is accounted as 17% of the body link length. Hence, the dynamic responses are considered as the large deformation analysis of compliant mechanism. The distinction of dynamic responses between the continuum model and the PRB model before correction is obvious. The dynamic responses of modified PRB closely approximate to the dynamic responses of the continuum model, especially in displacement and velocity analysis. And the average errors of displacement, velocity, and acceleration are between 3% and 8% correspondingly. Therefore, the mass distribution coefficient is also verified to be effective for the dynamics responses of the PRB model to the large deformation analysis.

## Case Study: A Double Parallel-Guiding Mechanism

Double parallel-guiding mechanism is a typical compliant mechanism application, which has been extensively used, such as atomic force microscopy, micro-assembly, and data storage. In this section, the dynamic characteristic of double parallel-guiding mechanism is analyzed. Since the planar compliant mechanisms have been investigated to obtain the mass distribution coefficient above, the double parallel-guiding mechanism is analyzed as a case study for validation of the new dynamic PRB model (Fig. 11).

Let the mass ratio of first stage be $χ1=Mo1/Mb1$, and the mass ratio of second stage be $χ2=Mo2/Mb2$. According to Eq. (5), the mass distribution coefficients can be calculated correspondingly. Suppose the mass ratio of first stage be a constant, the frequency comparison between the PRB model and continuum model of second stage are illustrated as following.

Figure 12 shows a series of frequency comparison of double parallel-guiding compliant mechanism for the second stage during dynamic analysis with PRB model and continuum model. When the mass distribution coefficient is applied for the dynamic research of the PRB model, the frequency matches that of the continuum model very well, however, the difference is large without the application of mass distribution coefficient. The frequency error analysis between the PRB model with/without the corrected mass distribution coefficient comparing with that of the continuum model is shown in Fig. 13. The frequency errors according to different mass ratios decrease obviously, where the absolute value of the error is less than 2%. At the same time, the correction effect is more obvious when the mass of motion stage is far less than that of the beams. Therefore, it is a validation that the mass distribution coefficient is necessary and effective for dynamic characteristics analysis of PRB model when the mass ratio is under 10.

The dynamic response of the modified PRB according to step excitation is analyzed as following. Suppose the step excitation F = 2000 N($t∈[00.05]$ s) and the simulation duration is $t∈[00.1]$ s. The parameters of the modified PRB model of compliant double parallel-guiding mechanism are fixed as the above analysis.

The according dynamic responses between the PRB model with and without the application of correction mass coefficient and continuum model, such as displacement, velocity, and acceleration are illustrated in Figs. 1416. As we can see, the displacement deformation of second stage range as $d∈[0100] mm$, which is considered as large deformation. The dynamic responses of modified PRB match very well with the dynamic responses of the continuum model. The effect of the corrected mass distribution coefficient is not only resultful in the amplitude, but also in the phase, especially in the forced vibration duration. The average errors of displacement, velocity, and acceleration are around 9%. Therefore, the mass distribution coefficient is verified to be effective for the dynamics responses of the PRB model of compliant double parallel-guiding mechanism to the large deformation analysis.

## Conclusions

In this paper, the effect of mass distribution on dynamics of the PRB model for compliant parallel-guided mechanism is studied. By comparing the natural frequencies of the PRB model and the continuum model, it is validated that the mass distribution does have an effect on the accuracy of the dynamics analysis of the PRB model. The results show that it is essential to adjust the mass distribution for the PRB model if the mass ratio of the motion stage over beams themselves is less than 10. A mass distribution coefficient is proposed to correct the PRB model. The dynamic responses are further analyzed. The effect of the corrected mass distribution coefficient is resultful in the dynamic response analysis including displacement, velocity, and acceleration. Finally, a double parallel-guided mechanism is employed as a case study for the verification, and the results show that the mass distribution coefficient is effective to improve the accuracy of the PRB model for dynamic analysis. The method presented in this paper is not only verified to be effective for the compliant parallel-guided mechanisms, but also to provide a general procedure how to assess and improve the accuracy of PRB models for dynamic analysis of compliant mechanisms.

## Acknowledgment

The first author would like to acknowledge the support by the Natural Science Fund Committee of China (Grant No. 51305125), Natural Science Foundation Committee of Hebei Province (Grant Nos. E2017204037 and E2013204110), and Foundation of Hebei Province College Technology Committee (Grant No. YQ2013007). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies.

## Funding Data

• National Science Foundation (Grant No. CMMI-1637656).

## Nomenclature

• $e$ =

error of first-order natural frequency between the PRB model and the continuum model

•
• $e¯$ =

error of first-order natural frequency in main motion direction in the nondimensional form between the PRB model and the continuum model

•
• $fc$ =

first-order natural frequency in main motion direction of the continuum model

•
• $fPRB$ =

first-order natural frequency in main motion direction of the pseudo-rigid-body model

•
• $f¯c$ =

nondimensional form of first-order natural frequency of the continuum model

•
• $f¯PRB$ =

nondimensional form of first-order natural frequency of the PRB model

•
• $η$ =

mass distribution coefficient of the PRB model for dynamics analysis

•
• $χ$ =

mass ratio of the motion stage over the beams

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