# Vibration Characteristics Analysis of Convalescent-Wheelchair Robots Equipped with Dynamic Absorbers

Authors: Song Wang; Leilei Zhao; Yanzhu Hu; Fuxing Yang
Journal: Shock and Vibration
 (2018)

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## Abstract

To provide a theoretical guidance for the vibration isolation design, the vibration responses characteristics for the convalescent-wheelchair robot with DA (dynamic absorber) undergoing the random road were revealed. Firstly, the vibration model of the convalescent-wheelchair system with DA was created. The frequency response functions of the road excitation velocity to the convalescent acceleration, the wheelchair body acceleration, and the tire dynamic deflection were deduced. Then, the numerical calculation method of the PSD (power spectral density) and the RMS (root mean square) responses were proposed. Thirdly, the vibration isolation performances of the wheelchair robot with DA and without DA were compared. Finally, the sensitivity analysis of the vibration responses to the mass ratios, the damping ratios, and the natural frequencies was carried out to reveal the effects of the parameters on the vibration responses. The results show that the DA can partly suppress the vibration of the convalescent and the wheelchair body, especially in the resonance area of the wheelchair body. However, the DA cannot successfully improve the tire contact behavior.


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## Body

## 1. Introduction

Convalescent-wheelchair robots belong to the rehabilitation medical machines [1]. They are important travel tools for convalescents. With the development of the society and the improvement of the living standard, the demand for them is increasing, for example, their ride comfort, their operational reliability, and so on [2]. When a wheelchair robot runs on the random road, its vibration caused by the road irregularity is inevitable. The harmful vibration produces adverse effects for the wheelchair and its users [3–5].In recent years, many scholars have studied the related vibration problems of wheelchair robots. The damping characteristics of the wheelchair cushion were studied on the basis of the standard ISO 16840-2:2007 in [6]. A new dynamic model for the wheelchair coupled with the user was developed and validated in [7]. The vibration transmission characteristics of wheelchairs and their effects on patients were analyzed in [8]. Hikmawan and Nugraha [9] simulated the vibration of a wheelchair under the sinusoidal excitation based on a 6 degree of freedom model of the wheelchair user. Wu et al. [10] proposed a new method of evaluating the ride comfort of wheelchair robots based on the Q-learning algorithm, which has a certain reference value for the vibration evaluation of wheelchair robots. Wang et al. [11] established a mathematical model of the wheelchair cushion comfortableness, which provides a reference for the wheelchair design. Miyawaki and Takahashi [12] designed a new wheelchair and analyzed the whole-body vibration of the user sitting on the wheelchair. Su et al. [13] designed a magnetic suspension vibrator to improve the vibration attenuation performance of electric wheelchairs. Ababou et al. [14] developed a test bench to provide harmful vibrations for wheelchairs and the test bench can be applied to analyze the wheelchair transmissibility. Kundu et al. [15] designed a four wheel-driven wheelchair and investigated its sliding and vibration characteristics. Garcia-Mendenz et al. [16] estimated the whole-body vibration of the wheelchair users in their communities. Requejo et al. [17] studied the effect of hand-rim wheelchairs equipped with rear suspension on seat forces. The influences of the user anatomy and wheelchair cushion type on the tissue deformation were analyzed in [18]. The aforementioned studies provide useful references for improving the vibration isolation performances of wheelchair robots. However, there are few studies on the comprehensive performance of wheelchair robots undergoing the random road.Most of convalescent-wheelchair robots do not have suspension systems due to the limited installation space. Their users’ vibration caused by the road excitation is mainly reduced by the tires and the cushion. The DA is a vibration reducing device connected to the vibration system by elastic and damping elements [19, 20]. It occupies a small space and provides a cheap and effective way to attenuate the wheelchair robots vibration.In the previous work [21], the impact responses and parameters sensitivity of electric wheelchairs were analyzed. In this paper, to provide a theoretical support for the design of convalescent-wheelchair robots, the vibration responses characteristics for the convalescent-wheelchair robot with DA undergoing the random road were revealed.


## 2. The Vibration Model of the Convalescent-Wheelchair System with DA

In the initial design stage of convalescent-wheelchair robots, the parameters needed to establish accurate vibration models are often unknown. Therefore, to easily guide their design in theory, the convalescent-wheelchair system should be simplified.In this paper, the cushion and tires are equivalent to a spring-damper, respectively. Moreover, the convalescent is regarded as a rigid body mass. The vibration model of the convalescent-wheelchair system with DA is created, as shown in Figure1 [21]. In Figure 1, m2, m1, and m0 represent the convalescent mass, the wheelchair body mass, and the DA mass, respectively; K2, K1, and K0 represent the cushion stiffness, the tire stiffness, and the DA stiffness, respectively; C2, C1, and C0 represent the cushion damping, the tire damping, and the DA damping, respectively; z2, z1, and z0 represent the convalescent vertical displacement, the wheelchair body vertical displacement, and the DA vertical displacement, respectively; and q represents the road displacement input. To facilitate the comparative analysis, the vibration model of the convalescent-wheelchair system without DA is also given in Figure 2.Figure 1
The vibration model of the convalescent-wheelchair system with DA.Figure 2
The vibration model of the convalescent-wheelchair system without DA.According to Newton’s second law, the motion equation for the vibration model of the convalescent-wheelchair system without DA can be expressed as(1)m2z¨2+C2z˙2−z˙1+K2z2−z1=0,m1z¨1+C2z˙1−z˙2+K2z1−z2+C1z˙1−q˙+K1z1−q=0.Analogously, the motion equation for the vibration model of the convalescent-wheelchair system with DA can be expressed as(2)m2z¨2+C2z˙2−z˙1+K2z2−z1=0,m0z¨0+C0z˙0−z˙1+K0z0−z1=0,m1z¨1+C2z˙1−z˙2+K2z1−z2+C0z˙1−z˙0+K0z1−z0+C1z˙1−q˙+K1z1−q=0.By comparing Equation (1) with Equation (2), when m0 equals zero, Equation (2) changes into Equation (1).To easily facilitate the wheelchair robots design, the following auxiliary variables are introduced:(3)r0=m0m1,r2=m2m1,f2=12πK2m2,f1=12πK1m1,f0=12πK0m0,ξ2=C22m2K2,ξ1=C12m1K1,ξ0=C02m0K0,where, r0 and r2 are the mass ratios; f0 is defined as the DA natural frequency; f1 is defined as the wheelchair body natural frequency; f2 is defined as the convalescent natural frequency; and ξ0, ξ1, and ξ2 are the damping ratios of the DA system, the tire system, and the cushion system.Based on the introduced variables, Equation (2) can be expressed as(4)m2z¨2+2m2ξ2ω2z˙2−z˙1+m2ω22z2−z1=0,m0z¨0+2m0ξ0ω0z˙0−z˙1+m0ω02z0−z1=0,m1z¨1+2m2ξ2ω2z˙1−z˙2+m2ω22z1−z2+2m0ξ0ω0z˙1−z˙0+m0ω02z1−z0+2m1ξ1ω1z˙1−q˙+m1ω12z1−q=0,where, ω0=2πf0, ω1=2πf1, and ω2=2πf2.Equation (4) can be simplified as(5)z¨2+2ξ2ω2z˙2−z˙1+ω22z2−z1=0,z¨0+2ξ0ω0z˙0−z˙1+ω02z0−z1=0,z¨1+2r2ξ2ω2z˙1−z˙2+r2ω22z1−z2+2r0ξ0ω0z˙1−z˙0+r0ω02z1−z0+2ξ1ω1z˙1−q˙+ω12z1−q=0.


## 3. System Amplitude Frequency Characteristics

For the free vibration of the convalescent-wheelchair system with DA, each mass vibrates at the same circular frequencyω and the same phase angle φ. Moreover, the corresponding amplitudes are z00, z10, and z20, respectively. Thus, the solution of the free vibration equation can be expressed as(6)z0=z00ejωt+φ,z1=z10ejωt+φ,z2=z20ejωt+φ.Substituting Equation (6) to Equation (5) produces(7)z2−ω2+2ξ2ω2ωj+ω22=z12ξ2ω2ωj+ω22,z0−ω2+2ξ0ω0ωj+ω02=z12ξ0ω0ωj+ω02,z1−ω2+2r0ξ0ω0+2ξ1ω1ωj+r2ω22+r0ω02+ω12=z22r2ξ2ω2ωj+r2ω22+z02r0ξ0ω0ωj+r0ω02+q2ξ1ω1ωj+ω12.Equation (7) can be further expressed as(8)A1z2=A2z1,A3z0=A4z1,A5z1=A6z2+A7z0+A8q.where(9)A1=−ω2+2ξ2ω2ωj+ω22,A2=2ξ2ω2ωj+ω22,A3=−ω2+2ξ0ω0ωj+ω02,A4=2ξ0ω0ωj+ω02,A5=−ω2+2r0ξ0ω0+2r2ξ2ω2+2ξ1ω1ωj+r2ω22+r0ω02+ω12,A6=2r2ξ2ω2ωj+r2ω22,A7=2r0ξ0ω0ωj+r0ω02,A8=2ξ1ω1ωj+ω12.From Equation (8), obtain the following:(10)z2q=A2A3A8A1A3A5−A2A3A6−A1A4A7,(11)z1q=A1A3A8A1A3A5−A2A3A6−A1A4A7.The amplitude frequency characteristics functionJ1 between the road velocity input q˙ and the wheelchair body acceleration z¨1 can be expressed as(12)J1=Hjωz¨1∼q˙=z¨1q˙=ωz1q.The amplitude frequency characteristics functionJ2 between the road velocity input q˙ and the convalescent acceleration z¨2 can be expressed as(13)J2=Hjωz¨2∼q˙=z¨2q˙=ωz2q.The amplitude frequency characteristics functionJ3 between the road velocity input q˙ and the TDD (tire dynamic deflection) fd=z1−q can be expressed as(14)J3=Hjωfd∼q˙=fdq˙=z1−qq˙=1ωz1−qq.Substituting Equation (11) in Equation (12) produces(15)J1=ωA1A3A8A1A3A5−A2A3A6−A1A4A7.Substituting Equation (10) in Equation (13) produces(16)J2=ωA2A3A8A1A3A5−A2A3A6−A1A4A7.Substituting Equation (11) in Equation (15) produces(17)J3=1ωA1A3A8−A1A3A5+A2A3A6+A1A4A7A1A3A5−A2A3A6−A1A4A7.


## 4. The PSD and RMS Responses under the Random Road

### 4.1. Random Road Model

The spatial frequency PSDGqn of the random road roughness q can be expressed as [22](18)Gqn=Gqn0nn0−w,where, n is the spatial frequency; n0 is the spatial reference frequency and n0=0.1m−1; Gqn0 is the road roughness coefficient; and w is the frequency index.When a wheelchair robot runs on the random road at a certain speedu, the time frequency of the road excitation is f=un. Thus, the time frequency PSD Gqf is(19)Gqf=1uGqn.Whenw=2, the time frequency PSD Gqf can be expressed as(20)Gqf=1uGqn0nn0−2=Gqn0n02uf2.The relationship between the time frequency PSDGqf of the random road roughness q and the time frequency PSD Gq˙f of the random road excitation velocity q˙ is given by(21)Gq˙f=2πf2Gqf=4π2Gqn0n02u.


### 4.2. The PSD and RMS Responses Calculation

The autocorrelation functionRq˙τ of q˙t is(22)Rq˙τ=limT⟶∞1T∫−T/2+T/2q˙tq˙t+τdt.The convalescent accelerationz¨2t can be expressed as(23)z¨2t=∫0+∞q˙τht−τdτ,where ht is impulse response function.The PSDGz¨2ω of the convalescent acceleration z¨2t can be expressed as(24)Gz¨2ω=Ez¨2tz¨2t+τ=HjωGq˙ωHjω=J2Gq˙ω.Substituting Equation (21) in Equation (24) produces(25)Gz¨2ω=4π2Gqn0n02uJ22.The convalescent RMS accelerationσz¨2 can be expressed as(26)σz¨2=12π∫0+∞Gz¨2ωdω.Substituting Equation (25) in Equation (26) produces(27)σz¨2=2πGqn0n02u∫0+∞J22ωdω.Analogously, the wheelchair body RMS accelerationσz¨1 can be expressed as(28)σz¨1=2πGqn0n02u∫0+∞J12ωdω.The TDD RMS can be expressed as(29)σfd=2πGqn0n02u∫0+∞J32ωdω.It is difficult to analytically calculate the Equations (27)–(29). The numerical integration method is adopted as(30)σz¨2=2πGqn0n02u∑i=1NJ222πiΔf×2πiΔf,σz¨1=2πGqn0n02u∑i=1NJ122πiΔf×2πiΔf,σfd=2πGqn0n02u∑i=1NJ322πiΔf×2πiΔf,where N is the number of the discrete frequency value; Δf is the frequency bandwidth; and i=1,2,3,…,N.


## 4.1. Random Road Model

The spatial frequency PSDGqn of the random road roughness q can be expressed as [22](18)Gqn=Gqn0nn0−w,where, n is the spatial frequency; n0 is the spatial reference frequency and n0=0.1m−1; Gqn0 is the road roughness coefficient; and w is the frequency index.When a wheelchair robot runs on the random road at a certain speedu, the time frequency of the road excitation is f=un. Thus, the time frequency PSD Gqf is(19)Gqf=1uGqn.Whenw=2, the time frequency PSD Gqf can be expressed as(20)Gqf=1uGqn0nn0−2=Gqn0n02uf2.The relationship between the time frequency PSDGqf of the random road roughness q and the time frequency PSD Gq˙f of the random road excitation velocity q˙ is given by(21)Gq˙f=2πf2Gqf=4π2Gqn0n02u.


## 4.2. The PSD and RMS Responses Calculation

The autocorrelation functionRq˙τ of q˙t is(22)Rq˙τ=limT⟶∞1T∫−T/2+T/2q˙tq˙t+τdt.The convalescent accelerationz¨2t can be expressed as(23)z¨2t=∫0+∞q˙τht−τdτ,where ht is impulse response function.The PSDGz¨2ω of the convalescent acceleration z¨2t can be expressed as(24)Gz¨2ω=Ez¨2tz¨2t+τ=HjωGq˙ωHjω=J2Gq˙ω.Substituting Equation (21) in Equation (24) produces(25)Gz¨2ω=4π2Gqn0n02uJ22.The convalescent RMS accelerationσz¨2 can be expressed as(26)σz¨2=12π∫0+∞Gz¨2ωdω.Substituting Equation (25) in Equation (26) produces(27)σz¨2=2πGqn0n02u∫0+∞J22ωdω.Analogously, the wheelchair body RMS accelerationσz¨1 can be expressed as(28)σz¨1=2πGqn0n02u∫0+∞J12ωdω.The TDD RMS can be expressed as(29)σfd=2πGqn0n02u∫0+∞J32ωdω.It is difficult to analytically calculate the Equations (27)–(29). The numerical integration method is adopted as(30)σz¨2=2πGqn0n02u∑i=1NJ222πiΔf×2πiΔf,σz¨1=2πGqn0n02u∑i=1NJ122πiΔf×2πiΔf,σfd=2πGqn0n02u∑i=1NJ322πiΔf×2πiΔf,where N is the number of the discrete frequency value; Δf is the frequency bandwidth; and i=1,2,3,…,N.


## 5. Vibration Characteristics Comparison

In order to facilitate the vibration characteristics analysis of the convalescent-wheelchair system with DA, the convalescent-wheelchair system without DA is taken as a comparison object. Based on a typical commercially available wheelchair and the selection method of the DA [19], the values of the parameters for the convalescent-wheelchair system is given in Table 1 [21]. The simulation condition is set as: u = 0.5∼2.5m/s, the C grade road with Gq(n0) = 256 × 10−6 m−3. For the numerical calculation, Δf = 0.1 Hz, and the upper limit frequency is set as 30 Hz. Based on the vibration models of the convalescent-wheelchair system with DA and without DA, the values of σz¨2, σz¨1, σfd, Gz¨2, Gz¨1, and Gfd were numerically calculated, respectively. The calculated results are shown in Figure 3.Table 1
The values of the parameters for the convalescent-wheelchair system.
Parameter
Value
m
2 (kg)
75
m
1 (kg)
25
m
0 (kg)
2.5
C
2 (Ns/m)
377
C
1 (Ns/m)
314
C
0 (Ns/m)
63
K
2 (N/mm)
11.84
K
1 (N/mm)
98.70
K
0 (N/mm)
9.87Figure 3
A comparison of the dynamic responses calculated from the vibration models of the convalescent-wheelchair with DA and without DA: (a) the wheelchair body RMS accelerationσz¨1 versus the running speed u; (b) the convalescent RMS acceleration σz¨2 versus the running speed u; (c) the TDD RMS σfd versus the running speed u; (d) the wheelchair body acceleration PSD Gz¨1 versus the frequency f; (e) the convalescent acceleration PSD Gz¨2versus the frequency f; (f) the TDD PSD Gfd versus the frequency f.
(a)
(b)
(c)
(d)
(e)
(f)Figures3(a)–3(c) show that the DA can effectively reduce the wheelchair body RMS acceleration σz¨1 and the convalescent RMS acceleration σz¨2, and the TDD RMS σfd is not improved obviously. Figure 3(d) shows that the peak value of the wheelchair body acceleration PSD Gz¨1 is decreased by 40.2% by using the DA. Figure 3(e) illustrates that the DA reduces the convalescent acceleration PSD Gz¨2 around 10 Hz but causes a slight increase in the range of 4∼7 Hz. Figure 3(f) depicts that the DA makes the peak value of the TDD PSD Gfd shift to the left. The comparison results show that the DA partly suppresses the vibration of the convalescent and the wheelchair body, especially in the resonance area of the wheelchair body. However, the DA does not successfully improve the tire contact behavior.


## 6. Sensitivity Analysis of Vibration Responses to System Parameters

In order to analyze the effects ofr2, r0, ξ1, ξ2, ξ0, f0, f1, and f2 on σz¨2, σz¨1, and σfd, the simulation calculation was carried out based on the vibration model of the convalescent-wheelchair system with DA. The simulation condition is set as u = 0.1 m/s, the C grade road with Gq(n0) = 256 × 10−6 m−3. To represent the curve changes in the same diagram, the values of σz¨2, σz¨1, and σfd are normalized on the basis of the baseline RMS values, respectively. The processing method is as follows: σ2=σz¨2/σz¨2b, σ1=σz¨1/σz¨1b, σ3=σfd/σfdb, where the subscript “b” represents “baseline.” In addition, to observe the performance changes in the local frequency band, the amplitude frequency characteristics functions J1, J2, and J3 are also calculated.When the influence of one of the eight parameters is analyzed, the baseline value is increased by 100% or decreased by 50%, and the other parameters are fixed. The values of the system parameters for the analysis are given in Table2 [21]. The baseline values are converted from Table 1.Table 2
The values of the system parameters for the analysis.
Parameter
Baseline
+100%
−50%
r
2
3.0
6.0
1.5
r
0
0.10
0.05
0.20
ξ
2
0.25
0.5
0.125
ξ
1
0.25
0.5
0.125
ξ
0
0.25
0.5
0.125
f
2
2.0
4.0
1.0
f
1
10.0
20.0
5.0
f
0
10.0
20.0
5.0


### 6.1. The Influences of the Mass Ratios

#### 6.1.1. The Influences of the Mass Ratior2

Figure4 depicts the influences of r2 on z¨1, z¨2, and fd. From Figure 4(a), it can be seen that increasing r2 can almost reduce the amplitude of J1 in the whole frequency domain. Therefore, increasing r2 is conducive to reducing the wheelchair body acceleration. Figure 4(b) illustrates that increasing r2 can reduce the amplitude of J2 in the range of 5∼15 Hz, but it causes the amplitude to increase around the low frequency resonance peak. From Figure 4(c), it can be seen that with the increase of r2, the low frequency resonance peak of J3 increases sharply, while the high frequency formant decreases. Therefore, too large r2 is not conducive to improving the tire grounding safety. As shown in Figure 4(d), σ1 and σ2 are in inverse proportion to r2, and σ3 has a minimum value for r2 in the range of 2.5∼3.5. Thus, the increase of r2 is mainly to reduce the values of σ1 and σ2.Figure 4
Influences ofr2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r2; (b) the amplitude frequency characteristics J2 at different values of r2; (c) the amplitude frequency characteristics J3 at different values of r2; (d) the RMS σ1, σ2, and σ3 versus r2.
(a)
(b)
(c)
(d)


#### 6.1.2. The Influences of the Mass Ratior0

Figure5 shows the influences of r0 on z¨1, z¨2, and fd. From Figure 5(a), it can be seen that increasing r0 can effectively reduce the peak value of J1. Figure 5(b) shows that increasing r0 can reduce the amplitude of J2 from 7 Hz to 15 Hz, but the amplitude in the range of 4∼7 Hz increases slightly. From Figure 5(c), it can be seen that the larger the value of r0 is, the higher the formant of J3 is. As shown in Figure 5(d), σ1 and σ2 are in inverse proportion to r0, σ3 is proportional to r0, and σ1 is most sensitive to the change of r0. Thus, the use of the DA with the larger mass helps to reduce the vibration of the convalescent and the wheelchair.Figure 5
Influences ofr0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r0; (b) the amplitude frequency characteristics J2 at different values of r0; (c) the amplitude frequency characteristics J3 at different values of r0; (d) the RMS σ1, σ2, and σ3 versus r0.
(a)
(b)
(c)
(d)


### 6.2. The Influences of the Damping Ratios

#### 6.2.1. The Influences of the Damping Ratioξ2

Figure6 depicts the influences of ξ2 on z¨1, z¨2, and fd. As shown in Figure 6(a), increasing ξ2 can help to reduce the peak value of J1. From Figure 6(b), it can be seen that increasing ξ2 can effectively reduce the low frequency resonance peak of J2, but the amplitude after 3.0 Hz increases. Figure 6(c) shows that the larger the value of ξ2 is, the smaller the peak values of J3 are, but the amplitude between the two formants increases obviously. Figure 5(d) illustrates that σ1 is almost in inverse proportional to ξ2, there is a minimum value of σ2 for the value of ξ2 between 0.15∼0.25, and σ3 has a minimum value when the value of ξ2 belongs to 0.15∼0.25. The ride comfort requires that the cushion system has the smaller value of ξ2, however, the tire grounding safety and the operational reliability need the larger value of ξ2. Thus, taking the smaller value of ξ2, about 0.2, is helpful to improve comfort.Figure 6
Influence ofξ2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ2; (b) the amplitude frequency characteristics J2 at different values of ξ2; (c) the amplitude frequency characteristics J3 at different values of ξ2; (d) the RMS σ1, σ2, and σ3 versus ξ2.
(a)
(b)
(c)
(d)


#### 6.2.2. The Influences of the Damping Ratioξ1

Figure7 depicts the influences of ξ1 on z¨1, z¨2, and fd. From Figure 7(a), it can be seen that increasing ξ1 can effectively suppress the peak value of J1, but the amplitude after 15 Hz increases. As shown in Figure 7(b), the increase of ξ1 can effectively suppress the low frequency and the high frequency resonance peak values of J2, and the suppression effectiveness around the high frequency resonance peak is more obvious. However, the amplitude after 15 Hz also increases. Figure 7(c) clearly shows that increasing ξ1 can almost reduce the amplitude of J3 in the full frequency domain. From Figure 7(d), it can be seen that both σ2 and σ3 are approximately in inverse proportion to ξ1, and when the value of ξ1 belongs to 0.1∼0.2, a minimum value of σ1 exists. Thus, the ride comfort and the tire grounding safety require that the tire system has the larger value of ξ1, however, the operational reliability needs the smaller value of ξ1.Figure 7
Influences ofξ1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ1; (b) the amplitude frequency characteristics J2 at different values of ξ1; (c) the amplitude frequency characteristics J3 at different values of ξ1; (d) the RMS σ1, σ2, and σ3 versus ξ1.
(a)
(b)
(c)
(d)


#### 6.2.3. The Influences of the Damping Ratioξ0

Figure8 depicts the influences of ξ0 on z¨1, z¨2, and fd. According to Figure 8(a), the amplitude of J1 in the region of 8∼13 Hz increases with the increase of ξ0, and the amplitude decreases after 13 Hz. Figure 8(b) shows that increasing ξ0 can effectively suppress the high frequency resonance peak value of J2. According to Figure 8(c), ξ0 has obvious effect on the high frequency resonance peak value of J3. From Figure 8(d), it can be seen that, with the increase of ξ0, the value of σ1 decreases. When ξ0 belongs to 0.1∼0.2, σ3 has a minimum value, and when ξ0 belongs to 0.15∼0.25, σ2 has a minimum value. When the value of ξ0 equals 0.2, the DA system can well give consideration to the three indexes: the ride comfort, the tire grounding safety, and the operational reliability.Figure 8
Influences ofξ0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ0; (b) the amplitude frequency characteristics J2 at different values of ξ0; (c) the amplitude frequency characteristics J3 at different values of ξ0; (d) the RMS σ1, σ2, and σ3 versus ξ0.
(a)
(b)
(c)
(d)


### 6.3. The Influences of the Natural Frequencies

#### 6.3.1. The Influences of the Natural Frequencyf2

Figure9 depicts the influences of f2 on z¨1, z¨2, and fd. From Figure 9(a), it can be seen that increasing f2 is beneficial to suppress the resonance peak value of J1. Figures 9(b) and 9(c) show that increasing f2 makes the low frequency formants of J2 and J3 increase sharply and shift rightward. As shown in Figure 9(d), σ1 is almost in inverse proportion to f2, σ2 is proportional to f2 and σ2 is most sensitive to the change of f2, and when f2 belongs to 1.5∼2.5 Hz, σ3 has a minimum value. Therefore, adopting a soft cushion to reduce f2 is an important measure to improve ride comfort.Figure 9
Influences off2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f2; (b) the amplitude frequency characteristics J2 at different values of f2; (c) the amplitude frequency characteristics J3 at different values of f2; (d) the RMS σ1, σ2, and σ3 versus f2.
(a)
(b)
(c)
(d)


#### 6.3.2. The Influences of the Natural Frequencyf1

Figure10 depicts the influences of f1 on z¨1, z¨2, and fd. Figure 10(a) shows that with the increase of f1, the resonance peak value of J1 increases sharply and moves to the right. Figure 10(b) illustrates that increasing f1 is beneficial to reduce the peak value of the low frequency resonance peak value of J2, but the amplitude after 10 Hz increases significantly. Figure 10(c) shows that the increase of f1 is conducive to significantly reducing the amplitude of J3 from 0 Hz to 18 Hz, although the amplitude increases slightly after 18 Hz. Figure 10(d) shows that σ1 and σ2 are proportional to f1, and σ3 nonlinearly decreases as f1 increases. Thus, the adoption of soft tires is conducive to improving ride comfort and operational reliability.Figure 10
Influences off1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f1; (b) the amplitude frequency characteristics J2 at different values of f1; (c) the amplitude frequency characteristics J3 at different values of f1; (d) the RMS σ1, σ2, and σ3 versus f1.
(a)
(b)
(c)
(d)


#### 6.3.3. The Influences of the Natural Frequencyf0

Figure11 depicts the influences of f0 on z¨1, z¨2, and fd. Figure 11(a) shows that the increase of f0 has little effect on J1 in the low frequency region 0∼5 Hz and the high frequency 25∼30 Hz. In the mid frequency region 5∼25 Hz, increasing f0 is beneficial to the amplitude reduction of J1. Figure 11(b) shows that f0 only obviously affects the amplitude of J1 from 5 Hz to 20 Hz. Figure 11(c) clearly states that f0 has a great influence on the high frequency resonance peak value of J3. Figure 11(d) illustrates that σ1 nonlinearly decreases as f0 increases, when the value of f0 belongs to 8∼12, σ2 has a minimum value, and when the value of f0 belongs to 6∼8, σ3 has a minimum value. Thus, when f0 approximately equals to f1, the DA system can well give consideration to the comprehensive isolation performance.Figure 11
Influences off0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f0; (b) the amplitude frequency characteristics J2 at different values of f0; (c) the amplitude frequency characteristics J3 at different values of f0; (d) the RMS σ1, σ2, and σ3 versus f0.
(a)
(b)
(c)
(d)


### 6.4. Sensitivity Calculation

Based on the influence trends analysis of the system parameters on the RMS responses in Sections6.1 to 6.3, sensitivity calculation was carried out by using the perturbation method. The calculation formulae are as follows:(31)sxi=σUxi+ΔUxi−σUxi/σUxiΔUxi/Uxi,(32)sx=1n∑i=1nsxi,where n is the number of the intervals, i represents the ith interval, ΔUxi refers to the incremental value of the system parameter x in the ith interval, and sx is the response sensitivity to the system parameter x.According to the calculated RMS responses based on the system parameters in Table2, the sensitivity values of each response are obtained by using Equations (31) and (32), as shown in Table 3. To easily observe the differences in sensitivity, the sensitivity values were reordered, as shown in Figure 12. Figure 12(a) shows that f1 is the most important influence factor on σz¨1, and ξ0 is the weakest influence factor. Other factors are ξ1, f2, ξ2, r2, r0, and f0 from strong to weak. From Figure 12(b), it can be seen that σz¨2 is most sensitive to f2, and the sensitivity is as high as 1.095. σz¨2 is most insensitive to ξ0. Other factors are f1, ξ2, r2, ξ1, f0, and r0 from strong to weak. Figure 12(c) illustrates that the greatest influence factor on σfd is f1, and the minimum influence factor is ξ0. Other factors affecting σfd in an order from intensely to slightly are f2, ξ1, r2, ξ2, f0, and r0. Thus, for the DA, ξ0 has the smallest influence on each RMS response, and f0 and r0 have relatively greater influences.Table 3
The sensitivity values of each response.
System parameter
s
z
¨
1
s
z
¨
2
s
f
d
r
2
0.362
0.324
0.330
r
0
0.198
0.107
0.099
ξ
2
0.367
0.460
0.269
ξ
1
0.498
0.242
0.505
ξ
0
0.110
0.057
0.092
f
2
0.376
1.095
0.511
f
1
1.247
0.560
0.912
f
0
0.173
0.113
0.184Figure 12
The vibration response sensitivity: (a) the wheelchair body RMS acceleration sensitivitysz¨1; (b) the convalescent RMS acceleration sensitivity sz¨2; (c) the TDD RMS sensitivity sfd.
(a)
(b)
(c)


## 6.1. The Influences of the Mass Ratios

### 6.1.1. The Influences of the Mass Ratior2

Figure4 depicts the influences of r2 on z¨1, z¨2, and fd. From Figure 4(a), it can be seen that increasing r2 can almost reduce the amplitude of J1 in the whole frequency domain. Therefore, increasing r2 is conducive to reducing the wheelchair body acceleration. Figure 4(b) illustrates that increasing r2 can reduce the amplitude of J2 in the range of 5∼15 Hz, but it causes the amplitude to increase around the low frequency resonance peak. From Figure 4(c), it can be seen that with the increase of r2, the low frequency resonance peak of J3 increases sharply, while the high frequency formant decreases. Therefore, too large r2 is not conducive to improving the tire grounding safety. As shown in Figure 4(d), σ1 and σ2 are in inverse proportion to r2, and σ3 has a minimum value for r2 in the range of 2.5∼3.5. Thus, the increase of r2 is mainly to reduce the values of σ1 and σ2.Figure 4
Influences ofr2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r2; (b) the amplitude frequency characteristics J2 at different values of r2; (c) the amplitude frequency characteristics J3 at different values of r2; (d) the RMS σ1, σ2, and σ3 versus r2.
(a)
(b)
(c)
(d)


### 6.1.2. The Influences of the Mass Ratior0

Figure5 shows the influences of r0 on z¨1, z¨2, and fd. From Figure 5(a), it can be seen that increasing r0 can effectively reduce the peak value of J1. Figure 5(b) shows that increasing r0 can reduce the amplitude of J2 from 7 Hz to 15 Hz, but the amplitude in the range of 4∼7 Hz increases slightly. From Figure 5(c), it can be seen that the larger the value of r0 is, the higher the formant of J3 is. As shown in Figure 5(d), σ1 and σ2 are in inverse proportion to r0, σ3 is proportional to r0, and σ1 is most sensitive to the change of r0. Thus, the use of the DA with the larger mass helps to reduce the vibration of the convalescent and the wheelchair.Figure 5
Influences ofr0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r0; (b) the amplitude frequency characteristics J2 at different values of r0; (c) the amplitude frequency characteristics J3 at different values of r0; (d) the RMS σ1, σ2, and σ3 versus r0.
(a)
(b)
(c)
(d)


## 6.1.1. The Influences of the Mass Ratior2

Figure4 depicts the influences of r2 on z¨1, z¨2, and fd. From Figure 4(a), it can be seen that increasing r2 can almost reduce the amplitude of J1 in the whole frequency domain. Therefore, increasing r2 is conducive to reducing the wheelchair body acceleration. Figure 4(b) illustrates that increasing r2 can reduce the amplitude of J2 in the range of 5∼15 Hz, but it causes the amplitude to increase around the low frequency resonance peak. From Figure 4(c), it can be seen that with the increase of r2, the low frequency resonance peak of J3 increases sharply, while the high frequency formant decreases. Therefore, too large r2 is not conducive to improving the tire grounding safety. As shown in Figure 4(d), σ1 and σ2 are in inverse proportion to r2, and σ3 has a minimum value for r2 in the range of 2.5∼3.5. Thus, the increase of r2 is mainly to reduce the values of σ1 and σ2.Figure 4
Influences ofr2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r2; (b) the amplitude frequency characteristics J2 at different values of r2; (c) the amplitude frequency characteristics J3 at different values of r2; (d) the RMS σ1, σ2, and σ3 versus r2.
(a)
(b)
(c)
(d)


## 6.1.2. The Influences of the Mass Ratior0

Figure5 shows the influences of r0 on z¨1, z¨2, and fd. From Figure 5(a), it can be seen that increasing r0 can effectively reduce the peak value of J1. Figure 5(b) shows that increasing r0 can reduce the amplitude of J2 from 7 Hz to 15 Hz, but the amplitude in the range of 4∼7 Hz increases slightly. From Figure 5(c), it can be seen that the larger the value of r0 is, the higher the formant of J3 is. As shown in Figure 5(d), σ1 and σ2 are in inverse proportion to r0, σ3 is proportional to r0, and σ1 is most sensitive to the change of r0. Thus, the use of the DA with the larger mass helps to reduce the vibration of the convalescent and the wheelchair.Figure 5
Influences ofr0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of r0; (b) the amplitude frequency characteristics J2 at different values of r0; (c) the amplitude frequency characteristics J3 at different values of r0; (d) the RMS σ1, σ2, and σ3 versus r0.
(a)
(b)
(c)
(d)


## 6.2. The Influences of the Damping Ratios

### 6.2.1. The Influences of the Damping Ratioξ2

Figure6 depicts the influences of ξ2 on z¨1, z¨2, and fd. As shown in Figure 6(a), increasing ξ2 can help to reduce the peak value of J1. From Figure 6(b), it can be seen that increasing ξ2 can effectively reduce the low frequency resonance peak of J2, but the amplitude after 3.0 Hz increases. Figure 6(c) shows that the larger the value of ξ2 is, the smaller the peak values of J3 are, but the amplitude between the two formants increases obviously. Figure 5(d) illustrates that σ1 is almost in inverse proportional to ξ2, there is a minimum value of σ2 for the value of ξ2 between 0.15∼0.25, and σ3 has a minimum value when the value of ξ2 belongs to 0.15∼0.25. The ride comfort requires that the cushion system has the smaller value of ξ2, however, the tire grounding safety and the operational reliability need the larger value of ξ2. Thus, taking the smaller value of ξ2, about 0.2, is helpful to improve comfort.Figure 6
Influence ofξ2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ2; (b) the amplitude frequency characteristics J2 at different values of ξ2; (c) the amplitude frequency characteristics J3 at different values of ξ2; (d) the RMS σ1, σ2, and σ3 versus ξ2.
(a)
(b)
(c)
(d)


### 6.2.2. The Influences of the Damping Ratioξ1

Figure7 depicts the influences of ξ1 on z¨1, z¨2, and fd. From Figure 7(a), it can be seen that increasing ξ1 can effectively suppress the peak value of J1, but the amplitude after 15 Hz increases. As shown in Figure 7(b), the increase of ξ1 can effectively suppress the low frequency and the high frequency resonance peak values of J2, and the suppression effectiveness around the high frequency resonance peak is more obvious. However, the amplitude after 15 Hz also increases. Figure 7(c) clearly shows that increasing ξ1 can almost reduce the amplitude of J3 in the full frequency domain. From Figure 7(d), it can be seen that both σ2 and σ3 are approximately in inverse proportion to ξ1, and when the value of ξ1 belongs to 0.1∼0.2, a minimum value of σ1 exists. Thus, the ride comfort and the tire grounding safety require that the tire system has the larger value of ξ1, however, the operational reliability needs the smaller value of ξ1.Figure 7
Influences ofξ1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ1; (b) the amplitude frequency characteristics J2 at different values of ξ1; (c) the amplitude frequency characteristics J3 at different values of ξ1; (d) the RMS σ1, σ2, and σ3 versus ξ1.
(a)
(b)
(c)
(d)


### 6.2.3. The Influences of the Damping Ratioξ0

Figure8 depicts the influences of ξ0 on z¨1, z¨2, and fd. According to Figure 8(a), the amplitude of J1 in the region of 8∼13 Hz increases with the increase of ξ0, and the amplitude decreases after 13 Hz. Figure 8(b) shows that increasing ξ0 can effectively suppress the high frequency resonance peak value of J2. According to Figure 8(c), ξ0 has obvious effect on the high frequency resonance peak value of J3. From Figure 8(d), it can be seen that, with the increase of ξ0, the value of σ1 decreases. When ξ0 belongs to 0.1∼0.2, σ3 has a minimum value, and when ξ0 belongs to 0.15∼0.25, σ2 has a minimum value. When the value of ξ0 equals 0.2, the DA system can well give consideration to the three indexes: the ride comfort, the tire grounding safety, and the operational reliability.Figure 8
Influences ofξ0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ0; (b) the amplitude frequency characteristics J2 at different values of ξ0; (c) the amplitude frequency characteristics J3 at different values of ξ0; (d) the RMS σ1, σ2, and σ3 versus ξ0.
(a)
(b)
(c)
(d)


## 6.2.1. The Influences of the Damping Ratioξ2

Figure6 depicts the influences of ξ2 on z¨1, z¨2, and fd. As shown in Figure 6(a), increasing ξ2 can help to reduce the peak value of J1. From Figure 6(b), it can be seen that increasing ξ2 can effectively reduce the low frequency resonance peak of J2, but the amplitude after 3.0 Hz increases. Figure 6(c) shows that the larger the value of ξ2 is, the smaller the peak values of J3 are, but the amplitude between the two formants increases obviously. Figure 5(d) illustrates that σ1 is almost in inverse proportional to ξ2, there is a minimum value of σ2 for the value of ξ2 between 0.15∼0.25, and σ3 has a minimum value when the value of ξ2 belongs to 0.15∼0.25. The ride comfort requires that the cushion system has the smaller value of ξ2, however, the tire grounding safety and the operational reliability need the larger value of ξ2. Thus, taking the smaller value of ξ2, about 0.2, is helpful to improve comfort.Figure 6
Influence ofξ2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ2; (b) the amplitude frequency characteristics J2 at different values of ξ2; (c) the amplitude frequency characteristics J3 at different values of ξ2; (d) the RMS σ1, σ2, and σ3 versus ξ2.
(a)
(b)
(c)
(d)


## 6.2.2. The Influences of the Damping Ratioξ1

Figure7 depicts the influences of ξ1 on z¨1, z¨2, and fd. From Figure 7(a), it can be seen that increasing ξ1 can effectively suppress the peak value of J1, but the amplitude after 15 Hz increases. As shown in Figure 7(b), the increase of ξ1 can effectively suppress the low frequency and the high frequency resonance peak values of J2, and the suppression effectiveness around the high frequency resonance peak is more obvious. However, the amplitude after 15 Hz also increases. Figure 7(c) clearly shows that increasing ξ1 can almost reduce the amplitude of J3 in the full frequency domain. From Figure 7(d), it can be seen that both σ2 and σ3 are approximately in inverse proportion to ξ1, and when the value of ξ1 belongs to 0.1∼0.2, a minimum value of σ1 exists. Thus, the ride comfort and the tire grounding safety require that the tire system has the larger value of ξ1, however, the operational reliability needs the smaller value of ξ1.Figure 7
Influences ofξ1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ1; (b) the amplitude frequency characteristics J2 at different values of ξ1; (c) the amplitude frequency characteristics J3 at different values of ξ1; (d) the RMS σ1, σ2, and σ3 versus ξ1.
(a)
(b)
(c)
(d)


## 6.2.3. The Influences of the Damping Ratioξ0

Figure8 depicts the influences of ξ0 on z¨1, z¨2, and fd. According to Figure 8(a), the amplitude of J1 in the region of 8∼13 Hz increases with the increase of ξ0, and the amplitude decreases after 13 Hz. Figure 8(b) shows that increasing ξ0 can effectively suppress the high frequency resonance peak value of J2. According to Figure 8(c), ξ0 has obvious effect on the high frequency resonance peak value of J3. From Figure 8(d), it can be seen that, with the increase of ξ0, the value of σ1 decreases. When ξ0 belongs to 0.1∼0.2, σ3 has a minimum value, and when ξ0 belongs to 0.15∼0.25, σ2 has a minimum value. When the value of ξ0 equals 0.2, the DA system can well give consideration to the three indexes: the ride comfort, the tire grounding safety, and the operational reliability.Figure 8
Influences ofξ0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of ξ0; (b) the amplitude frequency characteristics J2 at different values of ξ0; (c) the amplitude frequency characteristics J3 at different values of ξ0; (d) the RMS σ1, σ2, and σ3 versus ξ0.
(a)
(b)
(c)
(d)


## 6.3. The Influences of the Natural Frequencies

### 6.3.1. The Influences of the Natural Frequencyf2

Figure9 depicts the influences of f2 on z¨1, z¨2, and fd. From Figure 9(a), it can be seen that increasing f2 is beneficial to suppress the resonance peak value of J1. Figures 9(b) and 9(c) show that increasing f2 makes the low frequency formants of J2 and J3 increase sharply and shift rightward. As shown in Figure 9(d), σ1 is almost in inverse proportion to f2, σ2 is proportional to f2 and σ2 is most sensitive to the change of f2, and when f2 belongs to 1.5∼2.5 Hz, σ3 has a minimum value. Therefore, adopting a soft cushion to reduce f2 is an important measure to improve ride comfort.Figure 9
Influences off2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f2; (b) the amplitude frequency characteristics J2 at different values of f2; (c) the amplitude frequency characteristics J3 at different values of f2; (d) the RMS σ1, σ2, and σ3 versus f2.
(a)
(b)
(c)
(d)


### 6.3.2. The Influences of the Natural Frequencyf1

Figure10 depicts the influences of f1 on z¨1, z¨2, and fd. Figure 10(a) shows that with the increase of f1, the resonance peak value of J1 increases sharply and moves to the right. Figure 10(b) illustrates that increasing f1 is beneficial to reduce the peak value of the low frequency resonance peak value of J2, but the amplitude after 10 Hz increases significantly. Figure 10(c) shows that the increase of f1 is conducive to significantly reducing the amplitude of J3 from 0 Hz to 18 Hz, although the amplitude increases slightly after 18 Hz. Figure 10(d) shows that σ1 and σ2 are proportional to f1, and σ3 nonlinearly decreases as f1 increases. Thus, the adoption of soft tires is conducive to improving ride comfort and operational reliability.Figure 10
Influences off1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f1; (b) the amplitude frequency characteristics J2 at different values of f1; (c) the amplitude frequency characteristics J3 at different values of f1; (d) the RMS σ1, σ2, and σ3 versus f1.
(a)
(b)
(c)
(d)


### 6.3.3. The Influences of the Natural Frequencyf0

Figure11 depicts the influences of f0 on z¨1, z¨2, and fd. Figure 11(a) shows that the increase of f0 has little effect on J1 in the low frequency region 0∼5 Hz and the high frequency 25∼30 Hz. In the mid frequency region 5∼25 Hz, increasing f0 is beneficial to the amplitude reduction of J1. Figure 11(b) shows that f0 only obviously affects the amplitude of J1 from 5 Hz to 20 Hz. Figure 11(c) clearly states that f0 has a great influence on the high frequency resonance peak value of J3. Figure 11(d) illustrates that σ1 nonlinearly decreases as f0 increases, when the value of f0 belongs to 8∼12, σ2 has a minimum value, and when the value of f0 belongs to 6∼8, σ3 has a minimum value. Thus, when f0 approximately equals to f1, the DA system can well give consideration to the comprehensive isolation performance.Figure 11
Influences off0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f0; (b) the amplitude frequency characteristics J2 at different values of f0; (c) the amplitude frequency characteristics J3 at different values of f0; (d) the RMS σ1, σ2, and σ3 versus f0.
(a)
(b)
(c)
(d)


## 6.3.1. The Influences of the Natural Frequencyf2

Figure9 depicts the influences of f2 on z¨1, z¨2, and fd. From Figure 9(a), it can be seen that increasing f2 is beneficial to suppress the resonance peak value of J1. Figures 9(b) and 9(c) show that increasing f2 makes the low frequency formants of J2 and J3 increase sharply and shift rightward. As shown in Figure 9(d), σ1 is almost in inverse proportion to f2, σ2 is proportional to f2 and σ2 is most sensitive to the change of f2, and when f2 belongs to 1.5∼2.5 Hz, σ3 has a minimum value. Therefore, adopting a soft cushion to reduce f2 is an important measure to improve ride comfort.Figure 9
Influences off2 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f2; (b) the amplitude frequency characteristics J2 at different values of f2; (c) the amplitude frequency characteristics J3 at different values of f2; (d) the RMS σ1, σ2, and σ3 versus f2.
(a)
(b)
(c)
(d)


## 6.3.2. The Influences of the Natural Frequencyf1

Figure10 depicts the influences of f1 on z¨1, z¨2, and fd. Figure 10(a) shows that with the increase of f1, the resonance peak value of J1 increases sharply and moves to the right. Figure 10(b) illustrates that increasing f1 is beneficial to reduce the peak value of the low frequency resonance peak value of J2, but the amplitude after 10 Hz increases significantly. Figure 10(c) shows that the increase of f1 is conducive to significantly reducing the amplitude of J3 from 0 Hz to 18 Hz, although the amplitude increases slightly after 18 Hz. Figure 10(d) shows that σ1 and σ2 are proportional to f1, and σ3 nonlinearly decreases as f1 increases. Thus, the adoption of soft tires is conducive to improving ride comfort and operational reliability.Figure 10
Influences off1 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f1; (b) the amplitude frequency characteristics J2 at different values of f1; (c) the amplitude frequency characteristics J3 at different values of f1; (d) the RMS σ1, σ2, and σ3 versus f1.
(a)
(b)
(c)
(d)


## 6.3.3. The Influences of the Natural Frequencyf0

Figure11 depicts the influences of f0 on z¨1, z¨2, and fd. Figure 11(a) shows that the increase of f0 has little effect on J1 in the low frequency region 0∼5 Hz and the high frequency 25∼30 Hz. In the mid frequency region 5∼25 Hz, increasing f0 is beneficial to the amplitude reduction of J1. Figure 11(b) shows that f0 only obviously affects the amplitude of J1 from 5 Hz to 20 Hz. Figure 11(c) clearly states that f0 has a great influence on the high frequency resonance peak value of J3. Figure 11(d) illustrates that σ1 nonlinearly decreases as f0 increases, when the value of f0 belongs to 8∼12, σ2 has a minimum value, and when the value of f0 belongs to 6∼8, σ3 has a minimum value. Thus, when f0 approximately equals to f1, the DA system can well give consideration to the comprehensive isolation performance.Figure 11
Influences off0 on z¨1, z¨2, and fd: (a) the amplitude frequency characteristics J1 at different values of f0; (b) the amplitude frequency characteristics J2 at different values of f0; (c) the amplitude frequency characteristics J3 at different values of f0; (d) the RMS σ1, σ2, and σ3 versus f0.
(a)
(b)
(c)
(d)


## 6.4. Sensitivity Calculation

Based on the influence trends analysis of the system parameters on the RMS responses in Sections6.1 to 6.3, sensitivity calculation was carried out by using the perturbation method. The calculation formulae are as follows:(31)sxi=σUxi+ΔUxi−σUxi/σUxiΔUxi/Uxi,(32)sx=1n∑i=1nsxi,where n is the number of the intervals, i represents the ith interval, ΔUxi refers to the incremental value of the system parameter x in the ith interval, and sx is the response sensitivity to the system parameter x.According to the calculated RMS responses based on the system parameters in Table2, the sensitivity values of each response are obtained by using Equations (31) and (32), as shown in Table 3. To easily observe the differences in sensitivity, the sensitivity values were reordered, as shown in Figure 12. Figure 12(a) shows that f1 is the most important influence factor on σz¨1, and ξ0 is the weakest influence factor. Other factors are ξ1, f2, ξ2, r2, r0, and f0 from strong to weak. From Figure 12(b), it can be seen that σz¨2 is most sensitive to f2, and the sensitivity is as high as 1.095. σz¨2 is most insensitive to ξ0. Other factors are f1, ξ2, r2, ξ1, f0, and r0 from strong to weak. Figure 12(c) illustrates that the greatest influence factor on σfd is f1, and the minimum influence factor is ξ0. Other factors affecting σfd in an order from intensely to slightly are f2, ξ1, r2, ξ2, f0, and r0. Thus, for the DA, ξ0 has the smallest influence on each RMS response, and f0 and r0 have relatively greater influences.Table 3
The sensitivity values of each response.
System parameter
s
z
¨
1
s
z
¨
2
s
f
d
r
2
0.362
0.324
0.330
r
0
0.198
0.107
0.099
ξ
2
0.367
0.460
0.269
ξ
1
0.498
0.242
0.505
ξ
0
0.110
0.057
0.092
f
2
0.376
1.095
0.511
f
1
1.247
0.560
0.912
f
0
0.173
0.113
0.184Figure 12
The vibration response sensitivity: (a) the wheelchair body RMS acceleration sensitivitysz¨1; (b) the convalescent RMS acceleration sensitivity sz¨2; (c) the TDD RMS sensitivity sfd.
(a)
(b)
(c)


## 7. Conclusions

This paper compared the vibration isolation performances of the convalescent-wheelchair robot with DA and without DA and conducted the sensitivity analysis of the vibration responses to the important dynamic parameters. The parameters include the following: the mass ratior0 of the DA mass to the wheelchair body mass, the mass ratio r2 of the convalescent mass to the wheelchair body mass, the DA natural frequency f0, the wheelchair body natural frequency f1, the convalescent natural frequency f2, the DA system damping ratio ξ0, the tire system damping ratio ξ1, and the cushion system damping ratio ξ2. Some useful conclusions are as follows:(1)
The DA can partly suppress the vibration of the convalescent and the wheelchair body, especially in the resonance area of the wheelchair body. However, the DA cannot successfully improve the tire contact behavior.(2)
The increase ofr2 is mainly to reduce the values of the wheelchair body RMS acceleration σz¨1 and the convalescent RMS acceleration σz¨2. The use of the DA with the larger mass can help to reduce the vibration of the convalescent and the wheelchair. Taking the smaller value of ξ2, about 0.2, is helpful to improve comfort. The ride comfort and the tire grounding safety require that the tire system has the larger value of ξ1; however, the operational reliability needs the smaller value of ξ1. When the value of ξ0 equals 0.2, the DA system can well give consideration to the comprehensive performance. Adopting a soft cushion to reduce f2 is an important measure to improve ride comfort.(3)
f1 is the most important influence factor on σz¨1, and ξ0 is the weakest influence factor. Other factors are ξ1, f2, ξ2, r2, r0, and f0 from strong to weak. σz¨2 is most sensitive to f2. The greatest influence factor on σfd is f1. For the DA, ξ0 has the smallest influence on each RMS response, and f0 and r0 have relatively greater influences.



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*Source: 5393051-2018-09-24.xml*