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

To provide a theoretical guidance for the vibration isolation design, the vibration responses characteristics for the convalescentwheelchair robot with DA (dynamic absorber) undergoing the random road were revealed. Firstly, the vibration model of the convalescent-wheelchair system with DA was created. -e frequency response functions of the road excitation velocity to the convalescent acceleration, the wheelchair body acceleration, and the tire dynamic deflection were deduced. -en, the numerical calculation method of the PSD (power spectral density) and the RMS (root mean square) responses were proposed. -irdly, 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. -e 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.


Introduction
Convalescent-wheelchair robots belong to the rehabilitation medical machines [1]. ey 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. e harmful vibration produces adverse effects for the wheelchair and its users [3][4][5].
In recent years, many scholars have studied the related vibration problems of wheelchair robots. e 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]. e 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. e influences of the user anatomy and wheelchair cushion type on the tissue deformation were analyzed in [18]. e 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. eir users' vibration caused by the road excitation is mainly reduced by the tires and the cushion. e 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.

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. erefore, 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. e vibration model of the convalescent-wheelchair system with DA is created, as shown in Figure 1 [21]. In Figure 1, m 2 , m 1 , and m 0 represent the convalescent mass, the wheelchair body mass, and the DA mass, respectively; K 2 , K 1 , and K 0 represent the cushion stiffness, the tire stiffness, and the DA stiffness, respectively; C 2 , C 1 , and C 0 represent the cushion damping, the tire damping, and the DA damping, respectively; z 2 , z 1 , and z 0 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.
According to Newton's second law, the motion equation for the vibration model of the convalescent-wheelchair system without DA can be expressed as Analogously, the motion equation for the vibration model of the convalescent-wheelchair system with DA can be expressed as By comparing Equation (1) with Equation (2), when m 0 equals zero, Equation (2) changes into Equation (1).
To easily facilitate the wheelchair robots design, the following auxiliary variables are introduced: Wheelchair body m 1 m 0 Figure 1: e vibration model of the convalescent-wheelchair system with DA.
Wheelchair body m 1 Figure 2: e vibration model of the convalescent-wheelchair system without DA. (7) can be further expressed as where From Equation (8), obtain the following: e amplitude frequency characteristics function J 1 between the road velocity input _ q and the wheelchair body acceleration € z 1 can be expressed as e amplitude frequency characteristics function J 2 between the road velocity input _ q and the convalescent acceleration € z 2 can be expressed as e amplitude frequency characteristics function J 3 between the road velocity input _ q and the TDD (tire dynamic deflection) f d � z 1 − q can be expressed as Shock and Vibration Substituting Equation (11) in Equation (12) produces Substituting Equation (10) in Equation (13) produces Substituting Equation (11) in Equation (15) produces

Random Road Model.
e spatial frequency PSD G q (n) of the random road roughness q can be expressed as [22] where, n is the spatial frequency; n 0 is the spatial reference frequency and n 0 � 0.1 m −1 ; G q (n 0 ) is the road roughness coefficient; and w is the frequency index. When a wheelchair robot runs on the random road at a certain speed u, the time frequency of the road excitation is f � un. us, the time frequency PSD G q (f) is When w � 2, the time frequency PSD G q (f) can be expressed as e relationship between the time frequency PSD G q (f) of the random road roughness q and the time frequency PSD G _ q (f) of the random road excitation velocity _ q is given by

e PSD and RMS Responses Calculation. e autocorrelation function
e convalescent acceleration € z 2 (t) can be expressed as where h(t) is impulse response function. e PSD G € z 2 (ω) of the convalescent acceleration € z 2 (t) can be expressed as Substituting Equation (21) in Equation (24) produces e convalescent RMS acceleration σ € z 2 can be expressed as Substituting Equation (25) in Equation (26) produces Analogously, the wheelchair body RMS acceleration σ € z 1 can be expressed as e TDD RMS can be expressed as It is difficult to analytically calculate the Equations (27)-(29). e numerical integration method is adopted as where N is the number of the discrete frequency value; Δf is the frequency bandwidth; and i � 1, 2, 3, . . . , N. Table 1: e values of the parameters for the convalescentwheelchair system.

Parameter
Value

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 convalescentwheelchair system is given in Table 1 [21]. e simulation condition is set as: u � 0.5∼2.5m/s, the C grade road with G q (n 0 ) � 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 convalescentwheelchair system with DA and without DA, the values , and G f d were numerically calculated, respectively. e calculated results are shown in Figure 3. Figures 3(a)-3(c) show that the DA can effectively reduce the wheelchair body RMS acceleration σ € z 1 and the convalescent RMS acceleration σ € z 2 , and the TDD RMS σ f d is not improved obviously. Figure 3(d) shows that the peak value of the wheelchair body acceleration PSD G € z 1 is decreased by 40.2% by using the DA. Figure 3(e) illustrates that the DA reduces the convalescent acceleration PSD G €

Sensitivity Analysis of Vibration Responses to System Parameters
In order to analyze the effects of r 2 , r 0 , ξ 1 , ξ 2 , ξ 0 , f 0 , f 1 , and f 2 on σ € z 2 , σ € z 1 , and σ f d , the simulation calculation was carried out based on the vibration model of the convalescentwheelchair system with DA. e simulation condition is set as u � 0.1 m/s, the C grade road with G q (n 0 ) � 256 × 10 −6 m −3 . To represent the curve changes in the same diagram, the values of σ € z 2 , σ € z 1 , and σ f d are normalized on the basis of the baseline RMS values, respectively. e processing method is as follows: where the subscript "b" represents "baseline." In addition, to observe the performance changes in the local frequency band, the amplitude frequency characteristics functions J 1 , J 2 , and J 3 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. e values of the system parameters for the analysis are given in Table 2 [21].
e baseline values are converted from Table 1.   Figure 4 depicts the influences of r 2 on € z 1 , € z 2 , and f d . From Figure 4(a), it can be seen that increasing r 2 can almost reduce the amplitude of J 1 in the whole frequency domain. erefore, increasing r 2 is conducive to reducing the wheelchair body acceleration. Figure 4(b) illustrates that increasing r 2 can reduce the amplitude of J 2 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 r 2 , the low frequency resonance peak of J 3 increases sharply, while the high frequency formant decreases. erefore, too large r 2 is not conducive to improving the tire grounding safety. As shown in Figure 4(d), σ 1 and σ 2 are in inverse proportion to r 2 , and σ 3 has a minimum value for r 2 in the range of 2.5∼3.5. us, the increase of r 2 is mainly to reduce the values of σ 1 and σ 2 . Figure 5 shows the influences of r 0 on € z 1 , € z 2 , and f d . From Figure 5(a), it can be seen that increasing r 0 can effectively reduce the peak value of J 1 . Figure 5(b) shows that increasing r 0 can reduce the amplitude of J 2 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 r 0 is, the higher the formant of J 3 is. As shown in Figure 5(d), σ 1 and σ 2 are in inverse proportion to r 0 , σ 3 is proportional to r 0 , and σ 1 is most sensitive to the change of r 0 . us, the use of the DA with the larger mass helps to reduce the vibration of the convalescent and the wheelchair.

e Influences of the Damping Ratios
e Influences of the Damping Ratio ξ 2 . Figure 6 depicts the influences of ξ 2 on € z 1 , € z 2 , and f d . As shown in Figure 6(a), increasing ξ 2 can help to reduce the peak value of J 1 . From Figure 6(b), it can be seen that increasing ξ 2 can effectively reduce the low frequency resonance peak of J 2 , but the amplitude after 3.0 Hz increases. Figure 6(c) shows that the larger the value of ξ 2 is, the smaller the peak values of J 3 are, but the amplitude between the two formants increases obviously. Figure 5(d) illustrates that σ 1 is almost in inverse proportional to ξ 2 , there is a minimum value of σ 2 for the value of ξ 2 between 0.15∼0.25, and σ 3 has a minimum value when the value of ξ 2 belongs to 0.15∼0.25. e ride comfort requires that the cushion system has the smaller value of ξ 2 , however, the tire grounding safety and the operational reliability need the larger value of ξ 2 . us, taking the smaller value of ξ 2 , about 0.2, is helpful to improve comfort.

6.2.2.
e Influences of the Damping Ratio ξ 1 . Figure 7 depicts the influences of ξ 1 on € z 1 , € z 2 , and f d . From Figure 7(a), it can be seen that increasing ξ 1 can effectively suppress the peak value of J 1 , but the amplitude after 15 Hz increases. As shown in Figure 7(b), the increase of ξ 1 can effectively suppress the low frequency and the high frequency resonance peak values of J 2 , 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 ξ 1 can almost reduce the amplitude of J 3 in the full frequency domain. From Figure 7(d), it can be seen that both σ 2 and σ 3 are approximately in inverse proportion to ξ 1 , and when the value of ξ 1 belongs to 0.1∼0.2, a minimum value of σ 1 exists. us, the ride comfort and the tire grounding safety require that the tire system has the larger value of ξ 1 , however, the operational reliability needs the smaller value of ξ 1 .

6.2.3.
e Influences of the Damping Ratio ξ 0 . Figure 8 depicts the influences of ξ 0 on € z 1 , € z 2 , and f d . According to Figure 8(a), the amplitude of J 1 in the region of 8∼13 Hz increases with the increase of ξ 0 , and the amplitude decreases after 13 Hz. Figure 8(b) shows that increasing ξ 0 can effectively suppress the high frequency resonance peak value of J 2 . According to Figure 8(c), ξ 0 has obvious effect on the high frequency resonance peak value of J 3 . From Figure 8(d), it can be seen that, with the increase of ξ 0 , the value of σ 1 decreases. When ξ 0 belongs to 0.1∼0.2, σ 3 has a minimum value, and when ξ 0 belongs to 0.15∼0.25, σ 2 has a minimum value. When the value of ξ 0 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.

e Influences of the Natural Frequencies
e Influences of the Natural Frequency f 2 . Figure 9 depicts the influences of f 2 on € z 1 , € z 2 , and f d . From Figure 9(a), it can be seen that increasing f 2 is beneficial to suppress the resonance peak value of J 1 . Figures 9(b) and 9(c) show that increasing f 2 makes the low frequency formants of J 2 and J 3 increase sharply and shift rightward. As shown in Figure 9(d), σ 1 is almost in inverse proportion to f 2 , σ 2 is proportional to f 2 and σ 2 is most sensitive to the change of f 2 , and when f 2 belongs to 1.5∼2.5 Hz, σ 3 has a minimum value. erefore, adopting a soft cushion to reduce f 2 is an important measure to improve ride comfort.

6.3.2.
e Influences of the Natural Frequency f 1 . Figure 10 depicts the influences of f 1 on € z 1 , € z 2 , and f d . Figure 10(a) shows that with the increase of f 1 , the resonance peak value of J 1 increases sharply and moves to the right. Figure 10(b) illustrates that increasing f 1 is beneficial to reduce the peak value of the low frequency resonance peak value of J 2 , but the amplitude after 10 Hz increases significantly. Figure 10(c) shows that the increase of f 1 is conducive to significantly reducing the amplitude of J 3 from 0 Hz to 18 Hz, although the amplitude increases slightly after 18 Hz. Figure 10(d) shows that σ 1 and σ 2 are proportional to f 1 , and σ 3 nonlinearly decreases as f 1 increases. us, the adoption of soft tires is conducive to improving ride comfort and operational reliability.

6.3.3.
e Influences of the Natural Frequency f 0 . Figure 11 depicts the influences of f 0 on € z 1 , € z 2 , and f d . Figure 11(a) shows that the increase of f 0 has little effect on J 1 in the low frequency region 0∼5 Hz and the high frequency 25∼30 Hz. In the mid frequency region 5∼25 Hz, increasing f 0 is beneficial to the amplitude reduction of J 1 . Figure 11(b) shows that f 0 only obviously affects the amplitude of J 1 from 5 Hz to 20 Hz. Figure 11(c) clearly states that f 0 has a great influence on the high frequency resonance peak value of J 3 . Figure 11(d) illustrates that σ 1 nonlinearly decreases as f 0 increases, when the value of f 0 belongs to 8∼12, σ 2 has a minimum value, and when the value of f 0 belongs to 6∼8, σ 3 has a minimum value. us, when f 0 approximately equals to f 1 , the DA system can well give consideration to the comprehensive isolation performance.

Sensitivity Calculation.
Based on the influence trends analysis of the system parameters on the RMS responses in Sections 6.1 to 6.3, sensitivity calculation was carried out by