Abstract
To assess the effect of the massstiffness distribution of the cab and vibratory roller on the ride quality, the accelerationfrequency characteristics of the isolation systems are researched based on the nonlinear dynamic model of the vehicle. The complex domain is used to solve the vibration equations in the frequency region. The effect of the mass and stiffness parameters of the cab and vehicle is then analyzed via MATLAB software. The research results show that the resonant frequency of the cab's isolations and their accelerationfrequency is insignificantly affected by the massstiffness distributions in the case of without the excitation of the vibrator drum. Conversely, under an excitation 28 Hz of the vibrator drum, the massstiffness distributions strongly affect on the accelerationfrequency of the cab isolations, especially, when increasing the stiffness coefficient and reducing the cab mass with $\alpha =$ 1.5, the vehicle’s ride quality and structural durability of the vibratory rollers are greatly improved.
Highlights
 The vibration problem of the vibratory rollers is solved.
 The resonant frequency of the cab's isolations and their accelerationfrequency is insignificantly affected by the massstiffness distributions in the case of without the excitation of the drum.
 The massstiffness distributions strongly affect on the accelerationfrequency of the cab isolation under the excitation of drum.
 When increasing the stiffness coefficient x= 1.5 and reducing the cab mass x = 1.5, the vehicle's ride quality and structural durability of the vibratory rollers are greatly improved.
1. Introduction
The cab isolation system of the vibratory rollers was used to reduce the transmission vibration from the vibrator drum and the road surface to the driver [1]. In the previous researches of vibratory rollers showed that the vehicle's ride comfort was strongly affected by the vibration excitations of the drum and road surface [13], thus, the cab isolations were continuously researched and developed. The structure of the traditional rubber mounts equipped on the vibratory roller cab was redesigned and optimized to improve the ride comfort [2]. However, the obtained performance of those optimal rubber mounts was very small. Thus, the hydraulic and pneumatic mounts were researched [3]. The result showed that the performance of the hydraulic mounts (HM) was better than other isolations on improving the ride comfort. The HM was then controlled by using the optimal FLCPID controller to further improve the ride comfort [46]. Besides, a damper hydraulic mount was also added in the cab isolations to enhance the ride comfort [7, 8]. A damper mount in the horizontal direction was also added in the cab isolations [9]. Its damping coefficient was then controlled to reduce the cab shaking of vibratory rollers [10]. To further improve the ride comfort and control the cab shaking, the hydropneumatic mounts were researched for the vibratory roller cab [1113]. All the research results showed that the driver's ride comfort and cab shaking were greatly improved by applying the optimal control for the cab isolations. However, in all the existed researches, the vibration equations of the vibratory rollers and their isolations were mostly solved in the time region.
The ride quality was not only affected by the vibration excitation in the time region but also greatly affected by the frequency excitation below 10 Hz [14, 15]. To reduce the effect of the resonant under the lowfrequencies, the specific vibration frequency of the cab isolations should be designed in a range below 2.5 Hz [16]. However, this specific vibration frequency was strongly depended on the cab mass and stiffness of the isolation system. Thus, it is necessary to study the effect of the massstiffness distribution of the cab and isolations on the accelerationfrequency characteristics to reduce the vibration resonance and enhance the stability and durability. Therefore, a nonlinear vibratory roller dynamic model is established to analyze the effect of the massstiffness distribution on the characteristics of the accelerationfrequency of the cab isolations in the lowfrequency region. The complex domain is then applied to calculate the vehicle’s vibration equations. The mass and stiffness parameters of the vehicle and isolations are simulated to evaluate the results.
2. Vehicle model and vibration equation in the time region
Based on the actual structure of the vibratory roller, a nonlinear vehicle model is then established to calculate the different dynamic equations of the vehicle, as modeled in Fig. 1, where $\left\{{z}_{s},{z}_{c},{z}_{b}\right\}$ and $\left\{{\phi}_{c},{\phi}_{b}\right\}$ are the vertical motions of the driver’s seat, cab, vehicle body and the pitching motions of the cab and vehicle body. $\left\{{c}_{s},{c}_{c\mathrm{1,2}},{c}_{d},{c}_{t}\right\}$ and $\left\{{k}_{s},{k}_{c\mathrm{1,2}},{k}_{d},{k}_{t}\right\}$ are the damping and stiffness coefficients of the isolations of the driver’s seat, cab, vibrator drum, and tire, respectively. ${m}_{s}$, ${m}_{c}$, and ${m}_{b}$ are the mass of the driver’s seat, cab, vehicle body, respectively. ${l}_{17}$ is the longitudinal distances of the vehicle. ${z}_{d}$ and ${q}_{t}$ are the vibration excitations.
Fig. 1The nonlinear dynamic model of the vibratory roller
Based on the vehicle model, the different dynamic equations are given as follows:
In previous researches, the vibration of vibratory rollers was mainly solved in the time region on improving the ride comfort [13, 89], the accelerationfrequency characteristic of the isolations of the cab and the vehicle in the frequency region gets very little attention. In this study, the accelerationfrequency characteristic of isolations is studied to analyze the results.
3. Vibration equations in the frequency region
To research the characteristics of the amplitudefrequency, it must be to calculate via the transfer function. Thus, a variable $s=d/dt$ is applied to convert the unknown of the differential equations in the time region to the image function in the frequency region. Then, the numerical equations are calculated to search the transfer function of the Laplace. Finally, the transfer function of the frequency is determined by replacing $s=j\omega $, $\omega $ is the excitation frequency.
Based on the Laplace operator [14, 16], it is assumed that $f\left(t\right)$ with the variable $t\ge $ 0 is the original function and continuous in the range of [0, +∞], the image of $f\left(t\right)$ and its derivative $n$ via Laplace transformation is given by:
$\text{or}\mathrm{}f\left(t\right)\to F\left(s\right)\Rightarrow {f}^{\left(n\right)}\left(t\right)\to {s}^{n}F\left(s\right){s}^{n1}F\left(0\right)...F\left(0\right).$
It is assumed that at the initial time $t=$ 0, the displacement and velocity of the vehicle is also equal to zero, thus, based on Eq. (2), the unknowns in Eq. (1) are described as follows:
Thus, Eq. (1) described via Laplace "$s$" and replaced by $s=j\omega $ and ${s}^{2}={\left(j\omega \right)}^{2}={\omega}^{2}$, we have:
where:
${a}_{14}={a}_{15}={a}_{41}={a}_{51}=0,{a}_{21}={a}_{12},{a}_{34}={a}_{23},{a}_{43}={a}_{23},{a}_{53}={a}_{35},$
$\begin{array}{l}{a}_{22}={m}_{c}{\omega}^{2}+\left({c}_{s}+{c}_{c1}+{c}_{c2}\right)j\omega +\left({k}_{s}+{k}_{c1}+{k}_{c2}\right),\\ {a}_{23}=\left({c}_{c1}{l}_{2}{c}_{c2}{l}_{3}\right)j\omega +\left({k}_{c1}{l}_{2}{k}_{c2}{l}_{3}\right),{a}_{24}={a}_{42}=\left({c}_{c1}{c}_{c2}\right)j\omega +\left({k}_{c1}{k}_{c2}\right),\\ {a}_{25}={a}_{52}=\left({c}_{c1}{l}_{6}{c}_{c2}{l}_{4}\right)j\omega +\left({k}_{c1}{l}_{6}{k}_{c2}{l}_{4}\right),{a}_{31}={c}_{c1}{l}_{1}j\omega {k}_{c1}{l}_{1},\\ {a}_{32}=\left({c}_{s}{l}_{1}{c}_{c1}{l}_{2}+{c}_{c2}{l}_{3}\right)j\omega +\left({k}_{s}{l}_{1}{k}_{c1}{l}_{2}+{k}_{c2}{l}_{3}\right),\\ {a}_{33}={I}_{c}{\omega}^{2}\left({c}_{s}{l}_{1}^{2}+{c}_{c1}{l}_{2}^{2}+{c}_{c2}{l}_{3}^{2}\right)j\omega \left({k}_{s}{l}_{1}^{2}+{k}_{c1}{l}_{2}^{2}+{k}_{c2}{l}_{3}^{2}\right),\\ {a}_{35}=\left({c}_{c1}{l}_{2}{l}_{6}{c}_{c2}{l}_{3}{l}_{4}\right)j\omega +\left({k}_{c1}{l}_{2}{l}_{6}{k}_{c2}{l}_{3}{l}_{4}\right),\\ {a}_{44}={m}_{b}{\omega}^{2}+\left({c}_{c1}+{c}_{c2}+{c}_{d}+{c}_{t}\right)j\omega +\left({k}_{c1}+{k}_{c2}+{k}_{d}+{k}_{t}\right),\\ {a}_{45}=\left({c}_{c1}{l}_{6}+{c}_{c2}{l}_{4}{c}_{t}{l}_{5}+{c}_{d}{l}_{7}\right)j\omega +\left({k}_{c1}{l}_{6}+{k}_{c2}{l}_{4}{k}_{t}{l}_{5}+{k}_{d}{l}_{7}\right),\\ {a}_{54}=\left({c}_{c1}{l}_{6}+{c}_{c2}{l}_{4}+{c}_{t}{l}_{5}+{c}_{d}{l}_{7}\right)j\omega +\left({k}_{c1}{l}_{6}+{k}_{c2}{l}_{4}+{k}_{t}{l}_{5}+{k}_{d}{l}_{7}\right),\\ {a}_{55}={I}_{b}{\omega}^{2}+\left({c}_{c1}{l}_{6}^{2}+{c}_{c2}{l}_{4}^{2}+{c}_{t}{l}_{5}^{2}+{c}_{d}{l}_{7}^{2}\right)j\omega +\left({k}_{c1}{l}_{6}^{2}+{k}_{c2}{l}_{4}^{2}+{k}_{t}{l}_{5}^{2}+{k}_{d}{l}_{7}^{2}\right),\\ {b}_{41}={c}_{d}j\omega {k}_{d};{b}_{42}={c}_{t}j\omega {k}_{t},{b}_{51}={c}_{d}{l}_{7}j\omega +{k}_{d}{l}_{7},{b}_{52}={c}_{t}{l}_{5}j\omega +{k}_{t}{l}_{5}.\end{array}$
By dividing the Eq. (4) for the ${Q}_{1}\left(j\omega \right)$, the form of the matrix of Eq. (4) is then written by:
$A=\left[\begin{array}{lllll}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}& {a}_{15}\\ {a}_{21}& {a}_{22}& {a}_{23}& {a}_{24}& {a}_{25}\\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}& {a}_{35}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}& {a}_{45}\\ {a}_{51}& {a}_{52}& {a}_{53}& {a}_{54}& {a}_{55}\end{array}\right]$
where ${H}_{s}\left(j\omega \right)={Z}_{s}\left(j\omega \right)/{Q}_{t}\left(j\omega \right)\text{;}$${H}_{c}\left(j\omega \right)={Z}_{c}\left(j\omega \right)/{Q}_{t}\left(j\omega \right)\text{;}$${H}_{\phi c}\left(j\omega \right)={\mathrm{\Phi}}_{c}\left(j\omega \right)/{Q}_{t}\left(j\omega \right)\text{;}$${H}_{b}\left(j\omega \right)={Z}_{b}\left(j\omega \right)/{Q}_{t}\left(j\omega \right)\text{;}$${H}_{\phi b}\left(j\omega \right)={\mathrm{\Phi}}_{b}\left(j\omega \right)/{Q}_{t}\left(j\omega \right)$ are the vibration transfer functions from the road surface to the driver's seat, cab pitch angle, vehicle’s body, and vehicle pitch angle, respectively.
The matrix $A$ is defined based on the dynamic parameters of the vibratory roller, thus, the transfer functions of ${H}_{s}\left(j\omega \right)$, ${H}_{c}\left(j\omega \right)$, ${H}_{\phi c}\left(j\omega \right)$, ${H}_{b}\left(j\omega \right)$, and ${H}_{\phi b}\left(j\omega \right)$ are determined. The complex number values of ${H}_{i}$ in Eq. (5) are calculated by ${H}_{i}={A}_{i}+j{B}_{i}$, $i=s$, $c$, ${\phi}_{c}$, $b$, ${\phi}_{b}$, the absolute values of ${H}_{i}$ that expresses the relationship of the amplitudeexcitation frequency and their acceleration amplitudeexcitation frequency are given by:
4. Vibration excitations
In this study, two vibration types are considered including excitation of the vibrator drum at a frequency ${f}_{d}$ of 28 or 35 Hz and another vibration excitation from the road surface when the vehicle moving.
The excitation of the vibrator drum is determined by [3, 4]:
where ${m}_{e}$ and $e$ are the mass and eccentricity of the vibration excitation of the vibratory drum.
Besides, the vibration excitation from the road surface is also described as follows.
In the design of the isolation systems, the resonance frequency is always of particular concern. It is simulated and calculated based on the different vibration excitations of the system. The previous studies showed that the harmonicfunction profile of the road surface often causes resonant vibrations in the isolation system, and it was used to evaluate the driver's ride comfort and health as well as the safety of designed structures [14, 16]. In this study, to evaluate the effect of the massstiffness distribution of the cab isolations on the driver’s ride comfort and safety of designed structures, the harmonic function of the road surface in Eq. (8) is used.
where ${q}_{0}$ and $T$ are the high road surface and frequency excitation of the road surface.
From Eqs. (78), their Laplace transformation in complex number domain is written by:
Based on the vibration excitation in Eq. (9) and dynamic parameters of the vehicle in Ref. [3], the calculation in the complex number domain of Eq. (5) are then performed to evaluate the accelerationfrequency characteristics of the isolations of the cab and the vehicle.
5. Simulations and discussions
The resonant frequency of the cab isolations, $f={\left(k/m\right)}^{1/2}$, depends on the cab mass and stiffness parameter of the isolation system. Thus, to optimize the driver's seat comfort and safety; as well as enhance the stability and durability of the structures, the effect of the massstiffness distribution on the characteristics of the accelerationfrequency need to be simulated and analyzed under the vibration excitation of the road surface and the vibrator drum. The dynamic parameters of the vibratory roller in Ref. [3], an excitation 28 Hz of the vibrator drum, and the vehicle speed at 6 km/h is used for the simulation process. Three cases of the increase of the stiffness coefficient $k=\alpha \times {k}_{c\mathrm{1,2}}$ and reduction of the cab mass $m={m}_{c}/\alpha $, with $\alpha =$ 0.5, 1.0, 1.5 is simulated, respectively.
Fig. 2The accelerationfrequency responses without the excitation of the vibrator drum
a) The vertical driver’s seat
b) The pitching cab angle
Fig. 3The accelerationfrequency responses with an excitation of the vibrator drum, 28 Hz
a) The vertical driver’s seat
b) The pitching cab angle
Without the excitation of the vibrator drum, the simulation results of the accelerationfrequency of the driver’s seat and pitching cab angle are plotted in Fig. 2. The results show that the resonant frequencies with their high amplitude mainly appear in a range below 2.5 Hz. Thus, the specific vibration frequency of the cab isolations satisfies the design condition [15]. When the stiffness coefficient $k$ is increased by $\alpha =$ 0.5, 1.0, 1.5, concurrently, the cab mass is reduced by $\alpha =$ 0.5, 1.0, 1.5, respectively, the accelerationfrequency characteristic of the driver's seat is insignificantly changed, while that of the pitching cab angle appears at 1.273 Hz of $\alpha =$ 0.5 and 2.069 Hz at $\alpha =$ 1.0, 1.5 in a range below 2.5 Hz. However, these changes also insignificantly affect on the change of the high amplitude of the accelerationfrequency at the resonant frequencies. Thus, the change of the massstiffness distribution is insignificantly impacted on the ride quality as well as the safety and durability of the isolation’s structure.
With an excitation frequency, 28 Hz, of the vibrator drum, the simulation results in Fig. 3 indicate that the high amplitude of the accelerationfrequency characteristics of the driver's seat and pitching cab angle also mainly appear in a range below 2.5 Hz. However, under the effect of an excitation frequency 28 Hz and massstiffness distribution of the cab and cab’s isolations, the high amplitude of accelerationfrequency characteristics of the driver’s seat and pitching cab angle at $\alpha =$ 0.5 is strongly increased in comparison with the initial values $\alpha =$ 1.0. Thus, the reduction of the cab isolation's stiffness coefficient ${k}_{c\mathrm{1,2}}$ and the increase of the cab mass ${m}_{c}$ may have a negative impact on the ride quality and structural durability. Conversely, the increase of the stiffness coefficient and the reduction of the cab mass with $\alpha =$ 1.5, the high amplitude of accelerationfrequency characteristics of the driver’s seat and pitching cab angle is greatly reduced under various excitation frequencies. This research result has important implications for improving the ride quality and structural durability of the vibratory rollers, because the vibratory roller uses most of its time working on the workshop at an excitation 28 Hz of the vibrator drum [4, 5]. These results are similar to the results of vehicle's suspension systems in lowfrequency below 4 Hz [14, 17]. It showed that to reduce the amplitude of the accelerationfrequency, the suspension stiffness needs to be increased, conversely, in the frequency range above 10 Hz, the suspension stiffness needs to be reduced to decrease the noise. To solve this problem, a pneumatic suspension with the changed airbag stiffness was applied [11, 12, 17].
6. Conclusions
A new simulation method based on the Laplace transfer function to research the vibratory roller's vibration in the frequency region is applied to analyze the vehicle's ride quality.
Without the excitation of the drum, the massstiffness distribution of the cab and cab’s isolation insignificantly impacts on the accelerationfrequency of the cab isolations.
With the excitation 28 Hz of the drum, the accelerationfrequency characteristic are strongly reduced with the increases of the stiffness coefficient and the reduction of the cab mass at $\alpha =$ 1.5, thus, the ride quality and structural durability of vibratory rollers are greatly improved.
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About this article
This work has been supported by the Key Scientific Research Project of Hubei Polytechnic University, China (No. 18XJZ02A) and National College Student Innovation and Entrepreneurship Training Project, China (No. 201910920001, No. 202010920003).