Osmotolerant plant growth promoting bacteria mitigate adverse effects of drought stress on wheat growth

Naoual Bouremani; Hafsa Cherif-Silini; Allaoua Silini; Nour El Houda Rabhi; Ali Chenari Bouket; Lassaad Belbahri; Naoual Bouremani; Hafsa Cherif-Silini; Allaoua Silini; Nour El Houda Rabhi; Ali Chenari Bouket; Lassaad Belbahri

doi:10.3934/microbiol.2024025

AIMS Microbiology

2024, Volume 10, Issue 3: 507-541. doi: 10.3934/microbiol.2024025

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Research article

Osmotolerant plant growth promoting bacteria mitigate adverse effects of drought stress on wheat growth

1.
Laboratory of Applied Microbiology, Department of Microbiology, Faculty of Natural and Life Sciences, University Ferhat Abbas of Setif-1, 19000 Setif, Algeria
2.
Department of Natural Sciences and Life, Abdelhafid Boussouf University Center-Mila, 43000 Mila, Algeria
3.
East Azarbaijan Agricultural and Natural Resources Research and Education Centre, Plant Protection Research Department, Agricultural Research, Education and Extension Organization (AREEO), Tabriz 5355179854, Iran
4.
University Institute of Teacher Education (IUFE), University of Geneva, 24 Rue du Général-Dufour, 1211 Geneva, Switzerland

Received: 02 March 2024 Revised: 07 June 2024 Accepted: 25 June 2024 Published: 09 July 2024

Drought stress represents a major constraint with significant impacts on wheat crop globally. The use of plant growth-promoting bacteria (PGPB) has emerged as a promising strategy to alleviate the detrimental impacts of water stress and enhance plant development. We investigated 24 strains from diverse ecosystems, assessed for PGP traits and tolerance ability to abiotic stresses (drought, salinity, temperature, pH, heavy metals, pollutants, herbicides, and fungicides). The most effective bacterial strains Providencia vermicola ME1, Pantoea agglomerans Pa, Pseudomonas knackmussi MR6, and Bacillus sp D13 were chosen. Furthermore, these strains exhibited PGP activities under osmotic stress (0, 10, 20, and 30% PEG-6000). The impact of these osmotolerant PGPBs on wheat (Triticum durum L.) growth under drought stress was assessed at two plant growth stages. In an in vitro wheat seed germination experiment, bacterial inoculation significantly enhanced germination parameters. In pot experiments, the potential of these bacteria was evaluated in wheat plants under three treatments: Well-watered (100% field capacity), moderate stress (50% FC), and severe stress (25% FC). Results showed a significant decline in wheat growth parameters under increasing water stress for uninoculated seedlings. In contrast, bacterial inoculation mitigated these adverse effects, significantly improving morphological parameters and chlorophyll pigment contents under the stress conditions. While malondialdehyde (lipid peroxidation) and proline contents increased significantly with drought intensity, they decreased after bacterial inoculation. The antioxidant enzyme activities (GPX, CAT, and SOD) in plants decreased after bacterial inoculation. The increased root colonization capacity observed under water stress was attributed to their ability to favorable adaptations in a stressful environment. This study highlighted the potential of selected PGPB to alleviate water stress effects on wheat, promoting practical applications aimed at enhancing crop resilience under conditions of water shortage.

Keywords:

Citation: Naoual Bouremani, Hafsa Cherif-Silini, Allaoua Silini, Nour El Houda Rabhi, Ali Chenari Bouket, Lassaad Belbahri. Osmotolerant plant growth promoting bacteria mitigate adverse effects of drought stress on wheat growth[J]. AIMS Microbiology, 2024, 10(3): 507-541. doi: 10.3934/microbiol.2024025

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Abstract

1. Introduction

During the vehicle movement, the performance of the vehicle is affected by various vehicle structures and functions, such as power steering system, suspension system, braking system, etc. Moreover, in complex and high-speed environment, the vehicle's vertical, roll and pitch displacements contain a strong coupling relationship. Therefore, considering the motion and coupling characteristics of vehicle structures is meaningful to investigate vehicle dynamics. On the basis of geometric structure parameters of vehicle system and the nonlinear characteristics of shock absorber and leaf spring, the authors in ^[1] establish a nonlinear dynamic model for heavy vehicle. The correctness of the dynamic model is verified by testing the vertical acceleration data of the driver's seat, front wheel, middle wheel and rear wheel. To investigate the longitudinal driving behaviors of vehicle dynamics in the platoons, by taking the acceleration capability of heavy-duty vehicle into account, numerous heavy-duty vehicle platoon models are proposed ^{[2,3,4,5,6,7,8,9,10]}. Furthermore, by considering the lateral and longitudinal displacement characteristics of the vehicle, ^[11] represents a 2 DOF model of the vehicle and two diver cab models with time delays. In order to further describe the vehicle dynamics characteristics, in accordance with the two driver cab models in ^[11]. ^[15] and ^[16] further investigate the nonlinear lateral dynamics of a 2 DOF vehicle model. Based on longitudinal vehicle dynamics and by analyzing the dynamic of engine, torque converter, tire and capacitor pack, the authors of ^[17] present a dynamic model for a heavy-duty vehicle.

On the other side of research, the above mentioned vehicle models are mostly used to evaluate vehicle lateral and longitudinal dynamics characteristics, the influence of vehicle lateral and yaw dynamics characteristics are not considered enough. In practice, the tires not only provide horizontal and vertical forces to the vehicle, but also give vertical forces to the suspension system, especially in complex driving situations such as lane changes, cornering, or obstacle avoidance. In these cases, the vehicle's vertical, roll, and pitch dynamics are clearly coupled with lateral and yaw motion. Due to large inertia, high center of gravity and high roll center, heavy vehicles have poor stability when entering a turn or lane change, and the three-way coupling effect is large. Therefore, it is necessary to establish a three dimensional coupled vehicle model and study the influence of steering process on vehicle dynamics. More recently, more and more works focus on the coupling property of the vehicle. To reflect the steering influence on the overall response of the vehicle, ^[18] designs a novel 4 DOF hydraulic power steering (HPS) system. Simultaneously, ^[18] develops a 24 DOF model by taking the HPS system, the steering hand wheel angle, rack displacement, and hand wheel angles into account. According to the nonlinear characteristics of suspension damping and tire stiffness, ^[19] establishes a nonlinear three-way coupled lumped parameter model, and an improved nonlinear delay preview driver model was proposed based on ^[11], which was connected with the TCLP model to form a driver-vehicle closed-loop system. ^[20] establishes a complete vehicle model of a heavy truck, which not only investigates the nonlinear characteristics of suspension damping and tire stiffness, but also contains a modified preview driver model with nonlinear time delays to calculate the right front wheel steering angle for driving the vehicle along the desired route. In this paper, the kinematics and dynamics equations of cab and body are established by analyzing the three-way coupling effect of cab and body, as well as the dynamic characteristics of tire and suspension. Firstly, the dynamic relation of the tyre with deflection angle is introduced. Secondly, the coupling dynamics equation of cab was established by analyzing the three-way coupling effect of cab. Then, considering the dynamic characteristics of the vehicle suspension, the three-way coupling dynamic equation of the vehicle body is established. Finally, the kinematic and dynamic equations of cab and body are established based on the dynamic characteristics of tire and suspension and the Euler rotation theorem.

2. Nomenclature

Table 1. The symbols of the heavy-duty vehicle.

Definition	Symbol
forward traction (lateral traction) of the $i-th$ tire	$F_{xi}(i=1, \cdots, 6)(F_{yi})$
steering angle of the $i-th$ wheel	$\delta_i$
steering angular speed of the front axle tires	$\omega_t$
transverse (longitudinal) component of $i-th$ tire along the coordinate system $\{B\}$	$F_{Xi}(F_{Yi})$
suspension force, damping coefficient and spring constant of the $j-th$ spring between cab and body	$F_{cj}, H_{cj}$ and $K_{cj}$ $(j=1, 2, 3, 4)$
vertical displacement of the cab (body)	$z_c(z_b)$
pitching angle of the cab (body)	$\varphi_c(\varphi_b)$
roll displacements of the cab (body)	$\phi_c(\phi_b)$
longitudinal distance between origin of coordinates $\{C\}$ and cab rear (front) spring	$l_5(l_6)$
the distance between the origin of coordinates $\{C\}$ and $\{B\}$	$l_4$
the angle between the origin of coordinate $\{B\}$ and the sprung mass bar center of suspension	$\varphi_0$
transverse distance betweencab front spring and rear spring	$b_c$
resultant force of the cab inthe direction of axes $X_C$ , $Y_C$ and $Z_C$	$F_{xc}, F_{yc}$ and $F_{zc}$
resultant moment of the cab in the direction of axes $X_C$ , $Y_C$ and $Z_C$	$N_{xc}, N_{yc}$ and $N_{zc}$
total mass of the vehicle, cab and body	$m_s, m_t$ and $m_b$
velocity vectors of the cab in the coordinate system $\{C\}$ and $\{B\}$	$u_c, v_c$ and $w_c$ , $u_b, v_b$ and $w_b$
roll angle rate, pitch angle rate and yaw angle rate of the coordinate system $\{C\}$ and $\{B\}$	$p_c, q_c$ and $r_c$ , $p_b, q_b$ and $r_b$
vertical and transverse distance from the origin of $\{C\}$ and $\{B\}$ to the center of gravity of cab	$h_{os}$ and $e_{os}$ , $h_{ob}$ and $e_{ob}$
moment of inertia of a vehicle about axle $X_C, Y_C,$ $Z_C, X_B, Y_B$ and $Z_B$	$I_{xxc}, I_{yyc}$ and $I_{zzc},$ $I_{xxb}, I_{yyb}$ and $I_{zzb}$
moment of inertia of the cab about the axis $X_C$ and $Y_C$ ( $X_B$ and $Y_B$ )	$I_{xxsc}$ and $I_{yysc}$ $(I_{xxsb}$ and $I_{yysb})$
integral of the product of the $X_C$ and $Y_C$ ( $X_B$ and $Y_B$ ) deviation of an area element in a vehicle	$I_{xzc}(I_{xzb})$
compression displacement of the $j-th$ spring between cab and body	$x_{cj}$
cab center of gravity (body center of gravity) to cab front and rear spring transverse distance	$l_7$ and $l_8$ $(l_{15}$ and $l_{16})$
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 tires	$l_9, l_{10}$ and $l_{11}$ $(l_{12}, l_{13}$ and $l_{14})$
distance between the front axle and the rear axle in a suspension system	$l_3$
resultant force of the cab in the direction of axes $X_B, Y_B$ and $Z_B$	$F_{xb}, F_{yb}$ and $F_{zb}$
suspension force, damping coefficient and springconstant of the $j-th$ spring between the front axle or rear axle and the body of the suspension system	$F_{sj}, H_{sj}$ and $K_{sj}$
longitudinal transverse distance from the center of gravity of suspension system to the front axis of the suspension system	$l_1$
pitching angle of the left and right balance bars of the suspension system	$\varphi_{p1}$ and $\varphi_{p2}$
vertical displacement of the $j-th$ axle	$z_{uj}$
angle of inclination of the $j-th$ wheel shaft	$\phi_{uj}$
lateral distance between the left and right springs of the suspension system	$b_{s1}, b_{s2}$ and $b_{s3}$
damping forces of front suspension left and right springs	$F_{d1}$ and $F_{d2}$
resultant moment of the cab in the direction of axes $X_B, Y_B$ and $Z_B$	$N_{xb}, N_{yb}$ and $N_{zb}$
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 bearing spring	$l_{17}, l_{18}$ and $l_{19}$ $(l_{22}, l_{23}$ and $l_{24})$
transverse (longitudinal) distance of body center of gravity to set 1, set 2 and set 3 tires	$l_{25}, l_{26}$ and $l_{27}$ $(l_{28}, l_{29}$ and $l_{30})$
transverse distance (longitudinal distance) between the body center of gravity and the cab front (rear) spring	$l_{20}(l_{21})$
longitudinal distance from the center of gravity of the suspension system to the center of gravity of the rear axle of the suspension	$l_2$

| Show Table

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3. Model building

In this paper, the kinematic characteristics of a heavy-duty vehicle are considered to construct a 26 DOF vehicle body and a cab model. As shown in Figs. 1-4, the considered heavy vehicle has one front axle and two rear axles, which is called a three-axial vehicle. The degrees of freedom are vertical, roll and pitch displacements of the diver cab, vehicle body, the vertical and roll motion of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. To further study the coupling property with each part, the vertical, roll and pitch motion of cab and body is modeled independently. Before introducing the related coordinate systems, the Euler's laws of motion is firstly given.

Figure 1. The established reference frame.

DownLoad: Full-Size Img PowerPoint

Figure 2. The top view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Figure 3. The lateral view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Figure 4. The top view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Lemma 3.1. Observed from an inertial reference frame, the force applied to a rigid body is equal to the product of the mass of the rigid body and the acceleration of the center of mass, i.e.

$F^e = ma_c$

where $F^e$ is the resultant external force of the rigid body, $m$ is the rigid body mass, and $a_c$ is the acceleration of center of mass.

Lemma 3.2. . The fixed point $O$ (for example, the origin) of an inertial reference frame is set as the reference point. The net external moment applied to the rigid body is equal to the time rate of change of the angular momentum, i.e.

$M_O = \frac{dL_O}{dt}$

where $M_O$ is the is the external torque at point $O$ , $L_O$ is the angular momentum at point $O$ .

3.1. Establishing the related coordinate systems

To analyze the motion of heavy-duty vehicle, the corresponding coordinate frames are elaborated to describe the movement of the vehicle and indicated in . The moving coordinate frame $\{B\}$ is fixed to the vehicle's body and is called the body-fixed reference frame. The second coordinate frame $\{C\}$ is fixed to the cab and is called the cab-fixed reference frame. The third coordinate frame $\{E\}$ is fixed to the earth and is called the earth-fixed reference frame. The last coordinate frame $\{T_i\}$ is fixed to the $i-th$ , $(i = 1, 2, 3, 4, 5, 6)$ tire and is called the tire-fixed reference frame. In this paper, we assume that the body axes $X_B, Y_B, Z_B,$ the tire axes $X_{T_i}, Y_{T_i},$ and the cab axes $X_C, Y_C, Z_C,$ of heavy vehicle coincide with the principal axes of inertia, which are usually defined as:

$\bullet$ $X_B/X_{T_i}/X_C$ -longitudinal axis (directed from aft of the body/tire/cab to front).

$\bullet$ $Y_B/Y_{T_i}/Y_C$ -transverse axis (directed to right side of body/tire/cab).

$\bullet$ $Z_B/ Z_C$ -normal axis (directed from top to bottom).

3.2. Analysis of tire motion characteristics

To extract the kinetic model for the considered three-axial heavy-duty vehicle, the coordinate frame $T_{i}$ is designed for each tire, the corresponding schematic diagram is shown in Fig. 2. By taking the yaw angle into account, the forces produced by the engine are transformed into the forward traction and longitudinal traction on the suspension of heavy-duty vehicle. Based on the coordinate frame and Fig. 2, the forward and lateral traction of each tire can be expressed as

$\begin{equation} \begin{array}{lll} F_{Xi} = F_{xi}cos\delta_i-F_{yi}sin\delta_i,\\ F_{Yi} = F_{yi}sin\delta_i-F_{yi}cos\delta_i,\\ \dot{\delta_1} = \dot{\delta_2} = \omega_t,\delta_{3,4,5,6} = 0,\\ i = 1,\cdots,6. \end{array} \end{equation}$

(3.1)

3.3. Analysis of cab motion characteristics

In this subsection, the vertical, roll and pitch motion of the cab are considered to further accurately reflect the performance of spring suspension force between the cab and the body in the actual scenario. In accordance with the coordinate frame $\{C\}$ and Figs. 3-4, the spring force between the cab and body can be given as

$\begin{equation} \begin{array}{lll} F_{c1} = &K_{c1}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)-\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c1}(-\dot{z}_c-\dot{\varphi}_c l_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)-\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.2)

$\begin{equation} \begin{array}{lll} F_{c2} = &K_{c2}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)+\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c2}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z_b}+\dot{\varphi}_b(l_4+l_5)+\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.3)

$\begin{equation} \begin{array}{lll} F_{c3} = &K_{c3}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)-\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c3}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)-\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.4)

$\begin{equation} \begin{array}{lll} F_{c4} = &K_{c4}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)+\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c4}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)+\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.5)

Furthermore, by viewing the diver cab as a rigid body and employing the Lemmas 3.1 and 3.2, the kinematical equation of diver cab model is

$\begin{equation} \begin{array}{lll} F_{xc} = m_t(\dot{u}_c-v_cr_c)+m_s(w_cq_c-\dot{q}_ch_{os}-p_cr_ch_{os}-e_{os}q^2_c), \end{array} \end{equation}$

(3.6)

$\begin{equation} \begin{array}{lll} F_{yc} = m_t(\dot{v}_c-u_cr_c)+m_s(\dot{p}_ch_{os}-w_cp_c+q_c(p_ce_{os}-r_ch_{os})) \end{array} \end{equation}$

(3.7)

$\begin{equation} \begin{array}{lll} F_{zc} = m_s(\dot{w}_c+v_cp_c-u_cq_c+h_{os}(q^2_c+p^2_c)+e_{os}(p_cr_c-\dot{q}_c)), \end{array} \end{equation}$

(3.8)

$\begin{equation} \begin{array}{lll} N_{xc} = &I_{xxc}\dot{p}_c-I_{xzc}(\dot{r}_c+p_cq_c)+(I_{xxsc}-I_{yysc} -m_sh^2_{os})q_cr_c\\&+m_sh_{os}(\dot{v}_c+u_cr_c-w_cp_c), \end{array} \end{equation}$

(3.9)

$\begin{equation} \begin{array}{lll} N_{yc} = &I_{yyc}\dot{q}_c-I_{xzc}(p^2_c-r^2_c)+(I_{yysc}-I_{xxsc})p_cr_c-m_sh_{os}(\dot{u}_c\\ &-v_cr_c+w_cq_c)-m_se_{os}(\dot{w}_c+v_cp_c-u_cq_c), \end{array} \end{equation}$

(3.10)

$\begin{equation} \begin{array}{lll} N_{zc} = &I_{zzc}\dot{r}_c+I_{xzc}(q_cr_c-\dot{p}_c)+(I_{xxsc}-I_{yysc}+m_se^2_{os})p_cq_c\\ &-m_se_{os}w_cp_c, \end{array} \end{equation}$

(3.11)

When the vehicle is moving, the spring will produce spring force whether it is in a state of compression or tension. However, the direction of the force is opposite, so this paper considers the sign function and the direction of the spring displacement to determine the direction of the spring force. In order to obtain the dynamic force equation of the vehicle cab and body, we assume that the cab and body mass are evenly distributed, that is, the transverse distance between the cab and body center of gravity from the left and right tires is the same. By considering the definition of resultant force and resultant moment, the kinetic formula of longitudinal, transverse and vertical forces acting on the cab, as well as the yaw, pitch and roll moments is described as

$\begin{equation} \begin{array}{lll} F_{xc} = &c(\delta_1)c(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)s(\varphi_c)c(\delta_1)\\ &-s(\delta_1)c(\phi_c))\sum^6_{i = 1}{F_{Yi}}+(s(\delta_1)s(\phi_c)\\ &+c(\phi_c)s(\varphi_c)c(\delta_1))(F_{c1}+sign(x_{c2})F_{c2}\\ &+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ \end{array} \end{equation}$

(3.12)

$\begin{equation} \begin{array}{lll} F_{yc} = &s(\delta_1)c(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)s(\varphi_c)s(\delta_1)\\ &+c(\delta_1)c(\phi_c))\sum^6_{i = 1}{F_{Yi}}+(c(\phi_c)s(\phi_c)s(\delta_1)\\ &-c(\delta_1)s(\phi_c))(F_{c1}+sign(x_{c2})F_{c2}\\ &+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ \end{array} \end{equation}$

(3.13)

$\begin{equation} \begin{array}{lll} F_{zc} = &-s(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)c(\varphi_c))\sum^6_{i = 1}{F_{Yi}}\\ &+(c(\phi_c)c(\varphi_c)(F_{c1}+sign(x_{c2})F_{c2}) \end{array} \end{equation}$

(3.14)

$\begin{equation} \begin{array}{lll} N_{xc} = (F_{c1}+F_{c2})l_7-(F_{c3}+F_{c4})l_8, \end{array} \end{equation}$

(3.15)

$\begin{equation} \begin{array}{lll} N_{yc} = (F_{c1}-F_{c2})l_5+(F_{c3}-F_{c4})l_6, \end{array} \end{equation}$

(3.16)

$\begin{equation} \begin{array}{lll} N_{zc} = &(F_{X2}-F_{X1})l_9+(F_{X4}-F_{X3})l_{10}\\ &(F_{X5}-F_{X6})l_{11}+(F_{Y1}+F_{Y2})l_{12}\\ &-(F_{Y3}+F_{Y4})l_{13}-(F_{Y5}+F_{Y6})l_{14}, \end{array} \end{equation}$

(3.17)

where $sign(\cdot)$ denotes the symbolic function, $c(\cdot) = cos(\cdot)$ and $s(\cdot) = sin(\cdot)$ .

3.4. Analysis of suspension

For the considered heavy duty vehicle, two hydraulic dampers are fixed to the left and right front suspensions, and the balanced suspension does not have any shock absorbers. Thus, to represent the force situation of leaf spring in suspension system, the damping force of the two hydraulic dampers is considered for the front axle. The kinetic equation is given by

$\begin{equation} \begin{array}{lll} F_{s1} = K_{s1}(-z_b-(\varphi_b-\varphi_0)l_1-z_{u1}+\frac{(\phi_b-\phi_{u1})b_{s1}}{2})+F_{d1}, \end{array} \end{equation}$

(3.18)

$\begin{equation} \begin{array}{lll} F_{s2} = K_{s2}(-z_b-(\varphi_b-\varphi_0)l_1-z_{u1}-\frac{(\phi_b-\phi_{u1})b_{s1}}{2})+F_{d2}, \end{array} \end{equation}$

(3.19)

$\begin{equation} \begin{array}{lll} F_{s3} = &K_{s3}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p1}l_3}{2}-z_{u2}+\frac{(\phi_b-\phi_{u2})b_{s2}}{2})\\ &+H_{s3}(-\dot{z}_b+\dot{\varphi}_bl_2-\frac{\dot{\varphi}_{p1}l_3}{2}-\dot{z}_{u2}+\frac{(\dot{\phi}_b-\dot{\phi}_{u2})b_{s2}}{2}), \end{array} \end{equation}$

(3.20)

$\begin{equation} \begin{array}{lll} F_{s4} = &K_{s4}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p2}l_3}{2}-z_{u2}+\frac{(\phi_b-\phi_{u2})b_{s2}}{2})\\ &+H_{s4}(-\dot{z}_b+\dot{\varphi}_bl_2-\frac{\dot{\varphi}_{p2}l_3}{2}-\dot{z}_{u2}-\frac{(\dot{\phi}_b-\dot{\phi}_{u2})b_{s2}}{2}), \end{array} \end{equation}$

(3.21)

$\begin{equation} \begin{array}{lll} F_{s5} = &K_{s5}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p1}l_3}{2}-z_{u3}+\frac{(\phi_b-\phi_{u3})b_{s3}}{2})\\ &+H_{s5}(-\dot{z}_b+\dot{\varphi}_bl_2+\frac{\dot{\varphi}_{p1}l_3}{2}-\dot{z}_{u3}+\frac{(\dot{\phi}_b-\dot{\phi}_{u3})b_{s3}}{2}), \end{array} \end{equation}$

(3.22)

$\begin{equation} \begin{array}{lll} F_{s6} = &K_{s6}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p2}l_3}{2}-z_{u3}+\frac{(\phi_b-\phi_{u3})b_{s3}}{2})\\ &+H_{s6}(-\dot{z}_b+\dot{\varphi}_bl_2+\frac{\dot{\varphi}_{p2}l_3}{2}-\dot{z}_{u3}-\frac{(\dot{\phi}_b-\dot{\phi}_{u3})b_{s3}}{2}), \end{array} \end{equation}$

(3.23)

3.5. Analysis of body motion characteristics

This is analogous to the diver cab part, taking the body as a rigid body and according to the Lemmas 3.1 and 3.2, the force equation of body model can be expressed as

$\begin{equation} \begin{array}{lll} F_{xb} = m_t(\dot{u}_b-v_br_b)+m_b(w_bq_b-\dot{q}_bh_{ob}-p_br_bh_{ob}-e_{ob}q^2_b), \end{array} \end{equation}$

(3.24)

$\begin{equation} \begin{array}{lll} F_{yb} = m_t(\dot{v}_b-u_br_b)+m_b(\dot{p}_bh_{ob}-w_bp_b+q_b(p_be_{ob}-r_bh_{ob})), \end{array} \end{equation}$

(3.25)

$\begin{equation} \begin{array}{lll} F_{zb} = m_b(\dot{w}_b+v_bp_b-u_bq_b)+h_{ob}(q^2_b+p^2_b)+e_{ob}(p_br_b-\dot{q}_b)), \end{array} \end{equation}$

(3.26)

$\begin{equation} \begin{array}{lll} N_{xb} = &I_{xxb}\dot{p}_b-I_{xzb}(\dot{r}_b+p_bq_b)+(I_{xxsb}-I_{yysb}-m_bh^2_{ob})q_br_b\\ &+m_bh_{ob}(\dot{v}_b+u_br_b-w_bp_b), \end{array} \end{equation}$

(3.27)

$\begin{equation} \begin{array}{lll} N_{yb} = &I_{yyb}\dot{q}_b-I_{xzb}(p^2_b-r^2_b)+(I_{yysb}-I_{xxsb})p_br_b-m_bh_{ob}(\dot{u}_b\\ &-v_br_b+w_bq_b)-m_be_{ob}(\dot{w}_b+v_bp_b-u_bq_b), \end{array} \end{equation}$

(3.28)

$\begin{equation} \begin{array}{lll} N_{zb} = &I_{zzb}\dot{r}_b+I_{xzb}(q_br_b-\dot{p}_b)+(I_{yysb}-I_{xxsb}+m_be^2_{ob})p_bq_b\\ &-m_be_{ob}w_bp_b, \end{array} \end{equation}$

(3.29)

Recalling the problem of spring force direction and the assumption of uniform distribution of body mass, the longitudinal, transverse and vertical forces acting on the body are:

$\begin{equation} \begin{array}{lll} F_{xb} = &c(\delta_1)c(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)s(\varphi_b)c(\delta_1)\\ &-s(\delta_1)c(\phi_b))\sum^6_{i = 1}F_{Yi}+(s(\delta_1)s(\phi_b)\\ &+c(\phi_b)s(\varphi_b)c(\delta_1))(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.30)

$\begin{equation} \begin{array}{lll} F_{yb} = &s(\delta_1)c(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)s(\varphi_b)s(\delta_1)\\ &+c(\delta_1)c(\phi_b))\sum^6_{i = 1}F_{Yi}+(c(\phi_b)s(\varphi_b)s(\delta_1)\\ &-c(\delta_1)s(\phi_b))(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.31)

$\begin{equation} \begin{array}{lll} F_{zb} = &-s(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)c(\varphi_b)\sum^6_{i = 1}F_{Yi}\\ &+(c(\phi_b)c(\varphi_b)(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.32)

$\begin{equation} \begin{array}{lll} N_{xb} = &-(F_{c1}+F_{c2})l_{15}+(F_{c3}+F_{c4})l_{16}\\ &+(F_{s1}+F_{s2})l_{17}+(F_{s3}+F_{s4})l_{18}\\ &-(F_{s5}+F_{s6})l_{19}, \end{array} \end{equation}$

(3.33)

$\begin{equation} \begin{array}{lll} N_{yb} = &(F_{c2}-F_{c1})l_{20}+(F_{c4}-F_{c3})l_{21}\\ &+(F_{s1}-F_{s2})l_{22}+(F_{s3}-F_{s4})l_{23}\\ &+(F_{s5}-F_{s6})l_{24},\end{array} \end{equation}$

(3.34)

$\begin{equation} \begin{array}{lll} N_{zb} = &(F_{X2}-F_{X1})l_{25}+(F_{X4}-F_{X3})l_{26}\\ &+(F_{X5}-F_{X6})l_{27}+(F_{Y1}+F_{Y2})l_{28}\\ &-(F_{Y3}+F_{Y4})l_{29}-(F_{Y5}+F_{Y6})l_{30}, \end{array} \end{equation}$

(3.35)

3.6. Diver cab and body nonlinear model

In this final subsection, a common type of diver cab and body model for heavy-duty vehicle is proposed.

In accordance with 3.1-3.17 and invoking the Euler rotation theorem, the dynamic and kinetic equation of driver cab are designed as

$\begin{equation} \begin{array}{lll} \dot{\eta}_c = J_c(\eta_c)\upsilon_c,\\ \dot{\upsilon}_c = G_c(\eta_c,\upsilon_c)[F^*_{Xi},F^*_{Yi}]^T+F_c(\upsilon)+u_c, \end{array} \end{equation}$

(3.36)

where $\eta_c = [x_c, y_c, z_c, \phi_c, \varphi_c, \delta_1]^T,$

$J_c(\eta_c) = \left[\begin{array}{ccc} J_1(\eta_c) & 0_{3\times3}\\ 0_{3\times3} & J_2(\eta_c)\\ \end{array}\right],$

$J_2(\eta_c) = \left[ \begin{array}{ccc} 1 & s(\phi_c)t(\varphi_c) & c(\phi_c)t(\varphi_c) \\ 0 & s(\phi_c) & -s(\phi_c)\\ 0 & s(\phi_c)/c(\varphi_c) & c(\phi_c)/c(\varphi_c) \end{array}\right],$

$G_c(\eta_c, \upsilon_c) = \left[\begin{array}{ccc} J_3(\eta_c)\\ 0_{3\times2}\\ \end{array}\right],$

$J_3(\eta_c) = \left[ \begin{array}{ccc} \frac{c(\delta_1)c(\varphi_c)}{m_t} & -s(\delta_1)c(\phi_c)+\frac{s(\phi_c)s(\varphi_c)c(\delta_1)}{m_t}\\ \frac{s(\delta_1)c(\varphi_c)}{m_t} & +c(\delta_1)c(\phi_c)+\frac{s(\phi_c)s(\varphi_c)s(\delta_1)}{m_t}\\ \frac{-s(\varphi_c)}{m_s} & \frac{s(\phi_c)c(\varphi_c)}{m_s} \end{array}\right],$

$F_c(\upsilon_c) = [F_{c1}(\upsilon_c), F_{c2}(\upsilon_c), F_{c3}(\upsilon_c), F_{c4}(\upsilon_c), F_{c5}(\upsilon_c), F_{c6}(\upsilon_c)]^T,$

$J_1(\eta_c) = \left[\begin{array}{ccc} c(\delta_1)c(\varphi_c) & -s(\delta_1)c(\phi_c)+s(\phi_c)s(\varphi_c)c(\delta_1)\\ s(\delta_1)c(\varphi_c) & c(\delta_1)c(\phi_c)+s(\phi_c)s(\varphi_c)s(\delta_1)\\ -s(\varphi_c) & s(\phi_c)c(\varphi_c) \end{array}\right.$

$\left.\begin{array}{ccc} s(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)c(\delta_1)\\ -c(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)s(\delta_1)\\ c(\phi_c)c(\varphi_c) \end{array}\right],$

$\upsilon_c = [u_c, v_c, w_c, p_c, q_c, r_c]^T$ , $F^*_{Xi} = \sum^6_{i = 1}F_{Xi}$ , $F^*_{Yi} = \sum^6_{i = 1}F_{Yi}$ ,

$u_c = \left[\begin{array}{cc} 0_{5\times 1} \\ \frac{((F_{X2}-F_{X1})l_9+(F_{X4}-F_{X3})l_{10}+(F_{X5}-F_{X6})l_{11}}{I_{zzc}}\\ \frac{+(F_{Y1}+F_{Y2})l_{12}-(F_{Y3}+F_{Y4})l_{13}-(F_{Y5}+F_{Y6})l_{14}))}{} \end{array} \right],$ $t(\cdot)$ represents the tangent function. The expansion equation of the matrix $F_c(\upsilon_c)$ is

$\begin{equation} \begin{array}{lll} F_{c1}(\upsilon_c) = &\frac{1}{m_t}[(s(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)c(\delta_1))*(F_{c1}\\ &+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ &-m_s(w_cq_c-\dot{q}_ch_{os}-p_cr_ch_{os}-e_{os}q^2_c)+m_tv_cr_c], \end{array} \end{equation}$

(3.37)

$\begin{equation} \begin{array}{lll} F_{c2}(\upsilon_c) = &\frac{1}{m_t}[(c(\phi_c)s(\varphi_c)s(\delta)-c(\delta)s(\phi_c))*(F_{c1}\\ &+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ &-m_s(\dot{p}_ch_{os}-w_cp_c+q_c(p_ce_{os}-r_ch_{os}))\\ &+m_tu_cr_c], \end{array} \end{equation}$

(3.38)

$\begin{equation} \begin{array}{lll} F_{c3}(\upsilon_c) = &\frac{1}{m_s}[c(\phi_c)c(\varphi_c)(F_{c1}+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}\\ &+sign(x_{c4})F_{c4})]-(v_cp_c-u_cq_c+h_{os}(q^2_c+p^2_c)\\ &+e_{os}(p_cr_c-\dot{q}_c)), \end{array} \end{equation}$

(3.39)

$\begin{equation} \begin{array}{lll}F_{c4}(\upsilon_c) = &\frac{1}{I_{xxc}}(F_{c1}+F_{c2})l_7-(F_{c3}+F_{c4})l_8\\ &-m_sh_{os}(\dot{v}_c+u_cr_c-w_cp_c)+I_{xzc}(\dot{r}_c+p_cq_c)\\ &-(I_{xxsc}-I_{yysc}-m_sh^2_{os})q_cr_c,\end{array} \end{equation}$

(3.40)

$\begin{equation} \begin{array}{lll} F_{c5}(\upsilon_c) = &\frac{1}{I_{yyc}}(I_{xzc}(p^2_c-r^2_c)-(I_{yysc}-I_{xxsc})p_cr_c\\ &+m_sh_{os}(\dot{u}_c-v_cr_c+w_cq_c)\\ &+m_se_{os}(\dot{w}_c+v_cp_c-u_cq_c)\\ &+(F_{c1}-F_{c2})l_5+(F_{c3}-F_{c4})l_6), \end{array} \end{equation}$

(3.41)

$\begin{equation} \begin{array}{lll} F_{c6}(\upsilon_c) = &\frac{1}{I_{zzc}}[m_se_{os}w_cp_c-I_{xzc}(q_cr_c-\dot{p}_c)\\ &-(I_{yysc}-I_{xxsc}+m_se^2_{os})p_cq_c],\end{array} \end{equation}$

(3.42)

According to kinetic equations (18)-(42) and employing the Euler rotation theorem, the dynamic and kinetic equation of body are designed as

$\begin{equation} \begin{array}{lll} \dot{\eta}_b = J_b(\eta_b)\upsilon_b,\\ \dot{\upsilon}_b = G_b(\eta_b,\upsilon_b)[F^*_{Xi},F^*_{Yi}]^T+F_b(\upsilon)+u_b, \end{array} \end{equation}$

(3.43)

where $\eta_b = [x_b, y_b, z_, b\phi_b, \varphi_b, \delta_1]^T,$ $J_b(\eta_b) = \left[\begin{array}{ccc} J_1(\eta_b) & 0_{3\times3}\\ 0_{3\times3} & J_2(\eta_b)\\ \end{array}\right],$ $G_b(\eta_b, \upsilon_b) = \left[\begin{array}{ccc} J_3(\eta_b)\\ 0_{3\times2}\\ \end{array}\right],$ $\upsilon_b = [u_b, v_b, w_b, p_b, q_b, r_b]^T$ , $F^*_{Yi} = \sum^6_{i = 1}F_{Xi}$ , $F_b(\upsilon_b) = [F_{b1}(\upsilon_b), F_{b2}(\upsilon_b), F_{b3}(\upsilon_b), F_{b4}(\upsilon_b), F_{b5}(\upsilon_b), F_{b6}(\upsilon_b)]^T,$ $J_2(\eta_b) = \left[ \begin{array}{ccc} 1 & s(\phi_b)t(\varphi_b) & c(\phi_b)t(\varphi_b) \\ 0 & s(\phi_b) & -s(\phi_b)\\ 0 & s(\phi_b)/c(\varphi_b) & c(\phi_b)/c(\varphi_b) \end{array}\right],$ $F^*_{Xi} = \sum^6_{i = 1}F_{Xi}$ , $J_3(\eta_b) = \left[ \begin{array}{ccc} \frac{c(\delta_1)c(\varphi_b)}{m_t} & -s(\delta_1)c(\phi_b)+\frac{s(\phi_b)s(\varphi_b)c(\delta_1)}{m_t}\\ \frac{s(\delta_1)c(\varphi_b)}{m_t} & +c(\delta_1)c(\phi_b)+\frac{s(\phi_b)s(\varphi_b)s(\delta_1)}{m_t}\\ \frac{-s(\varphi_b)}{m_b} & \frac{s(\phi_b)c(\varphi_b)}{m_b} \end{array}\right],$ $u_b = \left[\begin{array}{cc} 0_{5\times 1} \\ \frac{((F_{X2}-F_{X1})l_{25}+(F_{X4}-F_{X3})l_{26}+(F_{X5}-F_{X6})l_{27}}{I_{zzb}}\\ \frac{+(F_{Y1}+F_{Y2})l_{28}-(F_{Y3}+F_{Y4})l_{29}-(F_{Y5}+F_{Y6})l_{30})}{} \end{array} \right],$ $J_1(\eta_b) = \left[\begin{array}{ccc} c(\delta_1)c(\varphi_b) & -s(\delta_1)c(\phi_cb)+s(\phi_b)s(\varphi_b)c(\delta_1)\\ s(\delta_1)c(\varphi_b) & c(\delta_1)c(\phi_b)+s(\phi_b)s(\varphi_b)s(\delta_1)\\ -s(\varphi_b) & s(\phi_b)c(\varphi_b) \end{array}\right.$

$\left.\begin{array}{ccc} s(\delta_1)s(\phi_b)+c(\phi_b)s(\varphi_b)c(\delta_1)\\ -c(\delta_1)s(\phi_b)+c(\phi_c)s(\varphi_b)s(\delta_1)\\ c(\phi_b)c(\varphi_b) \end{array}\right],$

The expansion equation of the matrix ${F_b(\upsilon_b)}$ is

$\begin{equation} \begin{array}{lll} F_{b1}(\upsilon_b) = &\frac{1}{m_t}[(s(\delta_1)s(\phi_b)+c(\phi_b)s(\varphi_b)c(\delta_1))*(-F_{c1}\\ &-sign(x_{c2})F_{c2}-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})\\ &+F_{s1}+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}\\ &+sign(z_{u4})F_{s4}+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &-m_b(w_bq_b-\dot{q}_bh_{ob}-p_br_bh_{ob}-e_{ob}q^2_b)+m_tv_br_b], \end{array} \end{equation}$

(3.44)

$\begin{equation} \begin{array}{lll} F_{b2}(\upsilon_b) = &\frac{1}{m_t}[(c(\phi_b)s(\varphi_b)s(\delta)-c(\delta_1)s(\phi_b))*(-F_{c1}\\ &-sign(x_{c2})F_{c2}-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})\\ &+F_{s1}+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}\\ &+sign(z_{u4})F_{s4}+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &-m_b(\dot{p}_bh_{ob}-w_bp_b+q_b(p_be_{ob}-r_bh_{ob}))\\ &+m_tu_br_b], \end{array} \end{equation}$

(3.45)

$\begin{equation} \begin{array}{lll} F_{b3}(\upsilon_b) = &\frac{1}{m_b}[c(\phi_b)c(\varphi_b)(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &+m_b(u_bq_b-v_bp_b-h_{ob}(q^2_b+p^2_b)\\ &-e_{ob}(p_br_b-\dot{q}_b)), \end{array} \end{equation}$

(3.46)

$\begin{equation} \begin{array}{lll}F_{b4}(\upsilon_b) = &\frac{1}{I_{xxb}}[(F_{c3}+F_{c4})l_{16}-(F_{c1}+F_{c2})l_{15}\\ &+(F_{s1}+F_{s2})l_{17}+(F_{s3}+F_{s4})l_{18}\\ &-(F_{s5}+F_{s6})l_{19}+I_{xzb}(\dot{r}_b+p_bq_b)\\ &-(I_{xxsb}-I_{yysb}-m_bh^2_{ob})q_br_b\\ &-m_bh_{ob}(\dot{v}_b+u_br_b-w_bp_b)],\end{array} \end{equation}$

(3.47)

$\begin{equation} \begin{array}{lll} F_{b5}(\upsilon_b) = &\frac{1}{I_{yyb}}[(F_{c2}-F_{c1})l_{20}+(F_{c4}-F_{c3})l_{21}\\ &+(F_{s1}-F_{s2})l_{22}+(F_{s3}-F_{s4})l_{23}\\ &-(F_{s5}-F_{s6})l_{24}+I_{xzb}(p^2_b-r^2_b)\\ &-(I_{yysb}-I_{xxsb})p_br_b\\ &+m_bh_{ob}(\dot{u}_b-v_br_b+w_bq_b)\\ &+m_be_{ob}(\dot{w}_b+v_bp_b-u_bq_b)], \end{array} \end{equation}$

(3.48)

$\begin{equation} \begin{array}{lll} F_{b6}(\upsilon_b) = &\frac{1}{I_{zzb}}[m_be_{ob}w_bp_b-I_{xzb}(q_br_b-\dot{p}_b)\\ &-(I_{yysb}-I_{xxsb}+m_be^2_{ob})p_bq_b], \end{array} \end{equation}$

(3.49)

4. Conclusions

In complex working conditions, there is a coupling relationship of the vertical, lateral and longitudinal dynamics of vehicles. By considering the kinetic character of the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. In this paper, a common model of three-axles heavy-duty vehicle with 26 DOF have been proposed to extrude the kinetic characterization diver cab and vehicle body.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants U22A2043 and 62173172.

Conflict of interest

The author declares that there is no conflicts of interest in this paper.

Acknowledgments

The study is supported by the Algerian Ministry of Higher Education and Scientific Research.

Conflict of interest

The authors declare no conflict of interest.

Author contributions

Conceived and designed the experiments: NB CHS, and AS. Performed the experiments: NB and CHS. Analyzed the data: CHS, ACB, NEHR, and AS. Wrote and enriched the literature: NB, CHS. Read and review the manuscript: CHS, AS, and LB. All authors read and approved the final manuscript.

References

[1]	Gowtham HG, Singh SB, Shilpa N, et al. (2022) Insight into recent progress and perspectives in improvement of antioxidant machinery upon PGPR augmentation in plants under drought stress: A review. Antioxidants 11: 1763. https://doi.org/10.3390/antiox11091763
[2]	Hossain A, Hassan Z, Sohag MH, et al. (2023) Impact of the endophytic and rhizospheric bacteria on crop development: prospects for advancing climate-smart agriculture. J Crop Sci Biotechnol 26: 405-431. https://doi.org/10.1007/s12892-023-00195-3
[3]	Papadopoulou A, Theodora M, Nathalie K, et al. (2022) Decoding the potential of a new Pseudomonas Putida strain for inducing drought tolerance of tomato (Solanum Lycopersicum) plants through seed biopriming. J Plant Physiol 271: 153658. https://doi.org/10.1016/j.jplph.2022.153658
[4]	Chieb M, Gachomo EW (2023) The role of plant growth promoting rhizobacteria in plant drought stress responses. BMC Plant Biol 23: 407. https://doi.org/10.1186/s12870-023-04403-8
[5]	Romero-Munar A, Aroca R, Zamarreño AM, et al. (2023) Dual inoculation with Rhizophagus irregularis and Bacillus megaterium improves maize tolerance to combined drought and high temperature stress by enhancing root hydraulics, photosynthesis and hormonal responses. Int J Mol Sci 24: 5193. https://doi.org/10.1186/s12870-023-04403-8
[6]	Shirmohammadi E, Alikhani HA, Pourbabaei AA, et al. (2020) Improved phosphorus (p) uptake and yield of rainfed wheat fed with p fertilizer by drought-tolerant phosphate-solubilizing fluorescent pseudomonads strains: A field study in drylands. J Soil Sci Plant Nutr 20: 2195-2211. https://doi.org/10.1007/s42729-020-00287-x
[7]	Drought and agricultureFood and Agriculture Organization of the United Nations. Available from: https://www.fao.org/land-water/water/drought/droughtandag/en/
[8]	Gao X, Luan J, Wang L, et al. (2023) Effect of the plant growth promoting rhizobacterium, Cronobacter sp. Y501, for enhancing drought tolerance in maize (Zea mays L.). J Soil Sci Plant Nutr 23: 2786-2797. https://doi.org/10.1007/s42729-023-01234-2
[9]	Arora S, Jha PN (2023) Drought-tolerant Enterobacter bugandensis WRS7 induces systemic tolerance in Triticum aestivum L. (wheat) under drought conditions. J Plant Growth Regul 42: 7715-7730. https://doi.org/10.1007/s00344-023-11044-6
[10]	Kour D, Yadav AN (2022) Bacterial mitigation of drought stress in plants: Current perspectives and future challenges. Curr Microbiol 79: 248. https://doi.org/10.1007/s00284-022-02939-w
[11]	Jha Y, Yadav KA, Mohamed HI (2023) Plant growth-promoting bacteria and exogenous phytohormones alleviate the adverse effects of drought stress in pigeon pea plants. Plant Soil . https://doi.org/10.1007/s11104-023-06155-8
[12]	Akhtar N, Ilyas N, Mashwani Z, et al. (2021) Synergistic effects of plant growth promoting rhizobacteria and silicon dioxide nano-particles for amelioration of drought stress in wheat. Plant Physiol Biochem 166: 160-176. https://doi.org/10.1016/j.plaphy.2021.05.039
[13]	Valizadeh-rad K, Motesharezadeh B, Alikhani HA, et al. (2023) Morphophysiological and nutritional responses of canola and wheat to water deficit stress by the application of plant growth-promoting bacteria, nano-silicon, and silicon. J Plant Growth Regul 42: 3615-3631. https://doi.org/10.1007/s00344-022-10824-w
[14]	Nikhil PT, Faiz U, Mohapatra S (2023) The drought-tolerant rhizobacterium, Pseudomonas putida AKMP7, suppresses polyamine accumulation under well-watered conditions and diverts putrescine into GABA under water-stress, in Oryza sativa. Enviro Exp Bot 211: 105377. https://doi.org/10.1016/j.envexpbot.2023.105377
[15]	Joshi B, Chaudhary A, Singh H, et al. (2020) Prospective evaluation of individual and consortia plant growth promoting rhizobacteria for drought stress amelioration in rice (Oryza sativa L.). Plant Soil 457: 225-240. https://doi.org/10.1007/s11104-020-04730-x
[16]	Gul F, Khan IU, Rutherford S, et al. (2023) Plant growth promoting rhizobacteria and biochar production from Parthenium hysterophorus enhance seed germination and productivity in barley under drought stress. Front Plant Sci 14: 1175097. https://doi.org/10.3389/fpls.2023.1175097
[17]	Umapathi M, Chandrasekhar CN, Senthil A, et al. (2022) Isolation, characterization and plant growth-promoting effects of sorghum [Sorghum bicolor (L.) Moench.] root-associated rhizobacteria and their potential role in drought mitigation. Arch Microbiol 204: 354. https://doi.org/10.1007/s00203-022-02939-1
[18]	Yaghoubian I, Modarres-Sanavy SAM, Smith DL (2022) Plant growth promoting microorganisms (PGPM) as an eco-friendly option to mitigate water deficit in soybean (Glycine max L.): Growth, physio-biochemical properties and oil content. Plant Physiol Biochem 191: 55-66. https://doi.org/10.1016/j.plaphy.2022.09.013
[19]	Ansari FA, Jabeen M, Ahmad I (2021) Pseudomonas azotoformans FAP5, a novel biofilm-forming PGPR strain, alleviates drought stress in wheat plant. Int J Environ Sci Technol 18: 3855-3870. https://doi.org/10.1007/s13762-020-03045-9
[20]	Singh D, Thapa S, Yadav J, et al. (2023) Deciphering the mechanisms of microbe mediated drought stress alleviation in wheat. Acta Physiol Plant 45: 81. https://doi.org/10.1007/s11738-023-03562-3
[21]	Ilyas N, Mumtaz K, Akhtar N, et al. (2020) Exopolysaccharides producing bacteria for the amelioration of drought stress in wheat. Sustainability 12: 8876. https://doi.org/10.3390/su12218876
[22]	Yasmin H, Rashid U, Hassan MN, et al. (2021) Volatile organic compounds produced by Pseudomonas pseudoalcaligenes alleviated drought stress by modulating defense system in maize (Zea mays L.). Physiol Plant 172: 896-911. https://doi.org/10.1111/ppl.13304
[23]	Cherif-Silini H, Silini A, Yahiaoui B, et al. (2016) Phylogenetic and plant-growth-promoting characteristics of Bacillus isolated from the wheat rhizosphere. Ann Microbiol 66: 1087-1097. https://doi.org/10.1007/s13213-016-1194-6
[24]	Kerbab S, Silini A, Bouket AC, et al. (2021) Mitigation of NaCl Stress in wheat by rhizosphere engineering using salt habitat adapted PGPR halotolerant bacteria. Appl Sci 11: 1034. https://doi.org/10.3390/app11031034
[25]	Balla A, Silini A, Cherif-Silini H, et al. (2022) Screening of cellulolytic bacteria from various ecosystems and their cellulases production under multi-stress conditions. Catalysts 12: 769. https://doi.org/10.3390/catal12070769
[26]	Boulahouat S, Cherif-Silini H, Silini A, et al. (2022) Critical Evaluation of biocontrol ability of bayoud infected date palm phyllospheric Bacillus spp. suggests that in vitro selection does not guarantee success in planta. Agronomy 12: 2403. https://doi.org/10.3390/agronomy12102403
[27]	Cherif-Silini H, Thissera B, Bouket AC, et al. (2019) Durum wheat stress tolerance induced by endophyte Pantoea agglomerans with genes contributing to plant functions and secondary metabolite arsenal. Int J Mol Sci 20: 3989. https://doi.org/10.3390/ijms20163989
[28]	Rabhi NEH, Cherif-Silini H, Silini A, et al. (2019) Alleviation of salt stress via habitat-adapted symbiosis. Forests 13: 586. https://doi.org/10.3390/f13040586
[29]	Borriss R, Chen XH, Rueckert C, et al. (2011) Relationship of Bacillus amyloliquefaciens clades associated with strains DSM 7T and FZB42T: A proposal for Bacillus amyloliquefaciens subsp. amyloliquefaciens subsp. nov. and Bacillus amyloliquefaciens subsp. plantarum subsp. nov. based on complete genome sequence comparisons. Int J Syst Evol Microbiol 61: 1786-1801. https://doi.org/10.1099/ijs.0.023267-0
[30]	Olsen SR, Sommers LE (1982) Phosphorus. Methods of Soil Analysis Part 2 Chemical and Microbiological Properties. Madison: American Society of Agronomy, Soil Science Society of America 403-430. https://doi.org/10.2134/agronmonogr9.2.2ed.c24
[31]	Schwyn B, Neilands JB (1987) Universal chemical assay for the detection and determination of siderophores. Anal Biochem 160: 47-56. https://doi.org/10.1016/0003-2697(87)90612-9
[32]	Slama HB, Triki MA, Bouket AC, et al. (2019) Screening of the high-rhizosphere competent Limoniastrum monopetalum' culturable endophyte microbiota allows the recovery of multifaceted and versatile biocontrol agents. Microorganisms 7: 249. https://doi.org/10.3390/microorganisms7080249
[33]	Dworkin M, Foster J (1958) Experiments with some microorganisms which utilize ethane and hydrogen. J Bacteriol 75: 592-603. https://doi.org/10.1128/jb.75.5.592-603.1958
[34]	Li Z, Chang S, Lin L, et al. (2011) A colorimetric assay of 1-aminocyclopropane-1-carboxylate (ACC) based on ninhydrin reaction for rapid screening of bacteria containing ACC deaminase. Lett Appl Microbiol 53: 178-185. https://doi.org/10.1111/j.1472-765X.2011.03088.x
[35]	Zeriouh H, de Vicente A, Pérez-García A, et al. (2014) Surfactin triggers biofilm formation of Bacillus subtilis in melon phylloplane and contributes to the biocontrol activity. Environ Microbiol 16: 2196-2211. https://doi.org/10.1111/1462-2920.12271
[36]	Li X, Peng D, Zhang Y, et al. (2020) Klebsiella sp. PD3, a phenanthrene (PHE)-degrading strain with plant growth promoting properties enhances the PHE degradation and stress tolerance in rice plants. Ecotoxicol Environ Saf 201: 110804. https://doi.org/10.1016/j.ecoenv.2020.110804
[37]	Saadaoui N, Silini A, Cherif-Silini H, et al. (2022) Semi-arid-habitat-adapted plant-growth-promoting rhizobacteria allows efficient wheat growth promotion. Agronomy 12: 2221. https://doi.org/10.3390/agronomy12092221
[38]	Dubois M, Gilles KA, Hamilton JK, et al. (1956) Colorimetric method for determination of sugars and related substances. Anal Chem 28: 350-356. https://doi.org/10.1021/ac60111a017
[39]	Lowry OH, Rosbrough NJ, Farr AL, et al. (1951) Protein measurement with the folin phenol reagent. J Bio Chem 193: 265-275. https://doi.org/10.1016/S0021-9258(19)52451-6
[40]	Murashige T, Skoog F (1962) A revised medium for rapid growth and bioassays with tobacco tissue cultures. Physiol Plant 15: 473-497. https://doi.org/10.1111/j.1399-3054.1962.tb08052.x
[41]	Cherif-Silini H, Silini A, Bouket AC, et al. (2021) Tailoring next generation plant growth promoting microorganisms as versatile tools beyond soil desalinization: A road map towards field application. Sustainability 13: 4422. https://doi.org/10.3390/su13084422
[42]	Rabhi NEH, Silini A, Cherif-Silini H, et al. (2018) Pseudomonas knackmussii MLR6, a rhizospheric strain isolated from halophyte, enhances salt tolerance in Arabidopsis thaliana. J Appl Microbiol 125: 1836-1851. https://doi.org/10.1111/jam.14082
[43]	Noha MA, Bothaina AA, Shereen AM, et al. (2022) Utilization of drought-tolerant bacterial strains isolated from harsh soils as a plant growth-promoting rhizobacteria (PGPR). Saudi J Biol Sci 29: 1760-1769. https://doi.org/10.1016/j.sjbs.2021.10.054
[44]	Al-Shwaiman HA, Shahid M, Elgorban AM, et al. (2022) Beijerinckia fluminensis BFC-33, a novel multi-stress-tolerant soil bacterium: Deciphering the stress amelioration, phytopathogenic inhibition and growth promotion in Triticum aestivum (L.). Chemosphere 295: 133843. https://doi.org/10.1016/j.chemosphere.2022.133843
[45]	Silambarasan S, Logeswari P, Vangnai AS, et al. (2022) Plant growth-promoting actinobacterial inoculant assisted phytoremediation increases cadmium uptake in Sorghum bicolor under drought and heat stresses. Environ Pollut 307: 119489. https://doi.org/10.1016/j.envpol.2022.11948
[46]	Khan A, Singh AV (2021) Multifarious effect of ACC deaminase and EPS producing Pseudomonas sp. and Serratia marcescens to augment drought stress tolerance and nutrient status of wheat. World J Microbiol Biotechnol 37: 198. https://doi.org/10.1007/s11274-021-03166-4
[47]	Latif M, Bukhari SAH, Alrajhi AA, et al. (2022) Inducing drought tolerance in wheat through exopolysaccharide-producing rhizobacteria. Agronomy 12: 1140. https://doi.org/10.3390/agronomy12051140
[48]	Gowtham HG, Brijesh SS, Murali M, et al. (2020) Induction of drought tolerance in tomato upon the application of ACC deaminase producing plant growth promoting rhizobacterium Bacillus subtilis Rhizo SF 48. Microbiol Res 234: 126422. https://doi.org/10.1016/j.micres.2020.126422
[49]	Rashid U, Yasmin H, Hassan MN, et al. (2022) Drought-tolerant Bacillus megaterium isolated from semi-arid conditions induces systemic tolerance of wheat under drought conditions. Plant Cell Rep 41: 549-569. https://doi.org/10.1007/s00299-020-02640-x
[50]	Zhao T, Deng X, Xiao Q, et al. (2020) IAA priming improves the germination and seedling growth in cotton (Gossypium hirsutum L.) via regulating the endogenous phytohormones and enhancing the sucrose metabolism. Ind Crops Prod 155: 112788. https://doi.org/10.1016/j.indcrop.2020.112788
[51]	Kasim WA, Osman MEH, Omar MN, et al. (2021) Enhancement of drought tolerance in Triticum aestivum L. seedlings using Azospirillum brasilense NO40 and Stenotrophomonas maltophilia B11. Bull Natl Res Cent 45: 95. https://doi.org/10.1186/s42269-021-00546-6
[52]	Batool T, Ali S, Seleiman MF, et al. (2020) Plant growth promoting rhizobacteria alleviates drought stress in potato in response to suppressive oxidative stress and antioxidant enzymes activities. Sci Rep 10: 16975. https://doi.org/10.1038/s41598-020-73489-z
[53]	Khalilpour M, Mozafari V, Abbaszadeh-Dahaji P (2021) Tolerance to salinity and drought stresses in pistachio (Pistacia vera L.) seedlings inoculated with indigenous stress-tolerant PGPR isolates. Sci Hort 289: 110440. https://doi.org/10.1016/j.scienta.2021.110440
[54]	Murali M, Singh SB, Gowtham HG, et al. (2021) Induction of drought tolerance in Pennisetum glaucum by ACC deaminase producing PGPR-Bacillus amyloliquefaciens through Antioxidant defense system. Microbiol Res 253: 126891. https://doi.org/10.1016/j.micres.2021.126891
[55]	Ma Y, Rajkumar M, Moreno A, et al. (2017) Serpentine endophytic bacterium Pseudomonas azotoformans ASS1 accelerates phytoremediation of soil metals under drought stress. Chemosphere 185: 75-85. https://doi.org/10.1016/j.chemosphere.2017.06.135
[56]	Zarei T, Moradi A, Kazemeini SA, et al. (2020) The role of ACC deaminase producing bacteria in improving sweet corn (Zea mays L. var. Saccharata) productivity under limited availability of irrigation water. Sci Rep 10: 20361. https://doi.org/10.1038/s41598-020-77305-6
[57]	Khan N, Bano A (2019) Exopolysaccharide producing rhizobacteria and their impact on growth and drought tolerance of wheat grown under rainfed conditions. PLoS One 14: 0222302. https://doi.org/10.1371/journal.pone.0222302
[58]	Sheteiwy MS, Abd Elgawad H, Xiong YC, et al. (2021) Inoculation with Bacillus amyloliquefaciens and mycorrhiza confers tolerance to drought stress and improve seed yield and quality of soybean plant. Physiol Plant 172: 2153-2169. https://doi.org/10.1111/ppl.13454
[59]	Sood G, Kaushal R, Sharma M (2020) Significance of inoculation with Bacillus subtilis to alleviate drought stress in wheat (Triticum aestivum L.). Vegetos 33: 782-792. https://doi.org/10.1007/s42535-020-00149-y
[60]	Han L, Zhang M, Du L, et al. (2022) Effects of Bacillus amyloliquefaciens QST713 on photosynthesis and antioxidant characteristics of alfalfa (Medicago sativa L.) under drought stress. Agronomy 12: 2177. https://doi.org/10.3390/agronomy12092177
[61]	Niu X, Song L, Xiao Y, et al. (2018) Drought-tolerant plant growth-promoting rhizobacteria associated with foxtail millet in a semi-arid agroecosystem and their potential in alleviating drought stress. Front Microbiol 8: 2580. https://doi.org/10.3389/fmicb.2017.02580
[62]	Rashid U, Yasmin H, Hassan M N, et al. (2022) Drought-tolerant Bacillus megaterium isolated from semi-arid conditions induces systemic tolerance of wheat under drought conditions. Plant Cell Rep 41: 549-569. https://doi.org/10.1007/s00299-020-02640-x
[63]	Singh RP, Jha PN (2017) The PGPR Stenotrophomonas maltophilia SBP-9 augments resistance against biotic and abiotic stress in wheat plants. Front Microbiol 8: 1945. https://doi.org/10.3389/fmicb.2017.01945

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Osmotolerant plant growth promoting bacteria mitigate adverse effects of drought stress on wheat growth

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3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

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1. Introduction

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3. Model building

3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

3.3. Analysis of cab motion characteristics

3.4. Analysis of suspension

3.5. Analysis of body motion characteristics

3.6. Diver cab and body nonlinear model

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Acknowledgment

Conflict of interest

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AIMS Microbiology

Osmotolerant plant growth promoting bacteria mitigate adverse effects of drought stress on wheat growth

Related Papers:

Abstract

1. Introduction

2. Nomenclature

3. Model building

3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

3.3. Analysis of cab motion characteristics

3.4. Analysis of suspension

3.5. Analysis of body motion characteristics

3.6. Diver cab and body nonlinear model

4. Conclusions

Acknowledgment

Conflict of interest

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Abstract

1. Introduction

2. Nomenclature

3. Model building

3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

3.3. Analysis of cab motion characteristics

3.4. Analysis of suspension

3.5. Analysis of body motion characteristics

3.6. Diver cab and body nonlinear model

4. Conclusions

Acknowledgment

Conflict of interest

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