Speed Oscillations of a Vehicle Rolling on a Wavy Road

Round 1

Reviewer 1 Report

The paper presents a study on the speed oscillation of a quarter car model on a wavy road. Analytical results are presented for several cases

Overall the paper is well written and the topic could be of interest to the readers of the journal. However some aspects should be improved. Specifically:

1) The author should expand the introduction section presenting more clearly the aim of the work and the novelty  in comparison to the literature 

2) throughout the manuscript The author should highlight the novel results obtained 

3) figures should be improved since they contain too many information and are difficult to be understood 

4) the part on the random road profile should be expanded and more clearly presented 

Author Response

Authors reply to Review Report Two

The author thanks the reviewer for his most valuable reviewing work which certainly help to improve the quality of the paper.

The corrections proposed by the reviewer in line 42, 52, 56, 73, 126, 175.202 318, 319, 335, 339 and 540 are inroduced in the paper marked by red colour in line 42, 52, 55, 76, 129, 181, 207, 340, 341, 344 and 540.

The comments of the reviewer with respect to the parameter values in Fig. 2a and Fig. 2b are realized in the figure captions in line 139 and 207, respectively. 

The first caption in Fig. 2a is extended by the wording „… for driving force values marked on the thick red characteristic by the green, cyan and yellow triangles, respectively”. One finds the numerical values 0.1 , 0.2 and 0.3 explicitly on the scale left of the Fig. 2a. The same holds in Fig. 2b for the numerical values 0.3, 0.6, 0.9, 1.2  in the under-critical speed range and the driving force values 0.7, 0.8, 0.9, 1.0 in the over-critical speed range. An explicit notation of all these 8 values will overload the Fig. 2b.   

Author Response File: Author Response.pdf

Reviewer 2 Report

Scientific content of the manuscript is of good quality. Reviewer recommends manuscript for publication.

Minor revisions are required or recommended:

- Line 42: “and al.: à “et al.”

- Line 52 –

“level z” – Maybe author means the vertical displacement of the road profile, “level” sounds unclear in the text;

“slope u” should be explained in the text or in Fig. 1;

Line 56 – Variable “s” should be of italic font.

Line 73 – “Ω is the road frequency“ – Denotation omega is specified in ISO 8608:2016 as „angular spatial frequency“. The correct terminology should be used. Road frequency sounds unclear.

Line 126 – typos error – “rspectively.”

Line 175 – “Figure 2 (a) shows three limit cycles in the stable under-critical speed range for the driving forces ?Ω?⁄=0.1,0.2 and 0.3 marked by green, cyan and yellow triangles,“

COMMENT: The parameters (0.1, 0.2, 0.3) of the curves in FIg. 2a should be inserted into Fig. 2a or in the figure caption.

Line 202 – „They are obtained for ?Ω?⁄=2.7,3.3,3.9 and 4.5 202 marked by green, yellow, red and cyan colour, respectively.“

COMMENT: The parameters (2.7,3.3,3.9 and 4.5) of the curves in FIg. 2b should be inserted into Fig. 2b or in the figure caption.

Line 318, 319 -

Text “??<? determines“ should be probably of normal font instead of bold.

Line 335-339 -

Variables „?“ „ Nn“ and  ?,  should be probably of normal font instead of bold.

Line 540 -

„[22] Wedig W.V. (2012) Simulation of Road-Vehicle Systems, Prob. Eng. Mech., 27, 82-87.“

Author Response

Authors reply to Review Report One

The author thanks the reviewer for his most valuable reviewing work which certainly helps to improve the quality of the paper.

Accordingly the following aspects are improved:

• The author expands the introduction section in order to present more clearly the aim and the novelty by means of the following text beginning in line 50 of the revised paper:  

Extensions to quarter car models with two degrees of freedom are made. The new speed amplitudes calculated in this paper show that the longitudinal speed oscillations of the vehicle are stable in the lower speed range before the resonance speed and in the upper higher speed range. In the middle range immediately after the resonance peak, stationary speeds are unstable and therefore physically not realizable. These stability properties correspond to the Duffing problem where vertical displacement vibrations of the vehicle possess three different amplitudes in the resonant speed range: the upper and lower displacement vibrations are stable and the middle ones are unstable, as well.

• throughout the manuscript the author should highlight the novel results obtained. This is done in line 264 by introducing the notation of multiplicative ergodicity into nonlinear vehicle dynamics.

Note that the four-dimensional time system (13) and (14) is not ergodic since the polar angle   is a state variable which grows infinitely by permanent rotation. This disadvantage is avoided by eliminating the time variable by means of the polar angle that leads to the three-dimensional angle system (15) and (16) where the polar angle now represents the independent integration variable restricted to one periodic interval. The solutions of this new time-free equation system are ergodic and multiplicative ergodic theorems are applicable in order to calculate characteristic numbers of the dynamic system of interest. 

3) figures should be improved since they contain too many information:

The paper has 12 figures. 6 Figures 1 (a), 3 (a) and (b), 4 (b), 5 (a) and 6 (a) are improved.  

4) the part on the random road profile is expanded. This is done in line 376 with the following text:

Note that Fig. 4 (a) and (b) present new results in nonlinear stochastic vehicle dynamics obtained applying noise perturbations which are bounded by means of sinusoidal. For small intensity σ, the limit flow in Fig. 4 (a) possesses periodic side limits in spite of the fact that all limit cycle realizations are random. This is a new effect presented in this paper. For growing intensity σ, the inner side limits of the limit cycle shown in Fig. 4 (a) become broader and disappear finally such that outer side limits remain, only, and the whole phase plane is covered by realizations of velocity and acceleration .

Author Response File: Author Response.pdf

Reviewer 3 Report

Comment

Introduction

1. In the introduction, the author concentrates on the problem without providing state of the art. The author uses a quarter vehicle model. Using such a model, it is difficult to achieve results, which would adequately repeat the vehicle dynamics. If the only longitudinal motion is taken into account, some simplification may be applied; however, even though the quarter car model does not take into account pitch. An additional explanation is required.
2. Line 40, 41 „First investigations of velocity jumps and turbulent speeds in nonlinear vehicle road dynamics are given by Wedig in [3, 4, 5, 6, 7] applying sinusoidal and random road models“the author cites five papers of his own without utilisation what was achieved, in current form it looks like self-citation
3. Line 49 prposed – should be proposed
4. 8 publications in the reference list are research papers of the author. It is almost 35%, in all these publications there are no co-authors.
5. More than 60% of the reference list is older than five years.

Section 2

1. The author uses different markings of elements than it is used in literature, c – stiffness and b – damping, in majority of literature k – is stiffness, and c – is damping
2. There is no citation where the author takes equations from
3. Equation 2 is not an equation of a quarter car. The quarter car model considers tyre stiffness, sometimes damping, unsprung mass, suspension damping and stiffness, a quarter of sprung mass. Equation 2 represents only part of the equation for unsprung mass acceleration. Suspension stiffness and damping are not taken into account; unsprung mass weight is not taken into account. In fig. 1 a) we can see that it is not a quarter car model as well. This part should be revised.
4. In equation 1, air drag force is not taken into account. It allows to simplify the solution; however, it gives an error if we compare them with experimental data. And as a result authors statement in line 128 „needs a linearly growing drivng force to reach higher speeds of operation.“ is not correct.

Summarising

The correctness of the solution of the problem is not questioned. However, assumptions made in the model make me doubt that it is applicable for vehicle dynamics.

Author Response

Authors reply to Review Report Three

The author thanks the reviewer for his most valuable reviewing work which certainly helps to improve the quality of the paper.

Introduction

According to the authors experience driving on dessert roads with wavy profile, there is a speed blockade immediately before the resonance speed where the car gets stuck similarly to the Sommerfeld effect in rotor dynamics. The wavy road profile comes up from heavy vehicle machines applied for the maintenance of dessert roads.  Accordingly, one can drive only slöwly with less than 50 km/h or much faster than the resonance speed in the range of 90 km/h. The drive through the resonance is critical because there are high vertical car vibrations and coupled with strong longitudinal speed oscillations.  Up to now, there are a few new papers which model this nonlinear coupling and investigate the resulting Sommerfeld effect that the car gets stuck before the resonance speed and can not pass over the resonance peak in case of sufficiently small car damping when the car dampers are burned out after a long driving on uneven roads.

Of course, this stucking effect of the car can not be explained by an additional air drag force which are  negligible small in the low speed range around 30 km/h in comparison with the full tracking force of  the car motor when the car gets stuck before resonance speed. The same is true for the pitch of cars.  The author tries to explain this Sommerfeld effect of cars by means of new quarter car models  with longitudinal coupling as shwon in Fig. 1 (a) of the paper.  This represents the minimal quarter car model with one and a half degree of freedom. The model has the advantage that all results are analytical so that the reader can understand all derivations in detail. In Fig. 5 (a) and 6 (a), the minimal quarter car model is extended  to higher models with suspension stiffness and damping which have the disadvantage that numerical results are availabe, only.

1. Additional explanation is introduced into the introduction marked by red colour.
2. In particular, it is noted that the new results, obtained, are similar to the Duffing problem.
3. Line 49 prposed is corrected by proposed and in the revised text marked by red colour.

Section 2

1. The author uses different markings of elements than it is used in literature, c – stiffness and b – damping, in majority of literature k – is stiffness, and c – is damping

The markings c – stiffness and β,b – damping are used in the Russian literature [13,14] in Procedia Engineering,199 and in Journal of Sound and Vibration, 405 as well as in the German literature [2, 4] in Journal Nonlinear Dynamics, 100 and in Journal of Dynamics and Control, 4 where the reader finds further basics of the applied new quarter car model with nonlinear along-across coupling.

2. There is no citation where the author takes equations from

The new quarter car model with nonlinear along-across coupling of motion is described by Eq. (1) and (2) which represent the two-dimensional dynamic equilibrium of mass-point applied in the literature mentioned above. Basic elements of both equations one finds in every text-book.

3. Equation 2 is not an equation of a quarter car. The quarter car model considers tyre stiffness, sometimes damping, unsprung mass, suspension damping and stiffness, a quarter of sprung mass. Equation 2 represents only part of the equation for unsprung mass acceleration. Suspension stiffness and damping are not taken into account; unsprung mass weight is not taken into account. In fig. 1 a) we can see that it is not a quarter car model as well. This part should be revised.

Equation 2 represents the minimal model of the quarter car  shown in Fig. 1 (a).  Stiffness and damping for the tyre and the suspension are extensions of this minimal model. Fig. 5 (a) shows this extended quarter car model. It is described by Eq. (30, 31) and adapted to the nonlinear along-across coupling by Eq.  (32).  Unsprung mass weight do not influence mean value and  amplitudes of the longitudinal speed oscillations of the driving car.

4. In equation 1, air drag force is not taken into account. It allows to simplify the solution; however, it gives an error if we compare them with experimental data. And as a result authors statement in line 128 „needs a linearly growing drivng force to reach higher speeds of operation.“ is not correct.

Air drag force grows quadratically with increasing speed. According to Fig. 1 (b) obtained by Eq. (12), damping resistance grows linearly in high speed range. Before the resonance in low speed ranges, however, there is a resonance resistance generated by the along-across motion coupling which grows by the rational peak much stronger than air resistance. In the limiting case of vanishing car damping, the resonance resistance becomes infinitly big, the car gets stuck before the resonance speed and can not pass the resonance peak. This effect can not be explained by means of air resistance.  Eq. (12) can be found in the literature mentioned above. New second order results of Eq. 12 are given in Eq. (23) which is derived by means of Fourier series and represents the drag force of the car needed to maintain its speed constant. .  

Authors final remark:

The enclosed revised form of the paper contains all proposals of reviewer 1,2 and 3 marked by red colour

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

This Reviewer's comments have been addressed and the paper can be considered for publication

Reviewer 3 Report

There is no experimental data to see if the theory works. 
 From my point of view, paper should be rejected.