Classification of Road Profiles on the basis of Degree of Roughness by using Displacement PSD Approach
AIM:
Using the PSD data given, determine the road quality using the ISO classification ISO 8608.These are your deliverables:
Download the PSD data from here: https://drive.google.com/file/d/0B2S2ZJKcURAmeDNFRlMtMm5MSTQ/view
1. Plot showing ISO classification for roads A through H (Use higher end of range for degree of roughness)
2. Plot showing the PSD profile
3. Overlay of PSD profile on ISO classification and determination of road quality
4. Error free Octave code
THEORY & CONCEPT:
Ride as a Quality Performance Parameter:
Automobiles travel at high speed, and as a consequence experience a broad spectrum of vibrations. These are transmitted to the passengers either by tactile(by touch), visual, or aural paths. The term "ride" is commonly used in reference to tactile and visual vibrations, while the aural vibrations are categorized as "noise."
Alternatively, the spectrum of vibrations may be divided up according to frequency and classified as ride (0-25 Hz) and noise (25-20,000 Hz). The 25 Hz boundary point is approximately the lower frequency threshold of hearing, as well as the upper frequency limit of the simpler vibrations common to all motor vehicles. The different types of vibrations are usually so interrelated that it may be difficult to consider each separately, noise is usually present when lower-frequency vibrations are excited.
The vibration environment is one of the more important criteria by which people judge the design and construction "quality" of a car. Being a judgment, it is subjective in nature, from which arises one of the greatest difficulties in developing objective engineering methods for dealing with ride as a performance mode of the vehicle.
The lower-frequency ride vibrations are manifestations of dynamic behaviour common to all rubber-tired motor vehicles. Thus, the study of these modes is an important area of vehicle dynamics. As an aid in developing a systematic picture of ride behavior, it is helpful to think of the overall dynamic system. The vehicle is a dynamic system, but only exhibits vibration in response to excitation inputs. The response properties determine the magnitude and direction of vibrations imposed on the passenger compart ment, and ultimately determine the passenger's perception of the vehicle.
Thus, understanding ride involves the study of three main topics:
• Ride excitation sources
• Basic mechanics of vehicle vibration response
• Human perception and tolerance of vibrations
There are multiple sources from which vehicle ride vibrations may be excited. These generally fall into two classes-road roughness and on-board sources. The on-board sources arise from rotating components and thus include the tire/wheel assemblies, the driveline, and the engine.
Road Roughness
Road roughness encompasses everything from potholes resulting from localized pavement failures to the ever-present random deviations reflecting the practical limits of precision to which the road surface can be constructed and maintained. Roughness is described by the elevation profile along the wheel tracks over which the vehicle passes. Road profiles fit the general category of "broad-band random signals" and, hence, can be described either by the profile itself or its statistical properties. One of the most useful representations is the Power Spectral Density (PSD) function.
PSD is nothing but a mean square value of a road signal with respect to siganal frequency(spatial frequency)
Like any random signal, the elevation profile measured over a length of road can be decomposed by the Fourier Transform process into a series of sine waves varying in their amplitudes and phase relationships. A plot of the amplitude versus spatial frequency is the PSD. Spatial frequency is expressed as the "wave number" with units of cycles/foot (or cycles/meter), and is the inverse of the wavelength of the sine wave on which it is based.
Road elevation profiles can be measured either by performing close interval rod and level surveys, or by high-speed profilometer. When the PSDs are determined, Although the PSD of every road section is unique, all roads show the characteristic drop in amplitude with wavenumber. This simply reflects the fact that deviations in the road surface on the order of hundreds of feet in length may have amplitudes of inches, whereas those only a few feet in length are normally only fractions of an inch in amplitude. The general amplitude level of the plot is indicative of the roughness level higher amplitudes implying rough roads.
The upper PSD in the figure is a deteriorating Portland Cement concrete (often called "rigid pavement") road surface. Note a marked periodicity in the range of wavenumber between 0.01 cycle/foot (0.03 cycle/meter) and 0.1 cycle/foot (0.3 cycle/meter), which is related to the fixed slab length used in construction of the road. The lowest PSD is a similar road overlaid with a bituminous asphalt surface layer yielding a much smoother surface (the PSD amplitude is reduced by an order of magnitude). The reduction is especially pronounced in the high-wavenumber range. The intermediate PSD is a typical asphalt road (often called "flexible pavement").
Approach:
To plot the ISO 8608 Class Road Profile, we require three spatial frequency values
- Starting frequency i.e. 0.01 cycles/m
- Break point frequency i.e. ( 1 / ( 2pi ) ) cycles/m
- Terminating frequency i.e. 10.0 cycles/m
Line equation is implemented to plot the Road Lines as per their quality of Roughness
y=mx+c">y=mx+cy=mx+c
logy=-logmx+logc">logy=−logmx+logclogy=-logmx+logc
logy=-mlogx+logc">logy=−mlogx+logclogy=-mlogx+logc
logy=logx-m+logc">logy=logx−m+logclogy=logx-m+logc
logy=log(c⋅x-m)">logy=log(c⋅x-m)">logy=log(c⋅x−m)logy=log(c⋅x-m)
The above expression equals to: Displacement PSD
Deduction:
As we all can see that this input PSD profile comes under Class D (cyan line) as per the ISO 8608 Standards. Also we can see that the amplitude is diminishes with frequency increase which means that as the frequency increases the Road Profile gets smoothen (Degree of Roughness reduces)
Further Development:
We could plot the elevation road profile as per spread length of recorded road. Now we have the actual road elevation and could consider it as a excitation input for our Degree of Freedom models (DOF) e.g. in Quarter car model or Bounce/Pitch Model. We could also check the frequency at which amplitude is maximum and consider it for mode shape analysis.
Elevation Road profile:
Approach:
To plot the Elevation of Road Profile w.r.t its spread length, we have to calculate the amplitude of sine wave from PSD data at each spatial frequency by using below given expression:
;
Where:
- Zx = Elevation as Height
- Ax = Amplitude of PSD Signal (sine wave)
- Omega = Spatial Frequency
- x = Length (Road Length) - Loop from 1 to length for instance/example
Matlab Code:
% Plotting the ISO Class Road Lines
% We need 3 frequency in x axis
% and corresponding PSD or Degree of Roughness at y axis
freq = [0.004
0.005
0.006
0.007
0.008
0.009
0.010
0.012
0.017
0.020
0.024
0.028
0.043
0.055
0.063
0.068
0.071
0.080
0.083
0.089
0.090
0.098
0.105
0.115
0.128
0.135
0.145
0.150
0.154
0.160
0.165
0.166
0.167
0.175
0.180
0.185
0.188
0.190
0.190
0.205
0.210
0.215
0.220
0.235
0.250
0.255
0.275
0.280
0.283
0.285
0.300
0.310
0.320
0.330
0.340
0.350
0.353
0.355
0.360
0.370
0.375
0.380
0.385
0.395
0.410
0.420
0.425
0.430
0.435
0.438
0.440
0.460
0.463
0.465
0.480
0.485
0.486
0.490
0.500
0.510
0.540
0.545
0.560
0.565
0.575
0.585
0.590
0.595
0.610
0.620
0.630
0.650
0.655
0.660
0.670
0.680
0.685
0.686
0.690
0.695
0.696
0.700
0.705
0.710
0.715
0.720
0.725
0.730
0.740
0.750
0.765
0.780
0.790
0.810
0.815
0.820
0.830
0.850
0.855
0.860
0.870
0.880
0.885
0.890
0.900];
amplitude = [0.006759586
0.013802191
0.012220044
0.012805345
0.010055096
0.008674203
0.009906454
0.013565614
0.038077859
0.039514035
0.039376401
0.029351388
0.011087978
0.007155759
0.004009582
0.005053475
0.005426501
0.002758627
0.003146703
0.004278752
0.003605323
0.001118033
0.004605136
0.001110531
0.001926103
0.000599825
0.000790703
0.00082178
0.001138063
0.000449034
0.000344602
0.000439721
0.000485891
0.000661237
0.000247849
0.000282707
0.000336575
0.000260436
0.000338
0.000334243
0.000339473
0.000258116
0.000296182
0.000286092
0.000377324
0.000235117
0.000158941
0.000351811
0.000475459
0.000269363
0.00013499
0.000410917
0.000243289
0.000183098
0.000224051
0.000614261
0.000355537
0.000239772
0.000288619
0.000268181
0.00037814
0.000279647
0.000210043
0.000159755
0.000224234
9.50236E-05
0.000197943
0.000138383
0.000143463
0.00012055
0.000161674
9.97741E-05
8.9633E-05
0.000122986
0.000172037
0.000102941
0.00012442
9.02789E-05
0.000143594
6.29552E-05
9.28956E-05
0.000144869
0.000181894
0.000119478
0.000128017
0.000154808
0.000109269
0.000107822
9.00107E-05
0.000123476
8.48896E-05
0.000116729
0.000156981
0.000189126
0.000140362
0.000287068
0.000162162
0.000229266
0.000138612
7.98678E-05
0.000149275
7.12492E-05
0.00022861
8.03851E-05
0.000140036
5.64396E-05
9.09341E-05
0.000118502
7.68087E-05
7.41522E-05
0.000101869
4.47478E-05
0.000103792
0.000164574
9.03105E-05
9.00065E-05
0.000148517
0.000169078
8.09882E-05
9.06266E-05
9.01182E-05
6.61969E-05
0.000101917
5.92651E-05
6.81185E-05
];
omega(1)=0.01
omega(2)=1/(2*pi)
omega(3)=10.0
s_a(2)=8e-06
s_b(2)=32e-06
s_c(2)=128e-06
s_d(2)=512e-06
s_e(2)=2048e-06
s_f(2)=8192e-06
s_g(2)=32768e-06
s_h(2)=131072e-06
s_a(1)=s_a(2)*((omega(1)/omega(2))^-2)
s_b(1)=s_b(2)*((omega(1)/omega(2))^-2)
s_c(1)=s_c(2)*((omega(1)/omega(2))^-2)
s_d(1)=s_d(2)*((omega(1)/omega(2))^-2)
s_e(1)=s_e(2)*((omega(1)/omega(2))^-2)
s_f(1)=s_f(2)*((omega(1)/omega(2))^-2)
s_g(1)=s_g(2)*((omega(1)/omega(2))^-2)
s_h(1)=s_h(2)*((omega(1)/omega(2))^-2)
s_a(3)=s_a(2)*((omega(3)/omega(2))^-1.5)
s_b(3)=s_b(2)*((omega(3)/omega(2))^-1.5)
s_c(3)=s_c(2)*((omega(3)/omega(2))^-1.5)
s_d(3)=s_d(2)*((omega(3)/omega(2))^-1.5)
s_e(3)=s_e(2)*((omega(3)/omega(2))^-1.5)
s_f(3)=s_f(2)*((omega(3)/omega(2))^-1.5)
s_g(3)=s_g(2)*((omega(3)/omega(2))^-1.5)
s_h(3)=s_h(2)*((omega(3)/omega(2))^-1.5)
subplot(2,1,1)
loglog(omega,s_a,'b')
hold on
loglog(omega,s_b,'g')
hold on
loglog(omega,s_c,'r')
hold on
loglog(omega,s_d,'c')
hold on
loglog(omega,s_e,'m')
hold on
loglog(omega,s_f,'yellow','linewidth',2)
hold on
loglog(omega,s_g,'k')
hold on
loglog(omega,s_h,'g','linewidth',2)
hold on
plot(freq,amplitude,'k','linewidth',2)
hold off
xlabel('Spatial Frequency (cycle/ft)','linewidth',20)
ylabel('Power Spectral Density PSD (in^2/cycles/ft)','linewidth',20)
legend('Class A','Class B','Class C','Class D','Class E','Class F','Class G','Class H')
title('Road Classification by ISO 8608 Standards','linewidth',20)
grid on
deflec(125)=freq(125)-freq(124);
for i=1:124
deflec(i)=freq(i+1)-freq(i);
end
for j=1:500
x(j)=j
TotalAMP(j)=0
for i=1:125
AMP(j)=sqrt(2*amplitude(i)*deflec(i))
TotalAMP(j)=TotalAMP(j)+(AMP(j)*sin(2*pi*x(j)*freq(i)))
end
end
subplot(2,1,2)
plot(x,TotalAMP,'b')
xlabel('Length in (ft)','linewidth',20)
ylabel('Elevation in Height (ft)','linewidth',20)
title('Road Elevation Profile deduced from PSD Data','linewidth',20)