Effect of Hub Motor Mass on Stability and Comfort ... - Protean Electric

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The two forces Fv and Fs, which are the forces acting on the sprung and unsprung mass respectively can be found through inspection <strong>of</strong> Fig. 1. The dynamic equations <strong>of</strong> the system become: = −K S ( z − y) − BS ( z − y) − M g (3) = K ( z − y) + B ( z − y) − K ( y − x) − B ( y − x) − M g (4) MV z V M S y S S T T S C. Wheel Hop A very real phenomenon is that <strong>of</strong> the tire losing contact with the road surface, also known as “wheel hop”. This phenomenon needs to be taken into account as it could happen during the simulations that the tire looses contact with the road due to either fast changing road conditions or the instability <strong>of</strong> the suspension system. The last point <strong>of</strong> contact between the tire and the road surface occurs when the unsprung mass and the road surface is equally displaced from their respective origins i.e. y-x=0. The force due to the tire spring and damper are only exerted when the wheel is in contact with the road surface. Incorporating wheel hop into the dynamic equations, (4) becomes: = K S ( z − y) + BS ( z − y) − KT ( y − x) − BS ( y − x) − M g (5) if ( y − x) ≤ 0 = K S ( z − y) + BS ( z − y) − M g (6) if ( y − x) ≥ 0 M S y S M S y S The wheel hop phenomenon adds a non-linearity to the system. It has been decided that it can be neglected during the frequency analysis <strong>of</strong> the system, but not for the simulations. It will have little or no effect on the frequency response <strong>of</strong> the system. It adds complexity to the analysis with little increase in the accuracy <strong>of</strong> the results. D. Vehicle Parameters Two vehicles are compared in the study. One is a standard vehicle and the other a vehicle with a hub motor place in the rear wheels. The same total mass i.e. sprung and unsprung mass combined, is used for both vehicles. A total mass <strong>of</strong> 1500 kg was chosen. This is the mass <strong>of</strong> a fully laden vehicle (vehicle mass, passengers and payload). All constants used, such as damping and spring coefficients, are kept the same for both vehicles. Table I gives a list <strong>of</strong> all the constants used. TABLE I VEHICLE PARAMETERS Standard Vehicle <strong>Hub</strong> Driven Vehicle Total Model Total Model Total <strong>Mass</strong> (kg) 1500 375 1500 375 Sprung mass (kg) 1340 335 1100 275 Unsprung mass (kg) 160 40 400 100 Ks (N/m) 36 000 36 000 36 000 36 000 Bs (Ns/m) 3000 3000 3000 3000 Kt (N/m) 110 000 110 000 110 000 110 000 Bt (Ns/m) 200 200 200 200 The standard vehicle will serve as the control for the investigation and the hub driven vehicle as the experiment. As the simulation uses the quarter vehicle suspension model, all masses are a quarter <strong>of</strong> the real values. III. FREQUENCY ANALYSIS A. Bode-plot Analysis It is important to verify that the suspension system and the vehicle are, through system frequency response analysis, stabile under changing road surface conditions. The simplest method to investigate the frequency response <strong>of</strong> the system is through a Bode-plot analysis. This can easily be done with the help <strong>of</strong> s<strong>of</strong>tware like MatLab. The transfer function is required to obtain the Bode-plot <strong>of</strong> the system. The transfer function can be mathematically derived from the dynamic equations or extracted from the linear model using MatLab. From MatLab the transfer function for the standard vehicle system is given as ( s) G ST 3 2 8. 527e −14s + 4. 093e −12s + 2. 463e4s + 2. 955e5 = (7) 4 3 2 s + 88. 96s + 3802s + 2. 516e4s + 2. 955e5 and the hub driven vehicle system as ( s) G HD 3 6. 395e −14s + 3. 638e −12s + 1. 2e4s + 1. 44e5 = (8) 4 3 2 s + 42. 91s + 1613s + 1. 226e4s + 1. 44e5 The transfer function can give an indication on the stability <strong>of</strong> the system. Both transfer functions have higher order poles than zeros. It can be seen that the second and third order zeros are small in comparison with the rest. These are good indicators that a system is stable. The Bode-plot will give an even better indication on the stability <strong>of</strong> the system. Fig. 2 Bode-plot <strong>of</strong> standard and hub driven vehicle. 2
A system is said to be unstable if it has a phase <strong>of</strong> -180 degrees at its crossover frequency. A system could also possibly be unstable if the magnitude is larger than 1 dB when the phase is equal to -180 degrees. From Fig. 2 and Table II it can be seen that the standard and hub driven vehicle do not meet these two criteria and thus are stable. The dominant natural frequency <strong>of</strong> the system can easily be seen from the Bode-plot. This natural frequency is where the Bode-plot reaches a maximum. As both systems are 2DOF systems, two natural frequencies occur. The first or lower frequency will be the dominant natural frequency, with the higher second frequency being the damped natural frequency. The damped natural frequency is difficult to distinguish, but can be found by looking at the shape <strong>of</strong> the Bode-plot. The natural frequency can be calculated more accurately. TABLE II BODE-PLOT INFORMATION Standard <strong>Hub</strong> Driven First Natural Frequency (rad/s) 9 10 Second Natural Frequency (rad/s) 60 40 Crossover Frequency (rad/s) 15 18 -180 deg Frequency (rad/s) 52 33 Mag. at natural frequency (dB) 7 7.5 Something to note is that the two natural frequencies move closer together as the mass is shifted from the body to the wheels. When the natural frequencies are far apart the second is extremely damped and plays virtually no part in the oscillation <strong>of</strong> the system. As the two moves closer together, the second frequency starts playing a larger role. The two frequencies could move so close together, super positioning on each other, causing larger and unwanted oscillations. B. Natural Frequency Analysis The natural frequency <strong>of</strong> a system is the frequency at which a driving force causes maximum oscillation amplitude or even unbounded oscillation. In multiple-degree-<strong>of</strong>-freedom systems, the system has n number <strong>of</strong> natural frequencies. It is possible that the system resonates at all, some or none <strong>of</strong> its natural frequencies. The natural frequencies <strong>of</strong> a 2DOF system are given as the square-root <strong>of</strong> its eigenvalues, that is ω n = λ n (9) The eigenvalues <strong>of</strong> the systems are derived from the state space equations. The eigenvalues are given as: 1 K S + KT K S K S + KT K S K S KT λ = + − + − 4 (10) 1 2 M M M M M M λ2 = 1 2 K S S + K M S T V K + M S V + K S S + K M S T V K + M S V 2 2 S K S K − 4 M M S V T V (11) The natural frequencies are calculated to be: Standard vehicle: ω = 80. 368 = 8. 96 rad/s or1. 43 Hz 1 ω = 3677. 09 = 60. 639 rad/s or 9. 65 Hz 2 <strong>Hub</strong> driven vehicle: ω = 96. 349 = 9. 82 rad/s or1. 56 Hz 1 ω2 = 1494. 56 = 38. 659 rad/s or 6. 15 Hz The calculated frequencies compare well with those given by the Bode-plot. The inaccuracy <strong>of</strong> the Bode-plot figures is due to the fact that the natural frequencies were obtained by inspection. The human body is sensitive to certain frequency ranges [4]. The vehicle will be classified as uncomfortable if the first natural frequency falls within these ranges. It has been found that frequencies between 0.5 and 1 Hz cause a high occurrence <strong>of</strong> motion sickness. The human head and neck is especially sensitive to vibrations between 18 and 20 Hz. The abdomen region <strong>of</strong> the body is sensitive to vibrations between 5 and 7 Hz. Research has shown that a system with a natural frequency higher than 3 Hz is perceived as a “harsh ride”. A ride is deemed to be comfortable near the 1.5 Hz mark. Taking the above mentioned frequency regions into account; it is safe to stipulate a guideline stating that a comfortable system would have a dominant frequency between 1 and 3 Hz. It can be seen that the calculated frequencies fall within this ranges. Furthermore they are close to 1.5 Hz which is perceived as the optimum natural frequency. C. Payload Analysis The analysis in the previous section was done on the suspension system <strong>of</strong> a fully loaded standard and hub driven vehicle. The next step is to investigate the effect <strong>of</strong> varying the payload on the natural frequency <strong>of</strong> the system. The payload range from empty to fully load. A vehicle’s curb weight is defined as the weight <strong>of</strong> the vehicle when it is fully operational plus one passenger. An electric vehicle’s curb weight is generally less than that <strong>of</strong> a standard vehicle, as is the case in this section. The curb weights for a standard and hub driven vehicle is chosen as 900 kg and 750 kg respectively. Fig. 3 shows the dominant natural frequency for both the standard and hub driven vehicles for a range <strong>of</strong> payloads. The natural frequencies are calculated by means <strong>of</strong> the equations used in the previous section. It can be seen that the varying payload has little effect on the natural frequency <strong>of</strong> the standard vehicle. On the other hand, the natural frequency <strong>of</strong> the hub driven vehicle shows significant variations due to the changing payload. This means that the hub driven vehicle will have a more varying ride response due to payload changes than the standard vehicle. However, both the vehicle’s natural frequencies stay within the 1 to 3 Hz range, although the hub driven vehicle’s frequency nears the 3 Hz limit when empty.
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Effect of Hub Motor Mass on Stability and Comfort ... - Protean Electric

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