Chapter-9 Bounce and Pitch Motions
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Bounce and Pitch Frequencies • Bounce and Pitch Vibration modes are important as far as the ride is concerned • Bounce and Pitch motions are coupled
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Pitch Plane Model for a Motor Vehicle • Consider a vehicle as shown in Fig: For simplicity in analysis, the tire and suspension will be considered as a single stiffness (the ride rate), and damping and unsprung masses will be neglected.
Fig: Pitch plane model for a motor vehicle. 3
For convenience in the analysis the following parameters are defined:
( K f
K r )
M
........................(1)
( K r c K f b)
........................(2)
M 2 2 ( K f b K r c ) 2
Mk
........................(3)
Where: K f = Front ride rate K r =
Rear ride rate b = Distance from the front axle to C.G c = Distance from the rear axle to C.G I y = Pitch moment of Inertia k = Radius of gyration = I y M 4
Bounce Degree of Freedom Z
M c
b
M Z M
K f
K r
K f b K f Z
K r Z
K r c
M Z ( K f K r ) Z ( K r c K f b) 0
Z
( K f K r ) M
Z
( K r c K f b) M
0
Z Z 0
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Pitch Degree of Freedom Z
M
2
Mk
c
b
M
K f
K r
K f bZ K f b 2 K c 2 r 2
K r cZ
Mk ( K f b 2 K r c 2 ) ( K r c K f b) Z 0
( K f b 2 K r c 2 ) 2
Mk 2 Z 0 k
( K r c K f b) 2
Mk
Z 0
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Then the differential equations for the bounce, Z, and pitch motions of a simple vehicle can now be written as: .. Z Z 0 ……… (4) ..
2
k
Z 0
……. (5)
Of the several coefficients in these equations, only appears in both and is appropriately called the coupling coefficient. When = 0 no coupling occurs, and the spring center is at the center of gravity. For this condition, a vertical force at the CG produces only bounce motion, and a pure torque applied to the chassis will produce only pitch motion. Without damping, the solutions to the differential equations will be sinusoidal in form. The vertical motion will be:
Z Z sin t
…………..(6)
and the pitch motion will be:
sin t
…………..(7) 7
When these are differentiated twice and substituted into Eq. (2), we obtain: 2
Z
•
sin t Z sin t sin t 0
…………..(8)
Since the terms must always equal zero regardless of the instantaneous value of the sine function:
( 2 ) Z 0
Z
2
( )
…………..(9)
…………..(10)
The same analysis applied to Eq. (5) yields:
Z
2
2
k ( )
…………..(11)
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• The above equations define conditions under which the motions can occur. The constraints are that the ratio of amplitudes in bounce and pitch must satisfy Eqs.(10) and (11). • Equating the right sides of Eqs.(10) and (11) yields the expressions for the natural frequencies of the two modes of vibration. 2
2
( )( ) 4
2
( ) (
2
2
k 2
2
…………..(12)
k
)0
…………..(13)
• The values of satisfying this equation are the roots representing the frequency of the vibration modes. Two of the roots will be imaginary and can be ignored. The others are obtained from the equations as follows:
( 1, 2 )
2
( ) 2
( ) 2 4
(
2
k 2
)
…………..(14)
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1, 2
2
1
2
( ) 2
( ) 2
( ) 2
( ) 2 4
( ) 2 4
( ) 2 4
2
2
2
k
…………..(16)
k 2
2
…………..(15)
2
k
…………..(17)
•
These frequencies always lie outside the uncoupled natural frequencies.
•
The oscillation centers can be found using the amplitude ratios of Eqs. (10) and (11) with the two frequencies 1 and 2 in Eqs.(16) and (17). When substituted it will be found that Z/ (1) and Z/ (2) will have opposite signs. 10
• When Z/ is positive, both Z and must be both positive or negative. Thus the oscillation center will be ahead of the CG by a distance x=Z/. Similarly, for the root with a negative value for Z/, the oscillation center will be behind the CG by distance x equal to Z/. • Likewise, one distance will be large enough that the oscillation center will fall outside the wheelbase, and the other will be small enough that the center falls within the wheelbase. • When the center is outside the wheelbase, the motion is predominantly bounce, and the associated frequency will be bounce frequency. For the center within the wheelbase, the motion will be predominantly pitch, and the associated frequency is the pitch frequency. These cases are illustrated in Fig: 11
Fig: The two vibration modes of a vehicle in the pitch plane.
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• The locations of the motion centers are dependent on the relative values of the natural frequencies of the front and rear suspensions, where those frequencies are defined by the square root of the ride rate divided by the mass. That is:
f f
f r
1
K f g
2
W f
1
K r g
2
W r
…………..(18)
…………..(19)
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• Fig (next slide): shows the locus of motion centers as a function of front/rear natural frequency. With equal frequencies, one center is at the CG location and the other is at infinity. • Equal frequencies correspond to decoupled vertical and pitch modes, and “pure” bounce and pitch motions result. With a higher front frequency the motion is coupled with the bounce center ahead of the front axle and the pitch center toward the rear axle.
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• A lower front frequency puts the bounce center behind the rear axle and the pitch center forward near the front axle. This later case was recognized by Maurice Olley in the 1930s as best for achieving good ride.
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Fig: Effect of natural frequency ratio on position of motion centers.
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What ride frequencies are common today? Front Suspension
Rear Suspen sion
95 BMW M3 2001 VW Passat 2000 Neon 2001 JR 99 LH Dodge Intrepid
Ride Rate wlo tire (Ib/in) 119 117 126 148 160 121 110 152 131 99 113 163 134 161 185
02 Jeep WG Grand Cherokee
197
1170
85
1085
1.33
184
1005
85
920
2000 VW Golf
107
797
85
712
1.21
105
586
85
501
Vehicle 99 Volvo V70 XC 2001 MB E320 4-Matic Jeep KJ Liberty 97 NS Chrysler T&C Pacifica 99 MB E320 4-Matic 97 Peugeot 306 GTI 99 Audi A6 Quattro 2001 MB E320 2WD
Corner Unsprung Sprung Frequency Weight Weight Weight (lb) (Ib) (lb) (hertz) 1032 100 932 1.12 991 100 891 1.13 1036 85 951 1.14 1173 85 1088 1.15 1286 85 1166 1.16 985 100 885 1.16 850 85 765 1.19 1070 100 970 1.24 907 85 822 1.25 907 85 822 1.09 783 85 698 1.26 1060 100 960 1.29 836 75 761 1.31 1009 85 924 1.31 1125 85 1040 1.32
Ride Rate wlo tire (Ib/in) 131 148 181 145 153 150 113 172
Corner Unsprung Sprung Frequency Weight Weight Weight (lb) (Ib) (lb) (hertz) 832 100 732 1.32 964 100 864 1.29 914 85 829 1.46 880 85 795 1.34 1074 85 989 1.23 960 100 860 1.31 1.7 468 85 383 864 100 764 1.48
Ride Ratio Rr/Frt 1.18 1.14 1.28 1.16 1.06 1.13 1.43 1.2
144
969
85
884
1.26
NA
159 136 127 136 152
790 670 510 607 651
85 100 65 85 85
705 570 445 522 566
1.48 1.53 1.67 1.6 1.62
1.18 1.19 1.27 1.22 1.23
1.4
1.05
1.43
1.18
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• Maurice Olley, one of the founders of modern vehicle dynamics, established guidelines back in the 1930s for designing vehicles with good ride (at least for the low-frequency, rigid-body modes of vibration). • These were derived from experiments with a car modified to allow variation of the pitch moment of inertia (his famous “k 2” rig). • Although the measure of ride was strictly subjective, those guidelines are considered valid rules of thumb even for modern cars.
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The Olley Criteria •
The front suspension should have a 30% lower ride rate than the rear suspension, or the spring center should be at least 6.5% of the wheelbase behind the CG. Although this does not explicitly determine the front and rear natural frequencies, since the front-rear weight distribution on passenger cars is close to 50-50, it will generally assure that the rear frequency is greater than the front.
•
The pitch and bounce frequencies should be close together: the bounce frequency should be less than 1.2 times the pitch frequency. For higher ratios, “interference kicks” resulting from the superposition of the two motions are likely. In general, this condition will be met for modern cars because the dynamic index is near unity with the wheels located near the forward and rearward extremes of the chassis. 19
The Olley Criteria •
Neither frequency should be greater than 1.3 Hz, which means that the effective static deflection of the vehicle should exceed roughly 6 inches. The value of keeping natural frequencies below 1.3 Hz is clearly understood.
•
The roll frequency should be approximately equal to the pitch and bounce frequencies. In order to minimize roll vibrations the natural frequency in roll needs to be low just as for the bounce and pitch modes.
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• The rule that rear suspensions should have a higher spring rate (higher natural frequency) is rationalized by the observation that vehicle bounce is less annoying as a ride motion than pitch. • Since excitation inputs from the road to a car affect the front wheels first, the higher rear to front ratio of frequencies will tend to introduce bounce.(Centre of percussion) • To illustrate this concept, consider a vehicle encountering a bump in the road. The time lag between the front and rear wheel road inputs at a forward speed, V, and a car wheelbase, L, will be: t = L/V …………..(20)
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Front
Rear
2
Front Suspension
Rear Suspension
Pitch 1.5
1 l e v a r T n o i s n e p s u S
l e v a r
T n o i s n e p s u S
0.5
0
) g e d ( h c t i P
-0.5
-1
Time Lag
-1.5
-2 Time
Time
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• The oscillations at the front and rear of the car for an input of this type are illustrated in Fig: Note that soon after the rear wheels have passed over the bump the vehicle is at the worst condition of pitching, indicated by the points A and B in the Fig: Point A corresponds to the front end of the car being in a maximum upward position, whereas the rear end (point B) is just beginning to move. Therefore, the car is pitching quite heavily. • With a higher rear frequency, after about one and one-half oscillations of the rear suspensions, both ends of the car are moving in phase. • That is, the body is now merely bouncing up and down until the motion is almost fully damped. At different speeds and for different road geometries, the vehicle response will change. • Thus the optimum frequency ratio of the front and rear ends of the car has to be determined experimentally. 23
Special Cases •
•
Most modern vehicles with substantial front and rear overhang exhibit a dynamic index close to unity. That is: DI = k 2/bc = 1 When the equality holds, the front and rear suspensions are located at conjugate centers of percussion (an input at one suspension causes no reaction at the other). In this case the oscillation centers are located at the front and rear axles. This is a desirable condition for good ride if Olley’s ride criteria are also satisfied. There is no interaction between the front and rear suspensions.
A: centre of percussion 24
Roll Degree of Freedom Z
M
2
Mk x
t/2
t/2
M
K l
K r K l (t / 2) Z
K r (t / 2)Z 2
K l (t / 2) 2 K r (t / 2)
Roll Frequency 1
( ) 2
( ) 2 4
2
k x
2
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