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Spring into Action.
The recent development of the Lotus active suspension system has proved that car body movements such as pitch and roll can be precisely controlled. Unfortunately it will take some time before we can all take advantage of these developments, and in the meantime we are stuck with the conventional springs and suspension that seems to have changed little since the days of the horse drawn carriage.
Every enthusiast knows that the roll characteristics of a car have considerable influence on its dynamic behaviour as well as the comfort of its occupants. The parameters that control these characteristics are numerous, terms such as roll centre, roll stiffness, roll axis and roll couple are often bandied about but seldom adequately explained. I have often seen, in books and magazine articles, geometric constructions that can be used to determine roll centres and axis for different suspension layouts, but I have never seen an explanation of why these methods actually provide this information. It seems to be taken for granted that the reader will just accept it on face value, or is it because the writer is unsure himself and is just repeating that which he has read elsewhere. Anyway let's break with tradition and see if we can untangle the subject.
Firstly, we must understand the terminology. The roll axis is an imaginary line running through the car from end to end; when the car rolls, whilst cornering on a smooth road, it rotates about this axis. Any part of the vehicle not on this axis bodily moves, either up or down, side to side or both. Fig 1 shows the motion, as well as the longitudinal position of the roll centres, these usually being points on the roll axis in line with the wheels. Note that there's no theoretical reason why the roll centre heights should be at the same level at each end, and indeed they rarely are.
Now before going into the details of determining the roll centre and axis location, it must be understood that these parameters are not fixed in relation to the car's chassis, but move about depending on the deflection of the suspension and therefore vary depending on the roll angle which is influenced by the cornering force. When the car rolls the suspension on one side is compressed and on the other it becomes extended, for the purposes of analysis it matters not whether we consider the wheels as fixed and the body as capable of movement relative to them, or the body as fixed with wheels capable of moving. But as it is easier to visualise the motion with a fixed chassis we will go with that.
Let's now con
sider the case of a double wishbone
suspension system as in Fig 2. If we allow a very small vertical wheel displacement to take place, then the path of this movement (at the wheel end of the wish- bones) is at right angles to the wishbones, therefore the length of the wishbone does not affect the geometry of movement of the upright and wheel (for small displacements). So if the wishbones were extended in length until their inner pivots coincide then the motion of the wheel would be unaffected, but as the two wishbones now pivot around the same axis, we could replace them with a single arm fixed to the wheel axle, thus in effect creating an equivalent swing axle suspension system.
Because this new swing axle is only a figment of our imagination, let's call it a 'virtual swing axle' and its pivot a 'virtual pivot'.
Now, we need to consider the motion of the wheel at the tyre contact patch because this is our only connection with terra-firma, the reference from which we measure the roll. The diagram shows that this motion is at right angles to the line connecting the virtual pivot and the contact patch. Again for small deflections, this motion is unchanged as long as the effective pivot is located anywhere along this line. A mirror image of this applies to the wheel at the other side also. Thus the only common pivot point that satisfies both sides is the one at the junction of these lines, through the contact patches to the respective virtual pivots.
So if there is one pivot point that can relate the motion of both tyre contact patches relative to the chassis, then that same point locates the chassis relative to the wheels. It is the point about which the body will pivot should the suspension on one side be compressed by the same amount as the suspension on the other side be extended, in other words it is the roll centre.
I have emphasised the point about small wheel displacements, this is very important because the roll centre may vary its position enormously throughout the range of normal wheel movement. As the car rolls, its roll centres may change not only in height but also from side to side, as Fig 3, demonstrates.
Even though we have used a double wishbone system to explain the method of determining the roll centre position, it is very easy to apply the method to any other suspension configuration. It is only necessary to determine the directions of movement of the contact patches, and draw lines at right angles to these through the contact patches, the point at which these last two lines cross is the roll centre. Fig 4, shows the method for several different systems.
All of the above makes one big assumption (anyone spot it?) that the effective spring rates at the wheels are equal side to side. But aren't they, I hear you ask. No, not always: what if you have progressive rate springing? The effective spring rate will be increasing on the side that is compressing and will be reducing on the other. To understand the effect that this may have, let's look at the extreme case of a car with a rigid suspension on one side only and with a normal spring on the other. The chassis is thus completely tied to one contact patch and so this is the only point about which it can roll. Thus the roll centre is at ground level directly under the wheel with the infinitely stiff springing. Obviously this situation is unreal but demonstrates how the actual roll centre moves away from the geometrically constructed one, if the springing is not symmetrical.
It is all very well knowing where the roll centres and thus the roll axis are but what use is that knowledge, how can we use it and where should they be anyway? To answer this, we need to look at the superficially obvious question, 'Just what causes roll?'.
As we negotiate a curve the car is subject to centrifugal force, which is equal to the lateral acceleration multiplied by the mass of the machine (for a 1000Kg car cornering at 0.5g. The centrifugal force is 500Kg). This force is distributed throughout the car but for most analysis purposes can be considered to be acting only through the centre of gravity (C of G). Fig 5, shows that unless the C of G is level with the roll axis, a torque or couple (the roll couple) will be created, tending to make the machine roll about the roll axis.
There is another equally valid way of considering the roll mechanism. The centrifugal force acting through the C of G produces a torque about ground level and is resisted by weight transfer to the outside wheels, that is, the outside wheels support a greater proportion of the car's weight and the inside wheels a lesser proportion.
This change of load on each wheel normally causes the suspension to adopt a new position, or to put it another way the car rolls.
It may have occurred to some of you that if the roll axis is made to coincide with the C of G then there will be no roll couple and hence no roll or, to take things a step further, if the roll axis is above the C of G then the roll couple will be in a direction which makes the car lean inward like a motorbike. Indeed it is quite possible to design the suspension layout to achieve these effects. If this is so, why do we need active suspension to do it for us? Well, because if we use the high roll axis necessary, then the suspension layout will cause a jacking up effect under cornering conditions, a phenomenon experienced with some swing axle designs in the past.
One important point to note, one which is often misunderstood, is that regardless of the amount of roll allowed by the suspension design, the actual degree of weight transfer remains unchanged. This is only affected by the track, C of G height and cornering acceleration.
So, as with most design features in anything mechanical, the selection of roll axis position is a compromise: too low and we get excessive roll, too high and other undesirable handling traits surface. In practice, the com- promise varies with different types of car but always such that some roll occurs. Lowering the C of G is another technically possible way of reducing the roll couple, but this can only be done to a certain extent, due to the boring necessity of leaving comfortable space for the occupants.
The roll couple that such compromise leaves must be resisted by the car's springs, which leads us to roll stiffness. This term is defined such that the degree of roll is equal to the roll couple divided by the roll stiffness. Stiff springing obviously reduces roll and hence increases the roll stiffness, but if this is the criterion for selecting spring rates we will usually end up with an uncomfortable ride over normal road irregularities, so the anti-roll bar was developed to ease the situation.
The anti-roll bar is a torsion bar (torsion spring) connecting the suspension systems on each side of the vehicle in such a way as to allow both wheels to respond unhindered to two-wheel bumps, such as a ridge across the road. But if the wheels try to move independently, as with a single-wheel bump, or in opposite directions when the car rolls then the anti-roll bar resists this tendency. Roll is reduced as intended but comfort suffers as the effective spring rate of each wheel is increased in the individual single-wheel bumps, although the combined spring rate of the two wheels is unchanged over joint disturbances.
Again a compromise must be reached between the requirements of minimum roll and good response to road shocks. Roll bars, unlike the springs, are undamped (theoretically damping could be incorporated, although the manufacturers have generally concluded that it is not worthwhile). This is another reason for limiting the influence of the anti-roll bar, as oscillations might occur if the undamped bar is too stiff.
It is well known that the under/over steering characteristics of a car can be substantially modified by tuning the springing and anti-roll bar stiffnesses, altering the roll stiffnesses of each end.
While the vehicle as a whole has a certain roll stiffness, this is made up of the separate roll stiffnesses at the front and back, which may be quite different. For example, let's consider the case of a beam axle pivotted on the chassis at its mid point, and devoid of any form of springing. As unlikely as this layout seems, you may see it fitted to the front of some tractors, because it has good terrain-following properties. Now, because the chassis is completely free to rotate about the pivot point of this beam axle, then no roll stiffness is provided at this end of the machine and so all the stiffness needed must be available from the other end.
This lack of any roll stiffness means that body roll cannot cause any weight transfer to the outside wheel, and hence as the total weight transfer must be the same anyway, the other end must obviously be subjected to proportionally more.
Tyres have the interesting property that although they are capable of supporting higher cornering forces when subject to higher vertical loading, this does not go up in proportion. In other words, the co-efficient of friction is reduced as more weight is placed on them. In practice this means that weight transfer reduces the combined cornering force capable of being developed by the pair of tyres at one end of the car. Now as we have seen, the weight transfer at either end of the vehicle can be controlled to some extent by altering the roll stiffnesses of one or other, or both ends. Therefore, the tyre slip angles needed to produce the required cornering forces can be adjusted by modifications to the wheel springing, thus giving us the means to alter the under/over steering properties.
Dampers, too, have their part to play in the extremely complex interrelations between the various forces acting on the car. During the transitional period between initiating a turn and it becoming fully established, the dampers will affect the dynamic roll stiffnesses. Because the dampers only contribute whilst the suspension is actually moving, however, they have no effect once the car has settled down to a steady state turn.
So now you know why racers spend so much time setting up the suspension rates on their machines, when at first sight it would appear that the suspension is there only to cushion the bumps. Many competition cars have facilities for altering roll bar stiffness whilst on the move, making it quicker to achieve the desired performance but also allowing adjustments during a race as tyres wear and fuel weight reduces.
So how will active suspension improve the situation? The computer program controlling the system could, for example, be set to apportion the front/rear roll stiffness.
Assuming, that the system is set to give zero roll (this is by no means certain to be the aim of future manufacturers) then it is very interesting to follow through the implications for geometric roll centres, etc. Basically if we have no roll then the term roll axis becomes irrelevant, as does the need to have suspension link layouts that try to keep the outside wheels upright under cornering roll rather than under all conditions. Perhaps active suspension means a return to parallel equal length wishbones or perhaps better still, true trailing or leading arm systems. Normally, these designs suffer because of vast changes in roll centre positions, and because the camber angle varies with the roll angle of the body, the wheels always being held parallel to it. Straight line stability would be improved as bump induced wheel deflections would not cause any of the bump steer which comes from changing wheel camber. That's a desirable state of affairs that is hard, if not impossible to achieve with conventional springs and dampers as we have seen.
彈性運動
近來蓮花主動懸架系統(tǒng)的發(fā)展證明了汽車車身像前傾后仰和反轉(zhuǎn)這樣的運動時是可以準確的控制的。令人遺憾的是在我們能夠利用所有的這些成果之前還將花費一些時間的。與此同時我們無法擺脫從四輪馬車時就有的只是看起來有些很小的變化的彈簧和懸架。
每一個愛好者都知道汽車的側(cè)傾性對汽車的動力性表現(xiàn)和乘坐舒適性有很大影響。這些參數(shù)控制這些特性是很多方面的，例如側(cè)傾中心、側(cè)傾剛度、側(cè)傾軸線和側(cè)傾力矩是經(jīng)常被討論的但是很少給出合適的解釋。我曾經(jīng)經(jīng)常在書中和雜志文章中看到運用幾何構(gòu)造學(xué)來決定對于不同懸架安排的側(cè)傾中心和側(cè)傾軸線，但是我從來沒有看見一種解釋為什么這種方法真正可以提供這些信息。這似乎被認為假定讀者只是表面認同這種方法，或者是不是因為作者本人也不確定，只是重復(fù)他從別處讀來的。無論如何讓我們打破傳統(tǒng)來看看我們是否能解開這個問題。
首先我們必需了解一些專用術(shù)語。側(cè)傾軸線是虛構(gòu)的一條從汽車的一端到另一端的直線；當汽車側(cè)傾時，當在一段光滑的路面上轉(zhuǎn)彎時，汽車繞著這條軸線旋轉(zhuǎn)。這條軸線上的汽車任何部分都不移動，既不向上也不向下，也不從一邊到另一邊運動或者兩者都有。圖1展示了這種運動，同時也展示了側(cè)傾中心的縱向位置，通常側(cè)傾中心在側(cè)傾軸線與車輪軸線的交點處。注意到?jīng)]有理論原因為什么側(cè)傾中心的高度在兩頭應(yīng)該在統(tǒng)一水平線上，而且事實上它們很少在同一水平線上。
現(xiàn)在在進入決定側(cè)傾中心和軸線的位置的細節(jié)之前，必須明白這些參數(shù)對于汽車底盤并不是固定不變的，而是根據(jù)懸架的傾斜而運動，因此是根據(jù)由側(cè)向力引起的側(cè)傾角而變化的。當汽車旋轉(zhuǎn)時一側(cè)的懸架壓縮另一側(cè)的懸架伸張，分析它的目的是關(guān)系到我們應(yīng)該認為車輪不動而車身相對車輪是可以運動的，還是應(yīng)該認為車身不動而車輪可以運動。但是當把固定的底盤想象成運動的是比較容易的時候我們就會這樣去做。
現(xiàn)在讓我們考慮如圖2的雙橫臂懸架系統(tǒng)的情況。如果允許垂直車輪發(fā)生一個很小的位移，那么這個運動的路徑（在橫臂末端的車輪上）相對于橫臂成直角，因此橫臂的長度不影響垂直車輪的運動幾何學(xué)（因為位移很小）。因此如果橫臂延長至它們的內(nèi)側(cè)軸相交那么車輪的運動就不會受影響，但是如果兩個橫臂繞一個軸轉(zhuǎn)動，我們就可以把它們設(shè)置成一個橫臂固定到車輪軸上，因此實際上等價于一個等長的擺動軸的懸架系統(tǒng)。
因為這種新的擺動軸只是我們設(shè)想的虛構(gòu)的，我們叫它虛擬擺動軸和它的軸線叫做虛擬擺動軸線。
現(xiàn)在我們需要考慮車輪輪胎邊緣處的運動因為這是我們唯一與泥土相接觸的，作為我們測量滾動的參考。上圖展示了這個運動與連接虛擬軸線和接觸邊緣的線成直角。
所以如果有一個軸線點能夠使兩個輪胎接地點的運動相對于懸架有關(guān)聯(lián)，也就是相對于車輪在懸架上的點。這一點就是車身繞其旋轉(zhuǎn)的點，懸架的一側(cè)伸張，另一側(cè)壓縮，換句話說這就是側(cè)傾中心。
我已經(jīng)強調(diào)國這一點關(guān)于車輪的小位移，這是非常重要的因為在車輪正常運動時側(cè)傾中心可能變化很大。當車側(cè)傾時它的側(cè)傾中心不僅可能在垂直方向改變也可以在水平方向改變，參見圖3所示。
雖然我們已經(jīng)用雙橫臂懸架解釋了如何確定側(cè)傾中心的位置，將這種方法運用到其它任何懸架結(jié)構(gòu)是很簡單的。只是需要確定接觸邊緣的運動方向，通過接觸邊畫直角線，兩條線的交點就是側(cè)傾中心。圖4展示了這種方法運用在一些不同的系統(tǒng)中。
以上的全部做了一個大膽的假設(shè)（有人能指出來么？）那就是兩邊車輪的彈性是一樣的。但是它們不一樣么，我聽見你問了。是的，不總一樣：除非你有非常先進的等比彈性材料。實
際的彈性特性是在被壓縮的一邊增加另一邊減少。為了理解可能導(dǎo)致的后果，讓我們看看極限情況，一邊的懸架是剛性的另一邊有正常彈簧。底盤因此完全和一邊的接觸邊緣相連，這
樣這就是唯一的懸架能夠繞其轉(zhuǎn)動的點。因此側(cè)傾中心在極其堅硬的剛性車輪的沿地平線的后面。顯然這種情況是假的但是卻論證了如果彈性不對稱，側(cè)傾中心是如何從原來的由幾何學(xué)確定的位置移動的。
這樣就很好的了解了側(cè)傾中心的位置也就知道了側(cè)傾軸但是這有什么用呢,我們怎么應(yīng)用它呢和在任何工況下它們的位置應(yīng)該是怎樣的呢？為了回答這個問題，我們需要看看一個顯而易見的問題“到底是什么引起側(cè)傾的呢”？
大家都知道轉(zhuǎn)彎時汽車受離心力影響，這個力等價于隨著質(zhì)量的增加而增加的側(cè)向加速度（1000千克的汽車在0.5g的情況下轉(zhuǎn)彎。離心力時500千克）。這個力被分解到汽車的每一個部分但是為了分析的目的可以認為這個力只作用在重心上。如圖5所示，除非重心與側(cè)傾軸平行，力矩或者轉(zhuǎn)矩（側(cè)傾轉(zhuǎn)矩）就會產(chǎn)生，就產(chǎn)生汽車繞側(cè)傾軸轉(zhuǎn)動的傾向。
由另一個等價的令人信服的方法可以討論這個側(cè)傾的機器。離心力作用通過重心產(chǎn)生一個關(guān)于地平線的力矩和轉(zhuǎn)移到外側(cè)車輪的重力的抵抗，這樣外側(cè)車輪支持汽車較大的重量而內(nèi)側(cè)車輪支持較小的重量。
每個車輪的這種載荷的變化引起懸架需要適應(yīng)一種新的位置，或者使汽車進入另一種側(cè)傾方式。
可能有的人會想如果使重心和側(cè)傾軸重合那么就不會有側(cè)傾力矩了因此就沒有側(cè)傾了，或者更進一步想如果側(cè)傾軸在重心之上那么側(cè)傾力矩就會使汽車像摩托車一樣向里傾斜。事實上設(shè)計一種懸架布置實現(xiàn)這些是很可能。如果這樣的話，為什么我們需要主動懸架系統(tǒng)呢？因為如果我們用較高的側(cè)傾軸的話，那么懸架布置在轉(zhuǎn)彎時將會提高，這是以前一些擺動臂設(shè)計的經(jīng)驗。
很重要的一點需要注意，一個經(jīng)常被誤解的問題，就是不管懸架設(shè)計所允許的擺動角度范圍，實際重量轉(zhuǎn)移的程度時不變的。這只受運動軌跡、重心和側(cè)向加速度的影響。
所以如同在機械中的大多數(shù)設(shè)計特征一樣，側(cè)傾軸線的位置的選擇時一個平衡：太低的話我們得到過度側(cè)傾，太高的話其它的不舒服的操縱特性將表形出來。實際上，不同型號的車這個平衡是不一樣的但是總是有側(cè)傾發(fā)生。降低重心是另一個技術(shù)上可能減小側(cè)傾力矩，但是這只能在一個特定的范圍內(nèi)做到，因為必需留出足夠的空間讓乘客下車。
這種平衡留下的側(cè)傾力矩必需有汽車彈簧來抵消，這使我們認識側(cè)傾剛度。這個名詞被定義為側(cè)傾角等于側(cè)傾力矩除以側(cè)傾剛度。剛性彈簧很顯然減少側(cè)傾因此增加側(cè)傾剛度，但是如果這是選擇彈簧彈性的標準我們將會很不舒服，在正常的不平的公路上，所以發(fā)展抗側(cè)傾的桿來緩解這種情況。
抗側(cè)傾的桿是扭轉(zhuǎn)桿（扭轉(zhuǎn)彈簧）連到懸架系統(tǒng)上在汽車的兩邊這樣車輪就可以表現(xiàn)不受阻礙的兩個碰撞，就像橫穿道路的脊。但是如果車輪試著獨立運動，像一個輪子碰撞，或者當汽車側(cè)傾時向相反的方向那么抗側(cè)傾桿就抵抗這種趨勢。側(cè)傾就將會減少但是舒適性變差因為實際車輪的彈性變硬在單獨的一個車輪碰撞時，雖然聯(lián)合兩個車輪的彈簧特性在干擾處沒有改變。
又一個平衡在最小的側(cè)傾需要和很好的道路沖擊感必需滿足。側(cè)傾保護桿，不像彈簧是等幅運動的（理論上阻尼可以合并，雖然制造商通常認為那是沒價值的）。這是另一個原因限制抗側(cè)傾桿的作用，因為如果這個桿過硬就會產(chǎn)生擺動。
很明顯汽車不足/過多轉(zhuǎn)向特性可以在本質(zhì)上改變通過協(xié)調(diào)彈性和抗側(cè)傾桿的剛度，改變兩邊的側(cè)傾剛度
當汽車作為一個整體有一個側(cè)傾剛度，這由獨立的前后側(cè)傾剛度，它們可能差很多。例如，讓我們考慮一個軸繞懸架轉(zhuǎn)動在其中點，沒有任何行駛的彈性。這種布置看似不可能，你能在一些牽引車上看到，因為它們由好的地面適應(yīng)能力?，F(xiàn)在，因為懸架是關(guān)于軸的轉(zhuǎn)動點完全自由轉(zhuǎn)動的，那么沒有側(cè)傾剛度在機械的末端被提供所以所有所需的剛度必需在另一端可用。
缺少任何側(cè)傾剛度意味著車身側(cè)傾不能引起任何重量轉(zhuǎn)移到外側(cè)車輪上，因此作為總的重量轉(zhuǎn)移必需在任何情況下一致，另一端必需明顯的成比例的增加
輪胎有一個有趣的特性是雖然它們能夠承受更大的離心力當它們能夠承受更大的垂直載荷，但這不是成比例增加的。換句話說，共同作用的摩擦力在更多的重量壓在車輪上時被減小了。實際上這意味著重量轉(zhuǎn)移減少了汽車一端車輪增加的離心力?，F(xiàn)在正如我們所看到的，重量轉(zhuǎn)移在汽車的任何一邊能夠由選擇這邊或另一邊活捉或者兩邊的側(cè)傾剛度控制在一定范圍內(nèi)。因此輪胎側(cè)偏角需要產(chǎn)生合適的離心力來適應(yīng)車輪彈性的改變，這樣給我們了選擇不足/過多轉(zhuǎn)向特性。
阻尼器也有參與極端復(fù)雜的的相互關(guān)聯(lián)的各種作用在汽車上的力。在剛開始變形到完全變形的這個過程中，減震器將影響動力側(cè)傾剛度。因為減震器只在懸架真正運動時起作用，但是它們在汽車穩(wěn)定轉(zhuǎn)動時不起作用。
所以現(xiàn)在你知道了為什么賽車手們花大量的時間在他們的賽車上設(shè)置懸架的剛度，當初看之下懸架只是防沖擊的墊子。許多競賽車都在運動中改變側(cè)傾穩(wěn)定桿剛度的能力，不僅使其更快的完成期望的表現(xiàn)也能允許對比賽時輪胎的磨損和燃油重量的減少進行調(diào)節(jié)。
所以主動懸架是怎么改進這種情況的？電腦程序控制系統(tǒng)能做到，例如分配到前后的側(cè)傾剛度。
假定，系統(tǒng)設(shè)置為零側(cè)傾(這沒準就是未來制造商的目的)那么實現(xiàn)幾何側(cè)傾中心等的關(guān)聯(lián)是很有趣的。根本上如果我們沒有側(cè)傾那么側(cè)傾軸就變得不恰當了，正如需要懸架連接布置來試著在轉(zhuǎn)彎側(cè)傾時保持外側(cè)車輪直立而不是在所有的情況下。也許主動懸架意味著回到了平行的等長雙橫臂或者單橫臂系統(tǒng)。通常，這些設(shè)計被否定了因為很多的側(cè)傾中心位置，和因為外傾角隨車身側(cè)傾角變化而變化，汽車總是平行橫臂。直線行駛穩(wěn)定性將會被提高因為沖擊引起的車輪側(cè)偏不會引起任何的由于改變車輪外傾角所引起的沖擊轉(zhuǎn)向。如果用我們已經(jīng)知道的通常的彈簧和阻尼減震器來完成不是不可能的話，那是令人滿意的很難做到的情況。