英语翻译:
汽车减振器参数化模型的发展和实验验证
                     作者：KIRK SHAWN RHOADES
     
摘 要
    这篇论文描述了汽车减振器的一个参数化模型的实现过程。研究的目标是创造一个可以准确地预测阻尼力的减振器模型来作为学生型方程式赛车团队的一个设计工具。这项关于单筒充气减振器研究适合于学生型方程式赛车的应用。
    这个模型考虑到了减振器中每一个单独的流通路径，并且建立了对每一个流通路径的流通阻力模型。阀片组的挠度由一个力平衡方程计算出并且与流通阻力相关。这些方程产生一个可以用牛顿的迭代方法求解的非线性方程组。
   这个模型的目标是创建准确的力-速度和力-位移关系并用于检验。应用一个震动测力计使模型与真实的减振器数据联系起来以验证准确性。通过一个有效的模型，组件包括常通孔、活塞孔、压缩和复原阀片是不同的以获得减振器阻尼力效果的了解。
 
一、减振器功能特性
1.减振器的构造
   要理解减振器的工作过程第一步是要弄清楚减振器的各个组成部件是如何相互作用产生阻尼力的。下面本文将对减振器的组成和功用做一个简单的介绍。减振器的参数特性通常由力-速度和力-位移曲线给出。关于这些图形的更详细的描述将在这一部分给出。
   有许多类型的汽车悬架减振器，其作用通常是用来缓和冲击。这其实是一个误称，因为减振器实际上并不能缓和冲击，这是悬架弹簧的作用。众所周知，一个弹簧振子系统在没有能量耗散时会做永久的简谐振动，其中弹簧与振子的势能与动能分别地相互转化。在这篇论文的目的中，减振器的术语将会被使用。减振器的功能就是消除系统动能并将其转化为内能。
   减振器的构造有许多类型：双筒减振器，单筒带或不带蓄能器的减振器，甚至中间有一个杆的减振器类型。在这篇论文的目的中，单筒的不带蓄能器的减振器将被用于实验。
   不同类型的减振器的另一个主要区别时其外部适应性的特征。有的减振器装配后仍可以被调节。汽车通常使用不可调节的减振器。相反地，在赛车中使用的减振器通常有一定程度的可调节性。既然这项研究的焦点是帮助赛车悬架设计，这种单筒减振器具有可调性。

                     图1   单筒减振器的组成
图1显示了单筒减振器的主要组成元件，外部可调减振器。这种减振器包含一个在充满油液的圆筒中运动的活塞总成。减振器的外罩包含了所有的内部元件。一个装配完全的减振器被分为三个压力腔：气室、复原腔和压缩腔。气室与压缩腔通过一个浮动活塞分开。这个浮动活塞将气室中的气体与液体分隔开来，在压缩腔与复原腔室中，典型的液体是油液。减振器中应用最多的气体是氮气，因为其不与油液发生反应。它对温度相对地不敏感并且不含水蒸气。
压缩腔是位于浮动活塞与连杆活塞之间的那一部分体积。复原腔是有活塞杆的那一部分体积。压缩腔与复原腔完全地被油液充满，在这里应用的是典型的是5W重的油液。
  活塞与活塞杆相连，活塞杆通过一个用来保持油液的密封装置。杆密封装置同时阻止灰尘和其他污染物进入复原腔影响内部油液的流动。活塞在其外罩上也有一个密封装置位于其外径和内径之间。这个密封装置将压缩腔与复原腔分隔开来。
图1所示的球型支座是用来将减振器安装在车体上。在未对减振器施加弯曲应力时，它们允许一定的装配误差。在赛车的应用上，减振器的活塞杆一般连接在车桥上，而套筒的另一面一般连接在车架上以减少不定质量的变化幅度。

2.减振器的一般工作过程
减振器有两个典型的工作行程：压缩行程与复原行程。这两个行程每一个都将被单独试验。图2所示的是压缩行程模型。

图2 压缩行程流通图
在压缩行程中，液体有压缩腔流入复原腔。由于油液具有很强的不可压缩性，活塞杆进入复原腔，复原腔和压缩腔中油液和活塞杆的体积之和必然增大。为了适应这种体积增大，浮动活塞在气室中压缩氮气，气体压缩的体积与活塞杆进入的体积相同。单筒减振器同时具有压缩气室以保持一个提升的油液压力的优点，这可以帮助阻止油液空穴的形成。模型分析显示活塞一英寸的位移只引起气室压力四到十磅/平方英寸的改变，根据气室初始压力而不同。这个小的压力改变意味着一个几乎相同的压力施加在压缩腔力的液压油液上。气室中的压力用Pg表示。
气室中的压力显示出一个气体弹簧效果。力等于活塞杆的面积与Pg的乘积，这个力一直作用在活塞杆上。气体弹簧效果是与活塞速度无关的，但与位移十分相关，并与加速度有微弱的关联。在压缩行程中气体弹簧力是不断增大的。
压缩行程中总的流量是三个流通路径的综合。这些流量与压力腔之间的压力差有关。复原腔中的压力用Pr表示，压缩腔中的压力用Pc表示。在压缩行程中，Pc大于Pr，这个压力差使油液由压缩腔进入复原腔，并产生阻尼力。流通路径和各腔压力在图2中显示并在下面解释。
第一条流通路径是常通孔。常通孔流通路径开始于压缩腔活塞杆的终点处，结束于复原腔活塞一面的活塞杆处。常通孔的尺寸是可以通过图2所示的活塞杆中的可动针阀调节的。针阀可以通过图1所示的常通孔调节器旋入或旋出。常通孔可以被调节成全开以减少阻尼至全闭增大阻尼。改变针阀的几何形状或尺寸也可以改变常通孔的流量。常通孔在低速减振中起首要作用因为这个孔常开，与活塞速度无关。
第二条流通路径是活塞孔流通路径。活塞孔流通路径通过活塞上的固定直径孔，再通过变形后允许流通的薄阀片组。活塞孔流通路径由压缩阀片或阀片组控制。为了简化，在图2中至显示了一个阀片，压缩阀中的液流通过复原阀片中的一个孔。复原阀中的孔取消了在活塞中开一个流通路径的必要，并且这是一个允许阀流通的简单的方式，降低了活塞制造的复杂性。
提高速度可以降低复原腔的压力和增大油液流通速度。不同的压力引起不同的阀片变形。压缩阀片，位于复原腔，根据活塞的速度限制液流的流通面积。速度增大，阀片变形增大，从而液流流通面积增大。Pv被定义为在活塞孔通道内部的压力。
第三条流通路径是在活塞与套筒内壁之间密封装置的泄露。泄露流通至少在重要性上不如前两种流通路径，但是很难将其完全消除。长时间的使用会使密封装置退化，增大泄露流通量，并且降低减振器的阻尼力。这种活塞套筒密封装置应该定期更换以使泄露流通量与其它流通方式比起来不会过多。

                        复原行程流通图
图3所示的是复原行程工作过程。在复原行程中，活塞杆在充满油液的套筒中被撤回，从而引起油液从复原腔流入压缩腔。压缩腔和复原腔中油液和活塞杆的体积之和因为活塞杆的撤出而减小，气室中的气体扩张。
复原行程的液流是从复原腔流入压缩腔。前面讨论的所有的阀，常通孔和泄露孔仍然存在，只是方向与原来相反。
常通孔现在开始于活塞杆上空的入口处，结束于活塞杆在压缩腔的终点处。所有的由常通孔引起的低速阻尼属性都可以有压缩行程移植到复原行程。
活塞孔流通路径在概念上与压缩行程一致，只不过具体的流通孔是不同的。复原行程的压力关系是Pr>Pv>Pc。阀内液流通过压缩阀片上适当的孔并引起压缩腔内复原阀片的变形。如前所述，复原速度的增大将会导致阀片变形和液流面积的增大。
泄露流量与前所述具有同样的重要性并且通过活塞和外套筒间相同的轴对称的缺口。只有方向复原行程与压缩行程是相反的。
通过测试复原行程与压缩行程，可以看到减振器的物理工作过程是复杂的。减振器具有不同的位移，速度和加速度。方程还与压力，阀片变形，油液流量等因素有关。这些都将成为建立减振器工作模型的基础。
3.减振器的工作特性
既然在任何汽车或赛车中的减振器活塞速度一直处于不断变化的状态，这就很难定义和解释减振器的工作情况。为了评估减振器的工作状况，在减振器测力计上测试成为一种规范。这项研究中使用的减振器测力计是一个Roehrig 2VS。这种减振器测力计是施加一个按正弦规律变化的位移。位移的振幅和频率是给定的。位移的一阶导数和二阶导数分别是速度和加速度。

                       图4 全过程力-速度特性

                    图5 与F-V图相应的减振器活塞位移-时间关系

图6 与F-V图相应的减振器活塞速度-时间关系
使减振器工作过程参数化的最初方法是输出力-速度关系。图4至6显示了基本的F-V图像与相应的运动曲线。
图4显示了全过程的力-速度曲线，包括压缩行程和复原行程。这有时候被称作连续的速度输出特性（CVP）。对力和速度给出常规的注释是重要的。压缩行程中速度是负的，而复原行程中，减振器度增大，速度是正的。在一些实例中，速度方向的定义可能是相反的。习惯上使用的是Roehrig测试测力计，在这篇报告中将会始终使用到它。
习惯上使用的力是减振器产生的力。复原力是负的，压缩力为正。有一段速度接近于零的区域，那里的情况并不真实。这是由于减振器的滞后效果造成的。图4中所示的滞后作用是当速度增大和速度降低时的力的差异。也就是说，当减振器加速和减速时其产生不同力是不同的。滞后作用这个词语通常用来指这种效果，在本文中将会一直使用这个概念表示在F-V图像上力的差异。然而，这种效果并不是传统的科学文献中定义的滞后性。这种现象的原因将会在文献回顾部分给出解释。
图4-6上还有标记有1-4的点。这些是减振器运动中的关键点。点1是循环的开始。减振器充分延伸，并且开始速度为零。从点1至点2减振器速率不断增大，进行的是压缩行程。在点2，达到最大负向速度。这通常对应于压缩行程中力的峰值。此时位移为零，这意味着全行程的一半已被压入减振器。从点2至点3，速率开始下降。点3标志着压缩行程的结束。这时的位移达到负的最大值，这意味着减振器被充分压缩，速度降至零。过了点3，复原行程立即开始，伴随着速度的不断增大。在点4，复原行程的力达到峰值，位移再次变为零，所以减振器扩张至复原行程的一半。循环而后从点4回到点1，随着活塞速率的降低，复原行程继续进行。在点1，减振器回到完全张开状态，速度减为零。
所有的图形通常排除了气体弹簧力。因此，这个力在速度为零时其值也为零。
其它的有时被用到的表征减振器工作状态的图像是力-位移图像。图7显示了典型的F-D曲线。这个曲线是减振器参数化后所有的机械设备被用来测量和测绘力-位移曲线的结果。

图7 全过程力-位移特性
F-D图像使用惯用的力符号，压缩时为正，复原时为负。在压缩行程和复原行程中力都不是关于y轴对称的。在F-V图像中同样的滞后作用是产生这种不对称性的原因。
为了获得进一步理解，可以用一个假想的理想弹簧，理想阻尼器，正弦运动来解释滞后性。一个理想线性弹簧在F-D图像中产生的刚度K是一条倾斜直线。在F-V图像中是一个椭圆（见附录A）。一个理想的线性阻尼器在F-V图像中会产生一条倾斜的刚度直线，在F-D图像中是一个椭圆。在一个实际减振器的F-V图像中滞后作用导致减振器产生像弹簧的力。
二、文献回顾
进行文献回顾有两个主要目的：第一个目的是通过研究减振器功能的参数化模型的发展过程，对单独的内部元件和内部液流在过去如何被参数化获得一个更好的理解。
文献回顾第二个目标是对发生在F-V图像中的滞后作用获得一个深刻的理解。理解产生这种现象的原因和如何使之最小化在减振器设计中具有起决定作用的重要性。所有这些概念将会在引用文献中被找到。
三、减振器规格
这项研究中所使用的减振器是Tanne赛车产品中的一个Tanner外部可调减振器Gen 2。它是一个充气的单筒构造，里面有一个浮动活塞将气室和油腔分割开来。Tanner Gen 2的最初用途是四分之一微型车竞赛中，但是它的尺寸，价格和可用的阻尼力范围使其也可以应用在学生型方程式赛车中。Tanner Gen 2质量轻，价格相对便宜，并且可以通过内部调节取得理想的阻尼力。图8显示了一个Tanner Gen 2减振器的三维模型。

 图8   Tanner Gen 2减振器

减振器伸张到最长时距球型支座的中心是10.33英尺。减振器行程大约是3英尺。减振器外罩和端盖是由铝制成，而镀铬的杆是由抛光的钢制成。端盖上面有螺纹可以拆除从而使得拆装容易。
活塞和阀片的设计用来控制活塞孔液流，允许这部分制造时成本比其他赛车低得多。活塞是由机械铝制成的，并且具有6个液流孔。活塞如图9所示。

        图9   Tanner Gen 2铝质活塞
  活塞液流孔具有0.038英寸的直径，用来将活塞装配到活塞杆上的孔直径是0.25英寸。位于圆筒的外径上的沟槽是用来装配活塞与圆筒之间橡胶密封装置的。这种活塞设计比起Ohlins和brand牌的减振器复杂性要小很多，并且这种简单的设计生产起来要便宜许多。
根据所需的减振器水平不同，可用的活塞的孔径从0.14英寸（软减振器）到0.038英寸（硬减振器）。在没有任何阀片时，6个孔在压缩和复原行程都允许液流通过。
Tanner赛车产品的一套阀片组的一个单独阀片也可以使用。如图10所示.

图10 Tanner Racing G2的一套碳纤维阀片组
    Tanner 赛车上的一套阀片组包含碳纤维阀片。这些阀片拥有与钢几乎相同的弹性模量和泊松比，但是它们的质量要轻得多。阀片有孔的位置与活塞上可以用来在压缩行程与复原行程中开通一条液流通路的孔是一致的。例如，如果两孔阀片被用在活塞压缩面而三孔阀片被用在活塞复原面，只要没有公用孔，复原行程中将会存在两个单独的液流通路而压缩行程中将会存在三条单独的液流通路。阀片的排列可以为Tanner Gen 2减振器创造无穷的可能。另外也可以利用不同厚度或不同材质的阀片来得到想要的阻尼特性。
用来调节常通孔的有螺纹的针阀可以旋转3.75圈。标记0圈的位置等效于一个全闭的常通孔。调节器旋转的圈数越大，常通孔开度越大。这是一个很实际的考虑，因为全闭活塞是容易辨认的。
减振器油液用的是Tanner Tuned振动油。这种油液的属性是未知的，所以典型的5W油将被用在模型上，其密度和粘度最为重要。

英语原文
DEVELOPMENT AND EXPERIMENTAL VERIFICATION OF A
PARAMETRIC MODEL OF AN AUTOMOTIVE DAMPER
               A Thesis by  KIRK SHAWN RHOADES

ABSTRACT
     This thesis describes the implementation of a parametric model of an automotive damper.  The goal of this research was to create a damper model to predict accurately damping forces to be used as a design tool for the Formula SAE racecar team.  This study pertains to monotube gas charged dampers appropriate to Formula SAE racecar applications.  
     The model accounts for each individual flow path in the damper, and employs a flow resistance model for each flow path.  The deflection of the shim stack was calculated from a force balance and linked to the flow resistance.  These equations yield a system of nonlinear equations that was solved using Newton’s iterative method.  
     The goal of this model was to create accurately force vs. velocity and force vs. displacement plots for examination.  A shock dynamometer was used to correlate the model to real damper data for verification of accuracy.  With a working model, components including the bleed orifice, piston orifice, and compression and rebound shims which were varied to gain an understanding of effects on the damping force.  

FUNCTIONAL DAMPER CHARACTERISTICS
The first step in understanding the operation of a damper is to understand how he components interact to create the damper force.  A brief discussion of damper components and functionality is given in this section.  The characteristics of damper are usually presented graphically in Force vs. Velocity and Force vs. Displacement graphs.  A detailed description of these graphs is contained in this section.  
GENERAL CONFIGURATION OF DAMPER 
There are many types of automotive suspension dampers, which are commonly referred to as shock absorbers.  This is a misnomer because the damper does not actually absorb the shock.  That is the function of the suspension springs.  As is well known, a spring/mass system without energy dissipation exhibits perpetual harmonic motion with he spring and the mass exchanging potential and kinetic energy, respectively.  For the purpose of this paper, the term damper will be used.  The function of the damper is to remove the kinetic energy from the system and to convert it into thermal energy.
There are numerous configurations of dampers: twin tube, monotube with or without reservoir, and even a rod through damper type.  For the purpose of this thesis, a monotube damper without a separate reservoir will be examined.  
Another major distinction in damper types is the feature of external adjustability, .e. if the damping can be adjusted after the damper is assembled.  Automotive applications generally use a nonadjustable damper.  In contrast, many dampers for racing applications have some degree of adjustability.  Since the main focus of this research is to aid in racecar suspension design, the monotube damper chosen has adjustable damping.  

      Figure 1 displays the major components of a monotube style, externally adjustable damper.  The damper is comprised of a piston assembly that moves inside a fluid filled cylinder.  The outer housing of the damper contains all internal components. A fully assembled damper is partitioned into three pressure chambers: gas, rebound and compression.  The gas chamber is separated from the compression chamber by a floating piston. This floating piston separates the gas in the gas chamber from the fluid, typically oil, in the compression and rebound chambers.  The gas used for most damper applications is dry nitrogen because it does not react with oil.  It is relatively insensitive to temperature and contains no water vapor.  
      The compression chamber is the volume between the floating gas piston and the piston attached to the rod.  The rebound chamber is the volume on the rod side of the piston.  The compression and rebound chambers are completely filled with oil, typically 5W weight oil designed for this application.  
      The piston is connected to the piston rod which exits the housing through a rod seal that retains the oil.  The rod seal also prevents dirt and other contaminates from entering the rebound chamber and affecting internal flow of oil.  The piston also has a seal between its outer diameter and the inner diameter of the outer housing.   This seal separates the compression and rebound chambers.  
      The spherical bearings shown in Figure 1 are for mounting the damper to the vehicle.  They allow for some degree of misalignment in mounting without imposing bending loads on the damper.  For racing applications, the piston rod of the damper is usually mounted to the wheel suspension, while the cylinder side is connected to the frame of the vehicle in order to minimize the unsprung weight.  
GENERAL OPERATION OF DAMPER
      There are two modes of operation in a damper: compression and rebound.  Each of these modes will be examined individually.  The compression operation mode is shown in Figure 2.  
             
  During the compression stroke, fluid flows from the compression chamber into the rebound chamber.  Since the oil is effectively incompressible, as the piston rod enters the rebound chamber the sum of the volumes of the oil and the rod in the rebound and compression chambers must increase.  To accommodate this volume increase, the gas piston compresses the nitrogen in the gas chamber to decrease the gas volume by an amount equal to the volume of the inserted rod.  Monotube dampers also have the advantage of pressurizing the gas chamber to maintain an elevated pressure on the oil, which helps prevent oil cavitation.  Model analysis has shown only a four to ten psi change in the gas chamber pressure for one inch of piston rod displacement, depending on initial gas pressure value.  This small pressure change means an almost uniform pressure exerted on the hydraulic oil in the compression chamber.  The pressure in the gas chamber is denoted Pg.
  A gas spring effect is also present due the pressure in the gas chamber. A force equal to the area of the rod times the gas pressure, Pg, will be on the rod at all times. Gas spring effect is independent of piston velocity, but strongly dependant on displacement and very weakly dependant on acceleration.  The gas spring force increases during the compression stroke.
  Total flow during compression is comprised of flow through three flow paths. These flows are related to the pressure differences in the pressure chambers.  Pressure in the rebound chamber is denoted as Pr and pressure in the compression chamber is denoted Pc. During compression Pc is greater than Pr; this pressure difference drives the flow from the compression chamber to the rebound chamber and generates the damping force.  Flow paths and chamber pressures are shown in Figure 2 and explained below.  
  The first path is the flow through the bleed orifice. The bleed orifice flow path begins at the end of the piston rod in the compression chamber and ends out of the side of the piston rod in the rebound chamber.  The bleed orifice size can be adjusted by moving the needle valve inside the piston rod in Figure 2.  The needle valve is adjusted in or out using the bleed adjustment shown in Figure 1.  The bleed flow orifice can be adjusted from fully open for less damping to fully closed for increased damping.  Modifications to the geometry of the needle value or size of the bleed orifice can change the bleed orifice flow also.  The bleed orifice dominates the low speed damping because this orifice is always open, regardless of piston velocity.  
  The second flow path is the valve orifice flow path.  Valve orifice flow travels through constant diameter holes in the piston and past thin washer-like shims that deflect to allow flow.  Valve flow is controlled by the compression shim or shims.  For simplicity, only one shim is shown in Figure 2.  The flow into the compression valve travels through a hole in the rebound shim.  This hole in the rebound shim eliminates the need to machine a flow path in the piston and is a simple way of allowing valve flow and decreasing complexity of piston manufacture.  
  Increased velocity decreases the pressure in the rebound chamber and increases the flow rate.  The pressure differential also triggers shim.  The compression shim, located in the rebound chamber, limits the area for flow depending on the velocity of the piston.  With increased velocity, shim deflection increases and valve flow area increases.  Pv is defined as pressure inside the exit of the orifice in the piston.  
  The third flow path is the leakage flow around the piston-cylinder wall seal. Leakage flow is at least an order of magnitude less then other two flows, but is difficult to eliminate completely.  With prolonged usage the seal may degrade, increase the leakage flow, and lessen the damping force from the damper.  The piston cylinder seal should be replaced periodically so that the leakage flow does not become significant in comparison to the other flow paths.  
       
  Rebound operation is shown in Figure 3.  During the rebound stroke, the piston rod is being withdrawn from the fluid filled cylinder, causing flow from the rebound to the compression chamber.  The combined volume of oil plus the rod in the compression and rebound chambers is now decreasing due to the removal of the rod, and the gas in the gas chamber expands.  
  The flow in rebound is from the rebound chamber to the compression chamber. All the valve, bleed, and leakage flow paths discussed previously still exist, only their directions have reversed.  
  The bleed orifice flow now begins at the side inlet hole in the piston rod, and exits out the end of the piston rod into the compression chamber.  All the properties of low speed damping dominated by the bleed are retained in the transition from compression to rebound.  
  The valve orifice flow path is conceptually the same as for compression, only the specific orifice is different.  During rebound the pressure relationships arePr>Pv>Pc . The valve flow now travels through the appropriate hole in the compression shim and initiates the deflection of the rebound shim in the compression chamber.  As before, an increase in rebound velocity will result in increased shim deflection and valve flow area.  
  The leakage flow is of the same magnitude and travels through the same axisymmetric gap between the piston seal and the outer cylinder.  Only the direction in rebound is opposite of that in compression.  
  After examination of the rebound and compression stroke, it can be seen that physical operation of the damper is complex.  Dampers are displacement, velocity and acceleration dependant.  The equations relating pressures, shims deflections, flows, etc. will be the basis for modeling the behavior of a damper.  

CHARACTERIZATION OF DAMPER OPERATION
  Since the position and velocity of a damper in any automotive or racing application is in constant state of change, it is hard to define and interpret damper performance.  To evaluate the performance of a damper, testing on a damper dynamometer has become the norm.  The damper dynamometer used in this research is a Roehrig 2VS.  This damper dynamometer imposes a sinusoidal input for displacement.  The displacement is defined by specifying the amplitude and the frequency.  The first and second derivatives of the displacement are the velocity and acceleration, respectively.  
               
             
The primary means used to characterize damper performance is the Force vs. Velocity (FV) plot.  Figures 4 through 6 show the basic FV plot and the corresponding motion profiles.  
  Figure 4 shows a Force vs. Velocity plot for a full cycle, compression and rebound strokes.  This is sometimes referred to as a Continuous Velocity Plot (CVP).  It is important to note the sign conventions for force and velocity.  Compression results in negative velocities, while rebound, increasing length, results in positive velocities.  In some instances [1], the velocity definitions may be opposite.  The convention shown here is used by the Roehrig test dynamometer, and will be used throughout this report. 
  The convention for forces is to record the force produced by the damper.  Rebound forces are negative while compression forces are positive.  There are small regions near zero velocities where this is not true.  This is due to the hysteretic effects of the damper.  The hysteresis shown in Figure 4 is the difference in the force at a given speed when the speed is increasing and when the speed is decreasing.  In other words, the damper produces a different force when it is speeding up than when it is slowing down.  The term hysteresis is commonly used to refer to this effect and will be used throughout this paper for the difference in forces in the FV plots.  However this effect is not the classical hysteresis defined in the scientific literature.  The cause of this phenomenon will be examined in the Literature Review section.  
  Figures 4-6 also have labeled points numbered one through four.  These are key points in the motion of the damper.  Point one is the beginning of the cycle.  The damper is at full extension and has zero starting velocity.  From point one to two the damper begins the compression stroke with increasing speed.  At point 2, the maximum negative velocity is achieved.  This usually corresponds to the peak force of the compression stroke.  The displacement is zero, which means half of the full stroke has been compressed into the damper.  From point two to three, the speed begins to decrease.  Point three represents the end of the compression stroke.  The displacement is at the full negative value, which means that the damper is fully compressed.  The speed has returned to zero.  Immediately after that point three, the rebound stroke begins with the speed increasing again.  At point four, the peak force of the rebound stroke is achieved.  The displacement is again at a zero value, so the damper is at extended to half of the total rebound stroke.  The cycle then goes from point four back to point one.  The rebound continues with the speed of the piston decreasing.  At point one, the damper returns to full extension and to zero velocity.  
  All plots generally remove the gas spring force.  Therefore, the force is equal to zero at velocity equal to zero.  
  The other plot sometimes used to characterize damper performance is the Force vs. Displacement (FD) plot.  Figure 7 shows a typical FD plot.  This plot is a carryover from the efforts to characterize dampers when all mechanical equipment used measured and charted only Force vs. Displacement.  
    
  FD plots use the same force sign convention; positive for compression, negative for rebound.  For both compression and rebound, the forces in Figure 7 are not symmetric about the y-axis.  The same hysteresis shown in the FV plots is the cause of this asymmetry.  
  In an attempt to gain understanding, hysteresis can also be examined using a hypothetical ideal spring, a hypothetical ideal damper, and sinusoidal motion input.  A hypothetical linear spring will produce a straight line with slope K in an FD plot and an ellipse in and FV plot (see Appendix A).  A hypothetical linear damper will produce a straight line with slope C in an FV plot and an ellipse in an FD plot.  Hysteresis in an FV plot for an actual damper results when the damper produces spring-like forces.  

LITERATURE REVIEW
  A literature review was conducted with two major goals.  The first goal was to obtain a better understanding of how individual internal components and internal flows had been characterized in the past by studying the development of parametric models for damper characterization.  
  The second goal of the literature review was to gain an insight into the hysteretic behavior that occurs in characteristic FV plots.  Understanding the causes of this phenomena and how it can be minimized are of crucial importance in damper design. Both of these concepts will be addressed in the cited literature.  
DAMPER SPECIFICATIONS
   The damper used for this research is a Tanner Externally Adjustable Gen 2 from Tanner Racing Products.  It is a gas charged monotube configuration with a floating piston which separates the gas and oil chambers.  The primary use for the Tanner Gen 2 s in quarter midget car racing, but their size, price, and range of available damping force make them appropriate for Formula SAE racecars as well.  The Tanner Gen 2 is ightweight, relatively inexpensive, and can attain the desired damping forces with nterior modifications.  Figure 8 shows a three dimensional model of the Tanner Gen 2 damper.  
       
  The length at maximum extension of the damper is 10.33 inches from centers of the spherical mounting bearings.  The stroke of the damper is approx three inches.  Outer housing of the damper and end caps are made of aluminum, while the chromed rod is made of polished steel.  The end caps are threaded for removal which makes disassembly easy for tuning or rebuild purposes.  
  The design of the piston and the shims used for controlling the piston orifice flow allow these parts to be manufactured much less expensively than most other racing dampers. The piston is made of machined aluminum and contains six straight orifice flow holes. The piston is shown in Figure 9.  
           
  The piston flow orifices have diameters of .038” and the center hole for mounting the piston on the rod is 0.25” diameter.  The groove on the outer diameter of the cylinder is for the rubber seal between the piston and the cylinder wall.  This piston design is less complex than that of an Ohlins or Penske brand damper and this simple design is much less expensive to produce.  
  Depending on the desired damping levels, pistons are available with the flow orifice diameters from 0.14” (soft damping) to 0.038” (hard damping).  Without any shims, the six orifices allow flow in both compression and rebound.  A separate shim tuning kit is also available from Tanner racing products; it is shown in Figure 10.  

           
  The shims kit from Tanner Racing includes carbon fiber shims.  The shims have almost an identical modulus of elasticity and Poisson’s ratio to that of steel, but are much lighter in weight.  The shims contain holes at locations corresponding to holes in the piston that can be used to create one way flow for compression or rebound.  For example, if a two hole shim is used for the compression side of the piston and a three hole shim for the rebound side of the piston, two one-way paths would exist for rebound and three paths would exist for compression as long as no holes are shared.  Also a combination of one way and two way flows can be created.  The arrangement of the shims can create numerous possibilities for tuning the Tanner Gen 2 damper.  It would also be possible to create shims of varying thicknesses or different materials to achieve
desired damping traits.  
  The threaded needle valve for adjusting the bleed orifice flow has 3.75 turns. The notation of zeros turns is analogous to a fully closed bleed orifice.  The larger the number of turns of adjustment, the more the bleed orifice is open.  This is a practical consideration since the fully closed position is easy to identify.  
  The damper fluid used was Tanner Tuned Shock Oil.  The properties were unknown for this oil, so typical 5W oil values were used for modeling purposes. Density and viscosity were of primary importance.