力、力矩、压力测量

7.1 Introduction 7.1.1 Force (1)What is the force? Force is one of the most important physical parameters .Various mechanical motion is virtually the transmission of force or torque. Force possesses the capability in both static and dynamic phenomenon: it can change the body’s motion status or cause body’s deformation.

(2)unit of force N(Newton): is the force which when applied to a mass of one kilogram gives it an acceleration of 1 m/s2. (3)Weight: is the force exerted on a body of mass m due to earth gravity. G=mg; g is the local gravity acceleration, typically about 9.8m/s2. Weight measurement is usually specified in terms of the mass.

(4) Torque (力矩、扭矩、转矩） Torque is sometimes called moment. It is the product of a force and its arm. For a rotating shaft of radius r, the torque T applied to the shaft to cause it to rotate is F×r, where F is the tangential force at the radius r. When a shaft is twisted as a result of the application of a torque, shear stresses and strain are produced in the shaft material.

(5) Pressure A force F perpendicular to an area A, the pressure P is P=F/A unit : Pascal(Pa) 1Pa=1N/m2 Atmospheric pressure: the atmosphere surrounding the earth exerts a force on the unit earth surface, that is the pressure caused by the weight of the atmosphere. It varies with weather condition, the height above sea level , latitude.

Absolute pressure: used for the pressure measured relative to zero pressure. Gauge pressure: the pressure measured relative to the atmospheric pressure. The differential pressure between the absolute pressure and the atmospheric pressure. Positive pressure: the gauge pressure higher than the atmospheric pressure.

Negative pressure: the gauge pressure lower than the atmospheric pressure. Vacuum: the absolute pressure lower than atmosphere.

7.2 Measurement methods 7.2.1 Force measurement methods (1) According to static or dynamic effect Static effect----- the static force causes mechanical deformation. By measuring the deformation or using some inner stress or force related physical effects. Dynamic effect---- according to the Newton 2nd law, the force is associated with acceleration. Knowing mass and acceleration measured, the force can be defined: F=ma

Force measurement (2) according to the methods using different structure & instruments Lever balance methods Based on the principle of moments i.e. at static equilibrium the algebraic sum of the clockwise moments about an axis equals the anticlockwise moments.The moment of a force about an axis is the product of the force and the perpendicular distance from its line of action to the axis.This method is typically used for weights up to 1000Kg. 2. Force balance methods 3. Elastic element methods 4. Pressure methods

一种伺服式测力系统。无外力作用时，系统处于初始平衡位置，光线全部被遮住，光敏元件无电流输出，力发生器不产生力矩。当被测力F作用在杠杆上时，系统失去平衡，杠杆发生偏转，窗口打开相应的缝隙。光线通过缝隙，照射到光敏元件上，光敏元件输出与光照成比例的电信号，经放大后加到磁电力矩发生器的旋转线圈上。载流线圈与磁场相互作用而产生电磁力矩，用于平衡被测力F与配重(标准质量m)力的力矩之差；使杠杆重新处于平衡状态。当杠杆处于新的平衡位置时，其转角与被测力F成正比，放大器输出电信号在采样电阻R上的电压U0与被测力F成比例。 磁电式力平衡测力系统 Force balance methods

Elastic element methods Sensors that are used for measurement of force, torque or pressure often contain an elastic element that converts the mechanical quantity into a deflection or strain which can then be transformed using another sensor into an electrical signal. Electrical resistance strain gauges are widely used in this capacity.

spring damping In general, elastic elements used for the measurement of force can be considered to represented by the model shown in Fig. When a force f(t) acts on the mass of element, a displacement is produced which varies with time before obtaining a steady-state value.The displacement variation with time is described by a second order differential equation. In practice, often it is not the displacement itself to be measured but the strain experienced by a strain gauge attached to what is equivalent of the spring.

Diagram of strain gauge

Various forms of elastic members are used Various forms of elastic members are used. The simplest is just a spring to make a device called spring balance. The extension of the spring represents the force applied. Load cells, i.e. elastic members which transform force into displacement or strains, can take many forms .

The structure of typical elastic element and its designing calculation as follows: Columnar load cell

Bending beam Applied force Proving ring 测力环

The relative extension of strain gauge in elastic element : Δl-------Total extension of strain gauge l--------the original length of strain gauge F------applied force A------working area of elastic element E------Yang model of elastic element σ------stress of elastic element

Sensitivity : ----poison constant of elastic element Total strain of elastic element: Output of electrical bridge: In general, select k=2 , Voltage sensitivity(mv/v):

Other principle of force measurement (1) Piezoelectric dynamometry 压电式测力仪利用压电材料(石英晶体、压电陶瓷）的压电效应，将被测力经弹性元件转换为与其成正比的电荷量输出，通过测量电路测出输出电荷，从而实现对力值的测量。弹性元件感受力F时压电材料产生电荷Q输出 dij: Piezoelectric constant 适用于动态力的测量

Piezomagnetic dynamometry 由工业纯铁、硅钢等铁磁材料制成的铁心在机械力的作用下磁导率发生变化称为压磁效应，或称磁弹性效应。如图a所示，在铁心上安置一对线圈一励磁线圈和测量线圈。两线圈的平面相互垂直, 若无外力作用，励磁线圈中心交流电流所建立的磁场对测量线圈没有输出，见图b。若外力作用在铁心上，铁心磁导率改变，则测量线圈被励磁线圈中的磁场交链而输出比例于外力大小的信号，见图c。

Torque measurement Torque measurement are used for rotating shafts to determine the power transmitted and also to monitor against failure as a result of shear stress. The measurement of torque is always linked with the power and rotating speed. The work done is the product of the force and the distance traveled in the direction of the force. Work done per second=F×2rn= (Power transmitted by a rotating shaft=2nT=T) F: Tangential force at a radius r T: Torque applied to a shaft to cause it to rotate.

Basic methods of torque measurement： （1）Transmission method (扭轴法） （2）Force balance method （3）Power transformation method

(1) When a torque applied to a n elastic shaft, as a result , the shaft is twisted and shear stress and strains are produced in the shaft material. G: shear model of material；: Torsion angle IP :polar second moment (极惯性矩)

When the torsion angle measured: When the shear stress  of shaft measured: When the strain of shaft measured: 45、135 : the strains along the direction of 45o、135o related to axis of the shaft.

1.电阻应变式转矩仪 在转轴上或直接在被测轴上，沿轴线的45。或135。方向将应变片粘贴上，当转轴受转矩M作用时，应变片产生应变，其应变量 与转矩M成线性关系。 对空心圆柱形轴： 对方形截面轴： G为转轴的弹性模量

When the torque is measured for a rotating shaft, it is necessary to obtain the output signal from the bridge connection of the strain gauges which are on the rotating shaft and provide the d.c. supply voltage for the bridge. One method of doing this is via slip rings. A problem with such an arrangement is the noise generated by variations in contact resistance between the rings and the brushes.

An alternative method is to mount the entire bridge, together with its voltage supply, on the rotating shaft and use the bridge output to modulate a radio signal which is then transmitted to nearby, stationary, equipment for display or recording.

(2)力平衡法 它是利用平衡转矩M0去平衡被测转矩M，从而求得M的方法。当转轴受转矩作用时，机体上必定同时作用着方向相反的平衡力矩(或称支座反力矩)，因此测量出机体上的平衡力矩就可以知被测转矩大小。

图为反力法测量小转矩的测试系统。由测量轴2、静压空气支承3、角位移传感器5、力矩电动机4和标准电阻Ro及电路组成。

(3)能量转换法 这是按能量守恒定律来测量力矩的仪器。它是通过测量其它与转矩有关的能量系数(如电能系数)来确定被测力矩大小的。 根据能量转换法制作的测力矩仪一般多用于测量各种电机的转矩。因其影响因素较多测量误差大，所以只有测量电参数的电机转矩测量仪应用较多。

Pressure measurement Together with temperature and flow, pressure is the most important parameters in industrial process control The unit of pressure is the Pascal (Pa) with 1Pa being 1N/m2 At the surface of the earth, the atmospheric pressure is generally about 100KPa. This is sometimes referred to as a pressure of 1bar.

1.Manometers（液体式压力计） U-tube manometer The cistern manometer The inclined tube manometer 2.Diaphragms(薄膜，弹性式压力计） Reluctance diaphragm gauge Capacitance diaphragm gauge Strain gauge diaphragm gauge Force-balance diaphragm gauge Piezoelectric diaphragm gauge 3. Bourdon tubes

Calibration of the pressure gauges in the region of 20Pa to 2000kPa is generally by means of the Dead-weight tester. Pressure is produced by winding in a piston. The pressure is determined by adding weights to the platform so that it remains at a constant height.                                                                          Dead-Weight Tester Schematic

Manometers The basic manometer consists of a U-tube containing a liquid.A pressure difference between the gases above the liquid in the two limbs produces a difference h in vertical heights of the liquid in the two limbs. If one of the limbs is open to the atmosphere then the pressure difference is the gauge pressure. P2 P1

Water, alcohol and mercury are commonly used manometric liquids Water, alcohol and mercury are commonly used manometric liquids. U-tube manometers are simple and cheap and can be used for pressure differences in the range 20 Pa to 140KPa. The accuracy is typically about 1%. Temperature affect---------liquid expansion

Thus the pressure when measured by a U-tube manometer at a temperature , when the manometer liquid density at 0°C is known, is given by:

Cistern manometer An industrial form of the U-tube manometer is cistern manometer. It has one of the limbs with a much greater cross-sectional area than the other.A difference in pressure between the two limbs causes a difference in liquid level with liquid flowing from one limb to the other. H h d P1 P2 A2 A1

This form of manometer thus only require the level of liquid in one limb to be measured from a fixed point.

The inclined tube manometer The inclined tube manometer is a U-tube manometer with one limb having a larger cross-section than the other and the narrower limb being inclined at some angle  to the horizontal. It is generally used for the measurement of small pressure differences and gives greater accuracy than the conventional U-tube manometer.

P2 P1 d x H h Initial zero level with no pressure difference  Since A2 is much greater than A1, the equation approximates to:

Bourdon tubes

The bourdon tube may be in the form of a “C”, a flat spiral, a helical spiral. In all forms, an increase in the pressure in the tube causes the tube to straighten out to an extent which depends on the pressure. This displacement may be monitored in a variety of ways, for example, to directly move a pointer across a scale, to move a slider of a potentiometer, to move the core of an LVDT.

Diaphragms With diaphragm pressure gauges, a difference in pressure between two sides of a diaphragm results in it blowing out to one side or the other. If the fluid for which the pressure is required is admitted to one side of the diaphragm and the other side is open to the atmosphere, the diaphragm gauge gives the gauge pressure. If fluids at different pressures are admitted to the two sides of the diaphragm, the gauge gives the pressure difference.

1 Reluctance diaphragm gauge (磁阻隔膜压力计） The displacement of the central part of the diaphragm increases the reluctance of the coil on one side of the diaphragm and decreases it on the other. With the two coils connected in opposite arms of an a.c. bridge, the out of balance voltage is related to the pressure difference causing the diaphragm displacement

Capacitance pressure transducers were originally developed for use in low vacuum research. This capacitance change results from the movement of a diaphragm element. The diaphragm is usually metal or metal-coated quartz and is exposed to the process pressure on one side and to the reference pressure on the other. Depending on the type of pressure, the capacitive transducer can be either an absolute, gauge, or differential pressure transducer.

The diaphragm is between two fixed plates and its movement thus increases the capacitance with respect to one fixed plate and decreases it with respect to the other.  As shown in Figure ，the deflection of the diaphragm causes a change in capacitance that is detected by a bridge circuit. The capacitor can also form part of the tuning circuit of a frequency modulated oscillator and so give an electrical output related to the pressure difference across the diaphragm.

式中，E为弹性膜片的弹性模量；u为材料的泊松比

Potentiometric Pressure Transducer (force balance bellow gauge)                                                                        

Measurement of low pressures (vacuum) Vacuum tends to be used for pressures less than the atmospheric pressure, namely 1.013×105 Pa. A unit that is often used for such pressure is the torr (托）, this being the pressure equivalent to that given by a column of mercury of height 1 mm. 1mmHg=133.322Pa=1 torr Atmospheric pressure is about 760mmHg The lower the absolute pressure is, the higher the degree of vacuum is.

按测量方法的原理来分可以分为： 现在真空技术在生产、科研中的用途愈来愈广，在整个真空范围内所采用的测量方法也是多种多样的。 基于力的作用原理：U形管，波登管式，波纹管式，膜片式； 基于压缩作用原理：麦氏真空计； 基于导热作用原理：电阻真空计，热电偶真空计； 基于电离作用原理：热阴极式，冷阴极式，放射性真空计。

U形管真空计是最简单的真空计。其结构原理如图所示。一根由玻璃管制成的U形管中盛上水银或油, U形管一端接到真空系统上，另一端为大气。随着系统中真空度的升高，U形管真空侧的液面在大气作用下随之上升。这样，依据两管中校面的高度差，就可以测得真空系统中的真空度，即p=pa-h，其中p为系统中真空度，pa为大气压值，h为两液面之差。

（2） 压缩真空计 玻义耳-马略特定律 PV=C（常数）（T不变时） （2） 压缩真空计     利用波义耳定律，将被测真空系统中一定的残余气体加以压缩，比较压缩前后体积、压力的变化，即能算出真空度。 玻义耳-马略特定律 它反映气体的体积随压强改变而改变的规律。对于一定质量的气体，在其温度保持不变时，它的压强和体积成反比；或者说，其压强P与它的体积V的乘积为一常量，即 PV=C（常数）（T不变时） 或 P1V1＝P2V2=…=PnVn 实际气体只是在压强不太高、温度不太低的条件下才服从这一定律。

麦氏计是一种绝对真空计，测量范围为1. 3xl03－1 麦氏计是一种绝对真空计，测量范围为1.3xl03－1.3×10－8帕，精度较高，可作为其它真空计的标准。它是于1874年由麦克劳研制的，故命名为麦克劳真空计，简称麦氏计。其结构如图3．2所示。主要由毛细管A、毛细管B、玻璃泡、导液管、水银瓶构成。 麦克劳真空计

毛细管A、玻璃泡及下面一段管子(到a-a面为止)的体积为V1。测量前需用油封机械泵将水银瓶抽真空，使水银面处于a-a位置。这时，玻璃泡与真空系统相通，其压强与真空系统相同，均为p。转动切换阀门，切断油封机械泵，使水银瓶与干燥空气相通。此时，水银瓶中水银在干燥空气压力作用下，沿导液管上升，充满玻璃泡．然后沿毛细管A上升，使玻璃泡和毛细管A中气体被压缩到毛细管A的顶部，其体积为V2。因管A中压强高于管B中压强，使两个毛细管产生高度差h，这时，根据物理学中波义耳定律就可计算出真空系统中的压强，即 pV1=(p+h)V2 其中p为真空系统中压强。

2 热导式真空计 热导式真空计是根据在低压强下气体热导率与其压力之间关系制成的一种测量低真空的相对真空计。 2 热导式真空计 热导式真空计是根据在低压强下气体热导率与其压力之间关系制成的一种测量低真空的相对真空计。 原理：基于大气体压强低于某一定值时，气体的导热系数 K 与 P 成正比，        即：     K = b P            b：比例常数

热导真空计的工作原理是假设灯丝由导热损失的热量与加热电流I所产生的热量平衡时，灯丝温度不变。其平衡方程为． 其中R为灯丝电阻；E1为气体分子迁移热量；E2为辐射迁移热量；E3为引出导线的迁移热量。若由于压力减小而使E1减小，则当I不变时，平衡方程将失去平衡，使灯丝温度变化。由此可根据灯丝温度来衡量压力的变化。所以热导真空计是通过测量灯丝温度来决定压力大小的。

根据测定气体热传导方法的不同，热导式真空计可分为电阻真空计和热电偶真空计两种。 (1)电阻真空计 又称皮喇尼真空计 它主要由电阻式规管和测量电路两部分组成。电阻式规管如图所示。在电阻规管内封装一只电阻温度系数较大的电阻丝，常用的有钨丝和铂丝。测量时规管与被测真空计系统相连。在较低的压力(小于13.3Pa)时，热电阻丝的电阻值取决于周围气体的压强。

(2) 热电偶真空计（相对真空计）     热偶真空计测量真空度的元件是热偶规管，其结构见图。它主要由玻璃壳、铂丝、热电偶构成。铂丝用于加热热电偶，通以恒定电流，其温度为100—200℃。热电偶是由镍铬—镍铝、或镍－康铜的丝制成的，其作用是在它的加热端与冷端(非加热端）温度不同时，会产生温差电动势。我们利用真空度不同，气体传热性不同，进而使温差电动势不同的特性来测量真空度的。

通过测量温差电动势，就间接地测得了真空度。热偶计只能测量低的真空度，真空度再高时，压强变化与气体热传导无关，故此真空计不能用于高真空测量。

3 电 离 真 空 计 （量程范围为：1×10-3～5×10-10托） 电离真空计诞生于1916年，由巴克利首先研制成功。以后不断地改进，使其测量范围达到1.3×10－2一1.3×10－5帕，变成了高真空区域广泛使用的真空计。这种真空计可以远距离测量，易于实现自动记录及控制，由于有炽热的灯丝，不适于测高氧气体。

电离真空计是通过在稀薄气体中引起电离然后利用离子电流测量压力。 在气体中如果有动能足够大的电子与气体分子相碰撞，它可以从气体分子中击出一个或几个电子使气体分子成为正离子。把这种正离子收集到一个电极上使其产生离子电流，在稀薄气体中，它与气体压力有关。电离真空计有三种形式：热阴极电离真空计、冷阴极电离真空计和放射性电离真空计。

热阴极电离真空计主要由圆筒式热阴极电离规管和测量电路两部分组成。 测量时规管与被测真空系统相连。通电后阴极加热所发射的电子在带正电的加速极作用下，以加速度运动，当电子的动能足够大，在飞向加速极的路途中与管内低压气体分子碰撞，即可使气体分子电离。电离产生的电子和正离子分别在加速极和收集极(带负电位)上形成电流Ie和离子流Ii。当压力足够低(低于10-1Pa)时，离子电流Ii与电子电流Ie之比正比于气体的压力p，即

电离规管由阴极(灯丝)、螺旋形栅极(加速极)和圆筒形收集极构成。

式中，k为规管的灵敏度。由于用电离真空计测量压力时，电子电流保持不变，故