Chapter 5 Force,Torque(转矩)，and Shaft Power(轴功率) Measurement by Yixin Ma 27/03/2013 2/50 Contents 1.Standards and Calibration 2.Basic methods of Force Measurement 3.Characteristics of Elastic Force Transducers 4.Resolution of Vector Forces and Moments into Rectangular Components 5.Torque Measurement on Rotating Shafts 6.Shaft Power Measurement(Dynamometers) 7.Vibrating-Wire Force Transducers
Chapter 5 Force, Torque(转矩), and Shaft Power(轴功率) Measurement by Yixin Ma 27/03/2013 Contents 1. Standards and Calibration 2. Basic methods of Force Measurement 3. Characteristics of Elastic Force Transducers 4. Resolution of Vector Forces and Moments into Rectangular Components 5. Torque Measurement on Rotating Shafts 6. Shaft Power Measurement (Dynamometers) 7. Vibrating-Wire Force Transducers 2/50
3/50 5.1 Standards and Calibration ■Force definition:F=M×A,inN=kgm/s2 M:mass is considered a fundamental quantity,in kg -International Kilogram Standards:cylinder of Platinum-Iridium,France A:acceleration,derived(from length and time,in m/s2 d2L A= dt2 ■Torque definition:T=F×R,inNm ■Shaft power definition:P=T×w,inw=Wm/s 4/50 5.1 Standards and Calibration The acceleration of gravity,g v=0U。=max v=0 U=max Fr=max KE=0 Fr=max KE =0 Fr=min Fr=min It is a convenient standard which can be determined with an accuracy of about 1/106 by 8=-8ma measuring the period and effective length of a 8=0 pendulum(摆锤)or by determining the change v=max U.=min with time of the speed of a freely falling body. F=0 KE max Fr=max >The actual value g varies with location and also slightly with time (in a periodic predictable fashion)at a given location.It may also change (slightly)unpredictable because of local T2m 6《1 geological activity. The so-called standard value of g refers to the value at sea level and 45 latitude and is numerically 9.80665m/s2
5.1 Standards and Calibration Force definition: ࡲ=ࡹ× ,in N=kg·m/s2 ¾ M: mass is considered a fundamental quantity, in kg - International Kilogram Standards: cylinder of Platinum-Iridium, France ¾ A: acceleration, derived(导出) from length and time, in m/s2 = ࡸࢊ ࢚ࢊ Torque definition: ࢀ=ࡲ×ࡾ , in N·m Shaft power definition: ࡼ=ࢀ×࣓ , in w=N·m/s 3/50 5.1 Standards and Calibration The acceleration of gravity, g ¾ It is a convenient standard which can be determined with an accuracy of about 1/106 by measuring the period and effective length of a pendulum (摆锤) or by determining the change with time of the speed of a freely falling body. ¾ The actual value g varies with location and also slightly with time (in a periodic predictable fashion) at a given location. It may also change (slightly) unpredictable because of local geological activity. ¾ The so-called standard value of g refers to the value at sea level and 45° latitude and is numerically 9.80665m/s2. 4/50
5/50 5.1 Standards and Calibration Basis of the "deadweight"calibration of force-measuring system: when the numerical value of g has been determined at a particular locality, the gravitational force (weight)on accurately known standard masses may be computed to establish a standard of force. Commercially available calibrating machine using deadweights,knife edges, and levers,covers the range of 0-50kN with an accuracy of +0.005%of applied load and a resolution of +0.0062%of applied load. Computerized calibration systems based on strain gage load cells and hydraulic load frames are also available. 6/50 5.2 Basic Methods of Force Measurement 1. Balancing it against the known gravitational force on a standard mass,either directly or through a system of levers(杠杆). >The analytical balance,which simple in principle, F requires careful design and operation to realize its Unknown Standard force maximum performance. mass Analytical balance >The beam deflection(梁的挠度)is a very sensitive Fig5.2(1)a indicator of unbalance.(due to its proper design) For the low end of a particular instrument's range,often the beam deflection is used as the output reading rather than attempting to null by adding masses or adjusting the arm length of a poise()weight.This approach is faster than nulling but requires that the deflection-angle unbalance relation be accurately known and stable. >Noise inputs:deformation of knife edges,buoyant force,temperature difference
5.1 Standards and Calibration Basis of the "deadweight" calibration of force-measuring system: ¾ when the numerical value of g has been determined at a particular locality, the gravitational force (weight) on accurately known standard masses may be computed to establish a standard of force. ¾ Commercially available calibrating machine using deadweights, knife edges, and levers, covers the range of 0-50kN with an accuracy of ±0.005% of applied load and a resolution of ±0.0062% of applied load. ¾ Computerized calibration systems based on strain gage load cells and hydraulic load frames are also available. 5/50 5.2 Basic Methods of Force Measurement 1. Balancing it against the known gravitational force on a standard mass, either directly or through a system of levers (杠杆). ¾ The analytical balance, which simple in principle, requires careful design and operation to realize its maximum performance. ¾ The beam deflection(梁的挠度) is a very sensitive indicator of unbalance. (due to its proper design) Fig 5.2 (1)a ¾ For the low end of a particular instrument’s range, often the beam deflection is used as the output reading rather than attempting to null by adding masses or adjusting the arm length of a poise(平衡) weight. This approach is faster than nulling but requires that the deflection-angle unbalance relation be accurately known and stable. ¾ Noise inputs: deformation of knife edges, buoyant force, temperature difference, … 6/50
7/50 5.2 Basic Methods of Force Measurement 1.Balancing it against the known ∠∠∠∠∠/∠∠1 gravitational force Tape >The pendulum scale,is a deflection- type instrument in which the unknown force is converted to a torque that is Sector then balanced by the torque of a fixed standard mass arranged as a pendulum. >A practical version of this principle utilizes specially shoed sectors and steel tapes to linearize the inherently nonlinear torque-angle relation of a Counter- pendulum. weights Steel An electrical signal proportional to tapes force is easily obtained from any angular-displacement transducer Pendulum Fig 5.2(1)b scale attached to measure the angle. 8/50 5.2 Basic Methods of Force Measurement 1.Balancing it against the known gravitational force. Standard ma552 >The platform(托盘)scale,utilizes ("Poise weight") Standard mass I a system of levers to allow ☐n weigh"T measurement of large forces in terms Platform of much smaller standard weights The beam is brought to null by a proper combination of pan weights and adjustment of the poise-weight lever arm along its calibrated scales. Fig 5.2(1)c Platform Scale Self-balance:add an electrical displacement pickup for null detection and amplifier-motor system to position the poise weight to achieve null. When a/b=c/d,the reading of the scale is independent of the location of force on the platform.(why?)
5.2 Basic Methods of Force Measurement 1. Balancing it against the known gravitational force. ¾ The pendulum scale, is a deflectiontype instrument in which the unknown force is converted to a torque that is then balanced by the torque of a fixed standard mass arranged as a pendulum. ¾ A practical version of this principle utilizes specially shoed sectors and steel tapes to linearize the inherently nonlinear torque-angle relation of a pendulum. ¾ An electrical signal proportional to force is easily obtained from any angular-displacement transducer attached to measure the angle. Fig 5.2 (1)b 7/50 ¾ Self-balance: add an electrical displacement pickup for null detection and amplifier-motor system to position the poise weight to achieve null. ¾ When a/b=c/d, the reading of the scale is independent of the location of force on the platform. （why?） 5.2 Basic Methods of Force Measurement 1. Balancing it against the known gravitational force. ¾ The platform (托盘) scale, utilizes a system of levers to allow measurement of large forces in terms of much smaller standard weights. ¾ The beam is brought to null by a proper combination of pan weights and adjustment of the poise-weight lever arm along its calibrated scales. Fig 5.2 (1)c Platform Scale 8/50
9/50 5.2 Basic Methods of Force Measurement 2.Measuring the acceleration of a Accelerometer body of known mass to which the unknown force is applied. The use of an accelerometer for force measurement,is of somewhat limited application since the force determined is the resultant force(合力) on the mass. Fig 5.2(2)Force Measurement via Acceleration measurement Often several unknown forces are acting,and they cannot be separately measured by this method. 10/50 5.2 Basic Methods of Force Measurement 3.Balancing it against a magnetic force developed by interaction(相互作用)of a current-carrying coil(线圈)and a magnet.. It utilizes a photoelectric ()(or other displacement sensor)null detector,an amplifier and a torquing coil in a servo-system(伺服系统)to balance the difference between the unknown force and the gravity force on a standard mass Its advantages relative to mechanical balances are ease of use,less sensitivity to environment,faster response,smaller size,and ease of remote operation. The electric output signal is convenient for continuous recording and/or automatic-control applications. Fig 5.2(3)a&b shows a design available in range from 22 to 405 grams,with resolutions from 2 to 100 ug
5.2 Basic Methods of Force Measurement 2. Measuring the acceleration of a body of known mass to which the unknown force is applied. ¾ The use of an accelerometer for force measurement, is of somewhat limited application since the force determined is the resultant force(合力) on the mass. ¾ Often several unknown forces are acting, and they cannot be separately measured by this method. Fig 5.2 (2) Force Measurement via Acceleration measurement 9/50 5.2 Basic Methods of Force Measurement 3. Balancing it against a magnetic force developed by interaction (相互作用) of a current-carrying coil (线圈) and a magnet. ¾ It utilizes a photoelectric (光电) (or other displacement sensor) null detector, an amplifier and a torquing coil in a servo-system (伺服系统) to balance the difference between the unknown force and the gravity force on a standard mass. ¾ Its advantages relative to mechanical balances are ease of use, less sensitivity to environment, faster response, smaller size, and ease of remote operation. ¾ The electric output signal is convenient for continuous recording and/or automatic-control applications. ¾ Fig 5.2 (3) a&b shows a design available in range from 22 to 405 grams, with resolutions from 2 to 100 ug. 10/50
11/50 5.2 Basic Methods of Force Measurement 3.Balancing against a magnetic force. >A flexure-pivot (lever system puts large input forces within the range of a relatively small magnetic force coil. The signal from the optical displacement sensor is the error signal in the servo- system,which provides a coil current to balance the unknown input force and restore the deflection to near zero. All motions are constrained with flexure bearings (to give the nearly frictionless (performance required for resolutions as small as 2 ug. Temperature effects (observed mainly in the magnetic field strength)are compensated in software. ③ ⑦ Pan ttttttt ② Suspension ③Parallel guide Flexible bearings Flexible bearing ⑤Coupling ① 3 Lever 7 Flexible fulcrum 8 Coil ⑤ ⑨Permanent magnet G Flux lines 包 Diaphragm ②Optical position indicator ③Temperature sensor (3a) (3b 12/50 5.2 Basic Methods of Force Measurement 3.Balancing against Operating principle Electromagnetic balance of the magnetic a magnetic force. suspension balance >A version that allows The controlled electromagnet exerts a magnetic force on the permanent magnet through the nonmagnetic the weighed sample to vessel wall,supporting the sample weight.This force Control system is measured by the electromagnetic balance be immersed in an Electromagnet atmosphere of controlled Set point controller Coupling housing temperature,pressure, Permanent magnet and fluid composition, PID controller completely sealed off Sensor core from the weighing Position transducer Sensor coil balance,for sensitive density,sorption(吸附)， Many versions of this basic balance are available for different pressure Measuring and chemical studies and temperature ranges and load decoupling measuring tasks Sample (3c
5.2 Basic Methods of Force Measurement 3. Balancing against a magnetic force. ¾ A flexure-pivot (挠性轴) lever system puts large input forces within the range of a relatively small magnetic force coil. ¾ The signal from the optical displacement sensor is the error signal in the servosystem, which provides a coil current to balance the unknown input force and restore the deflection to near zero. ¾ All motions are constrained with flexure bearings (轴承) to give the nearly frictionless (无摩擦) performance required for resolutions as small as 2 ug. ¾ Temperature effects (observed mainly in the magnetic field strength) are compensated in software. 11/50 5.2 Basic Methods of Force Measurement 3. Balancing against a magnetic force. ¾ A version that allows the weighed sample to be immersed in an atmosphere of controlled temperature, pressure, and fluid composition, completely sealed off from the weighing balance, for sensitive density, sorption (吸附), and chemical studies. 12/50
13/50 5.2 Basic Methods of Force Measurement 4.Transducing the force to a fluid pressure and then Load measuring the pressure. Loading head Hardened steel ball >Hydrostatic(流体静力学) cells:hydraulic cell is completely Soft rubber Clamp ring filled with oil and usually have a Flexible preload pressure of the order of boot 30 lb/in2. Box- -Clamp >Application of load increases Stayplate the oil pressure which is read on Preload an accurate gage. Gauging spring hole Bridge ring 》 Electrical pressure Base transducer can be used to Casing Hardened steel obtain an electrical signal. ring inserts The cells are very stiff. Hydrostatic load cell (4) 14/50 5.2 Basic Methods of Force Measurement SKETCH 4 SCHEMATIC HYDROSTATIC LOAD CELL 4.Transducing the force to a fluid pressure and STAYPLATE-FORCE then measuring the pressure. CYLINDER The figure illustrates the Emery Winslow Model WALL PISTON 136 -75,000 Ib.load cell used in truck scale DIAPHRAGN applications.This hydrostatic cell type is also used in high capacity tank weighing,floor scales and custom applications FL LOAD CELL BASE GROSS SECTION OF HYDROSTATIC LOAD CELL (MODEL 136) http://emerywieclow CENTERING K一BOOT CLAMP COLUMN +-CYLINDER 山和後器电 ■GAUGING ◆一HOLE EPLATE HYDROSTATIC FLUID DIAPHRAGM BRIDGE RING
5.2 Basic Methods of Force Measurement 4. Transducing the force to a fluid pressure and then measuring the pressure. ¾ Hydrostatic (流体静力学) cells: hydraulic cell is completely filled with oil and usually have a preload pressure of the order of 30 lb/in2. ¾ Application of load increases the oil pressure which is read on an accurate gage. ¾ Electrical pressure transducer can be used to obtain an electrical signal. ¾ The cells are very stiff. 13/50 5.2 Basic Methods of Force Measurement 4. Transducing the force to a fluid pressure and then measuring the pressure. ¾ The figure illustrates the Emery Winslow Model 136 - 75,000 lb. load cell used in truck scale applications. This hydrostatic cell type is also used in high capacity tank weighing, floor scales and custom applications http://emerywinslow.com/article_whosusing.html 14/50
15/50 5.2 Basic Methods of Force Measurement 4.Transducing the force to a fluid pressure and then measuring the pressure. >The pneumatic(气动式load cell uses a nozzle-flapper transducer as a high-gain amplifier in a servoloop. >Application of force F cause a diaphragm(横隔膜； 振动膜)deflection x,which in tum causes an increase in pressure po since the nozzle is more nearly shut off. This increase in pressure acting on the diaphragm area A produces an effective force F that tends to retum the diaphragm to its former position. >For any constant F,the system will come to equilibrium at a specific nozzle opening and corresponding pressure po. -Air supply >The static behavior is given by: (Fi-PoA)KaKn Po (5.3) 8☑ Where K is diaphragm compliance,in/lbf and K,is nozzle-flapper gain,(Ib/in2)/in. Pneumotic lood cell 16/50 5.2 Basic Methods of Force Measurement Column compression 5.Applying the force to some elastic(弹性) member and measuring the resulting deflection. All previously described force-measuring incular bending beam devices are intended mainly for static or slowly varying loads. >This method,the elastic deflection transducers, are widely used for both static and dynamic loads of Shear web frequency content up to many kHz. While all are essentially spring-mass systems Elostic with damping,they differ mainly in the geometric -to-do ection .ransducer form of "spring"employed and in the displacement transducer used to obtain an electrical signal. The parallelogram flexure is a displacement transducer arranged to measure motion in the a sensitive direction thus will measure only that component of an applied vector force which lies sT along the sensitive axis. Parollelogram flexure
5.2 Basic Methods of Force Measurement 4. Transducing the force to a fluid pressure and then measuring the pressure. ¾ The pneumatic(气动式)load cell uses a nozzle-flapper transducer as a high-gain amplifier in a servoloop. ¾ Application of force Fi cause a diaphragm(横隔膜; 振动膜) deflection x, which in turn causes an increase in pressure po since the nozzle is more nearly shut off. ¾ This increase in pressure acting on the diaphragm area A produces an effective force Fp that tends to return the diaphragm to its former position. ¾ For any constant Fi , the system will come to equilibrium at a specific nozzle opening and corresponding pressure po. ¾ The static behavior is given by: (3.5 ( = ܭௗܭ(ܣ − ܨ) Where Kd is diaphragm compliance, in/lbf and Kn is nozzle-flapper gain, (lb/in2)/in. 15/50 5.2 Basic Methods of Force Measurement 5. Applying the force to some elastic(弹性) member and measuring the resulting deflection. ¾ All previously described force-measuring devices are intended mainly for static or slowly varying loads. ¾ This method, the elastic deflection transducers, are widely used for both static and dynamic loads of frequency content up to many kHz. ¾ While all are essentially spring-mass systems with damping, they differ mainly in the geometric form of "spring" employed and in the displacement transducer used to obtain an electrical signal. ¾ The parallelogram flexure is a displacement transducer arranged to measure motion in the sensitive direction thus will measure only that component of an applied vector force which lies along the sensitive axis. S Type 16/50
17/50 5.3 Characteristics of Elastic Force Transducers 1.Idealized model of an elastic force transducer. The relationship between input force and output displacement is a simple 2nd- order form: Fi-Ksxo-Bio Mxo (5.8) (D) K D2/@3+25D/@n+1 (5.9) B where wn兰 Ks 2KsM andK1 Device of this type are also(unintentional)accelerometers and produce a spurious output in response to vibration inputs. For transducers that do not measure a gross displacement but rather use strain gages bonded to the"spring,"the output strain may be substituted for xo if K is reinterpreted 3K [8 as force per unit strain rather than force per unit deflection. 7777777777777777777 Fig 5.3 Elastic Force Transducer 18/50 5.3 Characteristics of Elastic Force Transducers 1.Idealized model of an elastic force transducer. For unusual shapes,or to check details such as stress concentrations, standard finite-element method(FEM)software can be used,but FEM does not provide any formulas,only specific numerical results for specific dimensions. >Most force transducers have no intentional damping(阻尼)；ξis due entirely to parasitic()effects and is not possible to predict theoretically. Transducers must be accurate to 1%or better,so they must always be calibrated after construction and before use;this calibration will easily get the actual damping ratio. >Example of FEM analysis regarding a"binocular'"(双筒望远镜)beam(K 横>梁)force transducer
5.3 Characteristics of Elastic Force Transducers 1. Idealized model of an elastic force transducer. ¾ The relationship between input force and output displacement is a simple 2ndorder form: ሶ࢞ − ࢙࢞ࡷ − ࡲ ሷ࢞ࡹ = ) 5.8) ࢞ ࡲ ࡷ = (ࡰ) ࣓ ࡰ (9.5 (࢘࣊ା ⁄࣓ ࡰࣈା ⁄ ࢙ࡷ ≜ ࣓ where ≜ ࣈ ,ࡹ ≜ ࡷ and, ࡹ࢙ࡷ ࢙ࡷ ¾ Device of this type are also (unintentional) accelerometers and produce a spurious output in response to vibration inputs. ¾ For transducers that do not measure a gross displacement but rather use strain gages bonded to the "spring," the output strain ε may be substituted for x0 if Ks is reinterpreted as force per unit strain rather than force per unit deflection. Fig 5.3 Elastic Force Transducer 17/50 5.3 Characteristics of Elastic Force Transducers 1. Idealized model of an elastic force transducer. ¾ For unusual shapes, or to check details such as stress concentrations, standard finite-element method (FEM) software can be used, but FEM does not provide any formulas, only specific numerical results for specific dimensions. ¾ Most force transducers have no intentional damping(阻尼); x is due entirely to parasitic(寄生) effects and is not possible to predict theoretically. ¾ Transducers must be accurate to 1% or better, so they must always be calibrated after construction and before use; this calibration will easily get the actual damping ratio. ¾ Example of FEM analysis regarding a "binocular"(双筒望远镜) beam (梁) force transducer. 18/50
19/50 5.3 Characteristics of Elastic Force Transducers 1.Idealized model of an elastic force transducer. ↓1.5in. The vertical support was L1 ILBS included to allow space under LBS the load cell for a deflection- measuring dial indicator. 2.0in. >The vertical load F is 4.0in. applied at the right link and the Thinnest section are 0.040 in. 5.25in. link have a simple translation x. Semicircles are 0.750 in.diameter 5-pound load shown at two locations FEM analysis based on some proper assumptions 7.0in. Fig 5.4a Finite Element Model of the Binocular Beam The meshed finite element model but will also serve as an"analysis sketch"for the strength of materials study. 20/50 5.3 Characteristics of Elastic Force Transducers 1.Idealized model of an elastic force transducer. The device is considered as a four-bar linkage where the usual pin joints are replaced by elastic hinges at the four thin sections. 以-0.00ae+000br-1.890e-0092-3.369e-002 Mode 1:0 Frequancy 588.0002 (rad/time) 93.5831 (cyeles/time Fig 5.4b Finite Element Deflection Results Fig 5.4c The First Natural Frequency
5.3 Characteristics of Elastic Force Transducers 1. Idealized model of an elastic force transducer. Fig 5.4a Finite Element Model of the Binocular Beam The meshed finite element model but will also serve as an “analysis sketch” for the strength of materials study. ¾ The vertical support was included to allow space under the load cell for a deflectionmeasuring dial indicator. ¾ The vertical load F is applied at the right link and the link have a simple translation x. ¾ FEM analysis based on some proper assumptions. 19/50 5.3 Characteristics of Elastic Force Transducers 1. Idealized model of an elastic force transducer. ¾ The device is considered as a four-bar linkage where the usual pin joints are replaced by elastic hinges at the four thin sections. Fig 5.4b Finite Element Deflection Results Fig 5.4c The First Natural Frequency 20/50
