STABILITY OF PNEUMATIC and HYDRAULIC VALVES

Transcription

1 STABILITY OF PNEUMATIC and HYDRAULIC VALVES These three tutorials will not be found in any examination syllabus. They have been added to the web site for engineers seeking knowledge on why valve elements sometimes go unstable and what can be done to revent it. TUTORIAL - FLUID SPRINGS AND DASHPOTS This tutorial show how analogies may be used to derive the sring rate for a fluid column and the daming characteristics of a dashot. These are imortant elements in any hydraulic or neumatic system. The analogous quantities throughout will be as follows. Pressure () - Voltage (V) Mass flow ( m& ) - Current(I or i) Mass (m) - Charge (Q). PNEUMATIC SPRING Examles of neumatic srings are found in susension systems and seats. Any linear neumatic actuator will have a sringiness that should be consider when analysing the ossibility of oscillations due to the interaction of the mass and the sring. The following shows the alication of tutorial to a neumatic sring. The diagram shows a volume of gas traed in a cylinder by a iston. The gas ressure is ''. If the iston is moved a small distance 'x' the ressure rises as the gas is comressed. For raid movement the comression is adiabatic. ( Al ) ( l ) ( l ) ( l ) ( l ) ( l x) V V l ( l ) x A D.J.Dunn The increase in the force due to the gas ressure is F A( - ) Substitute for. ( ) ( ) l l A A l x l x Differentiate with resect to x. df l A and this indicates that the sring rate deends on the osition of the dx l x l x d iston but when x 0, A Note the units are N/m dx l This is the neumatic sring rate k at the start of the change but may be alied over a range if x<<l. The exression is articularly useful when analysing vibrations with small amlitudes.. HYDRAULIC SPRING A hydraulic fluid is virtually incomressible and this deends on the bulk modulus K. Only at excetionally high ressures does this become an issue (e.g. in some aircraft undercarriage designs the elasticity of the hydraulic fluid is used to roduced a measure of sringing). The elasticity of the ies is more likely to be a factor in hydraulic circuits.

2 3. PNEUMATIC DASHPOT Pneumatic dashots are used on many devices to dam out oscillations. The examle shown here was used in conjunction with a ressure relief valve. It was found that the valve oscillated u and down at a high frequency when relieving air from some systems. This was due to the resonance of the connecting ie and volume interacting with valve. It is interesting to analyse fully why the valve oscillated but in this section we will examine the daming characteristics of the dashot. The urose of the dashot was to damen these oscillations. In the original design there was no daming orifice and it was thought that the clearance ga between the iston and cylinder would roduce daming. Research showed that the damer simly acted as a neumatic sring that added to the steel sring simly determined the resonant frequency. The daming orifice made quite a difference. Basically, when the iston moves u, air is ushed out of the chamber and when the iston moves down, air is sucked into the chamber. The ressure roduced by the restriction and inertance always acts to oose the motion of the iston and hence damens the movement. It is assumed that the changes in ressure are adiabatic. This is an accurate assumtion for frequencies above Hz. The ressure inside the dashot is and outside is atmosheric a. The ressure inside is equal with the ressure outside when the valve starts to oscillate starting from the rest osition x o. The neumatic or ressure force acting on the iston is F. Let us examine the case when the oscillations are small in amlitude. This aroach is called a SMALL PERTURBATION ANALYSIS. Consider the simlified diagram. The air inside the dashot has a mass m, volume V, ressure and temerature T. The characteristic gas law gives V mrt Differentiate with resect to time. d dv dt dm V + mr + RT dt dt dt dt dv dx V A (l - x) hence A dt dt It is reasonable to assume that the change in ressure is adiabatic so dt - T d dt dt D.J.Dunn T constant

3 a + δ where δ is the increase in ressure relative to the outside. d d(δ) dm δ The mass flow rate through the orifice is m& where G is the orifice restriction or dt G resistance. Combining the equations we have: d dv dt dm V + mr + RT dt dt dt dt d( δ) dx - T d δ V A mr RT dt dt dt G d( δ) dx - T d δ V A mr RT note mrt V dt dt dt G d( δ) dx - d δ V A V RT note d d(δ) dt dt dt G d( δ) ( -) dx δ V d V A RT dt ( δ) dx δ A RT dt G dt dt G V d( δ) δ dx + RT A It is convenient here to change to Lalace form. dt G dt V ( ) ( δ) V RT s δ + RT A s(x) δ s A s(x) G + G V RT V RT δa s + A s(x) note δ A F s A s(x) G + G F A s G A s V Pneumatic caacitance was defined as C x V RT s + G V s + R T RT G G A s G A s CG A s Define a time constant as τ CG x V G V s + V( CG s + ) VG s + C C τ A s τ A s Note V A x V( τ s + ) (l - x) x ( l x)( τ s + ) From the revious section we know that at the mean osition the rate of change of force with d A kτ s distance is k so if x is small comared with l dx l x ( τ s + ) kτ s x The neumatic force is conveniently defined as ( τ s + ) kτ jω x Making the Lalace substitution s becomes jω ( τ jω + ) This shows that at high frequencies k x and so behaves as a neumatic sring with negligible daming. At low frequencies k τ ω x and so behaves the same as a viscous damer where force is directly roortional to velocity (v ω x). In this case we usually define the force as F c v where c is the viscous daming coefficient and c k τ D.J.Dunn 3

4 ( ) kτ jω x τ jω + or x ( τ jω + ) kτ jω If this is turned into a comlex number we have x j k τ jω x ( ω τ ) From the vector hence F k τω x The hase angle between F and x is ENERGY DISSIPATION φ tan ωτ k τω ( ω τ ) If the dashot oscillates harmonically with an amlitude X the daming force is kτω kτω x X sin( ωt + φ) ( ω τ ) ( ω τ ) For a given set of arameters this may be written F KX sin( θ + φ) and the dislacement is x X sin(θ) If we lot F against x for a given set of arameters we get a loo and the area within the loo is the work done against the ressure and hence the energy dissiated by the dashot. For cycle the energy dissiated is E π ( θ φ) Fdx KXsin + 0 dx x X sin(θ) so dx X cos(θ) dθ π E KX sin + φ 0 ( θ ) cos( θ) dθ E Fdx KX E θsin( ) φ φ 0 E KX [ π sin( φ) ] From the vector we have sin( φ) and utting ω τ K ( + ω τ ) ( ) π cos(θ )cos( φ) sin(θin(( ) ( ) k τω the energy dissiated for each cycle is E k τ π ω X ( τ ω) In a viscous damer the energy dissiated is E c π ω X kτ If we equate we can establish an equivalent viscous daming coefficient such that ce ( ωτ) Maximum daming will occur at any given frequency when θ 45 o and ω τ in which case:- kτ π kx ce and E These are the design arameters for a dashot to roduce maximum daming at a given frequency. D.J.Dunn 4

5 CASE STUDY A neumatic dashot similar to that shown in the revious diagram has the following arameters. The volume of the air at the mean osition is V 0460 mm 3 and the effective length was 5 mm. The ambient conditions are 00 kpa, T 88 K. The gas constants for air are.4 and R 87 J/kg K. The relationshi between ressure dro and mass air flow through the orifice was measured and it was found that at low ressure values the neumatic resistance was reasonably linear with a value of 3.8 x 0 6 N/ m s. Determine the energy lost to daming and the equivalent daming coefficient when the dashot is oscillated at 75 Hz with a eak to eak amlitude of.35 mm. First calculate the neumatic caacitance of the dashot. 9 V 0460 x 0 C x 0 m s R T.4 x 87 x 88 The time constant is τ GC 3.8 x 0 6 x 90.3 x 0.5 x0-3 s X.35/.75 mm l 5 mm hence A V/ l 48.4 x 0-6 m F 75 Hz hence ω πf rad/s k A/ l.4 x 00 x 0 3 x 48.4 x 0-6 / N/m φ tan - (/ωτ) 44.6 o k τ π ω X 3 343x.5x 0 x π x 47.39x 0.05 E 5.08x 0 τ ω x.5x 0 ( ) ( ) k 3 τ 343 x.5x 0 c e ( ωτ) 3 ( x.5x 0 ).486 N s/m Tests to determine the actual values gave a result quite close to the redicted values. 3 J D.J.Dunn 5

6 3. HYDRAULIC DASHPOT A tyical hydraulic dashot is a iston in a cylinder with holes allowing the liquid to move from one side of the iston to the other. Many variations are ossible. Without derivation, it can be shown that since the force required to shear a Newtonian fluid is directly roortional to the rate of shear, then the daming force roduced by hydraulic dashot is directly roortional to the velocity of the iston. F v. Velocity v is the first derivative of distance so F dx/dt dx The basic law of a dashot is: F(t) c c is the daming coefficient. dt Changed into Lalace form. F c s x x Rearranged into a transfer function H (s) F cs c is the daming coefficient with units of Force/Velocity or N s/m. The dashot can be reresented by a simle transfer function as shown. δf In terms of rate of sring rate k cs δx When the iston is recirocated harmonically with amlitude X the energy dissiated is π kx π X E c Instantaneous ower dissiated is P Force x velocity D.J.Dunn 6