CN101871832A - Analysis method of stability of dehydration and vibration processes of impeller type full automatic washing machine - Google Patents

Abstract

The invention relates to an analysis method of the stability of the dehydration and vibration processes of an impeller type full automatic washing machine. The analysis method comprises the following steps of: constructing the coordinate system of a system, dispersing liquid in a balancing ring into a plurality of rigid spheres, constructing the vibration model of the system, transforming rotational coordinates, obtaining a balance point and analyzing the stability of the system. The method can be used for guiding an actual design process and determines the dehydration stability of the impeller type full automatic washing machine conveniently and quickly.

Description

A kind of analytical approach of stability of dehydration and vibration processes of impeller type full automatic washing machine

Technical field

The present invention relates to a kind of analytical approach, especially a kind of analytical approach of stability of dehydration and vibration processes of impeller type full automatic washing machine at the full-automatic pulsator washing machine dehydration and vibration.

Background technology

The dehydration and vibration of automatic washing machine is the problem that manufacturer pays close attention to always.Because clothing is eccentric big and distribute at random in the dehydration, therefore adopting dynamic balancing technique is the important measures that suppress the laundry machine dehydration vibration.At present, full-automatic pulsator washing machine extensively adopts the fluid balance ring mechanism to carry out transient equilibrium.But this mechanism is introduced in the complicacy that has increased the dehydration and vibration characteristic when shaking.Analysis to stability of dehydration and vibration processes of impeller type full automatic washing machine for a long time lacks effective ways always.

Summary of the invention

The objective of the invention is to overcome the deficiencies in the prior art, a kind of analytical approach that can be used for instructing the stability of dehydration and vibration processes of impeller type full automatic washing machine of actual design process is provided.

According to technical scheme provided by the invention, the analytical approach of described stability of dehydration and vibration processes of impeller type full automatic washing machine comprises the steps:

(a) set up the system coordinate system step

Set up reference frame X _rY _rZ _rWith moving coordinate system X _bY _bZ _b, reference frame X _rY _rZ _rBe consolidated in the earth, initial point is positioned at O _rMoving coordinate system X _bY _bZ _bBe consolidated in steel ladle, suppose that the following hitch point of four suspension rods is in same plane A, this plane follow that steel ladle moves and with steel ladle axis Z _bBe vertically intersected on an O _b, O _bBe moving coordinate system X _bY _bZ _bInitial point, adopt Bu Lien angle [α β γ] ^TMoving coordinate system X is described _bY _bZ _bAttitude;

(b) liquid in the gimbal being dispersed is some rigid spheres steps

Be N rigid spheres with the liquid in the gimbal is discrete, establishing i spheroid is φ with respect to the rotation angle of dehydration barrel _i, its radius of turn around central shaft is d _i, the surfaces of revolution is X moving _bY _bZ _bIn relative height be H _iThis spheroid is X moving _bY _bZ _bIn position vector can be described as formula (1):

r _i＝[d _icos(θ+φ _i) d _isin(θ+φ _i) H _i] ^T (1)

θ is the dehydration corner in the formula (1), and then this spheroid is at reference frame X _rY _rZ _rIn position vector can be described as formula (2):

s _i＝x+A ^rbr _i (2)

X=[x y z in the formula (2)] ^T, A ^RbBe moving coordinate system X _bY _bZ _bRelative reference coordinate system X _rY _rZ _rAttitude matrix, the velocity that can get spheroid to the following formula differentiate is formula (3):

B is in the formula (3)

With moving is X _bY _bZ _bTransition matrix between angular velocity,

$\frac{{\∂r}_i}{\∂\mathrm{\θ}}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\θ}+{\mathrm{\φ}}_i)& d_i\cos (\mathrm{\θ}+{\mathrm{\φ}}_i)& 0\end{array}\right]}^T---\left(4\right)$

$\frac{{\∂r}_i}{{\∂\mathrm{\φ}}_i}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\θ}+{\mathrm{\φ}}_i)& d_i\cos (\mathrm{\θ}+{\mathrm{\φ}}_i)& 0\end{array}\right]}^T=\frac{{\∂r}_i}{\∂\mathrm{\θ}}---\left(5\right)$

Order

$D_i=\frac{{\∂r}_i}{\∂\mathrm{\θ}}=\frac{{\∂r}_i}{{\∂\mathrm{\φ}}_i}---\left(6\right)$

Then (3) but the formula abbreviation is formula (7):

If spheroid mass is m _b, ignore the influence of spheroid inertia, derive spheroid kinetic energy:

$T_i=\frac{1}{2}{\stackrel{\·}{\mathrm{\κ}}}_i^T\left[\begin{array}{cccc}{m}_b{I}_3& -m_b{A}^{\mathrm{rb}}{\tilde{r}}_{i}B& m_b{A}^{\mathrm{rb}}{D}_i& m_b{A}^{\mathrm{rb}}{D}_i\\ & m_b{B}^T{\left({\tilde{r}}_i\right)}^T{\tilde{r}}_{i}B& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i\\ & & m_b{d}_i^2& m_b{d}_i^2\\ \mathrm{sym}& & & m_b{d}_i^2\end{array}\right]{\stackrel{\·}{\mathrm{\κ}}}_i---\left(8\right)$

In the formula (8) ${\stackrel{\·}{\mathrm{\κ}}}_i={\left[\begin{array}{cccccccc}\stackrel{\·}{x}& \stackrel{\·}{y}& \stackrel{\·}{z}& \stackrel{\·}{\mathrm{\α}}& \stackrel{\·}{\mathrm{\β}}& \stackrel{\·}{\mathrm{\γ}}& \stackrel{\·}{\mathrm{\θ}}& {\stackrel{\·}{\mathrm{\φ}}}_i\end{array}\right]}^T;$

(c) set up system vibration model step

Obtain the vibration equation of system by the Lagrange equation:

$M\stackrel{\·\·}{\mathrm{\ξ}}=\frac{1}{2}{\left[\frac{\∂M}{\∂\mathrm{\ξ}}\stackrel{\·}{\mathrm{\ξ}}\right]}^T\stackrel{\·}{\mathrm{\ξ}}-\left(\underset{i=1}{\overset{6+N}{\mathrm{\Σ}}}\frac{\∂M}{{\∂\mathrm{\ξ}}_i}{\stackrel{\·}{\mathrm{\ξ}}}_i\right)\stackrel{\·}{\mathrm{\ξ}}+\left[\begin{array}{c}Q\\ Q_{\mathrm{\τ}}\end{array}\right]-\frac{{\∂V}_g}{\∂\mathrm{\ξ}}---\left(9\right)$

Wherein N is the number of discrete spheroid, and M is the mass of system matrix; V _gBe system's gravitional force; Q is the suspension generalized force;

$Q_{\mathrm{\τ}}=-{\left[\begin{array}{cccc}{C}_{\mathrm{\τ}}{\stackrel{\·}{\mathrm{\φ}}}_1& C_{\mathrm{\τ}}{\stackrel{\·}{\mathrm{\φ}}}_2& ...& C_{\mathrm{\τ}}{\stackrel{\·}{\mathrm{\φ}}}_N\end{array}\right]}^T---\left(10\right)$

Be the suffered gimbal damping force of spheroid; C _τBe the gimbal ratio of damping.

(d) rotating coordinate transformation step

Introduce following rotating coordinate transformation:

$\left\{\begin{array}{c}\mathrm{\τ}=\mathrm{\Ωt}\\ \frac{\mathrm{d\τ}}{\mathrm{dt}}=\mathrm{\Ω}\\ {\mathrm{\ξ}}_1={\mathrm{\ϵ}}_1\cos \left(\mathrm{\τ}\right)-{\mathrm{\ϵ}}_2\sin \left(\mathrm{\τ}\right)\\ {\mathrm{\ξ}}_2={\mathrm{\ϵ}}_1\cos \left(\mathrm{\τ}\right)+{\mathrm{\ϵ}}_2\sin \left(\mathrm{\τ}\right)\\ {\mathrm{\ξ}}_3={\mathrm{\ϵ}}_3\\ {\mathrm{\ξ}}_4={\mathrm{\ϵ}}_4\cos \left(\mathrm{\τ}\right)-{\mathrm{\ϵ}}_5\sin \left(\mathrm{\τ}\right)\\ {\mathrm{\ξ}}_5={\mathrm{\ϵ}}_5\cos \left(\mathrm{\τ}\right)+{\mathrm{\ϵ}}_5\sin \left(\mathrm{\τ}\right)\\ {\mathrm{\ξ}}_6={\mathrm{\ϵ}}_6\\ ...\\ {\mathrm{\ξ}}_{6+N}={\mathrm{\ϵ}}_{6+N}\end{array}\right.---\left(11\right)$

(11) are expressed as the matrix-vector form:

ξ＝Hε (12)

Wherein

$H=\left[\begin{array}{ccccccccc}\cos \left(\mathrm{\τ}\right)& -\sin \left(\mathrm{\τ}\right)& 0& 0& 0& 0& 0& ...& 0\\ \sin \left(\mathrm{\τ}\right)& \cos \left(\mathrm{\τ}\right)& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 1& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& \cos \left(\mathrm{\τ}\right)& -\sin \left(\mathrm{\τ}\right)& 0& 0& ...& 0\\ 0& 0& 0& \sin \left(\mathrm{\τ}\right)& \cos \left(\mathrm{\τ}\right)& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 1& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 1& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 1\end{array}\right]---\left(13\right)$

Differentiate can get formula (14) to formula (12):

$\stackrel{\·}{\mathrm{\ξ}}=\stackrel{\·}{H}\mathrm{\ϵ}+H\stackrel{\·}{\mathrm{\ϵ}}---\left(14\right)$

Further differentiate gets to formula (14):

$\stackrel{\·\·}{\mathrm{\ξ}}=\stackrel{\·\·}{H}\mathrm{\ϵ}+2\stackrel{\·}{H}\stackrel{\·}{\mathrm{\ϵ}}+H\stackrel{\·\·}{\mathrm{\ϵ}}---\left(15\right)$

Order

And with formula (12), formula (14) and formula (15) substitution formula (9):

$\mathrm{MH}\stackrel{\·\·}{\mathrm{\ϵ}}=G(\mathrm{\ϵ},\stackrel{\·}{\mathrm{\ϵ}})-M\stackrel{\·\·}{H}\mathrm{\ϵ}-2M\stackrel{\·}{H}\stackrel{\·}{\mathrm{\ϵ}}---\left(16\right)$

Order And substitution formula (16) gets the single order autonomous system

$\left\{\begin{array}{c}\stackrel{\·}{\mathrm{\ϵ}}=z\\ \stackrel{\·}{z}={\left(\mathrm{MH}\right)}^{-1}(G(\mathrm{\ϵ},z)-M\stackrel{\·\·}{H}\mathrm{\ϵ}-2M\stackrel{\·}{H}z)\end{array}\right.---\left(17\right)$

(e) equilibrium point asks for step

Order Formula (17) formula can turn to

$\stackrel{\·}{q}=f\left(q\right)---\left(18\right)$

System is at equilibrium point q ^*The place satisfies f (q ^*The Newton-Raphson process of iteration for asking for the equilibrium point of system, can be adopted in)=0

$q_{n+1}^{*}=q_n^{*}-J^{-1}f\left(q_n^{*}\right)---\left(19\right)$

In the formula

$J=\frac{\∂f}{\∂q}{|}_{q=q_n^{*}}---\left(20\right)$

Be the Jacobi matrix of system at the equilibrium point place;

(f) system stability analysis step

By the iteration of previous step Newton-Raphson method, obtain system balancing point q ^*, adopt continuity algorithm keeps track system balancing point q ^*Stability draws the stable region or the zone of system design parameters with single or multiple parameter situations of change, instructs the practical design process, by Linux under the Unix AUTO or finish this step work based on the MatCont software of Matlab.

Method of the present invention can be used for instructing actual design process, determines the dewatering stability of full-automatic pulsator washing machine quickly and easily.

Description of drawings

Fig. 1 is a system coordinate system synoptic diagram of the present invention.

Fig. 2 is C of the present invention _a=50N s m ^-1The time Ω the one-parameter bifurcation graphs.

Fig. 3 is C among the present invention _a=50N s m ^-1The time, Ω and m _bTwo-parameter bifurcation graphs.

Fig. 4 is m among the present invention _bDuring=0.5kg, Ω-C _aTwo-parameter bifurcation graphs.

The acceleration plots of first group of experiment when Fig. 5 is Ω among the present invention=3hz.

The corresponding power spectrum chart of first group of experiment when Fig. 6 is Ω among the present invention=3hz.

The acceleration plots of first group of experiment when Fig. 7 is Ω among the present invention=1.5hz.

The corresponding power spectrum chart of first group of experiment when Fig. 8 is Ω among the present invention=1.5hz.

The acceleration plots of first group of experiment when Fig. 9 is Ω among the present invention=5.6hz.

The corresponding power spectrum chart of first group of experiment when Figure 10 is Ω among the present invention=5.6hz.

The acceleration plots of second group of experiment when Figure 11 is Ω among the present invention=3hz.

The corresponding power spectrum chart of second group of experiment when Figure 12 is Ω among the present invention=3hz.

Embodiment

The invention will be further described below in conjunction with specific embodiment.

In Fig. 1,1 is the fluid balance ring, and 2 is dehydration barrel, and 3 is suspension rod, and 4 is steel ladle, and 5 is motor, and 6 is casing.

The analytical approach of stability of dehydration and vibration processes of impeller type full automatic washing machine of the present invention comprises the steps:

1, sets up system coordinate system

At first set up two coordinate systems as shown in Figure 1: reference frame X _rY _rZ _rWith moving coordinate system X _bY _bZ _bReference frame X _rY _rZ _rBe consolidated in the earth, initial point is positioned at O _rMoving coordinate system X _bY _bZ _bBe consolidated in steel ladle 4.The following hitch point of supposing four suspension rods is in same plane A, this plane follow that steel ladle 4 moves and with steel ladle 4 axis Z _bBe vertically intersected on an O _b, O _bBe moving coordinate system X _bY _bZ _bInitial point.Here adopt Bu Lien angle [α β γ] ^TMoving coordinate system X is described _bY _bZ _bAttitude.

2, liquid in the gimbal being dispersed is some rigid spheres

For analyzing the stability of dehydration, the present invention disperses the liquid in the fluid balance ring 1 and is N rigid spheres.If i spheroid is φ with respect to the rotation angle of dehydration barrel 2 _i, its radius of turn around central shaft is d _i, the surfaces of revolution is X moving _bY _bZ _bIn relative height be H _iThis spheroid is X moving _bY _bZ _bIn position vector can be described as:

r _i＝[d _icos(θ+φ _i)?d _isin(θ+φ _i)?H _i] ^T (1)

θ is the dehydration corner in the formula.This spheroid is at reference frame X _rY _rZ _rIn position vector can be described as:

s _i＝x+A ^rbr _i (2)

X=[x y z in the following formula] ^T, A ^RbBe moving coordinate system X _bY _bZ _bRelative reference coordinate system X _rY _rZ _rAttitude matrix.The velocity that can get spheroid to the following formula differentiate is:

B is in the formula

With moving is X _bY _bZ _bTransition matrix between angular velocity.

$\frac{{\∂r}_i}{\∂\mathrm{\θ}}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\θ}+{\mathrm{\φ}}_i)& d_i\cos (\mathrm{\θ}+{\mathrm{\φ}}_i)& 0\end{array}\right]}^T---\left(4\right)$

$\frac{{\∂r}_i}{{\∂\mathrm{\φ}}_i}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\θ}+{\mathrm{\φ}}_i)& d_i\cos (\mathrm{\θ}+{\mathrm{\φ}}_i)& 0\end{array}\right]}^T=\frac{{\∂r}_i}{\∂\mathrm{\θ}}---\left(5\right)$

Order

$D_i=\frac{{\∂r}_i}{\∂\mathrm{\θ}}=\frac{{\∂r}_i}{{\∂\mathrm{\φ}}_i}---\left(6\right)$

Then (3) but the formula abbreviation be

If spheroid mass is m _b, ignore the influence of spheroid inertia, derive spheroid kinetic energy:

$T_i=\frac{1}{2}{\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i^T\left[\begin{array}{cccc}{m}_b{I}_3& -m_b{A}^{\mathrm{rb}}{\tilde{r}}_{i}B& m_b{A}^{\mathrm{rb}}{D}_i& m_b{A}^{\mathrm{rb}}{D}_i\\ & m_b{B}^T{\left({\tilde{r}}_i\right)}^T{\tilde{r}}_{i}B& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i\\ & & m_b{d}_i^2& m_{\mathrm{<figure-callout\; id="2"\; label="b\; d\; i"\; filenames="HSA00000160805700011.png,HSA00000160805700021.png"\; state="\{\{state\}\}">b</figure-callout>}}{d}_i^{}{}_2^{}\\ \mathrm{sym}& & & m_{\mathrm{<figure-callout\; id="2"\; label="b\; d\; i"\; filenames="HSA00000160805700011.png,HSA00000160805700021.png"\; state="\{\{state\}\}">b</figure-callout>}}{d}_i^{}{}_2^{}\end{array}\right]{\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i---\left(8\right)$

In the formula ${\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i={\left[\begin{array}{cccccccc}\stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{y}& \stackrel{\&CenterDot;}{z}& \stackrel{\&CenterDot;}{\mathrm{\&alpha;}}& \stackrel{\&CenterDot;}{\mathrm{\&beta;}}& \stackrel{\&CenterDot;}{\mathrm{\&gamma;}}& \stackrel{\&CenterDot;}{\mathrm{\&theta;}}& {\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_i\end{array}\right]}^T.$

3, set up the system vibration model

Consider the kinetic energy of other parts of system and suspension generalized force etc., can get the vibration equation of system by the Lagrange equation.

$M\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&xi;}}=\frac{1}{2}{\left[\frac{\&PartialD;M}{\&PartialD;\mathrm{\&xi;}}\stackrel{\&CenterDot;}{\mathrm{\&xi;}}\right]}^T\stackrel{\&CenterDot;}{\mathrm{\&xi;}}-\left(\underset{i=1}{\overset{6+N}{\mathrm{\&Sigma;}}}\frac{\&PartialD;M}{{\&PartialD;\mathrm{\&xi;}}_i}{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_i\right)\stackrel{\&CenterDot;}{\mathrm{\&xi;}}+\left[\begin{array}{c}Q\\ Q_{\mathrm{\&tau;}}\end{array}\right]-\frac{{\&PartialD;V}_g}{\&PartialD;\mathrm{\&xi;}}---\left(9\right)$

Wherein N is the number of discrete spheroid, and M is the mass of system matrix; V _gBe system's gravitional force; Q is the suspension generalized force;

$Q_{\mathrm{\&tau;}}=-{\left[\begin{array}{cccc}{C}_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_1& C_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_2& ...& C_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_N\end{array}\right]}^T---\left(10\right)$

Be the suffered gimbal damping force of spheroid; C _τBe the gimbal ratio of damping.

4, rotating coordinate transformation

Introduce following rotating coordinate transformation:

$\left\{\begin{array}{c}\mathrm{\&tau;}=\mathrm{\&Omega;t}\\ \frac{\mathrm{d\&tau;}}{\mathrm{dt}}=\mathrm{\&Omega;}\\ {\mathrm{\&xi;}}_1={\mathrm{\&epsiv;}}_1\cos \left(\mathrm{\&tau;}\right)-{\mathrm{\&epsiv;}}_2\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_2={\mathrm{\&epsiv;}}_1\cos \left(\mathrm{\&tau;}\right)+{\mathrm{\&epsiv;}}_2\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_3={\mathrm{\&epsiv;}}_3\\ {\mathrm{\&xi;}}_4={\mathrm{\&epsiv;}}_4\cos \left(\mathrm{\&tau;}\right)-{\mathrm{\&epsiv;}}_5\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_5={\mathrm{\&epsiv;}}_5\cos \left(\mathrm{\&tau;}\right)+{\mathrm{\&epsiv;}}_5\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_6={\mathrm{\&epsiv;}}_6\\ ...\\ {\mathrm{\&xi;}}_{6+N}={\mathrm{\&epsiv;}}_{6+N}\end{array}\right.---\left(11\right)$

(11) are expressed as the matrix-vector form:

ξ＝Hε (12)

Wherein

$H=\left[\begin{array}{ccccccccc}\cos \left(\mathrm{\&tau;}\right)& -\sin \left(\mathrm{\&tau;}\right)& 0& 0& 0& 0& 0& ...& 0\\ \sin \left(\mathrm{\&tau;}\right)& \cos \left(\mathrm{\&tau;}\right)& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 1& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& \cos \left(\mathrm{\&tau;}\right)& -\sin \left(\mathrm{\&tau;}\right)& 0& 0& ...& 0\\ 0& 0& 0& \sin \left(\mathrm{\&tau;}\right)& \cos \left(\mathrm{\&tau;}\right)& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 1& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 1& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 1\end{array}\right]---\left(13\right)$

Can get (12) differentiate:

$\stackrel{\&CenterDot;}{\mathrm{\&xi;}}=\stackrel{\&CenterDot;}{H}\mathrm{\&epsiv;}+H\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}---\left(14\right)$

(14) further differentiate is got:

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&xi;}}=\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}+2\stackrel{\&CenterDot;}{H}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}+H\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&epsiv;}}---\left(15\right)$

Order And with (12) (14) (15) substitutions (9):

$\mathrm{MH}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&epsiv;}}=G(\mathrm{\&epsiv;},\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}})-M\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}-2M\stackrel{\&CenterDot;}{H}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}---\left(16\right)$

Order

And substitution (16) gets the single order autonomous system

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=z\\ \stackrel{\&CenterDot;}{z}={\left(\mathrm{MH}\right)}^{-1}(G(\mathrm{\&epsiv;},z)-M\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}-2M\stackrel{\&CenterDot;}{H}z)\end{array}\right.---\left(17\right)$

5, equilibrium point asks for

Order

(17) formula can turn to

$\stackrel{\&CenterDot;}{q}=f\left(q\right)---\left(18\right)$

System is at equilibrium point q ^*The place satisfies f (q ^*The Newton-Raphson process of iteration for asking for the equilibrium point of system, can be adopted in)=0

$q_{n+1}^{*}=q_n^{*}-J^{-1}f\left(q_n^{*}\right)---\left(19\right)$

In the formula

$J=\frac{\&PartialD;f}{\&PartialD;q}{|}_{q=q_n^{*}}---\left(20\right)$

Be the Jacobi matrix of system at the equilibrium point place.

6, system stability analysis

By the iteration of previous step Newton-Raphson method, can get system balancing point q ^*System can analyze by (20) formula Jacobian's eigenvalue in the stability at equilibrium point place, when the real part that has an eigenwert at least greater than 0, autonomous system (17) instability then, original system (9) also can unstability.In design process, the variation of systematic parameter may cause that eigenwert passes through the imaginary axis, and with the difference of the mode of passing through, different bifurcations can take place in system.For automatic washing machine, mainly contain two kinds of Hopf fork and saddles.But adopt continuity algorithm tracking balance point stability with single or multiple parameter situations of change, draw the stable region (or zone) of system design parameters, instruct the practical design process.At present, can by ripe software such as Linux under the Unix AUTO or finish this step work based on the MatCont of Matlab etc.

7, analysis example

Set up the model of vibration of certain washing machine according to the method for step 1-4, and then adopt the method for step e to ask for the system balancing point.With spheroid number N=2 is example, finds that by iteration system has three groups of equilibrium solutions: wherein separate I and satisfy φ ₁With φ ₂Substantially symmetry is separated II and is satisfied φ ₁=φ ₂, separate III and satisfy φ ₁=φ ₂+ π.Because it is always unstable to separate III, hereinafter main the discussion separated I and separated II in the analytic process.Here in process, adopt numerical value fork software AUTO to system stability analysis.

Fig. 2 has provided as axial ratio of damping C _a=50N s m ^-1The time system the one-parameter bifurcation graphs.Solid line is represented stable solution among the figure, and dotted line is represented unstable solution.Subscript is pointed out the sequence number (I or II) understood among the figure, on be designated as the label of Hopf bifurcation point H.

Fig. 3 has provided as axial ratio of damping C _a=50N m s ^-1The time, stable state dehydrating speed Ω and spheroid mass m _bTwo-parameter bifurcation graphs.As can be seen, this moment, there be M and N between two ranges of instability in system.M is positioned near the full-automatic pulsator washing machine first rank hunting frequency between the range of instability, and N is positioned near the second rank hunting frequency of system between the range of instability.M place between the range of instability, the centrifugal acceleration r of system _bΩ ²=0.2 * (0.75 * 2 π) ²≈ 4.4m s ^-2, for the fluid balance ring, the suffered centrifugal force of liquid this moment is about half of gravity, thereby liquid can not form tangible gathering in the ring, and this moment, the unstability of gimbal was little to the influence of full-automatic pulsator washing machine; But different with M between the range of instability is, N is positioned at the right side of M between the range of instability, and this moment, full-automatic pulsator washing machine embodied the self-centering phenomenon of similar disc rotor, because this moment, dehydrating speed was higher, and the centrifugal acceleration r of system _bΩ ²=0.2 * (3 * 2 π) ²≈ 71m s ^-2Be about 7 times of acceleration of gravity, liquid in the gimbal is enough to overcome the influence of gravity and assembles, gimbal mechanism unstability herein can cause that steel ladle 4 significantly waves, thisly wave the violent bump that also can cause between steel ladle 4 and casing, thereby N can come into the picture more between the range of instability.Increase axial ratio of damping C _aCan effectively suppress the appearance of N between instability area.

Fig. 4 has provided to eliminating the required critical damping coefficient C of N between the range of instability _a

Here verify the feasibility of analytical approach of the present invention by experiment.Experiment is divided into two groups: axial damping C is adopted in first group of experiment _aLess suspension rod, second group is adopted axial damping C _aBigger suspension rod.

Fig. 5 and Fig. 6 have provided under the situation of Ω=3hz axial ratio of damping experimental result hour.As can be seen from Figure 5, the X of system to acceleration curve have tangible at times strong and at other times weak phenomenon.The existence of this phenomenon has reflected the instability of dehydration, and it can cause the generation of impingement phenomenon between parts 4 and casing.

According to the fork conclusion of Fig. 3, under the situation of little damping, when Ω=1.5hz or Ω=5.6hz gimbal should keep stable.Here these two speed points are tested.Empirical curve is seen Fig. 7, Fig. 8, Fig. 9 and Figure 10 respectively, as seen on these two speed points, system X to acceleration do not had tangible at times strong and at other times weak phenomenon.

According to the fork conclusion of Fig. 4, during big damping, should there be N between the range of instability in the fluid balance ring.In second group of experimentation of this paper Ω=3hz point is measured, as Figure 11, shown in Figure 12, visible system X to acceleration do not had tangible at times strong and at other times weak phenomenon.

Comprehensive above experimental result is not difficult to find out: analysis result of the present invention is consistent with experimental result.Thereby the feasibility and the validity of analytical approach of the present invention are described.

Claims (1)

1. the analytical approach of a stability of dehydration and vibration processes of impeller type full automatic washing machine, it is characterized in that: this analytical approach comprises the steps:

(a) set up the system coordinate system step

Set up reference frame X _rY _rZ _rWith moving coordinate system X _bY _bZ _b, reference frame X _rY _rZ _rBe consolidated in the earth, initial point is positioned at O _rMoving coordinate system X _bY _bZ _bBe consolidated in steel ladle, suppose that the following hitch point of four suspension rods is in same plane A, this plane follow that steel ladle moves and with steel ladle axis Z _bBe vertically intersected on an O _b, O _bBe moving coordinate system X _bY _bZ _bInitial point, adopt Bu Lien angle [α β γ] ^TMoving coordinate system X is described _bY _bZ _bAttitude;

(b) liquid in the gimbal being dispersed is some rigid spheres steps

Be N rigid spheres with the liquid in the gimbal is discrete, establishing i spheroid is φ with respect to the rotation angle of dehydration barrel _i, its radius of turn around central shaft is d _i, the surfaces of revolution is X moving _bY _bZ _bIn relative height be H _iThis spheroid is X moving _bY _bZ _bIn position vector can be described as formula (1):

r _i＝[d _icos(θ+φ _i)?d _isin(θ+φ _i)?H _i] ^T (1)

θ is the dehydration corner in the formula (1), and then this spheroid is at reference frame X _rY _rZ _rIn position vector can be described as formula (2):

s _i＝x+A ^rbr _i (2)

X=[x y z in the formula (2)] ^T, A ^RbBe moving coordinate system X _bY _bZ _bRelative reference coordinate system X _rY _rZ _rAttitude matrix, the velocity that can get spheroid to the following formula differentiate is formula (3):

B is in the formula (3)

With moving is X _bY _bZ _bTransition matrix between angular velocity,

$\frac{{\&PartialD;r}_i}{\&PartialD;\mathrm{\&theta;}}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\&theta;}+{\mathrm{\&phi;}}_i)& d_i\cos (\mathrm{\&theta;}+{\mathrm{\&phi;}}_i)& 0\end{array}\right]}^T---\left(4\right)$

$\frac{{\&PartialD;r}_i}{{\&PartialD;\mathrm{\&phi;}}_i}={\left[\begin{array}{ccc}-d_i\sin (\mathrm{\&theta;}+{\mathrm{\&phi;}}_i)& d_i\cos (\mathrm{\&theta;}+{\mathrm{\&phi;}}_i)& 0\end{array}\right]}^T=\frac{{\&PartialD;r}_i}{\&PartialD;\mathrm{\&theta;}}---\left(5\right)$

Order

$D_i=\frac{{\&PartialD;r}_i}{\&PartialD;\mathrm{\&theta;}}=\frac{{\&PartialD;r}_i}{{\&PartialD;\mathrm{\&phi;}}_i}---\left(6\right)$

Then (3) but the formula abbreviation is formula (7):

If spheroid mass is m _b, ignore the influence of spheroid inertia, derive spheroid kinetic energy:

$T_i=\frac{1}{2}{\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i^T\left[\begin{array}{cccc}{m}_b{I}_3& -m_b{A}^{\mathrm{rb}}{\tilde{r}}_{i}B& m_b{A}^{\mathrm{rb}}{D}_i& m_b{A}^{\mathrm{rb}}{D}_i\\ & m_b{B}^T{\left({\tilde{r}}_i\right)}^T{\tilde{r}}_{i}B& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i& -m_b{B}^T{\left({\tilde{r}}_i\right)}^T{D}_i\\ & & m_b{d}_i^2& m_b{d}_i^2\\ \mathrm{sym}& & & m_b{d}_i^2\end{array}\right]{\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i---\left(8\right)$

In the formula (8) ${\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_i={\left[\begin{array}{cccccccc}\stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{y}& \stackrel{\&CenterDot;}{z}& \stackrel{\&CenterDot;}{\mathrm{\&alpha;}}& \stackrel{\&CenterDot;}{\mathrm{\&beta;}}& \stackrel{\&CenterDot;}{\mathrm{\&gamma;}}& \stackrel{\&CenterDot;}{\mathrm{\&theta;}}& {\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_i\end{array}\right]}^T;$

(c) set up system vibration model step

Obtain the vibration equation of system by the Lagrange equation:

$M\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&xi;}}=\frac{1}{2}{\left[\frac{\&PartialD;M}{\&PartialD;\mathrm{\&xi;}}\stackrel{\&CenterDot;}{\mathrm{\&xi;}}\right]}^T\stackrel{\&CenterDot;}{\mathrm{\&xi;}}-\left(\underset{i=1}{\overset{6+N}{\mathrm{\&Sigma;}}}\frac{\&PartialD;M}{{\&PartialD;\mathrm{\&xi;}}_i}{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_i\right)\stackrel{\&CenterDot;}{\mathrm{\&xi;}}+\left[\begin{array}{c}Q\\ Q_{\mathrm{\&tau;}}\end{array}\right]-\frac{{\&PartialD;V}_g}{\&PartialD;\mathrm{\&xi;}}---\left(9\right)$

Wherein N is the number of discrete spheroid, and M is the mass of system matrix; V _gBe system's gravitional force; Q is the suspension generalized force;

$Q_{\mathrm{\&tau;}}=-{\left[\begin{array}{cccc}{C}_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_1& C_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_2& ...& C_{\mathrm{\&tau;}}{\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_N\end{array}\right]}^T---\left(10\right)$

Be the suffered gimbal damping force of spheroid; C _τBe the gimbal ratio of damping.

(d) rotating coordinate transformation step

Introduce following rotating coordinate transformation:

$\left\{\begin{array}{c}\mathrm{\&tau;}=\mathrm{\&Omega;t}\\ \frac{\mathrm{d\&tau;}}{\mathrm{dt}}=\mathrm{\&Omega;}\\ {\mathrm{\&xi;}}_1={\mathrm{\&epsiv;}}_1\cos \left(\mathrm{\&tau;}\right)-{\mathrm{\&epsiv;}}_2\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_2={\mathrm{\&epsiv;}}_1\cos \left(\mathrm{\&tau;}\right)+{\mathrm{\&epsiv;}}_2\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_3={\mathrm{\&epsiv;}}_3\\ {\mathrm{\&xi;}}_4={\mathrm{\&epsiv;}}_4\cos \left(\mathrm{\&tau;}\right)-{\mathrm{\&epsiv;}}_5\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_5={\mathrm{\&epsiv;}}_5\cos \left(\mathrm{\&tau;}\right)+{\mathrm{\&epsiv;}}_5\sin \left(\mathrm{\&tau;}\right)\\ {\mathrm{\&xi;}}_6={\mathrm{\&epsiv;}}_6\\ ...\\ {\mathrm{\&xi;}}_{6+N}={\mathrm{\&epsiv;}}_{6+N}\end{array}\right.---\left(11\right)$

(11) are expressed as the matrix-vector form:

ξ＝Hε (12)

Wherein

$H=\left[\begin{array}{ccccccccc}\cos \left(\mathrm{\&tau;}\right)& -\sin \left(\mathrm{\&tau;}\right)& 0& 0& 0& 0& 0& ...& 0\\ \sin \left(\mathrm{\&tau;}\right)& \cos \left(\mathrm{\&tau;}\right)& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 1& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& \cos \left(\mathrm{\&tau;}\right)& -\sin \left(\mathrm{\&tau;}\right)& 0& 0& ...& 0\\ 0& 0& 0& \sin \left(\mathrm{\&tau;}\right)& \cos \left(\mathrm{\&tau;}\right)& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 1& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 1& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 0\\ 0& 0& 0& 0& 0& 0& 0& ...& 1\end{array}\right]---\left(13\right)$

Differentiate can get formula (14) to formula (12):

$\stackrel{\&CenterDot;}{\mathrm{\&xi;}}=\stackrel{\&CenterDot;}{H}\mathrm{\&epsiv;}+H\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}---\left(14\right)$

Further differentiate gets to formula (14):

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&xi;}}=\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}+2\stackrel{\&CenterDot;}{H}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}+H\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&epsiv;}}---\left(15\right)$

Order And with formula (12), formula (14) and formula (15) substitution formula (9):

$\mathrm{MH}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&epsiv;}}=G(\mathrm{\&epsiv;},\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}})-M\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}-2M\stackrel{\&CenterDot;}{H}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}---\left(16\right)$

Order

And substitution formula (16) gets the single order autonomous system

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=z\\ \stackrel{\&CenterDot;}{z}={\left(\mathrm{MH}\right)}^{-1}(G(\mathrm{\&epsiv;},z)-M\stackrel{\&CenterDot;\&CenterDot;}{H}\mathrm{\&epsiv;}-2M\stackrel{\&CenterDot;}{H}z)\end{array}\right.---\left(17\right)$

(e) equilibrium point asks for step

Order

Formula (17) formula can turn to

$\stackrel{\&CenterDot;}{q}=f\left(q\right)---\left(18\right)$

System is at equilibrium point q ^*The place satisfies f (q ^*The Newton-Raphson process of iteration for asking for the equilibrium point of system, can be adopted in)=0

$q_{n+1}^{*}=q_n^{*}-J^{-1}f\left(q_n^{*}\right)---\left(19\right)$

In the formula

$J=\frac{\&PartialD;f}{\&PartialD;q}{|}_{q=q_n^{*}}---\left(20\right)$

Be the Jacobi matrix of system at the equilibrium point place;

(f) system stability analysis step

By the iteration of previous step Newton-Raphson method, obtain system balancing point q ^*, adopt continuity algorithm keeps track system balancing point q ^*Stability draws the stable region or the zone of system design parameters with single or multiple parameter situations of change, instructs the practical design process, by Linux under the Unix AUTO or finish this step work based on the MatCont software of Matlab.

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