CN106055749A - Method for increasing motion stability of link mechanism with clearance - Google Patents

Abstract

The present invention relates to a method for increasing motion stability of a link mechanism with a clearance. According to the multi-body system dynamics theory, based on multi-body system dynamics and kinematic pair clearance contact impact models, a link mechanism dynamics model taking regard of the kinematic pair clearance is established, the influence of the clearance on link mechanism dynamics behaviors is numerically simulated and analyzed, then to minimize the maximum peak of motion acceleration jitters of the link mechanism, the influence of the clearance is reduced by optimally designing and adjusting the mechanism rod length, and furthermore, the motion stability of the mechanism is increased. Through optimally designing and adjusting the mechanism rod length, the influence of the clearance on mechanism performance is reduced, and the motion stability of the mechanism with the clearance is increased. The method is simple, feasible and practical, clearance impact characteristics are considered, and furthermore, the method can be widely used in link mechanisms of various types.

Description

A kind of method improved containing gap connecting rod mechanism movement stability

Technical field

The present invention relates to field of mechanical technique, improve containing gap connecting rod mechanism movement stability in particular it relates to a kind of Method.

Background technology

Along with the development of precision optical machinery engineering, mechanical system is towards high accuracy, high efficiency, high reliability and long-life Target stride forward, in engineering reality, mechanical system realizes the kinetics transmission of system, movement needs etc. by mechanism etc., because of This mechanism is the important component part of mechanical system.Generally mechanism is the most complicated, component is the most, function is the most powerful, needs employing Kinematic pair is the most.But in practical set-up, due to the needs of dynamic cooperation, the reason such as foozle, fretting wear, mechanism transports Dynamic auxiliary air gap is inevitable.

The existence in gap can increase pair clearance impact force so that mechanism's acceleration is acutely shaken, dither amplitude and frequency Rate is the highest, produces serious noise and vibration, and then reduces the kinetic stability of mechanism, especially for high-speed mechanism Affect bigger.It is therefore desirable to reduce the pair clearance impact on mechanism kinematic stability, this is for improving precision optical machinery, boat Mechanism's service behaviour of the key areas such as empty space flight is significant.

In order to reduce the gap impact on mechanism dynamic performance, improving mechanism kinematic precision and stability, conventional grinds Study carefully many employing pair clearance lubrications, redistribute the method such as rod member quality, additional constant spring force, it is to avoid containing intermittent motion The separation of secondary accessory element, and then improve mechanism performance.Or pair clearance to be reduced to the rigid rod without quality, and then by former Organisation conversion containing gap is that gapless many bars many-degrees of freedom system carries out motion analysis and design, and the shortcoming of this method is Have ignored the elastic deformation of kinematic pair accessory element contact surface, it is impossible to the real contact-impact reflecting mechanism with clearance kinematic pair Characteristic, does not conforms to the actual conditions.Additionally, conventional research, how based on mechanism with clearance kinematics analysis, to carry out kinematic accuracy excellent Change design, but do not account for the dynamic characteristic of pair clearance contact-impact, do not meet the kinetics of mechanism with clearance originally Matter feature.

Summary of the invention

The present invention is based on dynamics of multibody systems theory, in conjunction with pair clearance contact-impact model, sets up and considers The linkage kinetic model of pair clearance, the impact of Numerical Simulation Analysis gap linkage dynamic behavior, enter And shake the minimum optimization aim of peak-peak with connecting rod mechanism movement acceleration, with a length of design variable of linkage bar, logical Cross optimization design guiding mechanism bar length to reduce the impact in gap, and then improve mechanism kinetic stability.Advantages of the present invention It is: long by optimizing design guiding mechanism bar, reduces the gap impact on mechanism performance, improve the motion of mechanism with clearance Stability, the method simple possible, it is contemplated that the contact-impact characteristic containing Clearance pair, meet reality, it is possible to widely It is applied in various types of linkage.

The technical solution adopted in the present invention is:

A kind of method improved containing gap connecting rod mechanism movement stability, comprises the steps of

Step one: set up the mathematical model Han Clearance pair；

Step 2: set up pair clearance normal direction Collision force model and tangential friction force model；

Step 3: set up ideal mechanism kinetic model based on dynamics of multibody systems theory, in setting up model process, The bar progress line parameter of linkage；

Step 4: set up the mechanism dynamic model considering pair clearance；

Step 5: set up the mathematical optimization models of mechanism with clearance kinetic stability；

Step 6: be optimized design, it is thus achieved that optimum linkage bar is long.

Wherein:

In described step one, gap length axle sleeve describes with the difference of axle radius, then radius clearance c is: c=r_B-r_J, Wherein r_BFor axle sleeve radius, r_JFor axle radius.E is axle and axle sleeve centre distance, and definition δ=e-c is the elastic deformation of contact point Amount, and then the condition that available axle and bearing come in contact collision is:

In described step 2, normal direction impact force F_nComputing formula beWherein K_nFor touching The contact stiffness coefficient of collision body, δ is collision process juxtaposition metamorphose amount, and n is index, takes 1.5,For relative impact velocity, c_eFor extensive Complex coefficient；Initial relative velocity for rum point.

In described step 2, tangential friction force F_tComputing formula beWherein F_nCollide for normal direction Power, μ (v_t) it is dynamic friction coefficient, v_tRepresent axle and the bearing relative sliding velocity at the point of impingement, i.e. the speed of tangential direction is divided Amount.

In described step 3, the kinetics equation of ideal mechanism is:

$\left\{\begin{array}{c}M\stackrel{\·\·}{q}+C\stackrel{\·}{q}+Kq+{\mathrm{\φ}}_q^T\mathrm{\λ}=F\\ \mathrm{\φ}(q,t)=0\end{array}\right.$

In formula, q is generalized coordinates array,For the q first derivative to the time,For the q second dervative to the time；M is mechanism Generalized mass matrix, C is the damping battle array of mechanism's broad sense, and K is the generalized stifflness battle array of mechanism, φ_qJacobi square for constraint equation Battle array,For φ_qTransposed matrix, λ is Lagrange multiplier array, and F is generalized velocity quadratic term and power battle array.

In described step 4, it is considered to the kinetics equation of the mechanism of pair clearance is:

$\left\{\begin{array}{c}M\stackrel{\·\·}{q}+C\stackrel{\·}{q}+Kq+{\mathrm{\φ}}_q^T\mathrm{\λ}=F+F_c\\ \mathrm{\φ}(q,t)=0\end{array}\right.$

F in formula_cComprise normal direction impact force F_nWith tangential friction force F_t。

In described step 5, for the Optimized model of the planar linkage mechanism containing gap, with a length of design variable of mechanism's bar, Shake the minimum target of peak-peak with mechanism with clearance acceleration, set up mathematical model of optimizing design as follows:

$\left\{\begin{array}{c}\begin{array}{cc}Minimize& F\left(X\right)=max\left({\mathrm{\α}}_i^c\right)\\ Subjectto& g_k\left(X\right)\≤0\end{array}\\ X=\[L_1,L_2,L_3,\mathrm{...},L_n\]\end{array}\right.$

Wherein X is design variable, and n is containing the number of component, L in the linkage of gap_nThe bar being the n-th component is long, F (X) it is optimization object function,For mechanism with clearance acceleration；g_k(X) it is constraint function, is only relevant with design variable X Definitiveness constraint function.

Accompanying drawing explanation

Fig. 1 is the flow chart of the present invention.

Fig. 2 is the structural representation containing gap hinge.

The toggle schematic diagram in gap in Fig. 3 embodiment of the present invention.

Response diagram before toggle acceleration optimizes in Fig. 4 embodiment of the present invention and after optimization.

Detailed description of the invention

Hinge is kinematic pair the most frequently used in toggle, below in conjunction with the accompanying drawings as a example by hinge movement pair, to this Invention is described further.As it is shown in figure 1, the present invention is first based on multisystem kinetic theory, hinge gap is entered Row definition, sets up gap normal direction Collision force model and gap tangential friction force model, thus sets up the hinge mathematical model in gap； Theoretical based on dynamics of multibody systems, set up preferable linkage kinetic model, gap former is embedded preferable linkage In kinetic model, set up dynamic model of mechanism with clearance connections.By definition design variable, set up object function, foundation constraint Condition and then set up mathematical model of optimizing design, specifically, the present invention is preferably with a length of design variable of mechanism's bar, with containing gap Mechanism's acceleration shake minimum target of peak-peak, with the long excursion of mechanism's bar as constraints, based on the brief ladder of broad sense Degree algorithm is optimized design to mechanism's bar length.Prioritization scheme is obtained according to mathematical model of optimizing design.

1 foundation containing gap hinge mathematical model

Fig. 2 is the structural representation containing gap hinge, and by the accessory element containing gap hinge, axle 1 and axle sleeve 2 are thought of as two Collision body, and the dynamics of gap hinge depends on clearance impact power, and this model has actually cut off original connection Hinge, is converted to impact force constraint by geometrical constraint.

Gap length axle sleeve describes with the difference of axle radius, then radius clearance is:

C=r_B-r_J (1)

Wherein r_BFor axle sleeve radius, r_JFor axle radius.E is axle and axle sleeve centre distance, and definition δ=e-c is the bullet of contact point Property deflection, and then the condition that available axle and bearing come in contact collision is:

The foundation of 2 gap normal direction Collision force models

Pair clearance can cause the interior collision of axle sleeve and axle, therefore gap hinge always to comprise certain contact and Collision process, needs to consider the correct description of gap-contact collision process.Pair clearance normal direction Collision force model uses non-thread Property spring damping model, expression formula is as follows:

$F_n=K_n{\mathrm{\δ}}^n+D\stackrel{\·}{\mathrm{\δ}}---\left(3\right)$

K in equation (3) formula_nFor the contact stiffness coefficient of collision body, D is the damped coefficient of collision process, and δ was for colliding Journey juxtaposition metamorphose amount, n is index, takes 1.5,For relative impact velocity.

Contact stiffness COEFFICIENT K_nCalculated by following formula:

$K_n=\frac{4}{3(\frac{1-v_B^2}{E_B}+\frac{1-v_J^2}{E_J})}{\[\frac{r_B{r}_J}{r_B-r_J}\]}^{\frac{1}{2}}$

Wherein v_BAnd E_BRepresent material Poisson's ratio and elastic modelling quantity, the v of axle sleeve respectively_JAnd E_JRepresent the material pool of axle respectively Pine ratio and elastic modelling quantity.

The damped coefficient of collision process is represented by:

$D=\frac{3K_n(1-{c_e}^2){\mathrm{\δ}}^n}{4{\stackrel{\·}{\mathrm{\δ}}}^{(-)}}$

Wherein c_eFor recovery coefficient；Initial relative velocity for rum point.

Gap normal direction Collision force model (3) formula is represented by further:

The foundation of 3 gap Frictional model

Use the Coulomb Frictional model revised to set up the frictional force containing clearance joints chain rivet Yu shaft room, repair at this Proposing the concept of dynamic friction coefficient in positive Frictional model, tangential friction force computing formula is

$F_t=-\mathrm{\μ}\left(v_i\right)F_n\frac{v_t}{\left|v_t\right|}---\left(4\right)$

Wherein μ (v_t) be dynamic friction coefficient, formula below it is calculated:

$\mathrm{\&mu;}\left(v_i\right)=\left\{\begin{array}{ccc}-{\mathrm{\&mu;}}_{d}sign\left(v_t\right)& for& \left|v_t\right|>v_d\\ -\{{\mathrm{\&mu;}}_d+({\mathrm{\&mu;}}_s-{\mathrm{\&mu;}}_d){\left(\frac{\left|v_t\right|-v_s}{v_d-v_s}\right)}^2\&lsqb;3-2\left(\frac{\left|v_t\right|-v_s}{v_d-v_s}\right)\&rsqb;\}sign\left(v_t\right)& for& v_s\&le;\left|v_t\right|\&le;v_d\\ {\mathrm{\&mu;}}_s-2{\mathrm{\&mu;}}_s{\left(\frac{v_t+v_s}{2v_s}\right)}^2\left(3-\frac{v_t+v_s}{v_s}\right)& for& \left|v_t\right|<v_s\end{array}\right.---\left(5\right)$

Wherein v_tRepresent axle and the bearing relative sliding velocity at the point of impingement, the i.e. velocity component of tangential direction, μ_dFor sliding Coefficient of friction, μ_sFor confficient of static friction, v_sFor static friction critical velocity, v_dFor maximum dynamic friction critical velocity.

The foundation of 4 dynamic model of mechanism with clearance connections

The existence in gap can cause the interior collision of connected links, and in gap, collision has two features: one is due to gap Existence, train of mechanism becomes variable topological structure.Because when there is gap in kinematic pair, the structure being connected by Clearance pair Constraint of kinematic pair free motion can be lost, hence into freely-movable state between part.Motion relative displacement when two bodies Having exceeded gap, gap hinge will collide with axle sleeve, and therefore mechanism kinematic state also changes, and becomes by impact force The contact-impact stage of constraint.Therefore, the method using " dynamic segmentation " processes mechanism with clearance structure changes characteristic.

(1) preferable linkage kinetic model is set up

When considering ideal mechanism, in the case of i.e. considering preferable hinge (without gap), according to method of Lagrange multipliers, machine The kinetics equation of structure is:

$\left\{\begin{array}{c}M\stackrel{\&CenterDot;\&CenterDot;}{q}+C\stackrel{\&CenterDot;}{q}+Kq+{\mathrm{\&phi;}}_q^T\mathrm{\&lambda;}=F\\ \mathrm{\&phi;}\left(q,t\right)=0\end{array}\right.---\left(6\right)$

In formula, q is generalized coordinates array,For the q first derivative to the time,For the q second dervative to the time；M is mechanism Generalized mass matrix, C is the damping battle array of mechanism's broad sense, and K is the generalized stifflness battle array of mechanism, φ_qJacobi square for constraint equation Battle array,For φ_qTransposed matrix, λ is Lagrange multiplier array, and F is generalized velocity quadratic term and power battle array.

(2) the mechanism dynamic model of consideration hinge gap is set up

According to practical situation, clearance joints is contained in mechanism, when clearance joints chain rivet will collide in occurring with axle sleeve, creates contact Impact force, thus introduce force constraint in systems, therefore this generalized force is mainly by the normal direction impact force during contact-impact Form with tangential friction force, be defined as F_c.Thus to actual mechanism, it is considered to during hinge gap, the kinetics equation of mechanism For:

$\left\{\begin{array}{c}M\stackrel{\&CenterDot;\&CenterDot;}{q}+C\stackrel{\&CenterDot;}{q}+Kq+{\mathrm{\&phi;}}_q^T\mathrm{\&lambda;}=F+F_c\\ \mathrm{\&phi;}\left(q,t\right)=0\end{array}\right.---\left(7\right)$

F in formula_cComprise normal direction impact force F_n, such as equation (3), and tangential friction force F_t, such as equation (4).Other are every contains Adopted and defined above identical.

The foundation of 5 mechanism with clearance mathematical model of optimizing design

(1) design variable

The planar linkage mechanism considering gap is optimized design studies, with a length of design variable of component bar.Then design Variable X is represented by:

X=(L₁,L₂,L₃,...,L_n)

N is containing the number of component, L in the linkage of gap_nThe bar being the n-th component is long.

(2) object function

Owing to the existence in gap can increase pair clearance impact force so that mechanism's acceleration is acutely shaken, dither amplitude The highest with frequency, mechanism kinematic stability there is is large effect, reduces mechanism kinematic stability, therefore to make gap Impact on mechanism kinematic stability is minimum, shakes the minimum target of peak-peak with mechanism with clearance acceleration, sets up and optimizes Object function is:

$\begin{array}{cc}Minimize& F\left(X\right)=\max \left({\mathrm{\&alpha;}}_i^c\right)\end{array}---\left(8\right)$

In formulaFor mechanism with clearance acceleration.

(3) constraints

Constraints is that linkage each component bar length is less than its corresponding bound.

(4) mathematical model of optimizing design

For the Optimized model containing gap linkage, with a length of design variable of mechanism's bar, with mechanism with clearance acceleration The shake minimum target of peak-peak, with the long excursion of component bar as constraints, sets up mathematical model of optimizing design as follows:

$\left\{\begin{array}{c}\begin{array}{cc}Minimize& F\left(X\right)=max\left({\mathrm{\&alpha;}}_i^c\right)\\ Subjectto& g_k\left(X\right)\&le;0\end{array}\\ X=\&lsqb;L_1,L_2,L_3,\mathrm{...},L_n\&rsqb;\end{array}\right.---\left(9\right)$

Wherein g_k(X) it is constraint function, is only relevant with design variable X qualitative constraint function really.

6. case study on implementation

The present invention is applicable to the multi-connecting-rod mechanisms such as quadric chain, five-bar mechanism, six bar mechanism, the present embodiment only with The present invention is further illustrated for quadric chain, the invention is not limited in this.

With the four-bar linkage containing gap as objective for implementation, as it is shown in figure 1, this four-bar mechanism is planar double cranks mechanism. Mechanism is made up of driving crank, connecting rod, driven crank and frame, comprises three preferable hinges, and a gap hinge, gap Hinge is between connecting rod and driven crank, i.e. hinge B exists gap.The four-bar linkage considering gap is optimized design Research, with a length of design variable of bar, shakes the minimum target of peak-peak with mechanism with clearance acceleration, by the brief ladder of broad sense Degree algorithm is optimized design to mechanism with clearance, and then is reduced the impact in gap by guiding mechanism bar length, improves mechanism Kinetic stability.

Initial geometric parameter and the long scope of bar of Fig. 3 midplane four-bar mechanism are as shown in table 1.Gap length is 0.5mm. Dynamics simulation process, crank rolling velocity is 600r/min, and original state is that crank is perpendicular to ground, i.e. initial angle is 90 °, just Beginning angular velocity is 0.

The long scope of table 1 linkage bar

After optimization, four-bar mechanism bar length is as shown in table 2, and mechanism's acceleration responsive is as shown in Figure 4.Optimum results shows, with machine After the optimization design of the structure acceleration shake minimum target of peak-peak, mechanism's acceleration shake peak value and shake number of times substantially drop Low, optimize post-acceleration shake peak-peak and reduce 77.5%.Visible long by optimizing the linkage bar in gap to contain The shake of clearance mechanism acceleration substantially reduces, and improves the stationarity of mechanism kinematic.

After table 2 optimizes, mechanism's bar is long

Claims (7)

1. improve the method containing gap connecting rod mechanism movement stability, it is characterized in that comprising the steps of

Step one, sets up the mathematical model Han Clearance pair；

Step 2, sets up pair clearance normal direction Collision force model and tangential friction force model；

Step 3, sets up ideal mechanism kinetic model based on dynamics of multibody systems theory, in setting up model process, to even The bar progress line parameter of linkage；

Step 4, sets up the mechanism dynamic model considering pair clearance；

Step 5, sets up the mathematical optimization models of mechanism with clearance kinetic stability；

Step 6, is optimized design, it is thus achieved that optimum linkage bar is long.

Method the most according to claim 1, is characterized in that in described step one, gap length axle sleeve and the difference of axle radius Describe, then radius clearance c is: c=r_B-r_J,

Wherein r_BFor axle sleeve radius, r_JFor axle radius, e is axle and axle sleeve centre distance, and definition δ=e-c is the elastic change of contact point Shape amount, and then the condition that available axle and bearing come in contact collision is:

Method the most according to claim 1, is characterized in that in described step 2, normal direction impact force F_nComputing formula beWherein K_nFor the contact stiffness coefficient of collision body, δ is collision process juxtaposition metamorphose amount, and n is Index, takes 1.5,For relative impact velocity, c_eFor recovery coefficient；Initial relative velocity for rum point.

Method the most according to claim 3, is characterized in that in described step 2, tangential friction force F_tComputing formula beWherein F_nFor normal direction impact force, μ (v_t) it is dynamic friction coefficient, v_tRepresent that axle and axle sleeve are at the point of impingement The velocity component of relative sliding velocity, i.e. tangential direction.

Method the most according to claim 1, is characterized in that in described step 3, and the kinetics equation of ideal mechanism is:

$\left\{\begin{array}{c}M\stackrel{\&CenterDot;\&CenterDot;}{q}+C\stackrel{\&CenterDot;}{q}+Kq+{\mathrm{\&phi;}}_q^T\mathrm{\&lambda;}=F\\ \mathrm{\&phi;}(q,t)=0\end{array}\right.$

In formula, q is generalized coordinates array,For the q first derivative to the time,For the q second dervative to the time；M is the wide of mechanism Justice Mass matrix, C is the damping battle array of mechanism's broad sense, and K is the generalized stifflness battle array of mechanism, φ_qFor the Jacobian matrix of constraint equation, For φ_qTransposed matrix, λ is Lagrange multiplier array, and F is generalized velocity quadratic term and power battle array.

Method the most according to claim 1, is characterized in that in described step 4, it is considered to the power of the mechanism of pair clearance Equation is:

$\left\{\begin{array}{c}M\stackrel{\&CenterDot;\&CenterDot;}{q}+C\stackrel{\&CenterDot;}{q}+Kq+{\mathrm{\&phi;}}_q^T\mathrm{\&lambda;}=F+F_c\\ \mathrm{\&phi;}(q,t)=0\end{array}\right.$

F in formula_cComprise normal direction impact force F_nWith tangential friction force F_t, q is generalized coordinates array,For the q first derivative to the time,For the q second dervative to the time；For φ_qTransposed matrix, λ is Lagrange multiplier array, F be generalized velocity quadratic term with And power battle array.

Method the most according to claim 1, is characterized in that in described step 5, for the connecting rod containing gap

The Optimized model of mechanism, with a length of design variable of mechanism member bar, shakes with mechanism with clearance acceleration

The big minimum target of peak value, sets up mathematical model of optimizing design as follows:

$\left\{\begin{array}{c}\begin{array}{cc}Minimize& F\left(X\right)=max\left({\mathrm{\&alpha;}}_i^c\right)\\ Subjectto& g_k\left(X\right)\&le;0\end{array}\\ X=\&lsqb;L_1,L_2,L_3,...,L_n\&rsqb;\end{array}\right.$

Wherein X is design variable, and n is containing the number of component in the linkage of gap, and F (X) is optimization object function,Between containing Gap mechanism acceleration；g_k(X) it is constraint function, is only relevant with design variable X qualitative constraint function really.