CN102829118B - Flexible mechanical arm energy consumption shock absorption method based on 2:1 internal resonance and realization device - Google Patents

Abstract

The invention relates to a flexible mechanical arm energy consumption shock absorption device based on 2:1 internal resonance, and the shock absorption device consists of a shock absorber installation seat, a torsion spring, a connecting turntable, an inertia adjusting ball, a swinging supporting rod and an adjustable damper, wherein the shock absorber installation seat is connected with a flexible mechanical arm through a screw, the adjustable damper is connected with the shock absorber installation seat through a bolt, the connecting turntable is tightened onto a rotating shaft of the adjustable damper, the torsion spring penetrates the rotating shaft of the adjustable damper, two ends of the torsion spring are respectively connected with the adjustable damper and the connecting turntable, the swinging supporting rod is connected with the connecting turntable in a threading way, and the inertia adjusting ball penetrates the swinging supporting rod via a through hole and is fixedly connected with the swinging supporting rod through a tightening screw; and a flexible mechanical arm energy consumption shock absorption method based on the 2:1 internal resonance comprises twelve steps. The flexible mechanical arm energy consumption shock absorption method and the device can be used for the vibration control of an aerospace flexible arm mechanism, a microelectronic manufacturing flexible arm, robot system flexible mechanical hand and the like and have characteristics of obvious shock absorption effect, simplicity in structure and good robustness.

Description

A kind of flexible mechanical arm passive energy dissipation method and implement device based on 2:1 internal resonance

Technical field

The present invention relates to a kind of flexible mechanical arm passive energy dissipation method and implement device based on 2:1 internal resonance, it can build mode energetic interaction passage between controlled flexible mechanical arm and absorbing fork, the vibrational energy of flexible mechanical arm is migrated to vibration damping equipment and by Damping work, belong to the technical fields such as mechanical vibration, precision optical machinery, robot motion's control.

Background technique

For meet lightweight, at a high speed, high capacity is from the performance requirement of anharmonic ratio, flexible mechanical arm is widely used in the technical fields such as Aero-Space, microelectronic manufacture, precision optical machinery.But the elastic vibration that flexible mechanical arm produces under name motion effect, has had a strong impact on spatial positioning accuracy and the working efficiency of mechanical arm.For solving on the vibration space of causing in flexible mechanical arm working procedure and temporal dual contradiction, the vibration control of flexible mechanical arm becomes the important topic that scientific research and technical applications are inquired into.

Up to now, people have carried out broad research from different visual angles to the vibration control of flexible mechanical arm.For example, aspect structural design, improve fundamental frequency by Rational choice physical dimension or shape, reduce plastic deformation and the vibration of robot; Aspect vibration damping Configuration Design, adopt the quality controllable rod member that slides to realize flexible robot's TRAJECTORY CONTROL; At control law design aspect, ACTIVE CONTROL progressively replaces traditional Passive Control, becomes the main development direction of vibration control.Wherein, when utilizing actuating motor control rigid motion, the smart materials such as additional piezoelectric constant, marmem are as actuator, and the method that suppresses flexible robot's elastic dynamic response has become a current study hotspot.Aspect the flexible robot's that current vibration control method is weak in flexible effect, can do linearization process General Oscillation, make remarkable progress, but under high-acceleration, complicated nominal movement environment, flexible mechanical arm linear dynamical effect significantly causes dynamic behavior to become very complicated, cause dynamic performance optimization problem taking linear vibration control as core make slow progress, difficult.

Different from General Oscillation, significantly nonlinear vibration has strengthened the negligible non-linear factor of script in system greatly, originally not only can cause that based on linearizing various controlling methods large error of calculations more can cause the mistake of essence, thereby lose due control effect.On the other hand, with in flexible structure research, its modal parameter can be considered as constant different, flexible robot be a class complexity time become multibody system, exist the coupling of rigid motion (the large displacement of entirety) and flexible vibration, not only mode is intensive but also modal parameter (as: frequency) changes with the variation of topological structure.In addition, the frequency of external excitation also can change along with the difference of task.Therefore, processing flexible robot significantly when nonlinear Problem of Vibration, originally place hope on structural design and will be faced with to subdue the way of vibration the predicament of " hard to guard against ".For example: application number is 200910071707.0 and the application number disclosed semi active vibration absorber based on magnetic rheology elastic body of patent that is 200510094882.3, for flexible structure, vibration control has reasonable effect, but will be difficult to carry out for the significantly nonlinear vibration control of the flexible mechanical arm of Coupled Rigid-flexible under name motion effect.Although various Active Control Methods make remarkable progress, in the time processing significantly nonlinear Problem of Vibration, be also faced with very big challenge.In essence, Active Control Method relies on the input of external energy to realize vibration suppression, directly changes external energy into control force and acts on controlled device.But significantly strong vibrational energy has been contained in nonlinear vibration conventionally, Active Control Method will have to consume more energy change into control force suppress this significantly vibration.Obviously, this way is not always appropriate, and its energy consumption is high, the deficiency of easy overload even makes it to lose more than gain.What is more important, the output power of general smart material is often limited, thereby is difficult to provide enough energy to overcome this judder, is even faced with the danger that overload is destroyed.As can be seen here, existing conventional controlling method is difficult to tackle flexible robot's significantly nonlinear Problem of Vibration.

On the other hand, from theory of non linear vibration, when the each order frequency of system or local a few order frequencies have approximate commensurability or have necessarily can commensurability time, will there is strong internal resonance in system.Now, between corresponding mode, will build the passage of mode energetic interaction, mode energy can periodically move to another rank mode from certain single order mode.If system exists damping, vibrational energy is by decay gradually in mutual process so.Therefore, from the intrinsic attribute of nonlinear vibration, utilize internal resonance mode energetic interaction passage, by flexible mechanical arm significantly nonlinear vibration energy transfer out and the method for utilizing damping to be dissipated, become us and solve the new approaches of significantly nonlinear vibration.

Therefore, for solving fields such as microelectronic manufacture, Aero-Space, robot, there is compliant mechanism, particularly there is the significantly nonlinear vibration control problem of the mechanical system of single flexible mechanical arm, the present invention proposes a kind of flexible mechanical arm passive energy dissipation method and the implement device based on 2:1 internal resonance taking the modal coupling of nonlinear vibration as theoretical foundation.

Summary of the invention

The object of the invention is for significantly shortcoming and the deficiency of nonlinear vibration flowing control method of current flexible mechanical arm, for solving flexible mechanical system, particularly there is the flexible mechanical arm of flexible single link, vibration control problem in the course of the work, provides a kind of flexible mechanical arm passive energy dissipation method and implement device based on 2:1 internal resonance.The present invention can be applied to the vibration control of Aero-Space flexible arm linkage, microelectronic manufacture flexible mechanical arm and robot system flexible manipulator etc., has the advantages that effectiveness in vibration suppression is obvious, simple in structure and robustness is good.

(1) a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance of the present invention, by bump leveller fitting seat A2, torsion spring A3, connect rotating disk A5, inertia regulates ball A6, swing pole A7 and adjustable damper A8 composition, wherein: bump leveller fitting seat A2 is connected with flexible mechanical arm by screw, adjustable damper A8 is connected with bump leveller fitting seat by bolt, connecting rotating disk A5 is fastened in the rotating shaft of adjustable damper A8, torsion spring A3 is through the rotating shaft of adjustable damper, the two ends of torsion spring A3 respectively with adjustable damper A8 and be connected rotating disk A5 connect, swing pole A7 and be connected rotating disk A5 and be threaded, inertia regulates ball A6 via through holes through swinging pole A7 and being connected by Cock screw and swing pole.

Described bump leveller fitting seat A2 is the erection support of one " L " shape, and one of them surface is provided with 4 through holes that are connected with flexible mechanical arm screw, and another surface is provided with a big hole and is convenient to be connected with adjustable damper A8 with two small through hole.Bump leveller fitting seat A2 is the link of this vibration damping equipment and flexible mechanical arm.

Described torsion spring A3, it respectively has a vertical cylindrical end up and down, respectively with the standing part of adjustable damper A8 with is connected rotating disk A5 connection.Torsion spring, for vibration damping equipment provides linear restoring torque, is the rigidity unit of vibration damping equipment;

Described connection rotating disk A5, its rotating disk axial direction has the through hole of non-circle, in order to coordinate with the non-cylndrical surface of adjustable damper A8 rotating shaft.The two mutually perpendicular radial directions that connect rotating disk A5, respectively have a set screw hole and tapped through hole.Connecting rotating disk A5 plays and locates with the rotating shaft of adjustable damper A8 and be connected the effect that swings pole.

Described inertia regulates bead A6, and bead is provided with one by the through hole of the centre of sphere, and the radial direction vertical with this through hole has a set screw hole.Can, by regulating inertia to regulate bead A6 in the position swinging on pole, change the rotary inertia of vibration damping equipment.

Described swing pole A7, is the lightweight cylinder rigid bar of a minor diameter, its one end have outside thread can be connected rotating disk A5 connect, rod member smooth surface can for inertia regulate bead slide location.

Described adjustable damper A8, its profile is column and can be connected with bump leveller fitting seat A2 with ear shape mounting interface.The large end of adjustable damper is damping adjusting knob, can damping size be set by the angle of rotation knob.The small end of adjustable damper A8 is the output shaft of damping, its with the interface that is connected rotating disk A5 and is located by connecting, can be vibration damping equipment damping be provided, its specifications and models are the FRN-P2 of FUJI SEIKI.

Wherein, the diameter of this torsion spring A3 is 1-2mm.

Wherein, the diameter range value of this swing pole A7 is 20-50mm.

Wherein, the specifications and models of this adjustable damper A8 are the FRN-P2 of FUJI SEIKI.

Working principle of the present invention and flow process brief introduction are as follows:

Bump leveller fitting seat A2 have 4 wait the end face of large through-holes and flexible mechanical arm side near, and fastening online by mechanical screw and flexible mechanical arm.The column outer ring of adjustable damper A8 is through the macropore of bump leveller fitting seat A2, and the ear shape mounting interface of adjustable damper A8 aligns with the mounting hole of bump leveller fitting seat A2 and to pass through bolt online fastening.Connect the through hole of rotating disk A5 through the rotating shaft end of adjustable damper A8, the inner side end that connects rotating disk A5 is close to the vertically small end face of rotating shaft middle part of adjustable damper A8, thereby screw screws in the little axle upper screwed hole of connection rotating disk A5 and adjustable damper A8, the two is fastenedly connected.Torsion spring A3 is through the cylinder gabarit of adjustable damper A8, and adjustable damper A8 is inserted respectively with in the circular hole being connected on rotating disk A5 in the vertical termination at torsion spring A3 two, thereby torsion spring is connected in to the outside of adjustable damper rotating shaft.Swing pole A7 by the tapped hole that connects rotating disk A5 that is threaded into of one end, and screw at the screw thread place that swings pole A7 nut be connected rotating disk A5 tapped hole place end face near, thereby will swing pole A7 and to be connected rotating disk A5 fastening.Inertia regulates the through hole of bead A6 through swing pole A7, and can on swing pole A7, slide, and swings pole A7 above by fastening inertia bead location thereby Cock screw screw-in inertia regulates bead tapped hole to be radially fixed on.

After vibration damping equipment is connected with flexible mechanical arm according to above-mentioned Placement, this system has specific structural parameter, therefore, flexible mechanical arm has specific modal parameter under intrinsic flexible characteristic and given name motion effect, it is now the primary condition that mode that the structural parameter of adjustable vibration damping equipment are followed the trail of flexible mechanical arm has reached internal resonance, thereby set up internal resonance mode energetic interaction passage, and realize migration and the dissipation of internal resonance energy.Be specially: first, under the prerequisite of given vibration damping equipment mounting point, rule of thumb select the torsion spring of reasonable, according to above-mentioned process assembling vibration damping equipment, now vibration damping equipment has certain stiffness.Secondly, thereby regulate inertia to regulate bead A6 swinging position on pole A7 and regulate the rotary inertia of vibration damping equipment, reach " determine rigidity and become inertia " and the object of adjusting vibration damping equipment frequency, thereby, the frequency condition of internal resonance is provided, has built the exchange channels of mode energy.Finally, select suitable damping due to rotation by the damping knob of adjustable damper A8, vibrational energy in the engineering of internal resonance mode energetic interaction is dissipated rapidly, thereby realize the effect that best vibrational energy dissipates.

(2) a kind of flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance of the present invention, the method basic principle and concrete steps are as follows:

The present invention by setting up vibration damping equipment on flexible mechanical arm, the natural frequency that changes vibration damping equipment is followed the variation of flexible mechanical arm natural frequency, thereby make vibration damping equipment and flexible mechanical arm meet internal resonance condition, and between builds the passage of mode energetic interaction, again by regulating the damping of vibration damping equipment, make the vibrational energy of flexible mechanical arm be migrated to rapidly vibration damping equipment, and by Damping work, thereby realize the vibration control of flexible mechanical arm.

Specific embodiment of the invention step as shown in figure 10, is specially:

Step 1: a kind of flexible mechanical arm passive energy dissipation equipment and system flexible mechanical arm based on 2:1 internal resonance is connected, guarantee relative tertiary location and the angle of absorbing fork and flexible mechanical arm, as shown in Figure 2.Attention must guarantee that rigidity absorbing fork is vertical with flexible mechanical arm, and now system dynamics equation is Inertia Decouple state.

Step 2: mockup in step 1 is carried out to theory and simplify, extract mathematical model.

As shown in Figure 1, but the Flexible Main system in the present invention is to wait the flexible mechanical arm that shakes, and this mechanical arm is at o ₁x ₁y ₁in plane, have rigid rotation (name motion), flexible mechanical arm load is reduced to end one mass block m _b.The name motion of flexible mechanical arm is determined by its driver element is unique, is characterized by the nominal moving corner of any time with respect to initial time, represents with q.Derive for convenience of theoretical, main system flexible mechanical arm is reduced to homogeneous quality Bernoulli-Euler beam (a lot of engineering calculation are also like this), and its length and cross section parameter are as shown in Fig. 2 a, b.

As shown in Fig. 2 a, Fig. 2 b, in the present invention, vibration damping equipment is made up of rigidity fork and flexible joint, and flexible joint comprises the torsion spring with linear restoring power and the rotation damper with viscous damping.Absorbing fork centroid distance flexible joint axle center r _ccharacterize.Absorbing fork is connected in the somewhere of main system flexible mechanical arm by flexible joint, concrete mounting point can be with it distance r sign with the nominal motion rotating shaft of main system.Absorbing fork has passive freedom degree, under the kinematic parameter effect of main system flexible mechanical arm, swings around flexible joint, and the spatial position of its swing can characterize with the angle β of current time and initial time rigidity fork.

Step 3: count the distortion of flexible mechanical sensitive direction in mathematical model.

As shown in Figure 3, consider the non-linear factor of main system flexible mechanical arm, note and flexible mechanical arm compared with the vibration deformation of sensitive direction, comprise that flexible mechanical departs from the transverse vibration δ (x of nominal moving direction ₁, t), angle of deflection and flexible mechanical arm be along the axial stretching distortion υ (x of self neutral line ₁).From the mechanics of materials and hypothesis modal method:

(1) transverse bending vibration:

u _i(x ₁) be the Mode Shape on i rank,

for i rank modal coordinate, n _ffor system need be considered rank number of mode; (because vibrational energy is mainly gathered in first step mode, the present invention mainly considers subduing of single order mode energy, therefore n _f=1.）

(2) corner that bending causes:

wherein

(3) axial stretching distortion:

wherein

Step 4: the dynamic model of setting up this system based on Kane method.

Single order modal coordinate based on Kane method with main system flexible mechanical arm

with the pivot angle β of the absorbing fork of damping device be that generalized coordinates can show that system vibration equation is:

Wherein, replacement parameter is as follows:

$m_1=B_2+m(u_{1\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">r</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="r\; c"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">r</figure-callout>}}_c^{}{\widetilde{\mathrm{\&beta;}}}_r^2)+m_B{u}_{1\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">l</figure-callout>}}^2;$ $B_2=\mathrm{\&rho;}{\&Integral;}_0^l{u}_1^2\left(x_1\right)d{x}_1;$ $u_{1r}=u_1\left(x_1\right){|}_{x_1=r};$ $u_{1l}=u_1\left(x_1\right){|}_{x_1=l};$ $k_1=\mathrm{EI}{\&Integral;}_0^{l_1}\frac{d^4{u}_1\left(x_1\right)}{d{x_1}^4}{u}_1\left(x_1\right)d{x}_1;$ $b_1=m{r}_c(r{\widetilde{\mathrm{\&beta;}}}_r^2-B_{1r});$ $B_{1r}=B_1\left(x_1\right){|}_{x_1=r};$ $b_2=m{r}_c{\widetilde{\mathrm{\&beta;}}}_r(u_{1r}{\widetilde{\mathrm{\&beta;}}}_r-B_{1r});$ b ₃=mr _cu _1r; $b_4=2m{r}_c{\widetilde{\mathrm{\&beta;}}}_r(u_{1r}{\widetilde{\mathrm{\&beta;}}}_r-B_{1r});$ $b_5=\mathrm{\&rho;}{\&Integral;}_0^l{u}_1\left(x_1\right)x_{1}d{x}_1+m(r{u}_{1r}+{\mathrm{<figure-callout\; id="2"\; label="r\; c"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">r</figure-callout>}}_c^{}{\widetilde{\mathrm{\&beta;}}}_r)+m_B{u}_{1l}l;$ $b_6=m{r}_c(u_{1r}-r{\widetilde{\mathrm{\&beta;}}}_r);$ $m_2=\mathrm{<figure-callout\; id="2"\; label="m\; r\; c"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">m</figure-callout>}{r}_c^{}{}_2^{}+I_{\mathrm{zz}}$

C in above-mentioned set of equation ₁for mechanism's damping of main system flexible mechanical arm, c ₂for inhaling the viscous damping of vibration damping equipment flexible joint, k ₂for the torsion spring rigidity of vibration damping equipment flexible joint.

Step 5: utilize Method of Multiple Scales to solve analysis to dynamic model.

Depending on generalized coordinates in whole system

and β, the speed of main system flexible mechanical arm name moving corner

main system flexible mechanical arm structural damping c ₁, the viscous damping c of vibration damping equipment flexible joint ₂for same order a small amount of.Through nondimensionalization processing, same order in a small amount after replacement and the replacement about many Size Equation solution of time, can draw vibration system have form that fast change time and single order become the time slowly into:

(1) about the vibration equation group of fast change time into:

${\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00002149328600012.png,HDA00002149328600013.png"\; state="\{\{state\}\}">D</figure-callout>}}_0^2{\mathrm{\&psi;}}_0+{\mathrm{\&psi;}}_0=0---\left(4\right)$

(2) about the vibration equation group of slow change time into:

The middle replacement parameter of deriving is as follows:

${\mathrm{\&omega;}}_{\mathrm{\&beta;}}=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}_2/{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2};$ η ₁＝c ₁/(ω _βm ₁);

d ₁＝b ₁/m ₁;d ₂=lb ₂/m ₁;d ₃＝b ₃/(m ₁l);

d ₄=lb ₄/m ₁;d ₅＝b ₅/(m ₁l);d ₆＝b ₆/(m ₁l);η ₂＝c ₂/(m ₂ω _β);d ₇＝b ₇/m ₂;d ₈=lb ₃/m ₂.

If the written form of above-mentioned fast change time corresponding equation group solution into:

ψ _0,k=A _2,k(T ₁)exp(jT ₀)+cc。（7）

Consider that the condition that 2:1 internal resonance occurs in nonlinear vibration is:

(be the frequency of vibration damping equipment and the frequency of main system approximate meet 2:1 relation), wherein σ is tuner parameters, ε be a small amount of.The amplitude of setting the solution of vibration equation has following form:

$A_{1,k}=\frac{1}{2}{a}_{1,k}\exp \left(j{\mathrm{\&theta;}}_{1,k}\right)---\left(8\right)$

$A_{2,k}=\frac{1}{2}{a}_{2,k}\exp \left(j{\mathrm{\&theta;}}_{2,k}\right)---\left(9\right)$

Wherein a _{1, k}, a _{2, k}, θ _{1, k}, θ _{2, k}the real function that becomes slowly the time, a _{1, k}, a _2kfor the mode amplitude of system, its size has reflected the size of corresponding mode energy.Slowly the solvability that becomes equation of time group according to single order can draw:

Wherein γ _k=θ _{1, k}-2 θ _{2, k}-ε σ T ₀.

Step 6: the solution of kinetic equations is carried out to Analysis of Internal Resonance.

Supposing that main system flexible mechanical arm does not exist mechanism's damping, there is not damping in vibration damping equipment yet, is drawn by formula (10), (11):

Draw through Mathematical treatment:

$a_{1,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}^2+\mathrm{\&upsi;}{a}_{2,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}^2=E_k=\mathrm{const}$ (integration constant relevant to primary power) wherein

obviously, this can illustrate the passage that has built energetic interaction between two rank mode corresponding in undamped situation, and υ >0 shows the two equal boundeds of rank mode energy and this those length that disappear.

Because not taking into account system damping, therefore only from mathematical meaning, the mutual permanent of this mode energy carries out.Consider that main system under truth has the viscous damping of structural damping and vibration damping equipment, system capacity is inevitable decay to disappearance under the effect of damping.

Step 7: kinetic equations solution is carried out to determination of stability.

For more scientific explanation mode energy process of decay gradually under the effect of damping, investigate in the stability of equinoctial point:

The Jacobian matrix of this system is so $\left[\begin{array}{ccc}{\mathrm{\&mu;}}_1& 0& 0\\ 0& {\mathrm{\&mu;}}_2& 0\\ 0& 0& 0\end{array}\right],$ Wherein

characteristic of correspondence value is: (μ ₁, μ ₂, 0).

Due to mechanism's dampingratioζ of main system flexible mechanical arm ₁>0, the dampingratioζ of flexible joint viscous damping ₂>0, has μ ₁<0, μ ₂<0.The characteristic root of system is negative value, therefore mode amplitude a _{1, k}, a _2kthere is continous-stable.So far show, use the vibration damping equipment of absorbing fork and flexible joint structure, can between main system and vibration insulating system, build the exchange channels of mode energy, and the decay of the vibrational energy of main system flexible mechanical arm continous-stable under vibration damping equipment flexible joint viscous damping (principal element) and flexible mechanical arm structural damping (secondary cause) effect.Feasibility and the science of using the flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance have so far been described.

Step 8: to dynamic model Simulation Example, to choose theoretical optimal damping for vibration damping equipment.

Explain superiority of the present invention for further directly perceived, and be the optimal damping of vibration damping equipment Choice Theory, given following serial example comparative illustration.Setting example initial conditions is: main system flexible mechanical arm lengths is l=1.0m, and flexible mechanical arm cross section is the rectangle of the wide b=0.003m of high h=0.05m, and end load is m _b=0.5kg, aluminium matter flexible arm Young's modulus is 71Gpa density 2710kg/m ³.Vibration insulating system rigidity absorbing fork mounting point r=0.5m, quality is m ₂=0.05kg centroid position is r _c=0.4m.

First, limit the name motion of main system flexible mechanical arm, do not set up vibration damping equipment, the end initial disturbance amount of given flexible mechanical arm is 0.1m.If the damping of main system flexible mechanical arm non-structure, flexible mechanical arm tip vibration situation as shown in Figure 4.On main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping, the mutual situation of mode energy is as Fig. 5.As shown in Figure 5, in undamped situation, between different modalities energy, build mutual passage, and this those long and boundeds that therefore disappear alternately, this fits like a glove with the theory analysis of the present invention's elaboration.

Secondly, the nominal characteristics of motion of giving main system flexible mechanical arm is: $\stackrel{\&CenterDot;}{q}=0.1\&times;\cos \left(\frac{\mathrm{\&pi;t}}{5}\right)+0.1\&times;\sin \left(\frac{\sqrt{2}}{2}t\right),$ (0≤t≤50s), the initial disturbance amount of given flexible mechanical arm 0.1m.If the damping of main system flexible mechanical arm non-structure, the Vibration Condition of flexible mechanical arm end is as Fig. 6.On main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping, the mutual situation of mode energy is as Fig. 7.Can be learnt by Fig. 7, giving after flexible mechanical arm name motion, in undamped situation, between different modalities energy, build mutual passage, and this mutual this that long and boundeds that disappear, this fits like a glove with the theory analysis of the present invention's elaboration.

The nominal characteristics of motion of then, giving main system flexible mechanical arm is: $\stackrel{\&CenterDot;}{q}=0.1\&times;\cos \left(\frac{\mathrm{\&pi;t}}{5}\right)+0.1\&times;\sin \left(\frac{\sqrt{2}}{2}t\right),$ (0≤t≤50s), the initial disturbance amount of given flexible mechanical arm 0.1m.Because the structural damping of ordinary circumstance flexible mechanical arm is very little, therefore setting the structural damping of main system flexible mechanical arm is 0, the viscous damping damping ratio of setting vibration damping equipment flexible joint is the damping ratio under 0.001(acts in a small amount), the Vibration Condition of flexible mechanical arm end is as Fig. 8, and the mutual situation of mode energy as shown in Figure 9.Just can be known by Fig. 9, in constructed energetic interaction passage, this disappears mode energy that is long alternately also, and the vibrational energy of flexible mechanical arm progressively decays to disappearance under the effect of damping.The tip vibration that can see clearly flexible mechanical arm in 10s by Fig. 8 has just decayed 80%.This example has been investigated worst working condition, it is the damping of main system flexible mechanical arm non-structure, but in fact flexible mechanical arm always has structural damping existence, even if very little this also has positive role to the decay of vibrating, this method and implement device that also further illustrates the present invention's proposition will produce better effect in actual applications.

Finally, based on set of equation (10), (11), (12), give each parameter values according to above-mentioned theory derivation, according to nominal laws of motion (being not limited only to the characteristics of motion of the mechanical arm that this patent enumerates), repeatedly revise and the flexible joint damping value that replaces carries out numerical Analysis, select mode viscosity or flexible mechanical arm tip vibration to become in figure damping corresponding when vibration attenuation is the fastest as theoretical optimal damping.

Step 9: by temporary fixed to rigidity absorbing fork and main system flexible mechanical arm, guarantee that the two relative position and angle are initial zero-bit angle, as shown in left in Figure 2.Application test mode measuring technology, obtains the natural frequency ω of the system including main system flexible mechanical arm and rigidity absorbing fork by hammering method ₁with damping ratio ξ ₁.

Step 10: according to the frequency match requirement of the non-linear internal resonance of structure 2:1, regulate the position of inertia adjusting bead in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance, make the natural frequency of vibration damping equipment meet ω ₂≈ 2 ω ₁.After torsion spring rigidity k is selected with device, show that according to the physical model of the physical pendulum with linear torsion restoring force vibration damping equipment frequency formula is

wherein J is the rotary inertia of absorbing fork around flexible joint.

Step 11: that in release steps nine, main system flexible mechanical arm and rigidity absorbing fork are implemented is temporary fixed, and regulate the size of rotation damper damping in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance of flexible joint, guarantee that rigidity absorbing fork is free state, make it passive swing under the kinematic parameter effect of main system flexible mechanical arm.

Near step 12: selected theoretical optimum damping parameter is carried out to actual test, and the theoretical parameter of selecting, damping is finely tuned, to obtain actual optimal parameter.So far just complete the flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance of the present invention's proposition and the implementation and operation process of device.

Advantage and effect: the present invention has broken through existing utilization linear vibration theory and solve the constraint of the theoretical method aspect of nonlinear Problem of Vibration, while taking full advantage of non-linear vibrating system generation internal resonance, mode energy mutual characteristic between different modalities, consider comparatively all sidedly to have stronger transverse bending vibration, bending corner and the axial stretching vibration of flexible armed lever receptance of name motion (rigid rotation), more nonlinear vibration reduction method and the implement device thereof of science and simple possible of a kind of theoretical foundation is provided.It is obvious that this invention has effectiveness in vibration suppression, and the feature of simple in structure and strong robustness, has important realistic meaning and wide application prospect in fields such as microelectronic manufacture, Aero-Space machinery and robots.

Brief description of the drawings

Fig. 1 is main system structure explanatory drawing of the present invention.

Fig. 2 a is vibration damping equipment of the present invention and main system scheme of installation

Fig. 2 b is that vibration damping equipment absorbing rigidity fork of the present invention swings schematic diagram

Fig. 3 is the distortion explanatory drawing that the present invention takes into account non-linear factor flexible mechanical arm sensitive direction.

Fig. 4 is for limiting the motion of main system name and not adding vibration damping equipment, main system flexible mechanical arm tip vibration deformation pattern.

Mode energetic interaction indicator diagram when Fig. 5 arranges for limiting the motion of main system name and add vibration damping equipment, not carrying out vibration damping equipment damping.

Fig. 6 is that given name motion does not add vibration damping equipment, the vibration deformation figure of main system flexible mechanical arm end.

Fig. 7 is given name motion and adds vibration damping equipment, mode energetic interaction indicator diagram when not carrying out vibration damping equipment damping and arranging.

Fig. 8 is given name motion, and main system adds vibration damping equipment and carries out vibration damping equipment damping rear main system flexible mechanical arm end distortion is set.

Fig. 9 is given name motion, and main system adds vibration damping equipment and carries out vibration damping equipment damping rear mode energetic interaction indicator diagram is set.

Figure 10 is that 2:1 internal resonance flexible mechanical arm passive energy dissipation method is carried out Vibrations of A Flexible Robot Arm control implementing procedure figure.

Figure 11 is the vibration damping equipment figure of 2:1 internal resonance flexible mechanical arm passive energy dissipation

Figure 12 is vibration damping equipment and the flexible mechanical arm mounting structure figure of 2:1 internal resonance flexible mechanical arm passive energy dissipation

In figure, symbol description is as follows:

X ₀y ₀z ₀the basis coordinates being connected with the earth is; x ₁y ₁z ₁with flexible mechanical arm name motion (rigidity corner) be connected with moving coordinate system; m _bflexible mechanical arm end load; The length of l flexible mechanical arm; Q flexible mechanical arm name motion (rigidity corner); r _cvariation pole barycenter is to the distance of flexible joint; R flexible joint mounting points is to the distance of flexible mechanical arm center of rotation; B flexible mechanical arm tranverse sectional thickness; H flexible mechanical arm cross-sectional height; β swings pole pivot angle; δ (x ₁, t) flexible mechanical arm transverse vibration amount of deformation; α flexible mechanical arm deflection angle amount of deformation; υ (x ₁) flexible mechanical arm dilatation amount; A1 flexible mechanical arm; A2 bump leveller fitting seat; A3 torsion spring; A4 end mechanism hand and load; A5 connects rotating disk; A6 inertia regulates ball; A7 swings pole; A8 adjustable damper; A9 robot driver element.

Embodiment:

(1) see Figure 11, a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance of the present invention, by bump leveller fitting seat A2, torsion spring A3, connect rotating disk A5, inertia regulates ball A6, swing pole A7 and adjustable damper A8 composition, wherein: bump leveller fitting seat A2 is connected with flexible mechanical arm by screw, adjustable damper A8 is connected with bump leveller fitting seat by bolt, connecting rotating disk A5 is fastened in the rotating shaft of adjustable damper A8, torsion spring A3 is through the rotating shaft of adjustable damper, the two ends of torsion spring A3 respectively with adjustable damper A8 and be connected rotating disk A5 connect, swing pole A7 and be connected rotating disk A5 and be threaded, inertia regulates ball A6 via through holes through swinging pole A7 and being connected by Cock screw and swing pole.Figure 12 is vibration damping equipment and the flexible mechanical arm mounting structure figure of 2:1 internal resonance flexible mechanical arm passive energy dissipation.

(2) see Figure 10, a kind of flexible mechanical arm passive energy dissipation installation method based on 2:1 internal resonance of the present invention, the method concrete steps are as follows:

Step 1: a kind of flexible mechanical arm passive energy dissipation equipment and system flexible mechanical arm based on 2:1 internal resonance is connected, guarantee relative tertiary location and the angle of absorbing fork and flexible mechanical arm, as shown in Fig. 2 a, Fig. 2 b.Attention must guarantee that rigidity absorbing fork is vertical with flexible mechanical arm, and now system dynamics equation is Inertia Decouple state.

Step 2: mockup in step 1 is carried out to theory and simplify, extract mathematical model.

As shown in Figure 1, but the Flexible Main system in the present invention is to wait the flexible mechanical arm that shakes, and this mechanical arm is at o ₁x ₁y ₁in plane, have rigid rotation (name motion), flexible mechanical arm load is reduced to end one mass block m _b.The name motion of flexible mechanical arm is determined by its driver element is unique, is characterized by the nominal moving corner of any time with respect to initial time, represents with q.Derive for convenience of theoretical, main system flexible mechanical arm is reduced to homogeneous quality Bernoulli-Euler beam (a lot of engineering calculation are also like this), and its length and cross section parameter are as shown in Fig. 2 a, Fig. 2 b.

As shown in Fig. 2 a, Fig. 2 b, in the present invention, vibration damping equipment is made up of rigidity fork and flexible joint, and flexible joint comprises the torsion spring with linear restoring power and the rotation damper with viscous damping.Absorbing fork centroid distance flexible joint axle center r _ccharacterize.Absorbing fork is connected in the somewhere of main system flexible mechanical arm by flexible joint, concrete mounting point can be with it distance r sign with the nominal motion rotating shaft of main system.Absorbing fork has passive freedom degree, under the kinematic parameter effect of main system flexible mechanical arm, swings around flexible joint, and the spatial position of its swing can characterize with the angle β of current time and initial time rigidity fork.

Step 3: count the distortion of flexible mechanical sensitive direction in mathematical model.

As shown in Figure 3, consider the non-linear factor of main system flexible mechanical arm, note and flexible mechanical arm compared with the vibration deformation of sensitive direction, comprise that flexible mechanical departs from the transverse vibration δ (x of nominal moving direction ₁, t), angle of deflection and flexible mechanical arm be along the axial stretching distortion υ (x of self neutral line ₁).From the mechanics of materials and hypothesis modal method:

(1) transverse bending vibration:

u _i(x ₁) be the Mode Shape on i rank, for i rank modal coordinate, n _ffor system need be considered rank number of mode; (because vibrational energy is mainly gathered in first step mode, the present invention mainly considers subduing of single order mode energy, therefore n _f=1.）

(2) corner that bending causes:

wherein ${\widetilde{\mathrm{\&beta;}}}_r=\frac{d{u}_1\left(x_1\right)}{d{x}_1}{|}_{x_1=r};$

(3) axial stretching distortion:

wherein ${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">B</figure-callout>}}_1={\&Integral;}_0^{x_1}{\left(\frac{d{u}_1\left(\mathrm{\&theta;}\right)}{\mathrm{d\&theta;}}\right)}^2\mathrm{d\&theta;}$

Step 4: the dynamic model of setting up this system based on Kane method.

Single order modal coordinate based on Kane method with main system flexible mechanical arm with the pivot angle β of the absorbing fork of damping device be that generalized coordinates can show that system vibration equation is:

Wherein, replacement parameter is as follows:

$m_1=B_2+m(u_{1r}^2+r_c^2{\widetilde{\mathrm{\&beta;}}}_r^2)+m_B{u}_{1l}^2;$ $B_2=\mathrm{\&rho;}{\&Integral;}_0^l{u}_1^2\left(x_1\right)d{x}_1;$ $u_{1r}=u_1\left(x_1\right){|}_{x_1=r};$ $u_{1l}=u_1\left(x_1\right){|}_{x_1=l};$ $k_1=\mathrm{EI}{\&Integral;}_0^{l_1}\frac{d^4{u}_1\left(x_1\right)}{d{x_1}^4}{u}_1\left(x_1\right)d{x}_1;$ $b_1=m{r}_c(r{\widetilde{\mathrm{\&beta;}}}_r^2-B_{1r});$ $B_{1r}=B_1\left(x_1\right){|}_{x_1=r};$ $b_2=m{r}_c{\widetilde{\mathrm{\&beta;}}}_r(u_{1r}{\widetilde{\mathrm{\&beta;}}}_r-B_{1r});$ b ₃=mr _cu _1r; $b_4=2m{r}_c{\widetilde{\mathrm{\&beta;}}}_r(u_{1r}{\widetilde{\mathrm{\&beta;}}}_r-B_{1r});$ $b_5=\mathrm{\&rho;}{\&Integral;}_0^l{u}_1\left(x_1\right)x_{1}d{x}_1+m(r{u}_{1r}+r_c^2{\widetilde{\mathrm{\&beta;}}}_r)+m_B{u}_{1l}l;$ $b_6=m{r}_c(u_{1r}-r{\widetilde{\mathrm{\&beta;}}}_r);$ $m_2=m{r}_c^2+I_{\mathrm{zz}}$

C in above-mentioned set of equation ₁for mechanism's damping of main system flexible mechanical arm, c ₂for inhaling the viscous damping of vibration damping equipment flexible joint, k ₂for the torsion spring rigidity of vibration damping equipment flexible joint.

Step 5: utilize Method of Multiple Scales to solve analysis to dynamic model.

Depending on generalized coordinates in whole system and β, the speed of main system flexible mechanical arm name moving corner

main system flexible mechanical arm structural damping c ₁, the viscous damping c of vibration damping equipment flexible joint ₂for same order a small amount of.Through nondimensionalization processing, same order in a small amount after replacement and the replacement about many Size Equation solution of time, can draw vibration system have form that fast change time and single order become the time slowly into:

(1) about the vibration equation group of fast change time into:

${\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00002149328600012.png,HDA00002149328600013.png"\; state="\{\{state\}\}">D</figure-callout>}}_0^2{\mathrm{\&psi;}}_0+{\mathrm{\&psi;}}_0=0---\left(4\right)$

(2) about the vibration equation group of slow change time into:

The middle replacement parameter of deriving is as follows:

${\mathrm{\&omega;}}_{\mathrm{\&beta;}}=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}_2/{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2};$ η ₁＝c ₁/(ω _βm ₁);

d ₁=b ₁/m ₁；d ₂=lb ₂/m ₁；d ₃=b ₃/(m ₁l)；

d ₄=lb ₄/m ₁;d ₅＝b ₅/(m ₁l);d ₆＝b ₆/(m ₁l);η ₂＝c ₂/(m ₂ω _β);d ₇＝b ₇/m ₂;d ₈=lb ₃/m ₂.

If the written form of above-mentioned fast change time corresponding equation group solution into:

ψ _0，k=A _2,k(T ₁)exp(jT ₀)+cc。（7）

Consider that the condition that 2:1 internal resonance occurs in nonlinear vibration is:

(be the frequency of vibration damping equipment and the frequency of main system approximate meet 2:1 relation), wherein σ is tuner parameters, ε be a small amount of.The amplitude of setting the solution of vibration equation has following form:

$A_{1,k}=\frac{1}{2}{a}_{1,k}\exp \left(j{\mathrm{\&theta;}}_{1,k}\right)---\left(8\right)$

$A_{2,k}=\frac{1}{2}{a}_{2,k}\exp \left(j{\mathrm{\&theta;}}_{2,k}\right)---\left(9\right)$

Wherein a _{1, k}, a _{2, k}, θ _{1, k}, θ _{2, k}the real function that becomes slowly the time, a _{1, k}, a _2kfor the mode amplitude of system, its size has reflected the size of corresponding mode energy.Slowly the solvability that becomes equation of time group according to single order can draw:

Wherein γ _k=θ _{1, k}-2 θ _{2, k}-ε σ T ₀.

Step 6: the solution of kinetic equations is carried out to Analysis of Internal Resonance.

Supposing that main system flexible mechanical arm does not exist mechanism's damping, there is not damping in vibration damping equipment yet, is drawn by formula (10), (11):

Draw through Mathematical treatment:

$a_{1,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}^2+\mathrm{\&upsi;}{a}_{2,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00002149328600022.png,HDA00002149328600032.png"\; state="\{\{state\}\}">k</figure-callout>}}^2=E_k=\mathrm{const}$ (integration constant relevant to primary power) wherein

obviously, this can illustrate the passage that has built energetic interaction between two rank mode corresponding in undamped situation, and υ >0 shows the two equal boundeds of rank mode energy and this those length that disappear.

Because not taking into account system damping, therefore only from mathematical meaning, the mutual permanent of this mode energy carries out.Consider that main system under truth has the viscous damping of structural damping and vibration damping equipment, system capacity is inevitable decay to disappearance under the effect of damping.

Step 7: kinetic equations solution is carried out to determination of stability.

For more scientific explanation mode energy process of decay gradually under the effect of damping, investigate in the stability of equinoctial point:

The Jacobian matrix of this system is so $\left[\begin{array}{ccc}{\mathrm{\&mu;}}_1& 0& 0\\ 0& {\mathrm{\&mu;}}_2& 0\\ 0& 0& 0\end{array}\right],$ Wherein

characteristic of correspondence value is: (μ ₁, μ ₂, 0).

Due to mechanism's dampingratioζ of main system flexible mechanical arm ₁>0, the dampingratioζ of flexible joint viscous damping ₂>0, has μ ₁<0, μ ₂<0.The characteristic root of system is negative value, therefore mode amplitude a _{1, k}, a _2kthere is continous-stable.So far show, use the vibration damping equipment of absorbing fork and flexible joint structure, can between main system and vibration insulating system, build the exchange channels of mode energy, and the decay of the vibrational energy of main system flexible mechanical arm continous-stable under vibration damping equipment flexible joint viscous damping (principal element) and flexible mechanical arm structural damping (secondary cause) effect.Feasibility and the science of using the flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance have so far been described.

Step 8: to dynamic model Simulation Example, to choose theoretical optimal damping for vibration damping equipment.

Explain superiority of the present invention for further directly perceived, and be the optimal damping of vibration damping equipment Choice Theory, given following serial example comparative illustration.Setting example initial conditions is: main system flexible mechanical arm lengths is l=1.0m, and flexible mechanical arm cross section is the rectangle of the wide b=0.003m of high h=0.05m, and end load is m _b=0.5kg, aluminium matter flexible arm Young's modulus is 71Gpa density 2710kg/m ³.Vibration insulating system rigidity absorbing fork mounting point r=0.5m, quality is m ₂=0.05kg centroid position is r _c=0.4m.

First, limit the name motion of main system flexible mechanical arm, do not set up vibration damping equipment, the end initial disturbance amount of given flexible mechanical arm is 0.1m.If the damping of main system flexible mechanical arm non-structure, flexible mechanical arm tip vibration situation as shown in Figure 4.On main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping, the mutual situation of mode energy is as Fig. 5.As shown in Figure 5, in undamped situation, between different modalities energy, build mutual passage, and this those long and boundeds that therefore disappear alternately, this fits like a glove with the theory analysis of the present invention's elaboration.

Secondly, the nominal characteristics of motion of giving main system flexible mechanical arm is: $\stackrel{\&CenterDot;}{q}=0.1\&times;\cos \left(\frac{\mathrm{\&pi;t}}{5}\right)+0.1\&times;\sin \left(\frac{\sqrt{2}}{2}t\right),$ (0≤t≤50s), the initial disturbance amount of given flexible mechanical arm 0.1m.If the damping of main system flexible mechanical arm non-structure, the Vibration Condition of flexible mechanical arm end is as Fig. 6.On main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping, the mutual situation of mode energy is as Fig. 7.Can be learnt by Fig. 7, giving after flexible mechanical arm name motion, in undamped situation, between different modalities energy, build mutual passage, and this mutual this that long and boundeds that disappear, this fits like a glove with the theory analysis of the present invention's elaboration.

The nominal characteristics of motion of then, giving main system flexible mechanical arm is: $\stackrel{\&CenterDot;}{q}=0.1\&times;\cos \left(\frac{\mathrm{\&pi;t}}{5}\right)+0.1\&times;\sin \left(\frac{\sqrt{2}}{2}t\right),$ (0≤t≤50s), the initial disturbance amount of given flexible mechanical arm 0.1m.Because the structural damping of ordinary circumstance flexible mechanical arm is very little, therefore setting the structural damping of main system flexible mechanical arm is 0, the viscous damping damping ratio of setting vibration damping equipment flexible joint is the damping ratio under 0.001(acts in a small amount), the Vibration Condition of flexible mechanical arm end is as Fig. 8, and the mutual situation of mode energy as shown in Figure 9.Just can be known by Fig. 9, in constructed energetic interaction passage, this disappears mode energy that is long alternately also, and the vibrational energy of flexible mechanical arm progressively decays to disappearance under the effect of damping.The tip vibration that can see clearly flexible mechanical arm in 10s by Fig. 8 has just decayed 80%.This example has been investigated worst working condition, it is the damping of main system flexible mechanical arm non-structure, but in fact flexible mechanical arm always has structural damping existence, even if very little this also has positive role to the decay of vibrating, this method and implement device that also further illustrates the present invention's proposition will produce better effect in actual applications.

Finally, based on set of equation (10), (11), (12), give each parameter values according to above-mentioned theory derivation, according to nominal laws of motion (being not limited only to the characteristics of motion of the mechanical arm that this patent enumerates), repeatedly revise and the flexible joint damping value that replaces carries out numerical Analysis, select mode viscosity or flexible mechanical arm tip vibration to become in figure damping corresponding when vibration attenuation is the fastest as theoretical optimal damping.

Step 9: by temporary fixed to rigidity absorbing fork and main system flexible mechanical arm, guarantee that the two relative position and angle are initial zero-bit angle, as shown in Figure 2 a.Application test mode measuring technology, obtains the natural frequency ω of the system including main system flexible mechanical arm and rigidity absorbing fork by hammering method ₁with damping ratio ξ ₁.

Step 10: according to the frequency match requirement of the non-linear internal resonance of structure 2:1, regulate the position of inertia adjusting bead in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance, make the natural frequency of vibration damping equipment meet ω ₂≈ 2 ω ₁.After torsion spring rigidity k is selected with device, show that according to the physical model of the physical pendulum with linear torsion restoring force vibration damping equipment frequency formula is

wherein J is the rotary inertia of absorbing fork around flexible joint.

Step 11: that in release steps nine, main system flexible mechanical arm and rigidity absorbing fork are implemented is temporary fixed, and regulate the size of rotation damper damping in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance of flexible joint, guarantee that rigidity absorbing fork is free state, make it passive swing under the kinematic parameter effect of main system flexible mechanical arm.

Near step 12: selected theoretical optimum damping parameter is carried out to actual test, and the theoretical parameter of selecting, damping is finely tuned, to obtain actual optimal parameter.So far just complete the flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance of the present invention's proposition and the implementation and operation process of device.

Claims (5)

1. the flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance, it is characterized in that: it is by bump leveller fitting seat, torsion spring, connect rotating disk, inertia regulates ball, swing pole and adjustable damper composition, wherein: bump leveller fitting seat is connected with flexible mechanical arm by screw, adjustable damper is connected with bump leveller fitting seat by bolt, connecting rotating disk is fastened in the rotating shaft of adjustable damper, torsion spring is through the rotating shaft of adjustable damper, the two ends of torsion spring respectively with adjustable damper and be connected rotating disk connect, swing pole and be connected rotating disk and be threaded, inertia regulates ball warp through hole through swinging pole and being connected by Cock screw and swing pole,

Described bump leveller fitting seat is the erection support of one " L " shape, one of them surface is provided with 4 through holes that are connected with flexible mechanical arm screw, another surface is provided with a big hole and is convenient to be connected with adjustable damper with two small through hole, and bump leveller fitting seat is the link of this vibration damping equipment and flexible mechanical arm;

Described torsion spring, it respectively has a vertical cylindrical end up and down, respectively with the standing part of adjustable damper with is connected rotating disk connection; Torsion spring, for vibration damping equipment provides linear restoring torque, is the rigidity unit of vibration damping equipment;

Described connection rotating disk, its rotating disk axial direction has the through hole of non-circle, in order to coordinate with the non-cylndrical surface of adjustable damper rotating shaft; The two mutually perpendicular radial directions that connect rotating disk, respectively have a set screw hole and a tapped through hole in each direction; Connecting rotating disk plays and locates with the rotating shaft of adjustable damper and be connected the effect that swings pole;

Described inertia regulates bead, is provided with one by the through hole of the centre of sphere, and the radial direction vertical with this through hole has a set screw hole, by regulating inertia to regulate bead in the position swinging on pole, changes the rotary inertia of vibration damping equipment;

Described swing pole, is the lightweight cylinder rigid bar of a minor diameter, and there is outside thread its one end and is connected rotating disk connection, and rod member smooth surface can regulate bead to slide for inertia and locate;

Described adjustable damper, its profile is column and is connected with bump leveller fitting seat with ear shape mounting interface, the large end of adjustable damper is damping adjusting knob, by the angle of rotation knob, damping size is set; The small end of adjustable damper is the output shaft of damping, its with the interface that is connected rotating disk and is located by connecting, for vibration damping equipment provides damping.

2. a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance according to claim 1, is characterized in that: the diameter of this torsion spring is 1-2mm.

3. a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance according to claim 1, is characterized in that: the diameter range value of this swing pole is 20-50mm.

4. a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance according to claim 1, is characterized in that: the specifications and models of this adjustable damper are the FRN-P2 of FUJI SEIKI.

5. a passive energy dissipation method for the flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance as claimed in claim 1, the method concrete steps are as follows:

Step 1: a kind of flexible mechanical arm passive energy dissipation equipment and system flexible mechanical arm based on 2:1 internal resonance is connected, guarantee relative tertiary location and the angle of absorbing fork and flexible mechanical arm, attention must guarantee that rigidity absorbing fork is vertical with flexible mechanical arm, and now system dynamics equation is Inertia Decouple state;

Step 2: mockup in step 1 is carried out to theory and simplify, extract mathematical model;

But Flexible Main system is to wait the flexible mechanical arm that shakes, and this mechanical arm is at o ₁x ₁y ₁in plane, have rigid rotation, flexible mechanical arm load is reduced to end one mass block m _b, the name motion of flexible mechanical arm is determined by its driver element is unique, is characterized by the nominal moving corner of any time with respect to initial time, represents with q; Derive for convenience of theoretical, main system flexible mechanical arm is reduced to homogeneous quality Bernoulli-Euler beam; Vibration damping equipment is made up of rigidity fork and flexible joint, and flexible joint comprises the torsion spring with linear restoring power and the rotation damper with viscous damping; Absorbing fork centroid distance flexible joint axle center r _ccharacterize, absorbing fork is connected in the somewhere of main system flexible mechanical arm by flexible joint, concrete mounting point is distance r sign with the nominal motion rotating shaft of main system with it, absorbing fork has passive freedom degree, under the kinematic parameter effect of main system flexible mechanical arm, swing around flexible joint, the spatial position of its swing characterizes with the angle β of current time and initial time rigidity fork;

Step 3: count the distortion of flexible mechanical sensitive direction in mathematical model;

Consider the non-linear factor of main system flexible mechanical arm, and flexible mechanical arm compared with the vibration deformation of sensitive direction, comprise that flexible mechanical departs from the transverse vibration δ (x of nominal moving direction ₁, t), angle of deflection and flexible mechanical arm be along the axial stretching distortion υ (x of self neutral line ₁); Known by the mechanics of materials and hypothesis modal method:

(1) transverse bending vibration:

u _i(x ₁) be the Mode Shape on i rank,

for i rank modal coordinate, n _ffor system need be considered rank number of mode;

(2) corner that bending causes:

wherein

(3) axial stretching distortion:

wherein

Step 4: the dynamic model of setting up this system based on Kane method;

Single order modal coordinate based on Kane method with main system flexible mechanical arm

with the pivot angle β of the absorbing fork of damping device be that generalized coordinates can show that system vibration equation is:

wherein, replacement parameter is as follows:

C in above-mentioned set of equation ₁for mechanism's damping of main system flexible mechanical arm, c ₂for inhaling the viscous damping of vibration damping equipment flexible joint, k ₂for the torsion spring rigidity of vibration damping equipment flexible joint;

Step 5: utilize Method of Multiple Scales to solve analysis to dynamic model;

Depending on generalized coordinates in whole system and β, the speed of main system flexible mechanical arm name moving corner

main system flexible mechanical arm structural damping c ₁, the viscous damping c of vibration damping equipment flexible joint ₂for same order a small amount of; Through nondimensionalization processing, same order in a small amount after replacement and the replacement about many Size Equation solution of time, draw vibration system have form that fast change time and single order become the time slowly into:

(1) about the vibration equation group of fast change time into:

(2) about the vibration equation group of slow change time into:

the middle replacement parameter of deriving is as follows:

η ₁=c ₁/(ω _βm ₁);

d ₁=b ₁/m ₁;d ₂=lb ₂/m ₁;d ₃=b ₃/(m ₁l);d ₄=lb ₄/m ₁;d ₅=b ₅/(m ₁l);d ₆=b ₆/(m ₁l);η ₂=c ₂/(m ₂ω _β);d ₇=b ₇/m ₂;d ₈=lb ₃/m ₂.

If the written form of above-mentioned fast change time corresponding equation group solution into:

Consider that the condition that 2:1 internal resonance occurs in nonlinear vibration is:

being that the frequency of vibration damping equipment and the frequency of main system are approximate meets 2:1 relation, and wherein σ is tuner parameters, and ε is in a small amount; The amplitude of setting the solution of vibration equation has following form:

wherein a _{1, k}, a _{2, k}, θ _{1, k}, θ _{2, k}the real function that becomes slowly the time, a _{1, k}, a _2kfor the mode amplitude of system, its size has reflected the size of corresponding mode energy; Slowly the solvability that becomes equation of time group according to single order draws:

wherein, γ _k=θ _{1, k}-2 θ _{2, k}-ε σ T ₀;

Step 6: the solution of kinetic equations is carried out to Analysis of Internal Resonance;

Supposing that main system flexible mechanical arm does not exist mechanism's damping, there is not damping in vibration damping equipment yet, is drawn by formula (10), (11):

draw through Mathematical treatment:

(integration constant relevant to primary power) wherein

Obviously, this can illustrate the passage that has built energetic interaction between two rank mode corresponding in undamped situation, and υ >0 shows the two equal boundeds of rank mode energy and this those length that disappear;

Because not taking into account system damping, therefore only from mathematical meaning, the mutual permanent of this mode energy carries out; Consider that main system under truth has the viscous damping of structural damping and vibration damping equipment, system capacity is inevitable decay to disappearance under the effect of damping;

Step 7: kinetic equations solution is carried out to determination of stability;

For more scientific explanation mode energy process of decay gradually under the effect of damping, investigate in the stability of equinoctial point:

the Jacobian matrix of this system is so

wherein

characteristic of correspondence value is: (μ ₁, μ ₂, 0);

Due to mechanism's dampingratioζ of main system flexible mechanical arm ₁>0, the dampingratioζ of flexible joint viscous damping ₂>0, has μ ₁<0, μ ₂<0; The characteristic root of system is negative value, therefore mode amplitude a _{1, k}, a _2kthere is continous-stable; So far show, use the vibration damping equipment of absorbing fork and flexible joint structure, can between main system and vibration insulating system, build the exchange channels of mode energy, and the decay of the vibrational energy of main system flexible mechanical arm continous-stable under vibration damping equipment flexible joint viscous damping and flexible mechanical arm structural damping effect; Feasibility and the science of using the flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance have so far been described;

Step 8: to dynamic model Simulation Example, to choose theoretical optimal damping for vibration damping equipment;

Explain superiority of the present invention for further directly perceived, and be the optimal damping of vibration damping equipment Choice Theory, given following serial example comparative illustration; Setting example initial conditions is: main system flexible mechanical arm lengths is l=1.0m, and flexible mechanical arm cross section is the rectangle of high h=0.05m, wide b=0.003m; End load is m _b=0.5kg, aluminium matter flexible arm Young's modulus is 71Gpa, density 2710kg/m ³; Vibration insulating system rigidity absorbing fork mounting point r=0.5m, quality is m ₂=0.05kg, centroid position are r _c=0.4m;

First, limit the name motion of main system flexible mechanical arm, do not set up vibration damping equipment, the end initial disturbance amount of given flexible mechanical arm is 0.1m; If main system flexible mechanical arm non-structure damping, on main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping and built mutual passage between different modalities energy, and this those long and boundeds that therefore disappear alternately, this and theory analysis fit like a glove;

Secondly, the nominal characteristics of motion of giving main system flexible mechanical arm is:

wherein, 0≤t≤50s, the initial disturbance amount of given flexible mechanical arm 0.1m; On main system flexible mechanical arm, add vibration damping equipment, but do not set in the situation of vibration damping equipment damping, the mutual situation of mode energy is to give after the motion of flexible mechanical arm name, in undamped situation, between different modalities energy, build mutual passage, and this mutual this disappears, and that is grown and bounded, and this and this theory analysis fit like a glove;

The nominal characteristics of motion of then, giving main system flexible mechanical arm is:

wherein, 0≤t≤50s, the initial disturbance amount of given flexible mechanical arm 0.1m; Because the structural damping of ordinary circumstance flexible mechanical arm is very little, therefore setting the structural damping of main system flexible mechanical arm is 0, the viscous damping damping ratio of setting vibration damping equipment flexible joint is 0.001, in constructed energetic interaction passage, this disappears mode energy that is long mutual, and the vibrational energy of flexible mechanical arm progressively decays to disappearance under the effect of damping; Finally, based on set of equation (10), (11), (12), give each parameter values according to above-mentioned derivation, according to nominal laws of motion, repeatedly revise and the flexible joint damping value that replaces carries out numerical Analysis, select mode viscosity or flexible mechanical arm tip vibration to become in figure damping corresponding when vibration attenuation is the fastest as theoretical optimal damping;

Step 9: by temporary fixed to rigidity absorbing fork and main system flexible mechanical arm, guarantee that the two relative position and angle are initial zero-bit angle, application test mode measuring technology, obtains the natural frequency ω of the system including main system flexible mechanical arm and rigidity absorbing fork by hammering method ₁with damping ratio ξ ₁;

Step 10: according to the frequency match requirement of the non-linear internal resonance of structure 2:1, regulate the position of inertia adjusting bead in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance, make the natural frequency of vibration damping equipment meet ω ₂≈ 2 ω ₁; After torsion spring rigidity k is selected with device, show that according to the physical model of the physical pendulum with linear torsion restoring force vibration damping equipment frequency formula is

wherein J is the rotary inertia of absorbing fork around flexible joint;

Step 11: that in release steps nine, main system flexible mechanical arm and rigidity absorbing fork are implemented is temporary fixed, and regulate the size of rotation damper damping in a kind of flexible mechanical arm passive energy dissipation device based on 2:1 internal resonance of flexible joint, guarantee that rigidity absorbing fork is free state, make it passive swing under the kinematic parameter effect of main system flexible mechanical arm;

Near step 12: selected theoretical optimum damping parameter is carried out to actual test, and the theoretical parameter of selecting, damping is finely tuned, to obtain actual optimal parameter; So far just complete a kind of flexible mechanical arm passive energy dissipation method based on 2:1 internal resonance and the implementation and operation process of device.

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