CN114386277B - Method for predicting parabolic track after liquid drop impacts on super-hydrophobic cantilever beam and application - Google Patents
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Abstract
The invention discloses a method for predicting a parabolic track after a water drop impacts a super-hydrophobic cantilever beam and application thereof, wherein parameters of impact conditions such as the impact position P of the water drop, the impact speed U of the water drop and the contact time t are used c Suspension for fixing a movable partThe statistics of parameters such as the maximum amplitude A and the vibration frequency omega of the arm beam are combined with the derivation of an Euler-Bernoulli equation and related theoretical knowledge, the post-impact track of the orientation bounce of the water drop impacting the super-hydrophobic flexible cantilever beam can be predicted, the track calculated by the method has high matching degree with the actual track, and guidance can be provided for research in the fields of directional transmission of substances, information, energy and the like of the orientation bounce of the water drop.
Description
Technical Field
The invention belongs to the technical field of physics, and particularly relates to a method and application for predicting a parabolic track after a liquid drop impacts a super-hydrophobic cantilever beam, which can accurately transmit the liquid drop or a substance in such a way or have guiding significance on the transmission of plant leaf pathogens, spores and the like.
Background
The phenomenon that liquid drops impact the surface of a solid is widely existed in the nature and daily life and is widely applied in the field of industrial and agricultural production, such as ink-jet printing, liquid drop power generation, self-cleaning, anti-icing, pesticide spraying and the like, and the operation of behavior and track of the liquid drops after impact has important guiding significance for the transmission of substances, information and energy. In the past, mass, energy and information have been transferred by constructing a base structure, wetting, charged gradient surface so that droplets can be directionally bounced after impact. However, the impact motion mechanism of the liquid drop in nature may be different, such as the wings of the ubiquitous insects and the leaves of plants, which are fixed at one end and suspended and move with the wind, the phenomenon that the liquid drop impacts such an elastic cantilever beam is ubiquitous in nature, the liquid drop can be bent with large flexibility and has resilience when being impacted, and the liquid drop can vibrate after being impacted, which increases the complexity and difficulty of research, and the difficulty and the challenge for controlling the behavior of the liquid drop after being impacted are increased.
In view of the complexity of the phenomenon of liquid droplets striking the surface of a cantilever beam, a method of predicting the parabolic trajectory of a liquid droplet striking a superhydrophobic cantilever beam after impact is a continuing goal sought by scientists and researchers.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method for predicting a parabolic track after a liquid drop impacts a super-hydrophobic cantilever beam and application thereof.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention provides a method for predicting a parabolic track after a liquid drop impacts a super-hydrophobic cantilever beam, which mainly comprises the following steps:
1) According to the Euler-Bernoulli beam transverse free vibration equation:
wherein E is the elastic modulus of the cantilever beam, I is the moment of inertia, after a general solution is obtained, the boundary condition of the beam is utilized to solve, and the deflection of the impact point position at the moment t is obtained as follows:
δ(t)＝Asin(ωt+θ 0 )+δ 0 (2)
wherein theta is 0 Is the initial phase, δ 0 The deflection of the balance position, A is the maximum amplitude,
m 0 the mass of the liquid drop, g is the gravity acceleration, P is the position of the liquid drop impact, and alpha is the inclination angle of the substrate;
2) And the deflection is subjected to dimensionless analysis, so that the following can be obtained:
depending on P, when P is at 12mm, U>0.059m s -1 When the relative equilibrium position delta is ignored 0 A and theta 0 Equation (2-5) can be simplified as:
formula (4) is a law equation of solid-liquid coupling motion in the process of impacting the flexible cantilever beam by water drops;
3) And the water drops impact on the flexible super-hydrophobic cantilever beam arm, and when the departure point is taken as a zero point, the water drops conform to parabolic motion in physics under most conditions, and the equation of a parabolic equation is as follows:
4) The fixed end of the substrate is used as a zero point, the parabolic track is decomposed into uniform motion (2-9) in the horizontal direction and uniform deceleration motion (2-10) in the vertical direction, and the track equation is as follows:
x＝v 0 cosθ+x 0 (6)
from the equations (6) and (7), it can be seen that to predict the parabolic motion trajectory of the drop, 4 unknowns x are obtained 0 、y 0 、v 0 A value of θ, wherein x 0 And y 0 Is the coordinate of the drop as it leaves the substrate, v 0 Is the initial velocity of the water drop when leaving the cantilever beam, and theta is the initial angle from the horizontal when the water drop leaves the cantilever beam.
a unknown quantity x 0 And y 0 Theoretical derivation of (1)
First, δ (t) can be calculated by equation (4) c )，δ(t c ) To indicate the deflection of the cantilever beam when the water drop leaves. This calculation is made dependent on the law of impact velocity and position on contact time, cantilever vibration frequency and maximum amplitude. The water drop has a contact angle of 155 deg., the size of 15mm × 5mm × 0.13mm, and the bending rigidity of 1.12 × 10 - 7 N m 2 On nylon cloth, except the impact position, against the free end (P)>12 mm) and impact velocity U>1.5m s -1 The contact time of the liquid drop and the cantilever beam is independent of the impact speed; the vibration frequency of the cantilever beam is irrelevant to the impact speed and only relevant to the impact position; as the impact velocity increases, the maximum amplitude produced by the cantilever beam and the impact velocity exhibit a linearly increasing relationship.
The following relationships are available:
and the coordinate x of the drop as it leaves the substrate 0 And y 0 Is expressed as
x 0 ＝Scos(α+β) (9)
y 0 ＝Ssin(α+β) (10)
Thus, by the derivation process described above, x can be theoretically calculated 0 And y 0 . Alpha is the inclination angle of the substrate, L is the length of the substrate, P is the distance from the fixed end to the impact position, the starting time t =0 of solid-liquid contact, and the time t = t of the water drop leaving the substrate c S is the distance between the water drop and the fixed end of the substrate when the water drop leaves the substrate, and beta is the included angle between the beam and the initial position when the water drop leaves the substrate.
b unknown quantity v 0 Empirical derivation of sum θ
Initial velocity v of water drop when leaving super-hydrophobic nylon cantilever beam 0 And the initial angle theta is obtained by an empirical equation. The water drop makes parabolic motion after flying away from the super-hydrophobic cantilever beam, and shows that the two quantities and waterThe impact velocity U of the water drop is in a linear relation, and the initial velocity v of the water drop flying away from the cantilever beam under unknown impact velocity U can be deduced according to the two linear relations 0 And an initial angle theta.
An application of the method for predicting the parabolic track after the liquid drop impacts the super-hydrophobic cantilever beam can be applied to a method for developing a liquid drop slingshot small program, and the method comprises the following steps: python language is used to mainly convert parameters P, t c 、δ、S、α、v 0 And the data of theta are programmed by a prediction method, a database is established, and a small program of the liquid drop slingshot is designed.
As applied above, the software of the slingshot applet mainly includes two interfaces, one is an input position (x, y) and output trajectory equation and result interface, and specifically includes: target Coordinates (x, y), a trajectory equation, U and P, coordinate points are input in x and y of the interface Target Coordinates, the range of x is (-0.01, 0.06), the range of y is (0, 0.04), and multiple solutions can be or cannot be obtained in the range; and the other is a parabolic track interface, after inputting x and y coordinates, clicking Draw to obtain 2 required interfaces, wherein one interface gives a track equation passing through the target coordinate and corresponding experimental conditions U and P, the other interface shows the track passing through the target coordinate and a corresponding track equation, and clicking any point on the track can display the experimental parameters (x) corresponding to the track equation 0 ,y 0 ,v 0 ,θ)。
Compared with the prior art, the invention has the following technical effects:
the method for predicting the parabolic track after the liquid drop impacts the super-hydrophobic cantilever beam has high coincidence degree of the experimental track and good prediction accuracy.
Drawings
Fig. 1 is a graph comparing theory and experiment for dimensionless deflection δ/a = sin (ω t); in FIG. 1, the lines for blue and red are the theoretical values of delta/A for P at 10mm and 12mm, respectively, and the blue and red points are the experimental values;
FIG. 2 is a diagram showing the influence law of impact speed and position on parameters of solid-liquid coupling motion. (a) The impact law of impact speed and position on contact time; (b) The impact speed and position on the vibration frequency; (c) The impact speed and position on the maximum amplitude.
FIG. 3 is a schematic diagram of an initial position of a water drop leaving after impacting a flexible super-hydrophobic cantilever beam and a motion trail after the water drop impacting the flexible super-hydrophobic cantilever beam;
in fig. 3, L is the length of the cantilever beam, P is the impact position and establishes a coordinate system with the fixed end of the cantilever beam, when t =0, the impact initial position is t = t c When the water drops leave the cantilever beam, x 0 And y 0 Is the coordinate of the water drop when leaving the cantilever beam, S is the distance of the water drop from the fixed end of the substrate when leaving the cantilever beam, v 0 The initial speed of the water drop leaving the cantilever beam is theta, and the initial angle of the water drop leaving the cantilever beam and the horizontal direction is theta;
FIG. 4 is a diagram of a post-impact trajectory of a water drop impacting an 11mm flexible superhydrophobic cantilever beam; in FIG. 4, (a) a trajectory curve of water drops after impacting a flexible super-hydrophobic cantilever beam missile at different impact speeds, wherein the impact position is 11mm; (b) The initial angle theta of the water drops leaving the cantilever beam changes along with the U; (c) Initial velocity v of water drop when leaving cantilever beam 0 Along with the change of U, the size of the flexible cantilever beam is 15mm multiplied by 5mm multiplied by 0.13mm, and alpha is 15 degrees;
FIG. 5 is a graph comparing an experimental trajectory with a theoretical trajectory; u in FIGS. 5 a-c is 0.78ms -1 、1.38m s -1 And 1.75 ms -1 Comparing the experimental trajectory with the theoretical calculation trajectory; d-f for the first point x on a-c basis 0 Carrying out comparison after correction;
FIG. 6 is a graph comparing the experimental and theoretical calculated trajectories of a-c of FIG. 4;
FIG. 7 is a results interface; (a) trajectory equations and experimental condition interfaces; (b) a trajectory equation interface;
FIG. 8 shows the result of a drop slingshot targeting experiment; inputting target position coordinates in a droplet slingshot small program; (b) the applet gives a predicted trajectory through the target; (c) the applet gives the predicted experimental conditions (U, P); (d) site-specific targeting experiments.
Detailed Description
The following embodiments of the present invention will be described in detail with reference to the accompanying drawings, so that the implementation process of the present invention, which adopts technical means to solve the technical problems and achieve the technical effects, can be fully understood and implemented.
Example 1
The invention discloses a method for predicting a parabolic track after a liquid drop impacts a super-hydrophobic cantilever beam, which mainly comprises the following steps:
1) According to the Euler-Bernoulli beam transverse free vibration equation:
after a general solution is obtained, solving is carried out by utilizing boundary conditions of the beam such as the initial phase of the fixed end, the bending moment of the impact point and the like, and the deflection of the impact point at the moment t is obtained as follows:
δ(t)＝Asin(ωt+θ 0 )+δ 0 (2)
wherein theta is 0 Is the initial phase, δ 0 The deflection of the balance position is realized,
2) And the deflection is subjected to dimensionless analysis, so that the following can be obtained:
depending on P, e.g. P at 12mm, U>0.059m s -1 When this occurs, large deformation bending of the beam is caused, so that the relative equilibrium position δ can be ignored 0 A and theta 0 Equation (2-5) can be simplified as:
the formula (4) is a regular equation of solid-liquid coupling motion in the process of impacting the flexible cantilever beam by water drops; referring to fig. 1, theory and experiment of dimensionless deflection δ/a = sin (ω t) are compared, and lines of blue and red are theoretical values of δ/a at 10mm and 12mm of P, respectively, and blue and red points are experimental values.
FIG. 1 is a comparison of the delta/A (line) theoretical values with the experimental values (points) over time, with the U chosen to be 0.7 ms -1 、1.05m s -1 、1.40m s -1 And 1.70 ms -1 The experimental process shows that the theory is well consistent with the delta/A of the experiment, the solid-liquid coupling motion of the water drop and the flexible cantilever beam conforms to the sine function relation, and only the solid-liquid coupling motion of the water drop and the omega of the beam and the t of the water drop are consistent with the sine function relation c It is relevant.
FIG. 3 is a schematic diagram of the initial position of a water drop leaving after impacting a flexible super-hydrophobic cantilever beam and the motion trail after the water drop impacts. L is the length of the substrate, P is the impact position, a coordinate system is established by the fixed end of the cantilever beam, when t =0, the initial position is the impact position, and t = t c When the water drops leave the cantilever beam, x 0 And y 0 Is the coordinate of the water drop when leaving the cantilever beam, S is the distance of the water drop from the fixed end of the substrate when leaving the cantilever beam, v 0 The initial velocity of the water drop leaving the cantilever beam, and theta is the initial angle of the water drop leaving the cantilever beam.
3) When the water drops impact on the flexible super-hydrophobic cantilever beam and the departure point is taken as a zero point, the water drops conform to the parabolic motion in physics under most conditions, and the equation of the parabola is as follows:
4) As shown in fig. 2, the fixed end of the substrate is used as a zero point, the parabolic track is decomposed into a uniform motion (6) in the horizontal direction and a uniform deceleration motion (7) in the vertical direction, and the track equation is as follows:
x＝v 0 cosθ+x 0 (6)
to predict the parabolic trajectory of the drop, four unknowns x are obtained 0 ，y 0 ，v 0 θ, where x 0 And y 0 Is the distance between the water drops in the horizontal and vertical directions when they leave, v 0 The speed of the water drops leaving the cantilever beam, and theta is the angle between the water drops leaving the cantilever beam and the horizontal direction;
wherein, a, unknown quantity x 0 And y 0 Is calculated as shown in FIG. 3 by δ (t) c ) To express the deflection of the cantilever beam when the water drop leaves, the following relationship can be obtained:
and the coordinate x of the drop as it leaves the substrate 0 And y 0 Is expressed as
x 0 ＝Scos(α+β) (10)
y 0 ＝Ssin(α+β) (11)
Thus, by the derivation process described above, x can be theoretically calculated 0 And y 0 ；
b. Unknown quantity v 0 Calculation of sum θ
Referring to fig. 3, a diagram of a post-impact trajectory of a water droplet impacting an 11mm flexible superhydrophobic cantilever beam; (a) a post-impact trajectory diagram of water droplets; (b) The initial angle theta of the water drops leaving the cantilever beam changes along with the U; (c) Initial velocity v of water drop when leaving cantilever beam 0 The flexible cantilever beam has the size of 15mm multiplied by 5mm multiplied by 0.13mm and alpha is 15 degrees along with the change of U.
V when water drop leaves flexible super-hydrophobic cantilever beam 0 And theta, as shown in fig. 4 (a), the water droplet makes a parabolic motion after flying off the cantilever beam, and fig. 4 (b and c) show that v is 0 Theta and U are in linear relation, and v of the water drop flying off the cantilever beam under unknown U can be deduced according to the two linear relations 0 And theta.
The specific solving process is as follows:
p isAt 11mm, omega is 178.5rad s -1 When U is 0.78ms -1 Then, t can be obtained c 21.2ms, A =4.06U-0.04=3.13mm, and the result is expressed by equation (4) as the δ (t) at which the water droplet leaves the substrate c ) Is-1.88 mm, delta (t) c ) Above the initial position of the cantilever beam, and therefore negative, in the right triangle, P 2 +δ(t c ) 2 ＝S 2 S was 11.16mm, so β =9.7 °, x 0 ＝Scos(α+β)＝10.14mm，y 0 ＝Ssin(α+β)＝4.66mm；
When P is 11mm, U is 0.78-1.85ms -1 The range, the equation of the after-collision trajectory thereof is shown in fig. 2 (a). As shown in FIGS. 4 (b) and (c), v 0 And theta is linear with impact position when U is 0.78ms -1 When, V 0 =0.46 ms-1, θ =102.3 °. Therefore, the four unknowns x0, y0, v0, θ can be obtained and substituted into equations (2-9) and (2-10) to obtain the trajectory thereof, as shown in fig. 4 (a). The equivalent substitution is 1.38ms into U -1 、1.75ms -1 Fig. 4 (b) and 4 (c) can be obtained.
FIGS. 5 (a-c) are comparisons of experimental and theoretical calculated trajectories, and differences in x-direction were found with some deviation, summarizing theoretical x 0E -x 0P And the relation between U and the equation of the rule of thumb is found to be x 0E -x 0P ＝0.004U 2 -0.004U +0.00025 as shown in FIG. 6, wherein x 0E Is the first trace point of the experiment, x 0P Is the first trace point of theoretical calculation, so the first point x of theory and experiment 0 Making correction to make x be added to theoretical calculated x 0E -x 0P A trace diagram with good experimental and theoretical agreement can be obtained, as shown in fig. 5 (d-f).
On the basis of a large amount of experimental data, P and t are converted by Python language c 、δ、S、α、v 0 And theta and the like are programmed by a semi-theoretical semi-empirical prediction method, a database is established, and a small program of the liquid drop slingshot is designed.
As shown in fig. 7 (a), the software mainly includes two interfaces, one is an input position (x, y) and output trajectory equation and result interface, which specifically includes: target coordinates(x, y), trajectory equation, U, P. Coordinate points are input in x and y of the interface Target Coordinates, the range of x is (-0.01, 0.06), the range of y is (0, 0.04), and a plurality of solutions can be in the range or not. The second is a parabolic trajectory interface, for example, after inputting x and y coordinates, click Draw, and the interface of fig. 7 (a) and (b) can be obtained. The interface of fig. 7 (a) gives the trajectory equation through the target coordinates and its corresponding experimental conditions U and P. Fig. 6 (b) shows the trajectory passing through the target coordinates and the corresponding trajectory equation, and the experimental parameters (x) corresponding to the trajectory equation can be displayed by clicking any point on the trajectory 0 ,y 0 ,v 0 ,θ)。
As shown in fig. 7 (a), the parabolic trajectories of four coordinate points are obtained from the database when x is 0.025 and y is 0.025, and the following points should be followed when selecting the parabolic trajectories and experimental conditions: the impact position is as close as possible to 12mm and the water droplet should pass through the target during descent.
By adopting the program, the design of the experimental device model shown in fig. 8 (a-c) is carried out according to the parabolic track and the experimental conditions obtained by inputting the coordinates of the target point, the target position is designed firstly, the experimental verification is carried out by combining the track equation and the experimental conditions of the target, the process of shooting the liquid drop targeting is carried out from two visual angles, the experimental result is shown in fig. 7 (d-g), and the detailed prediction conditions and the experimental conditions are shown in table 2-1. The liquid drop slingshot software is combined with an experiment, so that three aspects of the conditions before the collision, the collision process and the motion track after the collision are communicated with each other, and the motion track of the water drops after the collision can be accurately controlled.
TABLE 2-1 comparison of program prediction with actual targeting experiment conditions
The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The above-described embodiments of the invention are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims, and not by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.
Claims (4)
1. A method for predicting a parabolic track after a liquid drop impacts a super-hydrophobic cantilever beam is characterized by comprising the following steps:
1) According to the Euler-Bernoulli beam transverse free vibration equation:
wherein E is the elastic modulus of the cantilever beam, I is the moment of inertia, S is the distance between the water drop and the fixed end of the substrate when the water drop leaves the substrate, after a general solution is obtained, the boundary condition of the beam is used for solving, and the deflection of the impact point position at the moment t is obtained as follows:
δ(t)＝Asin(ωt+θ 0 )+δ 0 (2)
wherein theta is 0 Is the initial phase, δ 0 The deflection of the balance position, A is the maximum amplitude, omega is the vibration frequency of the cantilever beam,
m 0 the mass of the liquid drop, g is the gravity acceleration, P is the distance from the fixed end of the substrate to the impact position, and alpha is the inclination angle of the substrate;
2) And the deflection is subjected to dimensionless analysis, so that the following can be obtained:
depending on the impact of the drop on the cantilever beam P, the impact velocity U is at 12mm>0.059ms -1 When the relative equilibrium position delta is ignored 0 A and theta 0 Equation (3) can be simplified as:
formula (4) is a law equation of solid-liquid coupling motion in the process of impacting the flexible cantilever beam by water drops;
3) And the water drops impact on the flexible super-hydrophobic cantilever beam arm, and when the departure point is taken as a zero point, the water drops conform to the parabolic motion in physics under most conditions, and the parabolic equation is as follows:
4) The fixed end of the substrate is used as a zero point, the parabolic track is decomposed into uniform motion (6) in the horizontal direction and uniform deceleration motion (7) in the vertical direction, and the track equation is as follows:
x＝v 0 cosθ+x 0 (6)
to predict the parabolic trajectory of the drop leaving, four unknowns x are obtained 0 ，y 0 ，v 0 θ, where x 0 And y 0 Is the position coordinates of the water drop in the horizontal and vertical directions when it leaves, v 0 The speed of the water drops leaving the cantilever beam, and theta is an included angle between the water drops leaving the cantilever beam and the horizontal direction;
wherein the unknown quantity x 0 And y 0 Is calculated by delta (t) c ) To represent the deflection of the cantilever beam when the water drops leave, can be calculated from equation (4) with the following relationship:
and the coordinate x of the drop as it leaves the substrate 0 And y 0 Is expressed as
x 0 ＝S cos(α+β) (10)
y 0 ＝S sin(α+β) (11)
Wherein alpha is the inclination angle of the substrate, and P is the inclination angle of the right triangle 2 +δ(t c ) 2 ＝S 2 Solid-liquid contact start time t =0, and water droplet leaving base time t = t c Beta is the included angle between the beam and the initial position when the water drops leave;
unknown quantity v 0 Calculation of sum θ
V when water drop leaves flexible super-hydrophobic cantilever beam 0 And theta, the water drop makes a parabolic motion after flying off the cantilever beam, v 0 Theta and U are in linear relation, and v of the water drop flying away from the cantilever beam under the unknown U can be deduced according to the two linear relations 0 And theta.
2. The method for predicting the parabolic trajectory after the liquid drop impacts the super-hydrophobic cantilever beam as claimed in claim 1, wherein the specific solving process of the step 4) is as follows:
when P is 11mm, omega is 178.5rads -1 When U is 0.78ms -1 Then, t can be obtained c 21.2ms, A =4.06U-0.04=3.13mm, and the result is expressed by equation (4) as the δ (t) at which the water droplet leaves the substrate c ) Is-1.88 mm, delta (t) c ) Above the initial position of the cantilever beam, and therefore negative, in the right triangle, P 2 +δ(t c ) 2 ＝S 2 S was 11.16mm, so β =9.7 °, x 0 ＝Scos(α+β)＝10.14mm，y 0 ＝Ssin(α+β)＝4.66mm；
When P is 11mm, U is between 0.78 and 1.85ms -1 Extent of post-impact trajectory v 0 And θ is linear with impact position when U is 0.78ms -1 When, V 0 ＝0.46ms -1 θ =102.3 °, four unknowns x 0 ，y 0 ，v 0 Theta can be obtained byThe trajectory can be obtained by entering equations (6) and (7), and similarly, the U is 1.38ms -1 、1.75ms -1 The trajectory can be obtained.
3. A method for developing a droplet slingshot applet by applying the method for predicting a parabolic trajectory after a droplet impacts a superhydrophobic cantilever beam of claim 1, comprising the steps of: python language is used to convert parameters P, t c 、δ、S、α、v 0 And the data of theta are programmed by a prediction method, a database is established, and a small program of the liquid drop slingshot is designed.
4. The method of claim 3, wherein the slingshot applet software includes two interfaces, one interface for input position (x, y) and result of output trajectory equations, including: target Coordinates (x, y), a trajectory equation, U and P, coordinate points are input in x and y of the interface Target Coordinates, the range of x is (-0.01, 0.06), the range of y is (0, 0.04), and multiple solutions can be or cannot be obtained in the range; and the other is a parabolic track interface, after inputting x and y coordinates, clicking Draw to obtain 2 required interfaces, wherein one interface gives a track equation passing through the target coordinate and corresponding experimental conditions U and P, the other interface shows the track passing through the target coordinate and a corresponding track equation, and clicking any point on the track can display the experimental parameters (x) corresponding to the track equation 0 ,y 0 ,v 0 ,θ)。
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