JPH10268947A - Dumping starting method used for time variable oscillation mechanism - Google Patents

Abstract

PROBLEM TO BE SOLVED: To carry out ideal operation for preventing a device to drive from generating oscillation by previously setting a condition preventing oscillation at the time of finishing acceleration and deceleration and obtaining a starting system satisfying this. SOLUTION: A motion equation corresponding to a time variable oscillation mechanism expressing a time variable parameter by a time polynomial at a device with the time variable oscillation mechanism is set up. Next an environmental condition avoiding the generation of oscillation at the time of finishing the acceleration or deceleration of an object to be controlled driving by this device is set. In the case of a crane, e.g. when time before starting is 0, the run-out angle and the run-out speed of a rope 15 and the position of a trolley 14 and the speed of the trolley 14 are all 0 but at the time of finishing acceleration, the position and the speed of the trolley 14 are prescribed values and the run-out angle and the run-out speed of the rope 15 are 0 in the condition. Next, the equation of a degree capable of deciding a coefficient by the motion equation and the expression of a boundary condition is obtained and these expressions are solved to obtain the starting system.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention
The present invention relates to a vibration damping activation method used for a device having a time-varying vibration mechanism that requires acceleration or deceleration during driving, such as a crane that is conveyed by being moved in the horizontal direction while being lowered, or a carriage in a cutting machine. is there.

[0002]

2. Description of the Related Art Hitherto, control methods for preventing generation of vibration in various devices have been researched and developed. Among such control methods, for example, Japanese Patent Application Laid-Open No. Sho 63-31
There is a method disclosed in Japanese Patent Application Laid-Open No. 4606 or JP-A-63-314607 for suppressing vibration generated at the tip of a robot arm.

[0003] In each of these methods, a signal of an acceleration sensor attached to the tip of a robot arm is fed back as a disturbance torque, and the acceleration applied to the robot arm is controlled in accordance with the fed back data to control the vibration. Is what you do.

[0004] Therefore, according to this method, the vibration control is started after the vibration actually occurs in the robot, and there is a problem that the generation of the vibration itself cannot be prevented. Further, according to such a method, a CPU capable of high-speed operation is required, and there is a problem that the apparatus is expensive and complicated.

Further, without performing feedback control using a sensor or the like as in the above-described conventional example, the vibration mechanism is set as a control target, and the vibration mechanism is controlled in a time substantially equal to its own cycle.
In addition, as a method of obtaining a drive function that can be started and stopped without generating residual vibration, the Transactions of the Japan Society of Mechanical Engineers (C
Eds., Vol. 51, No. 469 (Showa 60-9), "Optimal high-speed start control of vibration mechanism system".

This control method takes into account the dynamic characteristics of the motor and the control circuit when the vibration mechanism system is driven by a servomotor, and as shown in FIG.
₁ shows a vibration damping method when a load is applied to a driving point 1 and a control target point 3 having a mass m ₂ is activated via a spring 2 having a spring constant k.

[0007] Incidentally, FIG. 10, the driving force f, the driving force f to the load on the driving force f ₁ and the control target point 3 to load the driving point 1
It shows that you can think of it as _two .
9 and 10, x indicates the displacement of the driving point 1 and y indicates the displacement of the control target point 3.

In this case, first, the displacement y (t) of the control target point 3 where no residual vibration occurs is expressed by the following equation.

(Equation 8) And a conditional expression that the control target point 3 becomes the velocity V at time T without residual vibration from the stopped state.

(Equation 9) , And an optimal driving function is obtained from these equations.

Then, a program for driving the motor so that the driving point 1 accelerates so as to follow this driving function is created, and the motor is controlled in accordance with this program, thereby causing vibration at the control target point 3. Driving can be performed without being performed. As a result, the vibration itself of the device can be prevented from occurring without using a complicated and expensive device.

[0010]

However, in the above-mentioned method, a vibration-inhibiting effect is exerted on a so-called time-invariant vibration mechanism, such as a robot arm, whose resonance frequency does not change with the passage of time. However, when moving an object to a destination, such as a crane, the vibration of a time-varying vibration mechanism whose resonance frequency changes over time, such as the action of extending and lowering the object by expanding and contracting a rope, is added. It cannot be effective.

For this reason, in a device having a time-varying vibration mechanism, there is still a problem that residual vibration is generated at the end of acceleration / deceleration even if the above method is used. SUMMARY OF THE INVENTION The present invention has been made in view of such circumstances, and prevents a device having a time-varying vibration mechanism in which a resonance frequency changes with time from occurring, thereby shortening the time from start to stop of the device. It is an object of the present invention to provide a vibration-damping activation method used for a time-varying vibration mechanism that can perform the method.

[0012]

In order to achieve the above object, a vibration damping activation method used for a time-varying vibration mechanism according to the present invention firstly uses an apparatus having a time-varying vibration mechanism whose resonance frequency changes with time. A motion equation corresponding to a time-varying vibration mechanism in which a time-varying parameter is represented by a polynomial of time t is established.

In this case, based on a general equation of motion,
Consider the equation of motion specific to the device used.
A polynomial is created using the element that changes the resonance frequency over time as a time-varying parameter, and this is included in the equation of motion. As a result, this equation of motion corresponds to the time-varying vibration mechanism.

Next, a boundary condition that does not cause vibration at the end of acceleration or deceleration of the controlled object driven by this device is set. For example, if the equipment used is a crane, if the time before starting is 0, the swing angle of the rope, the swing angular velocity, the position of the trolley, and the speed of the trolley are all 0. At the end of acceleration, the position of the trolley and The conditions are such that the speed becomes a predetermined value, the swing angle of the rope, and the swing angular speed become zero.

Next, an equation of an order whose coefficient can be determined by the equation of motion and the equation of the boundary condition is obtained,
The starting method is obtained by solving these equations. In the case of a crane as described above, the swing angle of the rope and the position of the trolley are expressed as a function of time, and the starting method is determined by the obtained expression and the above equations of motion and boundary conditions. can do.

In this case, the order of the above-mentioned equation of motion is obtained from the relationship between the number of equations obtained by the above-described method and the number of unknowns, as described later. Then, the apparatus is driven so as to conform to the determined starting method, and acceleration / deceleration control of the control target is performed.

As a result, the vibration of the controlled object at the end of acceleration / deceleration can be prevented unless other unexpected external factors occur, and the object is stopped almost simultaneously with the end of acceleration / deceleration. I will be. As a result, the time from starting to stopping of the control target can be reduced. Further, other problems caused by the vibration of the controlled object, for example, all problems such as occurrence of danger can be solved.

The device usable in the present invention can be any device having a time-varying vibration mechanism whose resonance frequency changes with time. In particular, a crane, a plotter, It is effective for a carriage of a cutting machine or the like. In this case, if the device is a crane, the time-varying parameter is the length of the telescopic rope, and if the device is a carriage, the time-varying parameter is the spring constant of the belt.

In another device having a time-varying vibration mechanism, the present invention is applied to all devices having a time-varying vibration mechanism by using a time-varying parameter as a characteristic element of the device for changing the resonance frequency with time. Can be used for equipment. In such a case, the device may be driven in any form, such as motor drive or hydraulic drive. Next, the vibration suppression starting method according to the present invention will be described in detail with reference to the drawings.

[0020]

FIG. 1 shows a driving unit 10 of a crane apparatus for performing a vibration suppression starting method according to the present invention. That is, in FIG. 1, reference numeral 11 denotes a frame-shaped support portion, and reference numeral 12 denotes a rail movably mounted on the support portion 11. The rail 12 can be moved along the longitudinal direction of the support portion 11 by driving the belt 13 engaged at both ends.

Reference numeral 14 denotes a trolley movably provided along the longitudinal direction of the rail 12, and a rope 15 which can be hoisted and lowered is hung below the trolley, and a load 16 is hung on the lower end of the rope 15. Have been. Reference numeral 17 denotes a belt for moving the trolley 14. Belt 13,1
7 run on a pulley (not shown) by driving a motor (not shown).

FIG. 2 shows a configuration of a main part of the crane device including the driving unit 10. In FIG. 2, reference numeral 18 denotes a CPU, which sends data, trapezoid information, and the like necessary for the vibration suppression activation method input via an input device (not shown) to a memory 19 for storage. Also, this CPU 1
8 operates according to the program stored in the memory 19 and drives the drive unit 10 while performing arithmetic processing based on the various data and information.

Reference numeral 20 denotes a D / A converter for converting an output signal sent from the CPU 18 into an analog signal, and reference numeral 21 denotes a servo driver. The servo driver 21 is connected to a motor (not shown) of the drive unit 10 and the D / A converter 2
The drive unit 10 is driven by rotating the motor in accordance with a signal sent from 0.

Using such a crane device, the luggage 1
In order to transport the trolley 6 without vibrating it, first, the equation of motion of the crane and the boundary conditions under which the luggage 16 does not swing and the trolley 14 reaches a predetermined speed and position based on the acceleration start time and the acceleration end time are determined. Considering this equation of motion and boundary conditions, the starting method is determined.

That is, the rope 1 in the crane device
As shown in FIG. 3, the equation of motion involving a change in the length of 5 is 1 [m] for the length of the rope 15 and m
[Kg], the deflection angle of the rope 15 is θ [rad], and the coefficient of viscous friction (in this case, friction due to air resistance to the luggage 16 and energy loss due to bending of the rope 15 are main) is D [Ns / m]. ], The gravitational acceleration is g [m / s
² ], and assuming that the position (moving distance) of the trolley 14 is x [m],

(Equation 10) It is represented by

Since the length 1 of the rope 15 changes at a constant speed, the initial length of the rope 15 is set to l ₀ [m],
The change in the length of the rope 15 per unit time is represented by l ₁ [m /
s], and l = l ₀ +1 ₁ t, rewriting the above equation (10),

## EQU1 ## At this time, ml is applied to both sides so that the term of t does not remain in the denominator of the equation. This gives equation (1)
Is represented by an integer polynomial of t.

In the crane apparatus, the boundary condition that the luggage 16 does not swing at the start and end of acceleration and the trolley 14 reaches a predetermined speed and position is defined as T [s] at the end of acceleration. When the speed of the trolley 14 at the end of acceleration is V [m / s],

## EQU2 ##

The above equation (2) indicates that there is no vibration at the start of acceleration and at the time when the time T has elapsed from the start of acceleration. When the time t is 0, that is, before the start, the deflection angle of the rope 15 is , Swing angular velocity, trolley 14 position, trolley 1
4 are both 0, and when the time T has elapsed from the start of acceleration, the swing angle and the swing angular velocity of the rope 15 become 0.
Indicates that the position of the trolley 14 is 0.5 VT [m] and the speed of the trolley 14 is V [m / s]. Here, 0.5 VT [m] means that the trolley 14 moves at an average speed of 0.5 V [m / s] for T seconds.

Next, the equation of the starting method to be obtained is the above equation of motion (1) and equation (2) of the boundary condition.
It satisfies. Then, the deflection angle θ of the rope 15 and the position x of the trolley 14 are respectively represented by a polynomial of time t.

[Equation 11] Then, when k = 6, n = 8, that is, when θ (t) is a sixth-order equation and x (t) is an eighth-order equation, both sides of the equation (1) become a seventh-order equation. At this time, eight simultaneous equations are obtained by comparing the following coefficients in the equation (1). By solving 16 simultaneous equations combined with the eight equations (2) of the boundary condition, 16 simultaneous equations are obtained. The unknowns ai and bi can be determined.
However, if the 16 simultaneous equations are not linearly independent, a coefficient that cannot be uniquely determined occurs. In that case, the coefficient is set to 0.

To explain this in more detail, first, in equation (1), the order of t on the right side and the left side must be equal in order for the equal sign on both the left and right sides to always hold regardless of t. For this purpose, the relationship must be k = n-2. This is because the highest order on the left side is obtained by multiplying θ by t, and the highest order on the right side is obtained by differentiating x twice by t.

It is also assumed that the degrees of the left and right sides of equation (1) are balanced by, for example, a j-th order equation. In this case, the condition that the equal sign on both sides is always satisfied regardless of t is as follows:
The coefficients of the left and right orders must also be the same. Thus, these coefficients yield j + 1 equations. Also, a total of j + 9 equations are obtained from the j + 1 equations and the eight equations (2) of the boundary conditions.

On the other hand, the number of unknowns in the above equation (11) is the total number of k + 1 and n + 1, and the condition for determining the unknowns must be j + 9 = k + 1 + n + 1. Also, since j is the order in which both sides in equation (1) are balanced, the order k + 1, n− in both sides in equation (1)
It is equal to 1. Therefore, n = 8 and k = 6,
Equation (1) is a seventh-order equation.

As a result, in equation (11), n = 8 and k = 6
By substituting

(Equation 12) Can be derived, and this equation (12) is substituted into equation (1) to obtain the equation

(Equation 13) Is obtained. Each of the following coefficients on both sides of this equation (13) is compared, and by taking into account the boundary conditions, each of the above ai, b
i can be determined.

In this case, since the viscous friction coefficient D is so small as to be negligible, if D = 0, k = 4, n = 6
Can represent the above equation. The reason why the degree is reduced in this way is that all of the 16 simultaneous equations that are satisfied are not linearly independent, and a coefficient that cannot be uniquely determined is set to 0. Furthermore, the ratio t between the sample time t and the acceleration end time T
When / T is represented by τ and normalized, the expression indicating the acceleration of the trolley 14 is

[Equation 14] It becomes a simple expression represented by

The roll-up / transport experiment of the luggage 16 was performed by using the derived starting method (14), and the results shown in FIGS. 4 and 5 were obtained. In FIG. 4, the horizontal axis represents time s, and the vertical axis represents acceleration m / s ² . The dashed line a indicates a curve representing the relationship between acceleration and time due to step acceleration, the thin line b indicates sine wave acceleration, and the solid line c indicates vibration suppression acceleration based on the above equation (14).

In this experiment, the trolley 14 was accelerated for 3 seconds by each of the acceleration methods described above. The parameters other than T = 3 [s] and D = 0 [Ns / m] are the mass m of the luggage 16 = 30 [kg] and the initial length l ₀ of the rope 15 = 0.3 [m]. , Rope 15
Change in length per unit time l ₁ = 0.18 [m /
s], the speed V of the trolley 14 at the end of acceleration V = 0.75
[M / s].

FIG. 5 shows a trolley 1 under the above conditions.
4 shows the deflection of each rope 15 when the acceleration is performed as shown in FIG. 4, the horizontal axis represents time s, and the vertical axis represents the deflection angle of the rope 15. As can be seen from this figure, three seconds after the start of the acceleration, the curve a 'indicating the step acceleration and the curve b' indicating the sine wave acceleration
It can be seen that the curve c 'due to the vibration suppression acceleration has hardly any vibration, and is in a stopped state, while the vibration still occurs.

In this way, when the acceleration by the step acceleration or the sine wave acceleration is performed, the vibration of the luggage 16 continues for a while after the end of the acceleration, but the vibration control acceleration by the equation (14) is applied to the trolley 14. In this case, it can be seen that the rope 15 does not swing at the end of the acceleration and the load 16 hardly vibrates.

In the above example, the trolley 1 in the stopped state is shown.
The case where the trolley 14 is accelerated has been described, but this can also be applied to the case where the traveling trolley 14 is decelerated and the load 16 does not shake when the trolley 14 is stopped. In this case, the equation for the boundary condition is

## EQU3 ##

In this case, the time t is 0, that is, before starting, the swing angle and swing angular velocity of the rope 15 and the position of the trolley 14 are all 0, and the speed of the trolley 14 is V [m / s]. When the time T has elapsed since the start of acceleration, the swing angle and swing angular velocity of the rope 15 are 0, the speed of the trolley 14 is 0, and the position of the trolley 14 is 0.5 VT.
[M]. By performing the same procedure as above based on these eight equations (3) and (1),
A vibration suppression starting method at the time of deceleration is obtained.

FIG. 6 shows an example in which the vibration suppression starting method of the present invention is used for a plotter or a carriage of a cutting machine. In FIG. 6, reference numerals 17 and 18 denote a pair of pulleys which are incorporated at predetermined intervals in a rail portion (not shown) which is movable in a direction orthogonal to the longitudinal direction of the pulley. A belt 19 is wound around the periphery.

A carriage 20 for holding a pen or a cutter is fixed to a predetermined portion of the belt 19, and a driving motor 21 is connected to the pulley 18. Therefore, when the pulley 18 is rotated by the drive of the motor 21, the belt 19 travels, and the carriage 20 also moves right and left following the belt 19.

To accelerate or decelerate such a carriage 20 without vibrating, first, as in the above example,
Consider the boundary condition that the carriage 20 has no vibration and reaches a predetermined speed and position based on the equation of motion of the carriage 20 and the acceleration start time and the acceleration end time, and a starting method is determined from the equation of motion and the above boundary condition. I do.

In this case, the distance between the pulleys 17 and 18 is 1
[M], the distance from the pulley 18 on the motor 21 side to the origin (initial position before the carriage 20 is started) O is l ₀
[M], the distance from the origin to the carriage 20 is x
[M], the radius × rotation angle of the pulley 18 is y [m], the spring constant per unit length of the belt 19 is k [N], the mass of the carriage 20 is m [kg], and the coefficient of viscous friction (mainly, Friction between the carriage 20) and D [Ns /
When m], when the radius of the pulley 17, 18 is sufficiently small compared to l, 1 _0, the belt 1 in the carriage 20
The equation of motion involving a change in the spring constant of 9 is

## EQU4 ##

This is a general equation of motion in the carriage 20.

(Equation 15) Can be obtained from That is, in FIG. 6, the length of the belt that pulls the carriage 20 to the right with respect to the pulley 18 can be represented by l ₀ + y, and the length of the belt that pulls the carriage 20 to the left is l + (l−l ₀ −y). Can be represented.

Therefore, assuming that the combined spring constant of the belt 19 is K, the relationship between k and K in the equation (15) is K = k / (l ₀ + y) + k / (2l-l ₀ -y ), And from this equation,

(Equation 16) Is led. Further, assuming that the average acceleration of the circumference of the pulley 18 at the time of starting the carriage is α [m / s ² ], and y is very close to the movement of displacing at the average acceleration α, y
Equation (16) can be rewritten as equation (4) when approximation is made by: = １ ／ αt ² . Where α = target speed V / acceleration time T
In a relationship.

The boundary condition that the carriage 20 does not vibrate at the time of starting acceleration and ending the acceleration and reaches a predetermined speed and position is such that the acceleration ending time is T [s] and the carriage at the end of the acceleration is T [s]. 20 as V [m / s]),

## EQU5 ##

That is, in this case, the time t is 0, that is, before the start-up, the movement distance of the carriage 20
When the speed of the carriage 20, the radius of the pulley 18, the rotation angle, and the rotation speed are all 0 and the time T has elapsed since the start of acceleration, the moving distance of the carriage 20 and the pulley 18
Are both 0.5 VT [m] and the speed of the carriage 20 and the peripheral speed of the pulley 18 are both V [m
/ S].

From the above equation of motion (4), equation (5) of the boundary condition, and equation of the time function obtained by transforming equation (11) to correspond to this case, the vibration suppression start-up adapted to the carriage 20 by calculation is performed. An equation can be obtained, and by rotating the motor 21 in accordance with the equation, it is possible to start the carriage 20 without vibration when the acceleration is completed.

In the process of calculating the combined spring constant of the belt 19, it is approximated that y = １ ／ · αt ² , which is true for the equation of motion (16) and the equation (5) of the boundary condition. Is different from the solution. Therefore, y of the equation of the activation method derived by this calculation procedure is strictly different from the true solution, and x obtained from the equation of motion (16) does not strictly satisfy the equation (5) of the boundary condition. However, if y is close enough to the true solution, x will also be approximately the same as equation (5). As a result, a practically sufficient vibration damping effect can be obtained.

For example, the conditions of each parameter are set as follows: the spring constant k per unit length of the belt 19 is set to 320 [N],
When the mass m of the carriage 20 is 1 [kg], the pulleys 17 and 1
The distance 1 between 8 is 0.3 [m] and the acceleration time T is 0.2
[S], the target speed V 0.4 [m / s], and the distance l ₀ from the pulley 18 to the origin O and 0.04 [m], the radius × time function of the rotation angle y of the pulley 18 ,

[Equation 17] become that way.

The acceleration shown in FIG. 8 is obtained by differentiating the equation (17) twice to obtain the acceleration, and controlling the motor 21 to drive as shown in FIG. Obtained. According to FIG. 8, it can be seen that the carriage 20 moves at a constant speed after approximately 0.2 seconds have elapsed. If the carriage 20 is vibrating, the speed should be uneven and form a zigzag curve. In this case, since there is no vibration, 0.2
After the second, it is indicated by a horizontal straight line.

Conversely, when the traveling carriage 20 is decelerated and stopped, the equation of the boundary condition is as follows.

## EQU6 ## Also in this case, a start-up method corresponding to this case can be obtained in the same manner as described above using Expression (6) and the like, whereby vibration of the carriage 20 at the time of stoppage can be prevented.

As described above, according to the vibration suppression starting method according to the present invention, it is possible to prevent the crane device and the carriage 20 from generating vibrations on the controlled object after acceleration / deceleration. In addition to the above-described crane device and carriage 20, the present invention can be applied to a device having all the mechanisms corresponding to the equation of motion of Expression (7).

That is, in the equation (7), m is the mass of the control target point, x is the displacement (generalized coordinates) of the control target point, c
Represents a viscous friction coefficient, k represents a restoring force, and f represents a displacement (external force) of a driving point. An apparatus to which the present invention can be used may include any element corresponding to these elements, and at least one of the elements has a time-varying vibration mechanism that is a time-varying parameter whose resonance frequency changes with time. .

[0056]

As described above, in the vibration damping starting method according to the present invention, the device to be used has a time-varying vibration mechanism whose resonance frequency changes with time. And
A motion equation expressing a time-varying parameter in the time-varying vibration mechanism as a polynomial in time is established, and a boundary condition that does not cause vibration at the end of acceleration of a controlled object driven by this device is set.

Then, equations of an order whose coefficients can be determined by the equations of motion and the equations of the boundary conditions are obtained, and the apparatus is driven in accordance with the starting method obtained from these equations. . In this way, acceleration
At the end of deceleration, a condition under which vibration does not occur is set in advance, and a start-up method that satisfies this condition is determined. Therefore, the driving device operates ideally to prevent vibration.

Therefore, the controlled object stops almost at the same time as the end of acceleration / deceleration. As a result, the time from start to stop of the controlled object can be shortened, and the working efficiency is greatly improved. . This also eliminates other problems caused by the vibration of the control target. Further, since the present invention can be used for all devices having a time-varying vibration mechanism, the practical effect is extremely large.

[Brief description of the drawings]

FIG. 1 is a perspective view of a driving unit of a crane device used in an example of the present invention.

FIG. 2 is a configuration diagram of a main part of the crane device.

FIG. 3 is an explanatory diagram showing a relationship between movement of a trolley and swing of a load in the crane device.

FIG. 4 is a curve diagram showing the trolley acceleration when the acceleration method is changed.

FIG. 5 is a curve diagram showing a swing angle of a rope when an acceleration method is changed.

FIG. 6 is a front view of a carriage used in another example of the present invention.

FIG. 7 is a curve diagram showing a relationship between circumferential acceleration and time when the pulley rotates.

FIG. 8 is a curve diagram showing a relationship between a circumferential speed and a time when the pulley rotates.

FIG. 9 is an explanatory diagram showing a relationship between a driving force and a control target point according to a conventional example.

FIG. 10 is an explanatory diagram showing the driving force of FIG. 7 divided into a force applied to a driving point and a force applied to a control target point.

[Explanation of symbols]

 10 Drive unit 14 Trolley 15 Rope 16 Luggage 17, 18 Pulley 19 Belt 20 Carriage 21 ················································ Curve indicating vibration due to vibration-damping acceleration

Claims (4)

[Claims] 1. A vibration damping starter for preventing vibration of a control object in a device having a time-varying vibration mechanism whose resonance frequency changes with time after the start, and for moving the control object while accelerating and decelerating. A method, wherein a motion equation corresponding to the time-varying vibration mechanism, in which a time-varying parameter is expressed by a time polynomial, is established, and in this motion equation, boundary conditions under which vibration does not occur at the end of acceleration / deceleration of the controlled object. After determining the equations of the order whose coefficients can be determined by the above equations of motion and the above boundary conditions, the starting method is obtained from these equations, and the acceleration / deceleration control of the controlled object is performed by the starting method. A vibration damping activation method used for a time-varying vibration mechanism. 2. The apparatus is a crane, and the time-varying parameter is the length of the rope that expands and contracts by hoisting and lowering. M: mass of the load [kg] l ₀ : initial rope length [m] l ₁ : unit of the rope Change in length per time [m / s] θ: Rope deflection angle [rad] D: Viscous friction coefficient [Ns / m] g: Gravitational acceleration [m / s ² ] x: Traveling distance of trolley [m] When, the equation of motion is And T: time until the end of acceleration [s] V: speed of the trolley at the end of acceleration [m / s] Or the formula The vibration suppression starting method used for the time-varying vibration mechanism according to claim 1. 3. A device, such as a cutting machine or a plotter, for reciprocating a carriage left and right by driving a motor through a belt or a pulley, wherein a time-varying parameter is a spring constant of the belt, and m is a mass of the carriage. kg] l: distance between pulleys [m] l ₀ : distance from pulley on motor side to origin [m] x: distance from origin to carriage [m] y: radius of drive pulley x rotation angle [m] k : Belt spring constant at unit length [N] D: Viscous friction coefficient [Ns / m] α: Average acceleration of pulley circumference [m / s ² ] At the time of motion, the equation of motion is the distance x from the origin to the carriage. , Without losing the characteristic of radius × rotation angle y of the driving pulley, And the boundary condition is given by the following equation. Or the formula The vibration suppression starting method used for the time-varying vibration mechanism according to claim 1. 4. The equation of motion of the apparatus to be used is as follows: m: mass of control target point [kg] x: displacement of control target point [m] c: viscous friction coefficient [Nm / s] k: restoring force [N / m] f: Equation [7] for the displacement [N] of the driving point A vibration damping activation method used for the time-varying vibration mechanism according to claim 1, which corresponds to (1).