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ОглавлениеDIFFERENTIAL EQUATIONS OF MOTION
A mechanical system’s equation of motion, also called the law of motion, represents the system’s displacement as a function of running time. Analyzing the equation of motion provides comprehensive information needed for the development, design, and improvement of the system. The equation of motion is the solution of the differential equation of motion for the system performing a certain working process.
The accuracy of the analysis can be evaluated by appropriate experiments. Results that disagree with the experiments tell us the differential equation does not closely enough reflect the actual conditions of the process of motion. In such cases, we revise the differential equation. We may need to carry out a few iterations to achieve the acceptable accuracy; however, in many practical cases, our first iteration should be enough. The considerations presented below may help us develop these equations.
1.1The Left Side of Differential Equations of Motion (Sum of Resisting Loading Factors Equals Zero)
From Dynamics, we know that the differential equation of motion is a second order differential equation. As it turns out, a second order differential equation also describes the processes in electrical circuits. The structure of such an equation is predetermined by principles of mathematics without any regard to either the characteristics of motion of a mechanical system or the characteristics of processes in electrical circuits. An ordinary linear second order differential equation reads:
| (1.0) |
where x is the function, t is the argument, c1, c2, and c3 are constant coefficients, P is a constant value, and f(t) is a certain known function of t.
Let’s examine the left side of this equation. The first term is the product of multiplying a constant coefficient by the second derivative. The second term is the product of multiplying another constant coefficient by the first derivative. The third term is the product of multiplying one more constant coefficient by the function, and finally the last term is a constant value. This constant value can be considered a product of multiplying a constant coefficient by the function (or argument) to the zeroth power.
The right side of this equation may include certain known variable and constant values. All these terms must either have the same units or be dimensionless. The solution of equation (1.0) represents an expression describing the dependence between the function x and the argument t.
Now let’s consider equation (1.0) from the viewpoint of Dynamics. Displacement, velocity, and acceleration are the three basic parameters of motion of a mechanical system. All other parameters are derived from these three. Hence, the left side of a second order differential equation helps describe the motion of a system because it contains the same basic structural parameters: the second derivative (acceleration), the first derivative (velocity), the function (displacement), and a constant value. Newton’s Second Law states that a body’s motion is caused by a force. This Second Law is expressed by the following well known formula:
F0 = m0a0
where F0 is the force, m0 is the mass of the body, and a0 is the acceleration of the motion of the body (the indexes “0” are given in order to avoid possible confusion with similar parameters in the text).
The first term of equation (1.0) contains the second derivative, which is the acceleration. According to Newton’s Law, the coefficient c1 in the differential equation of motion should be replaced by the mass m. Thus, the first term of the differential equation of motion is actually a force; all other terms of this equation should have the same units and they should be forces. The product of multiplying the mass by the acceleration (second derivative) represents the force of inertia. Because the mass is a constant value and in general the second derivative (the acceleration) is a variable quantity, we conclude that the force of inertia depends on the acceleration. Similarly, by multiplying the second and third terms of equation (1.0) by certain specific coefficients, we obtain respectively a force that depends on the velocity and a force that depends on the displacement.
The force that depends on the velocity is actually the reaction of a fluid medium to a movable body that interacts with this medium. This reaction represents a resisting damping force; the coefficient at the first derivative (the velocity) is called the damping coefficient.
The damping coefficient depends on both the type and condition of the fluid and also on the shape and dimensions of the movable object. Special hydraulic links (dashpots) are used in some mechanical systems in order to absorb impulsive loading. These links exert damping forces and are characterized by damping coefficients. Very often the damping coefficient depends on the velocity of the movable body.
When this coefficient does not depend on the velocity, or the dependence is negligible, the damping coefficient is considered to have a constant value. In this case, the differential equation of motion is linear — assuming, of course, that all other components of the equation are linear. If instead this coefficient has a variable value, the equation becomes non-linear.
There are no readily available formulas to calculate the damping coefficient. For each case, the characteristics and the value of the damping coefficient should be determined on the basis of experimental data. Note that in some cases a damping force becomes a part of the external loading factors (see Chapter 4).
The forces that depend on the function (the displacement) are exerted by the elastic media in the process of interacting with a movable object. By its nature, this force is the reaction of the medium to its deformation by a movable body. This force represents a resisting force and is called the stiffness force. The function’s coefficient is the stiffness coefficient; it depends on the type and condition of the elastic medium, the shape and dimensions of the body, and the peculiarities of the deformation.
For some elastic media, the stiffness coefficient depends on displacement of the movable object (deformer). If this dependence is negligible, the stiffness coefficient is considered to have a constant value. The corresponding differential equation of motion is linear. If, however, this dependence is significant, the stiffness coefficient is characterized by a variable value. The corresponding equation of motion is then non-linear.
Mechanical engineering systems often include elastic links in the shape of springs. The stiffness coefficient for the springs can be calculated using readily available formulas. Sometimes this coefficient is called the spring constant. For deformation of elastic media, there are no readily available formulas to calculate the stiffness coefficients; appropriate data is needed to determine the values and characteristics of these coefficients. In some cases, the stiffness force is considered an external active force (see Chapter 4).
The fourth term of equation (1.0) has a constant value. In the differential equation of motion, this value may be represented by certain constant resisting forces such as the force exerted during the deformation of a plastic medium, the dry friction force, or the force of gravity in case of an upward motion.
Consider the right side of equation (1.0) with respect to the differential equation of motion. This part may comprise a force that is a certain known function of time, velocity, or displacement — or a sum of all of them, including a constant force. In a very specific case (Chapter 4), the right side of the differential equation of motion contains a force that depends on acceleration. All these considerations let us conclude that the structure of a differential equation of motion is determined by equation (1.0).
With respect to Dynamics, the terms in the left side of equation (1.0) represent forces that resist the motion of a mechanical system, whereas the right side of the equation includes terms that cause the motion. The forces that resist the motion characterize the reaction of the system to its motion. Thus, the forces that have a reactive nature are the resisting forces. The forces in the right side of the differential equation of motion are applied to the system; they may be called the external forces or the active forces.
More considerations associated with loading factors (forces and moments) and with the structure of the differential equations of motion are discussed below.
Like any equation, a differential equation of motion consists of two equal parts. The components of the equation represent forces or moments applied to the mechanical system. Forces are used in equations of a particle’s rectilinear motion or a rigid body’s rectilinear translation, whereas moments are used for equations to describe the rotation of a body around its axis. The forces or moments can be classified into two groups:
1.Active forces and moments causing motion
2.Reactive forces and moments resisting the motion
It is justifiable to place all the resisting loading factors into the left side of the differential equation, and the active loading factors into the right side.
Based on all these considerations, it is possible to assemble the most general left side for an actual mechanical engineering system’s differential equation of motion. Let’s start with a system in the rectilinear motion. Assume that no external forces are applied to the system, which is moving in a horizontal direction. In this case, the right side of the equation equals zero. In the absence of external forces, the motion occurs due to the energy that the system possesses (kinetic, potential, or both). The initial conditions of motion contain information regarding the energy the system possesses at the beginning of the analysis. Based on these considerations and applying equation (1.0), we may write the differential equation of motion of a mechanical system, as seen in equation (1.1):
| (1.1) |
where x is the displacement (the function), t is the running time (the argument), m is the mass of the system, C is the damping coefficient, K is the stiffness coefficient, P is the constant resisting force of any nature except friction, and F is the constant dry friction force. The force P could represent the force of the system’s gravity in cases of upward motion, the resisting force exerted during a medium’s plastic deformation, or other. The gravity force becomes an active force when it acts in the direction of motion.
The first term of equation (1.1) is the force of inertia; it represents the product of multiplying mass m by acceleration The second term of this equation is the resisting damping force; it depends on the system’s velocity. This force equals the product of multiplying the damping coefficient C by the velocity The nature of the damping force is related to fluid (liquid and air) resistance, which represents the reaction of a viscous medium during its interaction with a movable object.
The third term of this equation is the stiffness resisting force that depends on the system’s displacement. This force equals the product of multiplying the stiffness coefficient by the displacement x. The nature of this force is the reaction of an elastic medium to its deformation by a movable body. By its nature, the resisting constant force associated with a medium’s plastic deformation and the dry friction constant force are also reactive forces. More details about the force of inertia, the damping and stiffness forces, and their coefficients are presented below.
From a mathematical point of view, assume that the left side of equation (1.1) could include a resisting force dependant on time. As a matter of fact, all forces in the differential equation of motion are functions of time, including the constant forces that are actually coefficients at the time that is to the zeroth power.
Let’s analyze a hypothetical case where a resisting force depends directly on time. Imagine a device programmed to increase a pressure force dependant on time. This device, which is attached to a movable system, applies a resisting force that is increasing in time. But this force is not reactive by nature. Instead, it is an external or active force; it should be included in the right side of the differential equation. Thus, a force that depends directly on time should not be included in the left side of a differential equation of motion. All this makes it clear that equation (1.1) represents the most general left side of a differential equation of motion that includes all possible resisting forces; it describes the rectilinear motion of a hypothetical mechanical system in the absence of external forces.
In order to solve the differential equation of motion for this system, first determine the initial conditions of motion:
| For | (1.2) |
where s0 and v0 are the initial displacement and velocity respectively.
According to these initial conditions, the system possesses the potential energy of the deformed medium — the deformation is proportional to the initial displacement s0. The system also possesses the kinetic energy that is proportional to the initial velocity v0. In this case, the system’s motion is caused by the combined action of potential and kinetic energy.
Equation (1.1) describes the motion in cases where the system possesses just the potential energy (for or just the kinetic energy (for For each of these cases, however, the solutions of the differential equation (1.1) will be different. (This will be demonstrated in Chapter 3.) When both the initial displacement and the initial velocity equal zero, there will be no motion.
The left side of equation (1.1) has five components or, in the general case of the differential equation of motion, these five resisting forces:
1.Force of inertia
2.Damping force
3.Stiffness force Kx
4.Constant force P
5.Dry friction force F
These forces represent the reaction of all possible factors to the system’s motion. Their characteristics depend on the structure of the mechanical system and on the nature of the environment in which the system is moving.
The structure of the left side of the linear differential equation of motion (1.1) corresponds to the structure of the second order linear differential equation. Not all resisting forces are present in each actual problem; in reality, the left side of the equation may include any number of components from one to five. However, the force of inertia associated with the second order derivative is always present in the differential equation of motion. Thus, the shortest and simplest expression of a differential equation of motion is that the force of inertia equals zero, and the motion is caused by the initial velocity. In this case, the body moves by inertia with a constant velocity (the acceleration equals zero). This case is discussed further in Chapter 3.
Now let’s compose a similar differential equation of motion for a body in rotation around its horizontal axis. We apply the same procedures as before:
| (1.1a) |
The initial conditions are:
| For | (1.2a) |
where J is the moment of inertia of the system, C and K are respectively the damping and stiffness coefficients in rotation, MP is a resisting constant moment, MF is a constant moment associated with dry friction, θ is the angular displacement, t is the running time, and Ω0 and θ0 are the initial angular velocity and initial angular displacement respectively. As indicated above, equations (1.1) and (1.1a) are absolutely similar as are expressions (1.2) and (1.2a). The solutions of these equations with their initial conditions are also absolutely similar.
The left side of the differential equation of motion comprises the resisting forces or moments whereas the right side consists of active or external forces or moments. These two parts are equal to each other.
Now let’s compose the differential equation of the rectilinear motion for a body that is subjected to all possible resisting and active forces. The left side of this equation is the same as in equation (1.1). Thus, we must compose the right side so that it includes all possible active or external forces. According to the structure of the second order differential equation, its components could represent
1.Constant values
2.Variables depending on the argument
3.Variables depending on the function
4.Variables depending on the derivatives of the function
In other words, the active forces include constant and variable forces. The variable forces may depend on the time, the displacement, the velocity, and the acceleration.
As an example of where the active force depends on the acceleration, consider the motion of a rear-wheel or front-wheel drive automobile. The automobile’s force of inertia is responsible for the redistribution of its force of gravity between the rear and front axels. This redistribution causes a certain increase of the rear axle’s loading, and a corresponding decrease of loading on the front axle.
For a rear-wheel drive automobile, the increase of loading on the rear axle leads to the increase of the friction force between the rear wheels and the ground. This results in the increase of the active force that causes the motion of the automobile. For a front-wheel drive automobile, the force of inertia plays an opposite role — the active force decreases. An example is provided in Chapter 4.
In this example, the value of the force of inertia from the right side of the automobile’s differential equation of motion represents just a small fraction of the total force of inertia from the left side of the equation. In the case of a four-wheel drive automobile, the redistribution of the weight between the axles does not influence the total resultant active force.
It is problematic to find other examples where an active force depends on the acceleration. Furthermore, in the rotational motion, it would probably be impossible to find a situation where the angular acceleration influences the applied active moment. Therefore, it is justifiable not to include in the right side of the equation a force that depends on acceleration.
Figure 1.1 shows a general case of a variable active force that is dependant on time. This graph approximates a random variable force. It represents the action of a random force by using a sinusoidal curve that has a maximum force Rmax, and a minimum force Rmin.
The mean force R divides the graph into two equal parts; it is calculated from the following formula:
| (1.3) |
Figure 1.1 The variable random active force.
The amplitude A of the sinusoidal force can be determined from equation (1.4):
| (1.4) |
The frequency ω1 of the sinusoidal force is:
| (1.5) |
where T is the period of fluctuation of the sinusoidal force.
A variable random force can be replaced by a superposition of a constant force R and a sinusoidal force A sin ω1t. The sinusoidal force is a harmonic function of time.
Active forces can be expressed as linear or non-linear functions of time. In most practical cases, these active forces are considered as linear functions of time. Active forces can also be presented as functions that are dependent on displacement or velocity. Based on all these considerations, we assemble equation (1.6), the most general differential equation of motion of a mechanical system that moves in the horizontal direction:
| (1.6) |
where R is a constant active force; A is the amplitude of a sinusoidal force; ω1 is the frequency of the sinusoidal force; Q is the constant value of an active force at the beginning of the motion; τ is the time that the motion can last; and μ is a constant dimensionless coefficient, for τ > 0, μ > 0, and t ≤ τ; and finally C1 and K1 are respectively the damping and stiffness coefficients of the active damping and stiffness forces.
The initial conditions for this case are arbitrary, which means that the parameters of motion at the beginning of the process could be equal to zero or could be different from zero. In general, the initial conditions of motion for equation (1.6) may be also presented by the expression (1.2).
The left side of equations (1.1) and (1.6) are identical, as expected. Thus, equation (1.6) represents the structure of the most general differential equation of a rigid body’s rectilinear motion. The complexity of the differential equations of motion in the actual situations is significantly lower than in equation (1.6).
The characteristics of a differential equation of motion’s forces determine the linearity or non-linearity of the equation.
It is very important for the overall analysis to clarify the peculiarities of the characteristics of the forces that should be included in the equation. There are two main concerns in determining the characteristics of the forces.
1.The first is associated with obtaining the most credible data about the particular forces for the particular real conditions.
2.The second is related to interpreting these characteristics in terms of their linearity or non-linearity.
Adequate information is obtainable from a comprehensive search of the relevant sources. The decision to categorize these characteristics as linear or non-linear depends on both the actual experimental data and the level of compromise that is justifiable in each particular case. The following analysis of these forces addresses these two concerns.
These considerations are all related to those differential equations of motion in the horizontal direction that are not affected by vertical forces such as gravity. In cases of vertical motion or an incline, the force of gravity plays a role. If a mechanical system is moving up vertically or on an incline, the force of gravity or its component represents a resistance and should be included in the left side of the differential equation of motion. When the system moves down, these forces represent active or external forces and should be included in the right side of the equation. According to equation (1.6), the right side of the differential equation of motion generally includes five active forces:
1.Constant force R
2.Sinusoidal force A sin ω1t
3.Force depending on time
4.Force depending on velocity
5.Force depending on displacement K1x
Detailed descriptions of the characteristic of forces included in the differential equations of motion (1.1) and (1.6) are presented in Chapter 2.
Now let us compose a differential equation of motion for a system that rotates around its horizontal axis and is subjected to all possible resisting and active moments. This equation is completely similar to equation (1.6) and is presented in the following shape:
| (1.6a) |
where M is a constant active moment; MA is the amplitude of a sinusoidal active moment; ω1 is the frequency of the sinusoidal moment; MQ is the constant value of an active moment at the beginning of the motion; and C1 and K1 are respectively the damping and stiffness coefficients; μ is a constant dimensionless coefficient, and τ > 0, μ > 0, and t ≤ τ.
The initial conditions of motion for equation (1.6a) are presented in expression (1.2a).
Because of the strong similarity between the characteristics of forces and moments, as well as between the appropriate differential equations of motion, we will consider only forces in this text while keeping in mind that all considerations related to forces are also applicable to moments.
It is important to realize that all components in the differential equation of motion should be functions of time. (For constant terms, time is to the zeroth power.) In certain real situations, the movable mechanical systems are subjected to forces that may depend on temperature. Changes of fluid temperature will cause changes of the damping force. The influence of temperature on forces cannot be directly incorporated into the differential equations of motion. If we could also incorporate forces that depend on temperature into these equations, we would obtain equations with two independent variables: time and temperature. There are no differential equations of motion with multiple arguments; there can be just one independent variable — and it must be the running time. Therefore, any forces that depend on temperature or other factors, except time, cannot be included in any differential equations of motion.
However, there is a way to account for the change of the forces due to temperature change. Consider the change of fluid viscosity due to temperature. As mentioned above, this change will cause the change of the damping force. In other words, temperature change will result in the change of the damping coefficient. The differential equation of motion and its solution are the same for different values of damping coefficients. These values should be determined for different temperatures and then used during the quantitative analysis of the parameters of motion. The results of this analysis will reveal the influence of temperature on the process of the system’s motion.
