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    Differential equation for dynamical systems

    Hi
    My question is elementary. I have seen that most of the solutions to DE are either exponential or sinusoidal. I do not really understand the need to express any dynamical event with DE. Why not express with the solutions directly. For example RC circuit, output voltage can be always represented as V(1-e(-t/tau)) and we can get any desired voltage with respect to time using this equation. Why should I first represent this my DE and then solve it to make generalization? Does the above equation not suffice to describe the model of the system?

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    Re: Differential equation for dynamical systems

    The simple exponential function is the solution of a first order RC circuit. If the solution is known a priori, you don't need to solve a differential equation, except as an exercise. But how do you determine the solution of a second or higher order RC circuit? How do you guess it without calculation?



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    Re: Differential equation for dynamical systems

    Well though experimental approach , i would curve fit and find the function . No need for DE



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    Re: Differential equation for dynamical systems

    My question is elementary
    Need to take a deep breadth.

    First you need to setup the equations that describe the physical phenomena. We have the famous Ohm's law that states a linear relation between current and voltage. But if you have a capacitor or an inductor in the same circuit, you need to consider instantaneous currents and voltages.

    For solving the voltages and currents in mesh containing only resistors, Ohm's law is enough and for simplicity you use Kirchoff's laws. But you will not need to use any differential equations.

    Now if you have a capacitor in the same circuit, the current is no more proportional to the voltage, it is directly proportional to dV/dt. In the same way, the voltage across an inductor is not proportional to the current, but to di/dt. This is where the differential equation kicks in.

    Solving a general differential equation is not easy: it is far from trivial except for simple cases. Simple cases mean dx/dt=x will give an exponential solution; d2x/dt2=x will give a sinusoidual solution. Any equation that cannot be converted to this form will have a different solution.

    Also remember that sin(x) and cos(x) form a basis set; many solutions can be expressed in terms of a series solution.

    Try the simple case of LCR and figure out different boundary conditions (filters for example).


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    Re: Differential equation for dynamical systems

    Well though experimental approach , i would curve fit and find the function . No need for DE
    O.K., what's e.g. the function prototype you'll fit to get the step response of this simple RC circuit. Hint: It's not V(1-e(-t/tau))

    Click image for larger version. 

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    Re: Differential equation for dynamical systems

    High school maths!! Ok. As I understand capacitor and inductor parameter such as I or V changes instantaneously, so we have no other option but to model the component with differential equation. So if we have non linear elements whose output change every instant, DE is in place. But I could not understand that why to model radioactive decay with DE? Are DE general representation of any events that change over time? I have serious difficulty in understanding difference between a function such as y=x^2 and a derivative because function also changes over time. So my question is why to care about the slope ,when I have a function that defines the trajectory?



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    Re: Differential equation for dynamical systems

    Quote Originally Posted by garimella View Post
    Hi
    My question is elementary. I have seen that most of the solutions to DE are either exponential or sinusoidal. I do not really understand the need to express any dynamical event with DE. Why not express with the solutions directly. For example RC circuit, output voltage can be always represented as V(1-e(-t/tau)) and we can get any desired voltage with respect to time using this equation. Why should I first represent this my DE and then solve it to make generalization? Does the above equation not suffice to describe the model of the system?
    A lot of things come into play in systems that are modelled with differential equations. Let's try and discuss it a little bit. Let's discuss the initial and final values and then the exponential response.

    Generally, expression of the form V(1-e^(-t/tau)) is a known solution of a system with one energy reservoir, one restriction and one 'empty' vessel if the empty vessel is to be filled with energy transferred from the reservoir through the restriction. The expression is a bit different if the vessel is not empty initially.

    Realize that V(1-e^(-t/tau)) is only valid if the vessel is much smaller than the reservoir else the 'V_reservoir' in the reservoir is going to decrease exponentially, significantly as the 'V_vessel' in the vessel increases exponentially. Recall that the transfer of energy stops when V_vessel = V_reservoir. This implies that if the size of the vessel is significant relative to the size of the reservoir, then when V_vessel = V_reservoir, their value would be different from the initial value of V_reservoir. Let's take a typical illustration with two cases.

    CASE 1: a constant voltage source and a capacitor
    If you are charging a capacitor (with V_cap_initial = 0V) with a constant voltage source (with V_source_initial) through a resistor, when V_cap_final = V_source_final, then then the charging ends. V_source_final = V_source_initial because the charges required to fill the capacitor is insignificant relative to the charges that the source has in stock.

    CASE 2: two capacitors.
    Let's take two capacitors of the same size for simplicity. If you are charging a capacitor (with V_cap_vessel_initial = 0V) with another capacitor of the same size (with V_cap_reservoir_initial) then V_cap_vessel_final = V_cap_reservoir_final = V_cap_reservoir_initial/2 ideally (assuming no losses). As energy is being transferred exponentially to the vessel capacitor, Energy level in the reservoir capacitor drops exponentially too.

    THE EXPONENTIAL RESPONSE
    Every such system has a exponential response (sine, cosine,.... can be expressed as exponential functions -- or better put, they are exponential functions). But why exponential? It's exponential because as you begin to fill the vessel with energy, that energy that's now in the vessel begins to oppose further transfer of energy into the vessel. Energy still gets transferred as long as the energy in the vessel is not equal to that in the reservoir. The opposition gets stronger and stronger with time because more and more energy gets deposited in the vessel with time and so energy transfer becomes more and more difficult with time resulting in a lower and lower transfer rate as time passes. This happens until the energy in the vessel equals that in the reservoir and the opposition is now enough to stop any further transfer. Also notice that without the restriction, the transfer would be instantaneous.

    Here, let's take a system with one reservoir, two restrictions and two vessels such that the energy transfer is as shown:

    reservoir -> restriction1 -> vessel1 -> restriction2 -> vessel2

    Energy from the reservoir will be transferred to vessel1 exponentially. As vessel1 is being filled up, part of the energy collected in vessel1 will be transferred exponentially to vessel2. Imagine what is happening in vessel1. It gets both exponential supply of energy and loss of energy at the same time. Also imagine what is happening vessel2. It gets an exponential supply from vessel1 whose energy level is not constant but is trying to rise exponentially (tying because it is not actually rising exponentially as part of it is collected into vessel2). This is a d2y/dx2 behaviour. As can be envisaged, analysing this complex behaviour would be a tedious task. The system described here is the system FvM showed in Post #5. If the components in the system were to be interconnected in a different way, its behaviour would be different.

    It is noteworthy that in any interconnected systems, the transfer of energy will end simultaneously in all vessels and that the more the number of components (restrictions and storage), the longer it will take to achieve this equilibrium for the same component sizes and the complex the behaviour of the system would be and the more tedious it would be to analyze it.

    Differential equation is a general, compact and systematic way to model a system. Solving the equation is results in the solution (with all necessary info contained) without having to bother about how the system behaves as energy is being transferred.

    It is the solution to differential equation that is useful. Even when a differential equation is described within a system, it is described in such a way that the solutions are extracted for use. If the behaviour of nown for a system, then it's okay to go ahead and use the solutions, else it is much easier to develop the differential equation than to think about the behaviour of the system directly.
    -------------
    --Akanimo.



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    Re: Differential equation for dynamical systems

    But I could not understand that why to model radioactive decay with DE?
    It is possible to give a long lecture on this but I shall try to be brief.

    The radioactive decay is a first order process. It is just like a chemical reaction in which the rate of reaction depends only on the concentration of one component. In equation terms, dc/dt=-kc. (the negative sign simply says that the rate decreases with time)

    The mechanism for the radioactive decay is statistical in nature. A nucleus that has not decayed can decay anytime from now to infinity (Poisson distribution applies). Hence the number of disintegration in a given time interval is directly proportional to the number of active nuclei at the given time.

    For the chemical reaction similar considerations apply. (see unimolecular and first order reactions for details).

    In dynamics, we have forces that cause acceleration (Newton's law); F=k dv/dt but it is best expressed in Lagrangian or Hamiltonian form. It becomes important when we consider canonical variables x and p (they are differential equations).

    Differential equations will come in play when we have a rate process that depends on the principal variables. (Heat conduction and diffusion processes are model examples)

    In brief, diff eqns are ubiquitous in real life.



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    Re: Differential equation for dynamical systems

    So my question is why to care about the slope ,when I have a function that defines the trajectory?
    This is important and relevant when you set up the description of the problem. For example, what is the shape of a perfectly flexible (non-extensible) rope fixed between two points?

    Another example: total energy is conserved but is composed of kinetic energy and potential energy. Kinetic energy is (1/2)m*v**2 and potential energy is (1/2)k*x**2. Hence E=(1/2)*m*(dx/dt)**2+(1/2)*k*x*x

    Hence E is not only a function of x but also of dx/dt; thus x, dx/dt, d2x/dt2 (acceleration) are independent (point wise) and completely describes the function x(t)

    In dynamical systems, we talk of x (position) and m(dx/dt) (momentum) as the two mutually conjugate variables. We need to consider them as independent quantities (position and velocity at a instant can be independent)



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    Re: Differential equation for dynamical systems

    Quote Originally Posted by garimella View Post
    Hi
    My question is elementary. I have seen that most of the solutions to DE are either exponential or sinusoidal. I do not really understand the need to express any dynamical event with DE. Why not express with the solutions directly. For example RC circuit, output voltage can be always represented as V(1-e(-t/tau)) and we can get any desired voltage with respect to time using this equation. Why should I first represent this my DE and then solve it to make generalization? Does the above equation not suffice to describe the model of the system?
    That equation you quoted is only good for a step function of a constant voltage. If the voltage is a ramp or some other time varying function, you will have to resort to differential methods to get the correct solution. If you have more than one storage element in your circuit, like a coil, then you cannot display the result by a simple expression like you put forth.

    Ratch
    Hopelessly Pedantic



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