# Differential Equation Of Simple Harmonic Motion And Its Solution

In this lecture we introduce the idea of using a power series to approximate the solution to a differential equation by showing a Taylor series approximation is very possible, though computationally intensive. Deriving the Equation for Simple Harmonic Motion. Simple Harmonic Motion • Differential equation: + =0 • Solutions can be written in various ways: = cos + = sin +˘ cos (and many others…) • Two constants of integration need to be determined from initial conditions or other information. 1 Stochastic differential equations. If the speed of the block is 40 m/s when the displacement from equilibrium is 3 m, what is the amplitude of the oscillations? Answer: 5m • A simple pendulum has a length L. Some multiple of the source gives me the solution to that differential equation. INTRODUCTION TO MECHANICAL VIBRATIONS 3 Mass m Spring k Damper c Excitation force F(t) 0 Displacement x Static equilibrium position Fig. A singular solution is a solution that can't be derived from the general solution. Notice that we're now back in configuration space!. The basic idea is that simple harmonic motion follows an equation for sinusoidal oscillations: x undamped=Acos(ωt+φ) We have added here a phase φ, which simply allows us to choose any arbitrary time as t = 0. From this equation, we see that the energy will fall by 1ê‰ of its initial value in time t g. Lectures Links to the individual lectures are listed below. The solution of the DE is represented as a power series. 0/ to determine the two constants c1 and c2 in the complete solution to my00 Cky D0: "Simple harmonic motion" y Dc1 cos r k m t! Cc2 sin r k m t!: (4). Thus, some aspects of motion can be addressed with fewer equations and without vector decompositions. Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. Wolfram Community forum discussion about Solve differential equation to describe the motion of simple pendulum. The model predicts that, in the absence of air resistance, the system will oscillate indeﬁnitely. And the most important step in determining the natural frequency of vibration of a system is the formulation of its differential equation. Find out the differential equation for this simple harmonic motion. The exponent s solves a simple equation such as As 2 + Bs + C = 0. Damped harmonic motion Edit. Adding this term to the simple harmonic oscillator equation given by Hooke's law gives the equation of motion for a viscously damped simple harmonic oscillator. 1 Origin of Di erential Equations: the Harmonic Oscillator as an Example We consider a particle of mass m that is moving along a straight line in x{direction. Differential Equation of Wave Motion. Now we consider some nontrivial external forces by entering various functions into the box on the right-hand side of the differential equation, creating an inhomogeneous differential equation. Check energy conservation for both the Euler and RK2. Check energy conservation for both the Euler and RK2. We solve it when we discover the function y (or set of functions y). This is the differential equation for SHM. How do I express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form? (see the attachment for the full question) x = -2x - y + 12t + 12, y = 2x - 5y - 5 How do I express the. The Attempt at a Solution I think the differential equation they're looking for is, a=-kx/m As a=d 2 x/dt 2 But from here I can't see where to go; integration of course leads to the wrong formula. It is an equation tying the behavior of the func-tion y(x) to its derivative, hence a basic differential equation to which the solution. where i = √(-1); c1, c2, C1, C2, A, φ and ψ are constants to be determined by the initial condition/s and/ or boundary condition/s appropriate to the physical problem. Superposition of two or more simple harmonic oscillators. This can be shown easily by differentiating x(t) twice with respect to time. In order to derive the simple pendulum equation and prove the dimensional analysis case about we show the following depiction. The harmonic oscillator solution: displacement as a function of time We wish to solve the equation of motion for the simple harmonic oscillator: d2x dt2 = − k m x, (1) where k is the spring constant and m is the mass of the oscillating body that is attached to the spring. dependent variable and its derivatives are of degree one, 2. Dividing out the exponential yields: Setting generates: which is the Hermite differential equation. motion of the trolley will be such that its acceleration ¨x satisﬁes mx¨ = −kx. In general, the equation of a simple harmonic motion may be represented by any of the following functions Although all the above three equations are the solution of the differential equation but we will be using x = A sin (w t + f) as the general equation of SHM. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. Here the equation of motion is mx x +=λ 0, which can be rewritten in terms of the velocity as mv v +=λ 0. This banner text can have markup. Under, Over and Critical Damping OCW 18. It is still true that in order to specify the full solution to our di er-ential equation, we must supply two initial conditions, and these will determine. $\begingroup$ For a systematic approach to this kind of problem (= linear differential equations with constant coefficients) there are special tools. Before we actually solve the differential equation of motion for this harmonic oscillator, let's take a look at the potential and kinetic energy associated with a given displacement of the mass from equilibrium. Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine. ) A particle is said to execute simple harmonic motion if it moves in a straight line such that its acceleration is always directed towards a fixed point in the line and is proportional to the distance of the particle from the fixed point. The displacement is given relative to the center of the path O and is represented by x = OC. There are many "tricks" to solving Differential Equations (if they can be solved. CHAPTER 4 Second-Order Linear Differential Equations 4. Notice that we're now back in configuration space!. DIFFERENTIAL EQUATIONS 111 Figure 5. However, solving it gives both the allowed values of the angular momentum discussed above and the allowed energies , which agree with the simpler Bohr model. Now that we have derived a general solution to the equation of simple harmonic motion and can write expressions for displacement and velocity as functions of time, we are in a position to verify that the sum of kinetic and potential energy is, in fact, constant for a simple harmonic oscillator. This is the Hamilton-Jacobi equation. An object of mass sitting on a frictionless surface is attached to one end of a spring. The short way F = ma gives ¡kx = m d2x dt2: (8) This equation tells us that we want to ﬂnd a function whose second derivative is. Note that the mass terms cancel out, suggesting that the motion of a pendulum is independent of its mass. The oscillation occurs with a constant angular frequency. Differential Equations Some Application of Differential Equation in Engineering 6. The equation for T is for harmonic motion. Using the trigonometric identity cos(α−β) =. Note: The following derivation is not important for a non- calculus based course, but allows us to fully describe the motion of a simple harmonic oscillator. Example 1: Find the solution of. EXERCISE: (See also here) Test various algorithms by applying them to an analytically solvable problem, as the harmonic oscillator or the 2-body Kepler problem. The differential equation obtained for harmonic motion is mx''+cx'+kx =0, where m is the mass of the object, k is the stiffness of the spring, and c is a dampening coefficient and x is the position of the object. motion that obeys a differential Equation of the form of Equation 11. It is relevant to a case of acoustics in a tube, with the pressure. MATLAB based numerical solution and visualization will be briefly covered. Dividing out the exponential yields: Setting generates: which is the Hermite differential equation. Slope Fields. 2 No friction; just the spring force; no external force Here the equation of motion is mx kx +=0. Using your harmonic oscillator program as the template, create a new program to analyze the motion of a large-amplitude pendulum for about 10 periods or so, for a starting amplitude of 2. simple harmonic motion, where x(t) is a simple sinusoidal function of time. Extra online content finder. The exponent s solves a simple equation such as As 2 + Bs + C = 0. to be given, just as in normal equations the coefficients are given. By studying oscillations in their simplest form, you will pick up important. Simple harmonic motion is defined by the differential equation, , where k is a positive constant. This remembering that the acceleration is the second. dependent variable and its derivatives are of degree one, 2. to the dimensionless variable ˝. Calculate the internal energy of a quantum simple harmonic oscillator at Calculate the motion of system of masses and springs Solve a differential equation. The buoy moves a total of 3. An object on a vertical spring oscillates up and down in simple harmonic motion with an angular frequency of 20. 3 Solving second-order differential equations a proposed solution to a differential equation by. Next the equations are written in a equation of motion for undamped free vibration (newton's second law of motion in this video i explain about equation of motion for undamped free vibration. Textbooks: Polking, Boggess and Arnold, Differential. Simple Harmonic Motion A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. The general solution of the above equation is rather complicated due to the large number of parameters involved, but the way of solving equation (1) is similar to the way a non - homogenous linear second order differential equation is solved, thus resulting in the general solution which is equation (2). But this general equation has not, in practice, led to solutions of real problems of any complexity. I am considering the equation for simple harmonic motion, which is $\ddot x +\omega ^2x=0$ To solve this, I have seen three approaches. Physics 1120: Simple Harmonic Motion Solutions 1. Energy density and energy transmission in waves. Defining Equation of Linear Simple Harmonic Motion: Linear simple harmonic motion is defined as the motion of a body in which. The solution of the DE is represented as a power series. This unit develops systematic techniques to solve equations like this. At this stage, many introductory physics courses will take the small-angle approximation in order to obtain the equation for simple harmonic motion, which can be solved analytically. 5 points) Imagine a vertical pendulum of length / and mass m. The harmonic oscillator is a canonical system discussed in every freshman course of physics. That is, a = -kd. Back to Configuration Space. previous home next PDF 20. The Physics Guide is a free and unique educational YouTube channel. The most general solution to this equation can be written as s(t) = A cos(ωt + φ) (3). It is a second-order diﬀerential equation whose solution tells us how the particle can move. Thegeneral solutionof a differential equation is the family of all its solutions. to the dimensionless variable ˝. The presentation of each section is fairly comprehensive and detailed, almost 6. SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 2. equation of motion of a simple harmonic oscillator. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period = /, the time for a single oscillation or its frequency = /, the number of cycles per unit time. There are also many applications of first-order differential equations. PHY191 Experiment 6: Simple Harmonic Motion 8/12/2014 Page 2 , 2 2 kx dt d x m (1) which can be written as 2 0, 2 0 2 x dt d x where m k 2 0 (2) We have studied the solution of this equation in Experiment 3. As an example, we demonstrate the power of the DTS algorithm for the following simple fourth-order linear differential equation for which the analytical solution exits for comparison,. To get a general idea of how a damped driven oscillator behaves under a wide variety of conditions, check out this spreadsheet for damped driven oscillator. We arrive at similar solutions for B(y) and C(z), so the general. The harmonic oscillator solution: displacement as a function of time We wish to solve the equation of motion for the simple harmonic oscillator: d2x dt2 = − k m x, (1) where k is the spring constant and m is the mass of the oscillating body that is attached to the spring. Initial Conditions. Its focus is primarily upon finding solutions to particular equations rather than general theory. Note that the mass terms cancel out, suggesting that the motion of a pendulum is independent of its mass. The other end of the spring is attached to a wall. All the solutions are given by the implicit equation Second Order Differential equations. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. We study ordinary differential equations, because we can study ordinary differential equations. If the displacement of the object is given by , then for an object with mass in simple harmonic motion, we can write: This is a differential equation. Calculation:. This is an example of simple harmonic motion. When we were describing simple harmonic motion, we called A the amplitude: the sine function varies from −1 to +1, so the motion varies from −A to +A. For instance, there is the notion of "Fourier transform": writing an unknown member of a fairly general class of functions as some kind of infinite linear combination of sines and cosines. differential equation that describes the simple harmonic oscillator naturally arises in a classical-mechanics setting. The solution is expressed in terms of functions not. This is a first order linear differential equation. Include in your code tests that do not rely on the existence of an analytical solution (energy conservation or such. Recently, Florin Diacu, Ernesto Pérez-Chavela, and Manuele Santoprete proposed a new setting for the problem, which allowed an easy derivation of the equations of motion for any n ￿ 2interms. In this section we will solve the second order differential equation that describes a damped harmonic oscillator with a sinusoidal forcing function. This lecture continues the topic of harmonic motions. , infinite] speed of propagation, and the parabolic and elliptic problems have solutions with C∞. In fact, it is much more common that there is no explicit formula for the solution. Simple harmonic motion is described by a second order differential equation:. The answer to “A 0. General solutions to differential equations and loss of information about eigenvalues. An equation relating a function to one or more of its derivatives is called a differential equation. 1 derivation as well as the following pages. A stiffer spring oscillates more frequently and a larger mass oscillates less frequently. Shutyaev ©Encyclopedia of Life Support Systems (EOLSS) 2. The presentation of each section is fairly comprehensive and detailed, almost 6. , determine what function or functions satisfy the equation. differential coefficient appearing in the polynomial form of the differential equation is called the degree of the differential equation. 9: Centrifugal and Coriolis Forces: 4. You may help yourself by substituting any of the three solution forms back into the differential equation to check that indeed they satisfy the equation. Cauchy-Euler Equations. The starting position of the mass. We will begin discussion of higher-order linear equations, and use the example of simple harmonic motion. If you are not familiar with differential equations, don’t worry. Spreadsheets can. An ideal spring with a spring constant $k$ is described by the simple harmonic oscillation, whose equation of motion is given in the form of a homogeneous second-order linear differential equation: $m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + k x = 0$. Equation (5) can now be written as two differential equations (Volterra, p. The next place this comes up in a standard curriculum is, of course, differential equations. 7 The Simple Harmonic Motion 6. When the mass is in motion and reaches the equilibrium position of the spring, the mechanical energy of the system has been completely converted to kinetic energy. Some multiple of the source gives me the solution to that differential equation. Solving the Harmonic Oscillator. For the moment, we will simply guess the solution and check that it works. A singular solution is a solution that can't be derived from the general solution. which is derived from the Euler-Lagrange equation, is called an equation of motion. We arrive at similar solutions for B(y) and C(z), so the general. Differential Equations and Linear Algebra, 6. simple harmonic motion, where x(t) is a simple sinusoidal function of time. Fourier Series Fourier expansion – statement of Dirichlet’s condition, analysis of simple waveforms with Fourier. It consists of a mass m,which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the mass's position x and a constant k. Problems are introduced and solved to explore various aspects of oscillation. < Example : Simple Harmonic Motion - Vertical Motion> This is one of the most famous example of differential equation. The Harmonic Oscillator. Assume that the object is. A good way to start is to move the second derivative over the to left-hand side of the equation, all by itself, and put all other terms and coefficients on the right-hand side. The above equation is known to describe Simple Harmonic Motion or Free Motion. mass m with a rod/string/cable that is not extensible and its weight is negligible, Hafez et al. Suppose mass of a particle executing simple harmonic motion is 'm' and if at any moment its displacement and acceleration are respectively x and a. This is known as simple harmonic motion and the corresponding system is known as a harmonic oscillator. Get this from a library! The differential equations problem solver : a complete solution guide to any textbook. For more videos of this type please SUBSCRIBE my channel so I'll. 1 Origin of Di erential Equations: the Harmonic Oscillator as an Example We consider a particle of mass m that is moving along a straight line in x{direction. The starting direction and magnitude of motion. the differential equation corresponding to a damped oscillator: x t x 2 x () =−γ −ω0. Series Solutions Near an Ordinary Point. We look at Simple Harmonic Motion in Physclips, first kinematically (i. Currrent Electricity. dependent variable and its derivatives are of degree one, 2. means position) at any instant. The general solution is a linear 0 0 0 m k y t A t B t = = + ω ω ω. Therefore, the angular frequency ω is 2π times the frequency ν of the simple harmonic motion, which is consistent with Equation (8-3). There, you actually get to look at the inner workings of the problem at hand (and learn to handle more complex situations like damped and driven oscillators). and are arbitrary constants that can be determined from the initial conditions of the problem. Before we actually solve the differential equation of motion for this harmonic oscillator, let's take a look at the potential and kinetic energy associated with a given displacement of the mass from equilibrium. And now comes the wonderful result: This is a simple harmonic oscillation. , are integration constant. Differential equation, Initial conditions, Initial-value prob-lem, Newton's second law of motion, Hooke's law, Spring constant, Simple harmonic motion, Simple harmonic oscil-lator, Newton's law of cooling, Orthogonal trajectories. And the first one was free harmonic motion with a zero, but now I'm making this motion, I'm pushing this motion, but at a frequency omega. The Simple Harmonic Oscillator. Linear Differential Equation A differential equation is linear, if 1. Solution of Differential Equations A function y = f (x) is a solution of a differential equation, if the substitution of f (x) and its derivative(s) in the differential equation reduces it to an identity. 3 Solving second-order differential equations a proposed solution to a differential equation by. Notice that we're now back in configuration space!. The starting position of the mass. e it is a STIFFNESS of the system (units = N/m). We arrive at similar solutions for B(y) and C(z), so the general. Since v(1) = 1, we get. simple harmonic motion of the mass. Chapter 8 Simple Harmonic Motion Activity 3 Solving the equation Verify that θ=Acos g l t +α is a solution of equation (3), where α is an arbitrary constant. This equation is then the solution to simple harmonic motion and can be transformed into an exponential equation of two components. 4) Because the constant coefficients a and b in Equation (4. This results in the differential equation. Thus, you might skip this lecture if you are familiar with it. +omega_0^2x=0, (1) where x^. There is yet another way to find the general solution to the wave equation which is valid in 1, 2, or 3 (or more!) dimensions. 1 Stochastic differential equations. [Hint: Contrast your equation of motion in part b of this problem to the equation of motion you solved in class. Comparing the Spring and Pendulum Periods. This can be verified by multiplying the equation by , and then making use of the fact that. Using the forces on the pendulum and applying Newton's second law, obtain a differential equation in terms of 0 (the angle with respect to the. we insert for the potential energy U the appropriate form for a simple harmonic oscillator: Our job is to find wave functions Ψ which solve this differential equation. Damped oscillations. Mathematics. This channel covers theory classes, practical classes, demonstrations, animations, physics fun, Puzzle and many more of the Physics. The general solution is of the form: x = Acos(Wt)+Bsin(Wt) The amplitude is sqrt(A^2+B^2) and the frequency is W. Find the equation of motion. CHAPTER 4 Second-Order Linear Differential Equations 4. Extra online content finder. We discuss various ways to solve for the position x(t), and we give a number of examples of such motion. Our basic model simple harmonic oscillator is a mass m moving back and forth along a line on a smooth horizontal surface, connected to an inline horizontal spring, having spring constant k, the other end of the string being attached to a wall. $\begingroup$ For a systematic approach to this kind of problem (= linear differential equations with constant coefficients) there are special tools. The period T of the motion is the time interval required for the particle to go through one full cycle of its motion. Once v is found its integration gives the function y. 1 Introduction In the last section we saw how second order differential equations naturally appear in the derivations for simple oscillating systems. $\begingroup$ @sangstar A way that always helped me to think about differential equations is that you are trying to find a solution for your original equation. Physics 1120: Simple Harmonic Motion Solutions 1. 10: The Top. Every physical system that exhibits simple harmonic motion obeys an equation of this form. The techniques for solving differential equations based on numerical. 06 (2001), pp. The harmonic oscillator is a canonical system discussed in every freshman course of physics. This solution method requires first learning about Fourier series. In addition to Differential Equations with Applications and Historical Notes, Third Edition (CRC Press, 2016), Professor Simmons is the author of Introduction to Topology and Modern Analysis (McGraw-Hill, 1963), Precalculus Mathematics in a Nutshell (Janson Publications, 1981), and Calculus with Analytic Geometry (McGraw-Hill, 1985). This is confusing as I do not know which approach is physically correct or, if there is no correct approach, what is the physical significance of the three different approaches. Although this result is standard, it provides an opportunity to discuss conservation of energy and motivate phase plots. Get Help to Solve Differential Equations More often than not students need help when finding solution to differential equation. Search the history of over 384 billion web pages on the Internet. Find the equation of motion if the spring is released from the equilibrium position with an upward velocity of 16 ft/sec. Once we start damping and/or. r Follow the Shadow: Simple Harmonic Motion But what if we just equate the real parts of both sides? That must be a perfectly good equation: it is. Solving the Simple Harmonic Oscillator 1. 2) Solve ordinary differential equations using the methods of undetermined. SIMPLE HARMONIC MOTION (S. COMPUTATIONAL METHODS AND ALGORITHMS – Vol. Simple harmonic motion is the simplest form of periodic motion. We will introduce the basic ideas for solving ODEs by looking at a very simple physical example: the bouncing ball with Newton's equations of motion where is the constant acceleration due to gravity 2 and is the position of the ball as a function of time (its trajectory). The descriptor "harmonic" in the name harmonic function originates from a pointon a taut string which is undergoing harmonic motion. Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. That is, it is a subject that is well amenable to study, a fairly accessible theory, and a wealth of methods of exact calculation. Solution: Since the general solution contains both e2x and a cos2x-sin2x pair, the original DE should have roots r = 2 and r = 2i. The solution of this equation is: where is and called the angular frequency. You know that d dt (sint) = cost and d dt (cost. Understanding the solutions of the differential equation is then of paramount interest. The harmonic oscillator is well behaved. 5 and is therefore simple harmonic motion in which ω = c I/. We have already noted that a mass on a spring undergoes simple harmonic motion. Suppose mass of a particle executing simple harmonic motion is ‘m’ and if at any moment its displacement and acceleration are respectively x and a. 5) due to the “square root” parts in the expression of m. Problems are introduced and solved to explore various aspects of oscillation. The starting position of the mass. Now we can solve the equation E=E (4. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t: 1. Use the Laplace transform together with its basic properties as a useful method to solve. This solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics. 6}\) are to simplify the equation by collecting constants in the parameter $$\beta$$. Damped oscillations. 3 The equation for simple harmonic motion: FLAP M6. Half-step + relaxation equation: $\Delta t \leq 2/\lambda$. At what time(s), to the nearest minute, in the 12 hours from 4am to 4pm would the temperature be:. Best Answer: Simple harmonic motion follows the differential equation x" + xW^2 = 0 where x is position as a function of time t and W gives the oscillation frequency in radians per second. Thursday, October 10 Special guest lecture from Read Jones Christoffersen Ltd. ( ) cos( ) sin( ) combination of sin and cos. Calculate the distance d by which the spring stretches from its unstrained length when the object is allowed to hang stationary from it. Now we consider some nontrivial external forces by entering various functions into the box on the right-hand side of the differential equation, creating an inhomogeneous differential equation. For you calculus types, the above equation is a differential equation, and can be solved quite easily. 2) Solve ordinary differential equations using the methods of undetermined. An equation of motion is a differential equation relating the variables of motion to one another, esp. A common problem in physics is to determine the motion of a particle in a given force field. Notice that we're now back in configuration space!. Its focus is primarily upon finding solutions to particular equations rather than general theory. SIMPLE harmonic motion occurs when the restoring force is proportional to the displacement. The equation is a second order linear differential equation with constant coefficients. General Solution of a Differential Equation A differential equationis an equation involving a differentiable function and one or more of its derivatives. That is, it is a subject that is well amenable to study, a fairly accessible theory, and a wealth of methods of exact calculation. Michael Fowler. This unit develops systematic techniques to solve equations like this. This tells us that if we can nd one particular solution to this di erential equa-tion, the most general solution is given by adding a solution to the homogeneous equation. Equation of motion under harmonic excitation can then be written as Now let us consider a 5 storeyed building. The mobile mass, however, experiences an acceleration given by Newton's equation Force = Mass*acceleration, or F = ma. 4) the equation becomes ¨x = −ω2x (1. Explain why the transient state is a valid part of the solution of the differential equation that describes forces and accelerations. If so, you simply must show that the particle satisfies the above equation. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. The characteristic equation is m r 2 + k = 0. The reason the equation includes angular velocity is that simple harmonic motion is very similar to circular motion. Solving the Simple Harmonic Oscillator 1. A stiffer spring oscillates more frequently and a larger mass oscillates less frequently. for constants a 1 and a 2. Key Mathematics: We will gain some experience with the equation of motion of a. ( ) cos( ) sin( ) combination of sin and cos. Is it realy simple like this to get a solution for any differential equations (ordinary differential equation, more specifically)? If it is the case, why our numerical method text book is so thick ?" Good Question. The full step-by-step solution to problem: 2 from chapter: 15 was answered by Sieva Kozinsky, our top Physics solution expert on 09/09/17, 04:30AM. For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in. The analysis that follows here is fairly brief. Differential Equations of Motion; Air Drag; Free Fall with Drag; Propulsion and Machines; Vehicle Resistive Forces; Solving Differential Equations of Motion; PROBLEMS; Lab Exercise - Drag Coefficient Measurement; Harmonic Motion. Damped oscillations. Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. Differential Equation of the simple harmonic motion. Simple harmonic motion If d=0 (no damping) then the previous equation becomes d2x dt2 +w2 0 x(t)=0 which is said to describe simple harmonic motion. Simple Harmonic Motion A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever.