Massspringdamper systems the theory the unforced massspring system the diagram shows a mass, m, suspended from a spring of natural length l and modulus of elasticity if the elastic limit of the spring is not exceeded and the mass hangs in equilibrium, the spring will extend by an amount, e, such that by hookes law the tension in the. In the case of the mechanical systems, energy was stored in a spring or an inertia. Coupled electrical oscillators physics 3600 advanced physics lab summer 2019 don heiman, northeastern university, 5302019. For example, in many applications the acceleration of an object is known by some physical laws like newtons second law of motion f ma. An example of a system that is modeled using the basedexcited massspringdamper is a class of motion sensors sometimes called seismic sensors. Springs and dampers are connected to wheel using a flexible cable without skip on wheel. Find the transfer function for a single translational mass system with spring and damper. Newtons law for mechanical systems and kirchhoffs laws for electrical systems. Mass pulley system a mechanical system with a rotating wheel of. These systems mainly consist of three basic elements.
Consider a torsional spring system with applied torque tat. Masspulley system a mechanical system with a rotating wheel of. It is normal for the direction of motion to be expressed as the x direction regardless of the actual direction. Determination of natural frequenrapidly developing interest in the sured.
Therefore, the salt in all the tanks is eventually lost from the drains. Capacitive storage electrical capacitance translational mass. The mass could represent a car, with the spring and dashpot representing the cars bumper. The compliance of a spring is the reciprocal of its spring constant.
Mechanical systems electrical systems electronic systems thermal systems hydraulic systems chemical systems first off we need to understand why do we need to model these systems in the first place. Free response of undamped 2nd order system for an undamped system. Mathematical modeling of systems school of electrical. Massspring system by the indirect analogy mobility 17. Dynamic response of second order mechanical systems with.
Block diagrams block diagrams are usually part of a larger visual programming environment. A linear spring is considered to have no mass described by. We will study the motion of a mass on a spring in detail. When the spring mass system is subjected to linear acceleration, a force equal to mass times acceleration acts on the proofmass, causing it to deflect. An understanding of the behavior of this simple system is. The velocity of the car doesnt change instantaneously, so it starts at zero, and follows a saturating exponential up to v0, with a time constant mb. Writing a transfer function for a mechanical system the following example illustrates how to write a transfer function to describe a mechanical system. Therefore the electrical system and the mechanical system are said to be analogous. The simplest springdamper system can be modeled as follows. How to design two mass damper spring system in simulink.
A guide to mechanical impedance and structural response techniques by h. Mathematical modelling of control system mechanical. In mechanics, the subject of vibrations is considered a subset of dynamics, in which one is concerned with the motions of bodies subjected to forces and moments. Randall introduction in recent years there has been a that the comfort of passengers is en 1. Pdf dynamic response control for a massspringviscous damper.
Security system contractor license required for this work. Equivalent mass inertia elements the mass of a body is a fundamental material property and thought as the amount of matter within a body. Model the whole system and predict efficiencyperformance multiple drive cycles hil test. The following table gives formulas for the spring that is equivalent to a system of two springs, in series or in parallel, whose spring constants are and. In mechanical systems, mass m is the equivalent property for linear motion and moment of inertia i for angular motion. Those are force voltage analogy and force current analogy. Suppose that the masses are attached to one another, and to two immovable walls, by means of three identical light horizontal springs of spring constant, as shown.
External force applied to a massspring system 12 if we consider the mechanical system illustrated in the figure below, which consists of an external operator applying a force on a massspring. The input vint is a step functionzero until the engine couples, and then at a constant velocity v0 thereafter. Now lets summarize the governing equation for each of the mass and create the differential equation for each of the massspring and combine them into a system matrix. Where modal truncatim is employed, the rcsidual mass metrix is used to preserve the correctness of the rigidbody mass properties. Since the mass and spring displacements are physically the same, we can define. Classification of vibration degrees of freedom of the system, 7 differential equation of motion of the system, 78, 7f input, 34 belt friction system, 4f mass spring system, 3f unbalanced disk, 4f output, 46, 4f, 5f, 6f coherence function, experimental modal analysis. The pressure p given by 1 includes the electrical force, f e. There are two types of electrical analogies of translational mechanical systems. This deflection is sensed by a suitable means and converted into an equivalent electrical signal. Similarity means that mathematically the parameters and components will be indistinguishable from each other. Electrical systems why should mes care about all of this electrical stuff. Since the spring is massless, the sum of the torques acting on the spring itself must be zero, or. Inductive storage electrical inductance translational spring rotational spring fluid inertia chp3 4.
Massspring system by the indirect analogy mobility. Coupled oscillations two springcoupled masses consider a mechanical system consisting of two identical masses that are free to slide over a frictionless horizontal surface. Mass enters the system dynamics through the fundamental. Equivalence can be very useful in understanding how complex systems work. Larger values for c increase the amount of damping so the object will come to rest more quickly, c 0. How to use lumped elements to model a mechanical system.
Frvso university of michigan by considering each mass in a linear mechanical system as having two terminals, one fixed in the mass and one fixed to the frame of reference, every linear mechanical system is reduced to a multiplicity of closed mechanical circuits to which. Electrical systems and mechanical systems are two physically different systems. Write all the modeling equations for translational and rotational motion, and derive the translational motion of x as a. Application for permit to perform electrical work all work to be performed in accordance with the massachusetts electrical code mec, 527 cmr 12. Spring mass system an overview sciencedirect topics. It is usual to obtain the effective mass by analyzing results of tm in the limit m 0. Equivalent systems can be used to change large and expensive mechanical, thermal, and fluid systems into a simple, cheaper electrical system. That is, displacement is equivalent to quantity of electrical charge, mass to inductance, and spring constant to the reciprocal of capacitance. Onward to equivalent circuits in electrical circuits, voltage is physically across and current is physically. The problem given the mechanical system f t 1 1 2 dashpot mechanical resistance mass output 2 2 m k 3 k 2 b 2 m k 2 2 3 1 b x x 2 0 input force 2 b 1 0 k b 2 spring. The equivalent circuit has their electrical analogs connected in series, as shown in fig. Electrical analogies of mechanical systems tutorialspoint. Engs 22 systems system analogies page 2 m b vint frictionless fig. The input is a disturbing force f which is a function of time.
Now lets summarize the governing equation for each of the mass and create the differential equation for each of the mass spring and combine them into a system matrix. An understanding of the behavior of this simple system is the. An analogous electrical and mechanical system will have differential. Let ut denote the displacement, as a function of time, of the mass relative to its equilibrium position. Engineering acousticstransducers loudspeaker wikibooks. Coupled electrical oscillators northeastern university. Now lets add one more springmass to make it 4 masses and 5 springs connected as shown below. Active vibration control using mechanical and electrical.
Block diagram models consist of two fundamental objects. Apr 05, 20 introduction to free undamped motion spring system mathispower4u. Motion of the mass under the applied control, spring, and damping forces is governed by the following second order linear ordinary differential equation ode. First, move the branch point of the path involving hi outside the loop involving h, as shown in figure 343a. The transmission and distribution network include substations, lines and distribution transformers. Introduction to free undamped motion spring system youtube. Electrical circuit analog to springmassdamper system. Zak school of electrical and computer engineering ece 680. Mathematical modelling of control system there are various types of physical systems, namely we have. Relate this electrical exchange to that in the massspring system in the lab room.
Mathematical modeling of mechanical systems and electrical. Lets consider an example of a loudspeaker driver system comprised of a massspringdamper system, where the diaphragm and voice coil form the mass, spider, and surround as shown in the loudspeaker driver diagram below constitutes to the spring element. Mathematical models of translating mechanical systems. A spring mass system, used to isolate vibrating equipment from its support structure, is based on a theory that assumes that the support system is very stiff.
In force voltage analogy, the mathematical equations of translational mechanical. Since the energy of the mass in a mechanical 1 analogy is measured relative to mechanical ground i. Electrical system bureau of energy efficiency 3 service. Introduction to free undamped motion spring system mathispower4u. To apply this analogy, every node in the electrical circuit becomes a point in the mechanical system. The diagram shows a mass, m, suspended from a spring of natural. In the case of electrical systems, energy can be stored either in a capacitance or. Finally, the characteristic responses of this system will be discussed, and a parameter called the amplification ratio ar will be introduced that will help in plotting the resonance of the forced, simple springmass system. Translational mass with spring and damper the methodology for finding the equation of motion for this is system is described in detail in the tutorial mechanical systems modeling using. The mass m is a constant at velocities well below the speed of light and not to be confused with its weight w mg. Consider a simple system with a mass that is separated from a wall by a spring and a dashpot. In mechanics, the subject of vibrations is considered a subset of dynamics, in. Using these numbers, the equivalent capacitance is.
Modelling of springmassdamper system, part i, differential equation, 101020. Mechanical vibrations pennsylvania state university. Let us obtain mathematical models of this system by assuming that the cart is standing still for t xt k electrical equivalent circuit. Modeling mechanical systems california state university. The frequency response function is the system transfer function evaluated along. Assuming that the time between the front and rear axles of vehicle passing through the speed bump is t v, then 11 4 v nv t ff d 2. The equivalent electrical circuit is shown in figure 9. As for the current source, i did a little experiment. External force applied to a mass spring system 12 if we consider the mechanical system illustrated in the figure below, which consists of an external operator applying a force on a mass spring. The equivalence between them is exhibited in x q, m l, k 1c. A guide to mechanical impedance and structural response. In practice it is important to construct support systems that are stiff, compared to the deflection of the isolators, and to minimize radiation from lightweight diaphragms. An example of a system that is modeled using the basedexcited mass spring damper is a class of motion sensors sometimes called seismic sensors.
This is a preliminary inexpensive way for engineers to test that their complex system performs the. Then the electrical system can be analyzed to validate that the system dynamics will work as designed. The spring and damper elements are in mechanical parallel and support the seismic mass within the case. Sep 28, 2009 springmassdamper system example consider the following springmass system.
Translational mechanical systems move along a straight line. We suppose added to tank a water containing no salt. Other parts of the environment may include numerical algorithms for integration, realtime interfacing, code generation, and hardware interfacing for highspeed applications. Design, modeling and test of a novel speed bump energy harvester prakhar todaria, lirong wang, abhishek pandey, james oconnor, david mcavoy. Torsional spring follows the same relationship f k f k x 1 x 2 k.
Finally, the characteristic responses of this system will be discussed, and a parameter called the amplification ratio ar will be introduced that will help in plotting the resonance of the forced, simple spring mass system. From the electrical equivalent we know that if we dont connect anything to the wires of the motor, the only thing that drains energy out of the system is the current source that represents friction. Suppose we want to measure the torque tst transmitted to the mass m. In summary, we used the laplace transform to solve for the motion of a simple physical system an ideal mass in response to initial conditions no external driving forces. In this, the charge on the capacitor, c, varies in time in the same way as the displacement of the mass on a spring. If a force is applied to a translational mechanical system, then it is opposed by opposing forces due to. For this spring mass system, motion only occurs in one direction so the system has a single degree of. Masspulley system a mechanical system with a rotating wheel of mass m w uniform mass distribution. Jun 15, 2017 in a first approximation, a moving coil loudspeaker may be thought of as a mass spring system where the diaphragm and the voice coil constitute the mass and the spider and surround constitute the spring element.
Mechanical vibrations a mass m is suspended at the end of a spring, its weight stretches the spring by a length l to reach a static state the equilibrium position of the system. Lower the mass, let go of it and describe its motion. Losses in the suspension can be modeled as a resistor. Now lets add one more spring mass to make it 4 masses and 5 springs connected as shown below. Assuming that the spring has mass ms per unit length z, use the equivalent system method to find the equivalent mass of the system and determine its effect on the natural frequency. Mathematical modeling of systems in this chapter, we lead you through a study of mathematical models of. The principles and the methods by which, through changes of mechanical andor electric parameters, an effective control of the dynamic.