It is therefore convenient to introduce certain transformation factors to convert the mass and existing loads of the structure to equivalent mass, and load for the equivalent sdof system. That motion will be centered about a point of equilibrium where the net force on the mass is zero rather than where the spring is at its rest position. The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. Forced vibration of a mass spring system with damping november 26, 2014 september 20, 2018 engineeering projects in foundation soil system damping is always present in. Abstract the purpose of the work is to obtain natural frequencies and mode shapes of 3 storey building by an equivalent mass spring system, and demonstrate the modeling and simulation of this mdof mass spring system to obtain its first 3 natural frequencies and mode shape. 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. The transformation factors are a function of the deflected shape of the structure under dynamic impacts. In the springmass system shown in its unstrained position in fig. Now lets add one more springmass to make it 4 masses and 5 springs connected as shown below. Spring stores potential energy by its deformation kx22. 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. Evaluation of methods for analysis of multidegreeoffreedom. The spring and damper elements are in mechanical parallel and support the seismic mass within the case. Equivalent mass example k no slip 3d4 d cm b meff x b m.
Derive the equivalent spring constant for the system. Spring mass system an overview sciencedirect topics. W mg is the weight of the mass m, e youngs modulus, and i moment of inertia of crosssection of beam spring constant is given by. Where modal truncatim is employed, the rcsidual mass metrix is used to preserve the correctness of the rigidbody mass properties. Use of equivalent mass method for free vibration analyses of. As before, we can write down the normal coordinates, call them q 1 and q 2 which means substituting gives. In the case of electrical systems, energy can be stored either in a capacitance or an inductance. In this video we simplify a rotational vibrating system to a simple translational springmass system using the energy method. Equivalent mass, stiffness, and loading for offcentre.
Equivalent stiffness example k l l2 p keff methods for calculating equivalent stiffness. Forced vibration of a mass spring system with damping. The mass m is a constant at velocities well below the speed of light and not to be confused with its weight w mg. The strain energy for the spring is 2 3 2 4 1 g ku y3. When all energy goes into ke, max velocity happens. Note that the effective damping for a particular mode varies directly as the natural. The remainder of the energy goes into the internal energy associated with the oscillations of the massspring system, u int mv o 2. In this lesson, we present two more examples of distributed systems. Evaluation of methods for analysis of multidegreeof.
Forced vibration of a mass spring system with damping november 26, 2014 september 20, 2018 engineeering projects in foundation soil system damping is always present in one form or another. Mechanical system elements nyu tandon school of engineering. Introduction all systems possessing mass and elasticity are capable of free vibration, or vibration that takes place in the absence of external excitation. Mechanical system elements three basic mechanical elements.
In a similar way, hitting a bell for a very short time makes it vibrate freely. All systems possessing mass and elasticity are capable of free vibration, or vibration that takes place in the absence of external excitation. 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 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. Springmass analogs any other system that results in a differential equation of motion in the same form as eq. Solving the massspringdamper second order differential equation. From physics, hookes law states that if a spring is displaced a distance of y from its equilibrium position, then the force exerted by the spring is a constant k 0 multiplied by the displacement of the y. It is shown how an equivalent sdof model with one equivalent mass, one equivalent spring. Oct 27, 2019 in general, a spring mass system will undergo simple harmonic motion if a constant force that is colinear with the spring force is exerted on the mass in this case, gravity. Apr 03, 2018 equivalent mass and stiffness by energy method.
Then, an sdof system with many energy storage and dissipation elements, which are not independent, is considered. Dynamics of simple oscillators single degree of freedom systems. On the representation of a cantilevered beam carrying a tip mass by an equivalent springmass system article pdf available in journal of sound and vibration 2821. The compliance of a spring is the reciprocal of its spring constant. In this video we simplify a rotational vibrating system to a simple translational spring mass system using the energy method. Now lets add one more spring mass to make it 4 masses and 5 springs connected as shown below. A typical mechanical massspring system with a single dof is shown in fig. Examples of systems analogous to a spring mass system fig. In this video we find the equivalent mass for a rather complicated system involving a rotating disc and rod, a pully, a spinning cylinder and a.
Review two common massspringdamper system models and how. An example of a system that is modeled using the basedexcited massspringdamper is a class of motion sensors sometimes called seismic sensors. Spring mass analogs any other system that results in a differential equation of motion in the same form as eq. In other words, it possesses energy by virtue of its position. The following are a few examples of such single degree of freedom systems. Determine the equivalent spring constant of the system. Moreover, many other forces can be represented as an infinite. Cleaner match between physical system and circuit velocity is naturally across e. 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. Find the equivalent spring constant and equivalent mass of the system shown in fig. Pdf on the representation of a cantilevered beam carrying a.
This is because external acceleration does not affect the period of motion around the. Since the mass an initial velocity of 1 ms toward equilibrium to the left y00. Structural dynamics of linear elastic singledegreeof. Gavin fall, 2018 this document describes free and forced dynamic responses of simple oscillators somtimes called single degree of freedom sdof systems. Understanding the coefficient of restitution cor using mass. The dynamic parameters of the equivalent system are evaluated on the basis of an assumed shape. The effective mass of the spring in a springmass system when using an ideal spring of uniform linear density is of the mass of the spring and is independent of the direction of the springmass system i. This kinetic energy of the center of mass is equivalent to the macroscopic kinetic energy of the object. Since the mass is displaced to the right of equilibrium by 0. 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. 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. Mass or inertia elements using mathematical model to represent the actual vibrating system e.
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. Since the energy of the mass in a mechanical 2 analogy is measured relative to a fixed reference i. Solve by decoupling method add 1 and 2 and subtract 2 from 1. Use of equivalent mass method for free vibration analyses. The effective mass of the spring in a spring mass system when using an ideal spring of uniform linear density is of the mass of the spring and is independent of the direction of the spring mass system i. Once the mass is released, it starts vibrating freely. The prototype single degree of freedom system is a springmassdamper system in which the spring has no damping or mass, the mass has no sti. Mass stiffness damping ft ut, t ft t ut the simple frame is idealized as a sdof massspringdashpot model with a timevarying applied load. Massspring system an overview sciencedirect topics. Study the response of the mass spring system to various initial conditions using the matlab file springmassinit. Sdof massspring system replacing the actual structure.
The function ut defines the displacement response of the system under the loading ft. This type of excitation is common to many system involving rotating and reciprocating motion. Of primary interest for such a system is its natural frequency of vibration. Furthermore, the mass is allowed to move in only one direction.
Determine the equivalent spring constant of the system shown in fig. Understanding the coefficient of restitution cor using. The static deflection of a simple massspring system is the deflection of. Observe the open loop pole locations and system response for a keep 0. Secondorder massspring system ideally undamped, damping ratio. The spring force acting on the mass is given as the product of the spring constant k nm and displacement of mass x m according to hooks law.
Where modal truncatim is employed, the rcsidual mass metrix is used to preserve the correctness of. To avoid integration in the energy method for continuous systems, the mass is assumed to be lumped at few points. For a simple springmass system, we have rayleighs method for continuous systems needs integration to compute equivalent mass and stiffness. The energy lost per cycle in a damper in a harmonically forced system may be expressed as w d i f ddx 1 where f d represents the damping force. 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. A typical mechanical mass spring system with a single dof is shown in fig. Spring elastic element damper frictional element mass inertia element translational and rotational versions these are passive nonenergy producing devices driving inputs force and motion sources which cause elements to respond.
In this system, a damping factor is neglected for simplicity. The natural frequencies of the pneumatic cylinder system are calculated in the same way as the load mass spring system k 0. The static deflected shape is computed by applying concentrated loads on those points. Combining the two springs in this way is thus equivalent to having a single spring, but with spring. Dynamics of simple oscillators single degree of freedom. Examples of systems analogous to a springmass system fig. Dynamics of simple oscillators single degree of freedom systems cee 541. Sep 28, 2009 study the response of the mass spring system to various initial conditions using the matlab file springmassinit. The mass of the system is 10 kg and the spring stiffness.
Each cycle of an oscillation is equivalent to one rotation of the wheel and 1 revolution is an angle of 2. For example,in the analysis of a system consisting of a mass and a spring, it is commonly assumed that the mass of the spring is negligible so that its only effect is to exert a force between the mass and the support to which the spring is attached,and that the mass is perfectly rigid so that it does not 7. Lecture 6 mechanical system modeling equivalent mass gears. Find the equivalent spring constant of the system shown in. Apr 18, 2018 in this video we find the equivalent mass for a rather complicated system involving a rotating disc and rod, a pully, a spinning cylinder and a mass. Mass enters the system dynamics through the fundamental. To apply this analogy, every loop in the electrical circuit becomes a point in the mechanical system. 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. Equivalent mass, stiffness, and loading for offcentre missile impact evaluations asadollah bassam1, mohammad amin2. Equivalent mass inertia elements the mass of a body is a fundamental material property and thought as the amount of matter within a body. The properties of the structure can be completely defined by the mass, damping, and stiffness as shown. Structural dynamics department of civil and environmental engineering duke university henri p. In the basic linear models considered here, thermal systems store energy in thermal capacitance, but there is no thermal equivalent of a second means of storing energy.
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