Summary of modal mass, stiffness, and damping given the three equations 20, 24, and 28 for diagonalizing the mass, stiffness and damping matrices, it is now a straightforward task to define. As stated in the sap2000 analysis reference manual, csi 2009 for. Program will take a description of a system containing masses and springs with proportional dampening and will output the massm, dampeningd and stiffnessk matrices for that system with respects to its system of equations as shown below. Computation of rayleigh damping coefficients for large systems.
Damping technical knowledge base computers and structures. Model of a tuned mass damper on a simple structure youtube. Gavin a rayleigh damping matrix is proportional to the mass and sti. Finite element steadystate vibration analysis considering. For modal response history analysis, sap2000 allows an explicit damping ratio to be used in each mode. However, the question fielded concerns modal damping. Barkans theory barkan 1962, sec prakash, 1981a studied the dynamic behavior of foundations and base soils and developed a theory. The system modal damping ratio for uniform subsystem damping is thus a b a b a a b b s u k k m m k m k m 12 the two mass and stiffness values were varied randomly in an informal numerical study using 100,000 trials. Classical damping, nonclassical damping and complex modes. Therefore, sliding and rocking stiffness and damping arc discussed only. The stiffness and damping properties of a massspring damper system are to be determined by a free vibration test. The c viscous damping force is a cosine function, and the structural damping force is the product of the coulomb square wave force times. Effect of mass and springs on the damping of mass spring. And i understand damping as the energy dissipative properties of an objectsystem e.
Stiffness and damping video simulation khan academy. Both the sap2000 and gtstrudl ctg models consist of. Then i can find the spring stiffness k by sett all the derivatives to zero. Structural analysis for performance based earthquake. Pdf computation of rayleigh damping coefficients for large. The coefficients a 2p, b 0 were used to create a proportional damping matrix, and the damped fea model was solved for its modes. At the end of 20 complete cycles, the time is 3 sec and the amplitude is 0. Effect of data incompleteness and noise on the coupling between the two stages is investigated. Structural dynamic modification of vibrating systems. Rayleigh damping, mass proportional damping, elasticplastic, time history analysis. Damping coefficients technical knowledge base computers and. During steadystate analysis, csi software uses hysteretic damping. Mass and stiffnessproportional damping, normally referred to. In cases such as automatic control system design, accurate estimates of the mass, stiffness, and damping properties for only a few dofs may be all that is required in order to build an effective controller.
It is shown in the paper that correct expressions for complex stiffness are different in the cases of forced and free oscillations. There are concerns raised by the mass proportional part of rayleigh damping. Derivation of mass, stiffness and damping parameters from. What i would like to do is size a rotational spring located at the pivot point and a damping system, such that it that will damp the spring force. Rayleigh s proportional damping is generalized the generalized proportional damping expresses the damping matrix in terms of any nonlinear function involving specially arranged mass and stiffness matrices so that the system still posses classical normal modes this enables one to model practically any type of variations in the modal damping. There are many ways to compute a classical damping matrix from mass and stiffness matrices.
The critical damping ratio of a system is often used to compare system damping to that which would result in a critically damped case i. Remove the carriage extension bracket and dashpot from the second mass carriage, replace the low stiffness spring with a. I understand stiffness as the extent to which an object e. A new direct method for the finite element fe matrix updating problem in a hysteretic or material damping model based on measured incomplete vibration modal data is presented. However, for the sliding mode of motion, the vertical geometrical damping relationship is applied for the backfill. Mass, stiffness, and damping matrix technical papers. Damping devices may also be modeled as a structural subsystem, as described in the tuned mass damper tutorial. Analysis of structural response with different forms of. Report pacific earthquake engineering research center. This property is proportional to mass and stiffness.
As far as the stiffness proportional part of rayleigh damping is. Mod02 lec development of mass, stiffness and damping. Updating stiffness and hysteretic damping matrices using. Structural analysis for performancebased earthquake engineering. Damping faq technical knowledge base computers and.
Understanding and using dynamic stiffness a tutorial. Hysteretic proportional damping, also mass and stiffnessproportional, is used for. Concept of complex stiffness applied to problems of. Materialbased damping is now available for steadystate and psd analysis. Hysteretic proportional damping, also mass and stiffness proportional, is used for steadystate and powerspectraldensity analyses. For hinges modeled as links, the stiffnessproportional damping used for directintegration timehistory analyses may now be specified by the user to be proportional to the initial stiffness of the hinge the current behavior, the tangent stiffness, or a mixture of the two. The f is the composition of two forces, the elastic one, proportional to the elastic constant k and the position, and the damping force, proportional to the velocity and a. Consider the case in which the damping matrix d is of the form dm k 3 where. F d dc v the force of friction is proportional and. These assumptions greatly simply the equations for calculating mass, stiffness, and damping matrices, but a matrix inversion in this case the mode shape matrix is still required. Repeat the above for the first mass carriage, spring and damping m c1, c m1 and k respectively.
Nonlinear static analysis using sap 2000 mitesh surana. Method of relating modal damping to local dampers in lumpedparameter systems by harry j. A weight w80lb suspended by a spring with k 100 lbin. Rayleigh damping, mass proportional damping, elasticplastic, timehistory analysis. Model of a tuned mass damper on a simple structure. Topic 6 structural dynamics iii analysis of elastic mdof. Mass and stiffnessproportional damping, normally referred to as rayleigh damping, is commonly used in nonlineardynamic analysis. Changing the effective mass to control resonance problems. During formulation, the damping matrix is assumed to be proportional to the mass and stiffness matrices as follows. Coupling between the two stages of updating is investigated through a numerical study. A twostage approach to updating of mass, stiffness and. Mass proportional damping in nonlinear timehistory.
If the approximation that the effective mass is 90% of the total mass had been used, the determined weight 0. This energy dissipation may depend on velocity general viscous damping, displacement hysteresis damping or a combination of both. Stiffness and damping parameters for dynamic analysis of. Please note that setting a constant hysteretic damping with both mass and stiffness proportional damping set to 0. Stiffnessproportional damping now uses initial stiffness instead of tangent stiffness to improve the consistency of results and convergence behavior.
Hysteretic proportional damping, also mass and stiffnessproportional, is used for steadystate and powerspectraldensity analyses. If the system is lightly damped, authors propose a simple and ef. Structural dynamics department of civil and environmental engineering duke university henri p. Damping devices may also be modeled as a structural subsystem, as described.
Paper 15 mass, stiffness, and damping matrix estimates. Suitability for an incremental approach to numerical solution merits its use. The physical connectivity of the original model is preserved and the measured modal data are embedded. Free vibration of a mass spring system with damping damper. Proportional damping is the most common approach to model dissipative forces in complex engineering structures and it has been used in various dynamic problems for more than 10 decades. A two stage approach to updating of mass, stiffness and damping matrices is proposed. Here you can control damping of each spring, number of hairs, number of springs per hair, and spring stiffness. If a 0, then viscous damping isa only proportional to the stiffness distribution, and eq. For this structure, a damping of 5 percent of critical was. Free vibration of a mass spring system with damping november 22, 2014 september 20, 2018 engineeering projects fig. When the materials, vibration control devices or vibration isolators in a structure have different damping ratios, elements with different damping ratios can be.
Sap2000 allows users to either specify coefficients. Drain lewis research center summary the longitudinal dynamic characteristics of a linear structure may be approximated by a lumpedparameter model, which usually includes only the stiffness and mass distri bution of the system. So the trick is to find a nice balance between spring stiffness and damping. Damping constant 1nsm spring constant k 1nm m 1kg mass x 0 position from the point of equilibrium there are a total of 3 forces acting on mass m. Program can handle both numeric and symbolic equations.
Determine the vibration response, if the system is given an initial displacement of 2 inches and. To maintain the same behavior as in previous versions of the software, set the modification factor to unity and use the initial stiffness for stiffnessproportional viscous damping. Pdf parametric study on equivalent damping ratio of different. Method of relating modal damping to local dampers in. With this method, the optimally approximated stiffness and hysteretic damping matrices can be easily constructed. The concept of complex stiffness in problems of oscillations with viscous, or structural hysteretic damping is often used in a wrong way, leading to erroneous solutions. Learn about the sap2000 structural analysis and design program. Damping modelling using generalized proportional damping. This tutorial requires familiarity with vector calculations and concepts, conversions between rectangular and polar coordinates, and the basics of complex number operations, i. See if you can come up with something that behaves a bit more realistically. Pdf initial versus tangentstiffness based rayleigh damping in. A hybrid data based approach is suggested to deal with data. Mass proportional damping in nonlinear timehistory analysis. Topic 6 structural dynamics iii analysis of elastic mdof systems.
In these models damping must be introduced to account for energy dissipation during motion. Pdf in the inelastic time history analyses of structures in seismic motion. Materialbased damping is now available for linear and nonlinear directintegration timehistory analysis. Gavin fall 2018 1 preliminaries this document describes the formulation of sti. Viscous proportional damping is used for directintegration timehistory analysis. One of the main limitation of the mass and stiffness proportional damping approximation comes from the fact that the arbitrary variation of damping factors with. In fact, setting the rayleigh damping to be both mass and stiffness proportional, as usually done in. The resulting coulomb damping force is obviously a square wave of period.
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