Finite Element Method - Chapter 4: Dynamic analysis - Nguyễn Thanh Nhã

analyses: – Modal – Transient 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 A. Definition & Purpose • A dynamic analysis is a technique used to determine the dynamic behavior of a structure or component. • It is an analysis involving time, where the inertia and possibly damping of the structure play an important role. • “Dynamic behavior” may be one or more of the following: – Vibration characteristics: + How the structure vibrates and at what frequencies. – Effect of harmonic loads. – Effect of seismic or shock loads. – Effect of random loads. – Effect of time-varying loads. 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 A. Definition & Purpose • A static analysis might ensure that the design will withstand steady-state loading conditions, but it may not be sufficient, especially if the load varies with time. • The famous Tacoma Narrows bridge (Galloping Gertie) collapsed under steady wind loads during a 42-mph wind storm on November 7, 1940, just four months after construction. 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A dynamic analysis usually takes into account one or more of the following: – free vibrations • natural vibration frequencies and shapes – forced vibrations • e.g. crank shafts, other rotating machinery – seismic/shock loads • e.g. earthquake, blast – random vibrations • e.g. rocket launch, road transport – time-varying loads • e.g. car crash, hammer blow • Each situation is handled by a specific type of dynamic analysis. A. Definition & Purpose 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Consider the following examples: – An automobile tailpipe assembly could shake apart if its natural frequency matched that of the engine. How can you avoid this? – A turbine blade under stress (centrifugal forces) shows different dynamic behavior. How can you account for it? B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A modal analysis can be used to determine a structure’s vibration characteristics. • A harmonic-response analysis can be used to determine a structure’s response to steady, harmonic loads. – Rotating machines exert steady, alternating forces on bearings and support structures. These forces cause different deflections and stresses depending on the speed of rotation. B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A random-vibration analysis can be used to determine how a component responds to random vibrations. – Spacecraft and aircraft components must withstand random loading of varying frequencies for a sustained time period. B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A response-spectrum analysis can be used to determine how a component responds to earthquakes. – Skyscrapers, power-plant cooling towers, and other structures must withstand multiple short-duration transient shock/impact loadings, common in seismic events. B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A transient analysis can be used to calculate a structure’s response to time varying loads. – An automobile fender should be able to withstand low-speed impact, but deform under higher-speed impact. – A tennis racket frame should be designed to resist the impact of a tennis ball and yet flex somewhat. B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Choosing the appropriate type of dynamic analysis depends on the type of input available and the type of output desired. B. Types of Dynamic Analysis 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Equation of Motion • The linear general equation of motion, which will be referred to throughout this course, is as follows (matrix form): • Note that this is simply a force balance: Mu + Cu + Ku = F :M structural mass matrix :C structural damping matrix :K structural damping matrix :u nodal acceleration vector :u nodal velocity vector :u nodal displacement vector :F applied force vector damping appliedinertial internal F FF F Mu + Cu + Ku = F C. Basic Concepts and Terminology 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Equation of Motion Different analysis types solve different forms of this equation. – Modal • F(t) set to zero; C usually ignored. – Harmonic Response • F(t) and u(t) assumed to be sinusoidal. – Response Spectrum • Input is a known spectrum of response magnitudes at varying frequencies in known directions. – Random Vibration • Input is a probabilistic spectrum of input magnitudes at varying frequencies in known directions. – Transient • The complete, general form of the equation is solved Mu + Cu + Ku = F C. Basic Concepts and Terminology 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Modeling Considerations - Geometry and Mesh • Generally same geometry and meshing considerations for static analysis apply to dynamic analysis: – Include as many details as necessary to sufficiently represent the model mass distribution. – A fine mesh will be needed in areas where stress results are of interest. If you are only interested in displacement results, a coarse mesh may be sufficient. C. Basic Concepts and Terminology 4.1. Introduction Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Modeling Considerations - Material properties • Mass properties [M] – e.g. density, point mass – required for all dynamic analysis types – specify mass density when using metric units, and – specify weight density when using British units • Damping properties [C] – e.g. viscous, material (discussed later) – required for mode-superposition harmonic – optional but recommended for all other dynamic analysis types • Stiffness (elastic) properties [K] – e.g., Young’s modulus, Poisson’s ratio, shear modulus – required for all flexible analysis types Mu + Cu + Ku = F C. Basic Concepts and Terminology Department of Engineering Mechanics – HCMUT 2016 4.2. Modal analysis 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 A. Define modal analysis and its purpose. B. Discuss associated concepts, terminology, and mode extraction methods. C. Learn how to do a modal analysis. Objectives 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • A modal analysis is a technique used to determine the vibration characteristics of structures: – natural frequencies • at what frequencies the structure would tend to naturally vibrate. – mode shapes • in what shape the structure would tend to vibrate at each frequency.. – mode participation factors • the amount of mass that participates in a given direction for each mode • Most fundamental of all the dynamic analysis types. Description & Purpose 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Benefits of modal analysis • Allows the design to avoid resonant vibrations or to vibrate at a specified frequency (speaker box, for example). • Gives engineers an idea of how the design will respond to different types of dynamic loads. • Helps in calculating solution controls (time steps, etc.) for other dynamic analyses. Recommendation: Because a structure’s vibration characteristics determine how it responds to any type of dynamic load, it is generally recommended to perform a modal analysis first before trying any other dynamic analysis. Description & Purpose 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 A “mode” refers to the pair of one natural frequency and corresponding mode shape. - A structure can have any number of modes, up to the number of DOF in the model. Terminology Description & Purpose 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • The structure is linear (i.e. constant stiffness and mass). • There is no damping. • The structure has no forces, displacements, pressures, or temperatures applied (free vibration). Assumptions & Restrictions Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Start with the linear general equation of motion: Development • Assume free vibrations, and ignore damping: • Assume harmonic motion: Mu + Cu + Ku = F Mu + Cu + Ku = F Mu + Ku = 0      2 sin cos sin i i i i i i i i i i i t t t             u =Φ u = Φ u = Φ Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Substitute and simplify Development • This equality is satisfied if (trivial, implies no vibration) or if • This is an eigenvalue problem which may be solved for up to n eigenvalues and n eigenvectors where n number of DOF. This is an n-th order polynomial of 2, from which we can find n solutions (roots) or eigenvalues i. Mu + Ku = 0    2 sin sini i i i i i it t      MΦ +KΦ = 0  2i i M + K Φ = 0 iΦ = 0  2det i iK M Φ = 0 2 i iΦ Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Note that the equation: Extraction & Normalization has one more unknown than equations; therefore, an additional equation is needed to find a solution. – The addition equation is provided by mode shape normalization. • Mode shapes can be normalized either to the mass matrix: or to unity, where the largest component of the vector {}i is set to 1. • Because of this normalization, only the shape of the solution has real meaning.  2i iK M Φ = 0 1Ti iΦ MΦ = Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • The square roots of the eigenvalues are ωi, the structure’s natural circular frequencies (rad/s). Eigenvalues & Eigenvectors • Natural frequencies fi can then calculated as fi = ωi/2π (cycles/s). • The eigenvectors represent the mode shapes, i.e. the shape assumed by the structure when vibrating at frequency fi. iΦ Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • The equation: Equation Solvers can be solved using one of two solvers available in Workbench Mechanical: – Direct (Block Lanczos): • To find many modes (about 40+) of large models. • Performs well when the model consists of shells or a combination of shells and solids. • Uses the Lanczos algorithm where the Lanczos recursion is performed with a block of vectors. Uses the sparse matrix solver. – Iterative (PCG Lanczos): • To find few modes (up to about 100) of very large models (500,000+ DOFs). • Performs well when the lowest modes are sought for models that are dominated by well-shaped 3-D solid elements. • Uses the Lanczos algorithm, combined with the PCG iterative solver. • In most cases, the Program Controlled option selects the optimal solver automatically.  2i iK M Φ = 0 Theory 4.2. Modal analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 [V,D] = eigs(K,M,k,'sm') MATLAB command return the k largest magnitude eigenvalues V: full matrix whose columns are the corresponding eigenvectors D: diagonal matrix of generalized eigenvalues Theory Department of Engineering Mechanics – HCMUT 2016 4.3. Transient analysis 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Transient structural analysis provides users with the ability to determine the dynamic response of the system under any type of time-varying loads. – Unlike rigid dynamic analyses, bodies can be either rigid or flexible. For flexible bodies, nonlinear materials can be included, and stresses and strains can be output. – Transient structural analysis is also known as time-history analysis or transient structural analysis. Overview 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Transient structural analyses are needed to evaluate the response of deformable bodies when inertial effects become significant. – If inertial and damping effects can be ignored, consider performing a linear or nonlinear static analysis instead. – If the loading is purely sinusoidal and the response is linear, a harmonic response analysis is more efficient. – If the bodies can be assumed to be rigid and the kinematics of the system is of interest, rigid dynamic analysis is more cost- effective. – In all other cases, transient structural analyses should be used, as it is the most general type of dynamic analysis. A. Introduction to Transient Structural Analyses 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • In a transient structural analysis, Workbench Mechanical solves the general equation of motion: Some points of interest: – Applied loads and joint conditions may be a function of time and space. – As seen above, inertial and damping effects are now included. Hence, the user should include density and damping in the model. – Nonlinear effects, such as geometric, material, and/or contact nonlinearities, are included by updating the stiffness matrix.    tMu + Cu + K u u = F A. Introduction to Transient Structural Analyses 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 • Transient structural analysis encompasses static structural analysis and rigid dynamic analysis • However, one of the important considerations of performing transient structural analysis is the time step size: – The time step should be small enough to correctly describe the time varying loads. – The time step size controls the accuracy of capturing the dynamic response. Hence, running a preliminary modal analysis is suggested. – The time step size also controls the accuracy and convergence behavior of nonlinear systems. A. Introduction to Transient Structural Analyses 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 In 1959 Newmark presented a family of single-step integration methods for the solution of structural dynamic problems for both blast and seismic loading. During the past 40 years Newmark’s method has been applied to the dynamic analysis of many practical engineering structures. In addition, it has been modified and improved by many other researchers. NATHAN M. NEWMARK September 22, 1910-January 25, 1981 B. Newmark algorithm 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 In order to illustrate the use of this family of numerical integration methods consider the solution of the linear dynamic equilibrium equations written in the following form: t t t tMu + Cu + Ku = F The direct use of Taylor’s series provides an approach to obtain the following two additional equations: 2 3 2 2 6 2 t t t t t t t t t t t t t t t t t t t t t                   u = u u u u u = u u u B. Newmark algorithm 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Newmark truncated these equations and expressed them in the following form: 2 3 2 2 t t t t t t t t t t t t t t t t t                  u = u u u u u = u u u If the acceleration is assumed to be linear within the time step, the following equation can be written: t t t t   u u u Rewrite the Newmark’s equations in standard form   2 21 2 1 t t t t t t t t t t t t t t t t t t t                           u = u u u u u = u u u B. Newmark algorithm 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Rewrite the Newmark’s equations     1 2 3 4 5 6 t t t t t t t t t t t t t t t t b b b b b b             u = u u u u u = u u u u where   1 2 3 4 12 0.51 1 ; ; ;b b tb t t                Subtitute to the dynamic equilibrium equations       1 4 1 2 3 4 5 6 t t t t t t t t t t t t t t b b b b b b b b                M C K u F M u u u C u u u  5 2 6 31 ; 1b tb b t b         B. Newmark algorithm 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 For zero damping Newmark’s method is conditionally stable if max 1 1 1 , 2 2 / 2 and t          Newmark’s method is unconditionally stable if 1 2 2    0.25 and 0.5   are usually usedIn practice, C. Stability of Newmark algorithm 4.3. Transient analysis Finite Element Method Department of Engineering Mechanics – HCMUT 2016 I. Initial calculation: - Form static stiffness matrix K , mass matrix M and damping matrix C and - Specify integration parameters - Form effective stiffness matrix 1 4b b  K M C K - Specify initial conditions: 0 0, andu u   1 0 0 0 0C   u M F u Ku II. For each time step: 1 t t u K F - Calculate effective load vector    1 2 3 4 5 6t t t t t t t t t t t t t tb b b b b b           F F M u u u C u u u - Solve for node displacement vector at time t: - Calculate nodal velocities and accelerations at time t: and constants 1 2 6, ,...,b b b     4 5 6 1 2 3 t t t t t t t t t t t t t t t t b b b b b b               u u u u u u u u u u - Continue next step with , 2 , 3 ,...,t t t t n t     t t t   D. Summary of the Newmark Method for Direct Integration