1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction

Transcription

1 1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction Lesson Objectives: 1) List examples of MDOF structural systems and state assumptions of the idealizations. 2) Formulate the equation of motion for MDOF systems and describe its elements. 3) Quantitatively compute the natural frequency and mode shapes for a MDOF system. 4) Quantitatively compute the response of a damped MDOF system under various vibrations. Background Reading: 1) Read/review. Introduction: 1) In real world applications, structures have multiple degrees-of-freedom (MDOF). 2) Examples of MDOF structural systems include: a. b. c. d. 3) The simplest idealization of MDOF structure is a. 4) Assumptions of a typically include: a. b. i. c. d. MDOF Intro Richard L Wood, 2018 Page 1 of 16

2 5) Sketch of a (with damping neglected): 6) How can shear building can be analyzed? a. Equation of motion: b. Solution methods? i. Depend on damping and other parameters in general. MDOF Intro Richard L Wood, 2018 Page 2 of 16

3 MDOF Damping Matrix: 1) Can be defined as either or. 2) damped matrix of a linear system satisfies the following identity: 3) This is termed because the natural modes of the damped system are real-valued and identical to those of the associated undamped system. a. Therefore solved using the : 4) The classical damped matrix,, will be and. 5) If the damping is considered to be, the natural modes of vibration are no longer, the matrix is, and is not applicable. a. For this case, analytical solutions can be developed as discussed in. 6) If is, the equation of motion can be decoupled MDOF Intro Richard L Wood, 2018 Page 3 of 16

4 Decoupled MDOF Equation of Motion: 1) The equation of motion for a damped MDOF system can be written as: 2) The assumed solution can be written as 3) In this assumed solution, modal expansion simplifies the expression for the displacement response, where is the. a. Other names include: and. b. Function of the structural system: and. 4) The assumed solution can be differentiated for velocity and acceleration expressions: 5) Via substitution, the equation of motion for the damped MDOF system can be written as: MDOF Intro Richard L Wood, 2018 Page 4 of 16

5 6) Premultiplication of the above equation by, yields: 7) If classical damping exists, the equation of motion can be decoupled into because the modes are : 8) As shown in the undamped case, the matrix equation of motion simplifies into a set of. 9) In each equation of the set of, the damping ratio can be defined for each mode similar to a previously described SDOF system. 10) Where the damped natural frequency of the system is given by: MDOF Intro Richard L Wood, 2018 Page 5 of 16

6 Free and Forced Vibration of Damped MDOF Systems: 1) For a damped system, the matrix system of equations can be decoupled into an equation of a representative SDOF in. 2) For the free vibration component, a closed form solution exists of the form: 3) Where the modal coordinates are defined as: 4) If an externally applied force exists, the solution can be found using,, or. a. These solution strategies would directly correspond to the formulation illustrated in earlier sections of the lesson notes. b. Review Chopra or CIVE 842 (or its equivalent) notes for additional details. MDOF Intro Richard L Wood, 2018 Page 6 of 16

7 Damping Matrices in MDOF Systems: 1) A completely defined damping matrix is needed when: a. The damping matrix is. b. The analysis is beyond the. i. As a result, is not valid. ii. Why? iii. As a result: require modifications. 2) The damping matrix is not calculated from the structural properties of the building. 3) Classical damping is an okay approximate idealization when similar damping mechanisms are throughout the structure. Rayleigh Damping: 1) Classical damping matrices can be developed with three types: a. b. c. 2) In most commonly assumed type of damping is, damping. This is proportional to both and. 3) The constants of and have units of and respectively. 4) Stiffness is considered since it is interpreted that damping can model the energy dissipation arising from. 5) Mass is considered since it can be argued to physically represent, which is negligibly small for most structures. 6) However by themselves, neither nor are appropriate for practical application. MDOF Intro Richard L Wood, 2018 Page 7 of 16

8 7) To find the damping ratios of all modes, typically the value is assumed in two modes and can solve to find the constants of and. 8) This can be shown graphically as: MDOF Intro Richard L Wood, 2018 Page 8 of 16

9 Mode Shapes and Natural Frequencies of MDOF Systems: 1) This concept can be related back to. a. Where the total displacement response is equal to the summation of many. b. Recall: 2) So in short, mode shapes are that will define the total. 3) Another way to consider these are as shapes that exist for a given structure that if displaced in one of these shapes or profiles and then, it will vibrate in a simple harmonic motion that maintains the. 4) Mode shapes are also known as. 5) These mode shapes are a function of the, namely and. a. Assuming that the damping is. 6) Example of mode shapes: First Mode: MDOF Intro Richard L Wood, 2018 Page 9 of 16

10 Second Mode: 7) During a the vibration of a mode shape, the time required for of simple harmonic motion in one of these natural modes is known as the. a. Recall: b. For a system, natural vibration frequencies exist. c. The of the natural periods is known as the denoted as. d. The shorter one(s) is known as the, or in this specific case denoted as. 8) So how are these calculated? This is will result in a problem. MDOF Intro Richard L Wood, 2018 Page 10 of 16

11 9) Equation of motion for a MDOF system: 10) Modes are a function of and under, therefore let s simplify to: 11) The assumed solution to this 2 nd order ordinary differential equation is: 12) Substitution into the above equation for acceleration: 13) Substitution into the above equation for displacement: 14) Factoring and clean-up to identify the problem to solve: MDOF Intro Richard L Wood, 2018 Page 11 of 16

12 15) Therefore the generalized eigenvalue problem to solve is: 16) Note this is the identical for undamped and MDOF systems. 17) Therefore to find the natural frequencies of the system: 18) For n-degrees of freedom: a. A order polynomial exists with unique solutions. b. This polynomial is of order in terms of. c. Each frequency corresponds to a mode shape: 19) So for a particular mode/frequency (or ), the mode shape can be computed via: MDOF Intro Richard L Wood, 2018 Page 12 of 16

13 Example: Forced Vibration of a Damped MDOF 1) A two story reinforced concrete shear building shown below is excited by a forced vibration at the 1 st story, sin. Determine the steady state response of the system. Assume damping is 5% of critical in each mode. Note the driving frequency is not in resonance with any of the natural frequencies. 2) The equation of motion considering the structure above: MDOF Intro Richard L Wood, 2018 Page 13 of 16

14 3) The solution strategy using modal superposition. a. Find the natural frequencies and the mode shapes: b. Uncouple the system of equations: c. Determine the generalized force vector: MDOF Intro Richard L Wood, 2018 Page 14 of 16

15 d. Consider each modal coordinate solution: i. Recall that the complete response for an SDOF system is: MDOF Intro Richard L Wood, 2018 Page 15 of 16

16 e. The final solution can be written as: MDOF Intro Richard L Wood, 2018 Page 16 of 16