The University of New South Wales School of Civil and Environmental Engineering CVEN 4308 Structural Dynamics Assignment Due date: 5pm, Friday, 22/11/2019 Total Marks Available: 50 First Name: Last Name: zID:

Free Vibration Response of MDOF-Systems A 5-storey building is idealized as a shear frame structure with rigid floors and consequently, 5 degrees of freedom as depicted in Fig. 1. All storeys have the same height h , but the columns have different stiffness coefficients k i . Experiments have found that damping is small and can therefore be neglected. The stiffness of a column on the i-th floor is given as k i ? 12 EI C i h 3 ; @ i ? 1 ; 2 :::; 5 : (1) Note that axial deformations of the columns are also neglected ( EA Ñ 8 ). The mass of each storey is lumped at the floor level as indicated in Fig. 1. The set of parameters describing the system is solely based on the zID of each student. The zID consists of a seven digit number as z C 1 C 2 C 3 C 4 C 5 C 6 C 7 ; where each constant C i defines a parameter of the system. Note that each digit in the zID that is equal to 0 should be replaced by C i ? 5 . In the following, the system properties are listed: Data: I C i ? C i ? 1000 r cm 4 s m ? C 6 ? 1000 r kg s h ? C 7 r m s E ? 200 r kN/mm 2 s . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... . ........... ........... - - - - - u 1 ( t ) u 2 ( t ) u 3 ( t ) u 4 ( t ) u 5 ( t ) ? ? ? ? ? 6 6 6 6 6 h h h h h k 1 k 2 k 3 k 4 k 5 k 1 k 2 k 3 k 4 k 5 5 m 3 m 4 m m 2 m 1 Figure 1: 5-storey shear frame. 1

Questions 1. Derive the equations of motion for the 5-storey shear frame structure. (10 Marks) (a) Provide the mass matrix m . (5 Marks) (b) Provide the stiffness matrix k . (5 Marks) 2. Solve the general eigenvalue problem to determine the mode shapes and eigenfrequencies of the frame structure. (10 Marks) (a) Calculate the circular natural frequencies ! i and the natural periods T i of each mode. (5 Marks) (b) Determine the mode shapes ' i and normalize each mode such that the maximum value is 1.0. (5 Marks) 3. Determine the analytical solution u p t q for the free vibration response based on the modal expansion technique. (18 Marks) (a) Calculate the generalized mass matrix. (5 Marks) (b) Determine the vector of the modal contribution factors and its temporal derivative at t ? 0 s, i.e. q p 0 q and 9 q p 0 q . The initial conditions of the frame structure are given as: (5 Marks) u 0 ? ? ? ? ? ? ? ? ? C 1 ? C 2 ? C 3 C 4 C 5 ˝ ˚ ˚ ˚ ˚ ˚ ˛ ; 9 u 0 ? 0 : (c) Provide the expression for the free vibration response of the system. (5 Marks) (d) Evaluate the analytical solution for the time interval 0 ¤ t ¤ r 5 T 1 s 1 using the time increment ? t 1 and determine the maximum and minimum values for the displace- ment of floor 5, i.e max r u ? t 1 5 p t qs and min r u ? t 1 5 p t qs . (3 Marks) ? t 1 ? T 5 20 : (2) Round the value of the time increment to two significant digits! 4. Determine the numerical solution ~ u p t q for the free vibration response based on New- mark’s method (use the constant average acceleration method). (12 Marks) (a) Compute the numerical solution with the time increment ? t 1 in the same time in- terval ( 0 ¤ t ¤ r 5 T 1 s ) and determine the maximum and minimum values for the displacement of floor 5, i.e max r ~ u ? t 1 5 p t qs and min r ~ u ? t 1 5 p t qs . (4 Marks) (b) Plot the solution for u 5 p t q and ~ u 5 p t q in one diagram. (2 Marks) (c) Discuss the differences in the results. (2 Marks) 1 The mathematical operator r ? s denotes the so-called ceiling function. This function rounds the value to the next greater integer number. 2

(d) Determine a suitable time increment ? t 2 for the numerical solution such that the relative error ? in the maximum displacement is less than 0.1%. (4 Marks) ? ? | max r u ? t 1 5 p t qs? max r ~ u ? t 1 5 p t qs| max r u ? t 1 5 p t qs ? 100% (3) Use the following approach to determine ? t 2 : ? t 2 ? T 5 20 n (4) where n P N 2 . Round the value of the time increment to two significant digits! Note: ? To determine the solution either a spreadsheet developed in Excel or a computer pro- gram written in Matlab, Fortran, C++, etc. has to be used. This has to be explained and documented in the submitted report. ? Late submissions of the assignment will be penalised at a rate of 20% per day after the due time and date have expired 3 . ? Submit 1 pdf file through the Moodle assignment item containing all relevant data (as- signment task, report, source code). 2 N denotes the set of natural numbers. 3 Note that even if the submission is only seconds too late the penalty applies! 3