Dynamics of Structures

**Description of the Course**

Lecture, four hours; outside study, eight hours. Principles of dynamics. Determination of normal modes and frequencies by differential and integral equation solutions. Transient and steady state response. Emphasis on derivation and solution of governing equations using

matrix formulation. Letter grading.

**Background students will need**

Although prerequisites are not enforced for graduate students, it is strongly recommended that each student has taken courses equivalent to “Introduction to Mechanical Vibrations.”

**About the Instructor**

**Jiann-Wen Ju: ** My current research interests concern mechanics of materials and constitutive modelings of advanced engineering materials. In particular, I am interested in micromechanical damage mechanics of brittle composites, micromechanics of random heterogeneous elastic and inelastic fibrous and particulate composites, continuum elastoplastic damage mechanics, plasticity and viscoplasticity theories and computational algorithms, advanced constitutive modeling of materials, nonlinear computational solid mechanics and nondestructive testing. Applications can be made to high-strength plain concretes and fiber reinforced concretes, geological materials, advanced ceramics and ceramic composites, metals and metal matrix composites, and so on.

**Syllabus**

- Free vibration analysis of SDOF equation of motion (Lecture 1)
- Components of the basic SDOF dynamic system
- Equation of motion (linear system)
- Influence of gravity
- Influence of support excitation
- Undamped free vibrations; Damped free vibrations
- Logarithmic decrement (for underdamped system)
- Numerical examples

- Dynamic response to harmonic loading (Lecture 2)
- Response to harmonic loading of a SDOF system
- Undamped system, Damped system
- Steady-state vibration response
- Resonant response
- Accelerometers and displacement meters
- Vibration isolation
- Numerical examples

- Dynamic response to periodic loading and impulsive loads (Lecture 3)
- Fourier series expression of the loading
- Response to Fourier series loading: undamped and damped
- Response to impulsive loads: Rectangular impulse, triangular impulse, sine-wave impulse
- Numerical examples

- Dynamic response to general dynamic loading (Lecture 4)
- Time domain: Duhamel integral for an undamped system
- Numerical integration of Duhamel integral
- Response of damped system
- Numerical examples

- Formulation of MDOF EOM (Lecture 5)
- Formulation of the MDOF equation of motion. Matrix formulation
- Evaluation of elastic properties
- Evaluation of mass properties
- Evaluation of damping properties
- External loading

- MDOF Undamped free vibrations (Lecture 6)
- Vibration frequencies and corresponding mode shapes
- Orthogonality conditions
- Modal superposition method
- Numerical examples

- MDOF Analysis of damped system: modal superposition (Lecture 7)
- Damped uncoupled equations of motion
- Conditions for damping orthogonality; Rayleigh and Caughey damping
- Numerical examples

- Vibrations of undamped continuous systems (strings) (Lecture 8)
- Problem overview and summary of general solution procedure
- Free vibration analysis
- Orthogonality of mode shapes
- Modal expansion theorem
- Forced vibration – normal mode method
- Initial conditions

- Vibrating strings

- Problem overview and summary of general solution procedure
- Examples on strings, and longitudinal vibrations of rods (Lecture 9)
- Examples – vibrating strings
- Free vibrations of a string supported on both ends
- Free vibrations of a string with a spring/mass system on one end
- Forced vibrations of a string fixed at both ends

- Longitudinal vibrations of rods

- Examples – vibrating strings
- Examples on rods, and transverse vibrations of beams (Lecture 10)
- Examples – longitudinal vibration of rods
- Free vibrations with one end fixed and other end free
- Free vibrations with both ends fixed
- Free vibrations of two bars with fixed and free ends3
- Free vibrations of bar with a concentrated mass Mc at free End

- Transverse vibrations of beams

- Examples – longitudinal vibration of rods
- Examples on transverse vibrations of beams (Lecture 11)
- Examples – transverse vibrations of beams
- Free vibrations of a simply supported beam on both ends
- Free vibrations of a beam fixed on both ends
- Free vibrations of a free-free beam
- Two beams hinged together with ends simply–supported

- Examples – transverse vibrations of beams
- Vibrations of structural systems (Lecture 12)
- Vibrations of structural systems
- Examples – vibrations of structural systems
- Two beams with an additional massless spring
- Two beams with an axial force member
- Two beams interconnected by springs with an attached mass

- Hamilton’s principle on dynamic structural systems I (Lecture 13)
- Hamilton’s principle
- Kinetic energy
- Strain energy

- Hamilton’s principle on dynamic structural systems II (Lecture 14)
- Potential energy of external forces
- Potential energy due to axial shortening
- Equation of motion via Hamilton’s principle
- Axial force members
- Examples on axial force members

- Hamilton’s principle on dynamic structural systems III (Lecture 15)
- Equation of motion via Hamilton’s principle
- Circular cylinders – torsion members
- Example on torsional shaft system
- Bernoulli-Euler beams
- Example on Bernoulli-Euler beam with spring and lumped mass at one end
- Bernoulli-Euler beam under an axial force

- Hamilton’s principle on dynamic structural systems IV (Lecture 16)
- Equation of motion via Hamilton’s principle
- Examples on Bernoulli-Euler beams
- Forced vibration of a beam – column supporting a mass
- L shape frame
- L shape frame/re-derivation considering axial motions
- Beam with a rigid appendage supporting a lumped mass4

- Examples on Bernoulli-Euler beams

- Equation of motion via Hamilton’s principle
- Global approximation methods (Lecture 17)
- Preliminary remarks
- Rayleigh’s Quotient
- Numerical examples
- Upper boundedness of Rayleigh’s Quotient
- An additional comment

- The Ritz method I (Lecture 18)
- The Ritz method
- Numerical examples:
- Clamped bar attached to massless linear spring
- Vibration of a simply-supported beam with an additional spring

- The Ritz method II (Lecture 19)
- Numerical examples:
- Use of more coordinate functions in example 1
- Free vibration of a frame
- Structural system with a member under compression.

- Numerical examples: