Chapter 7. Free and Forced Response of Single-Degree-of-Freedom Linear Systems

1. Chapter 7. Free and Forced Response of Single-Degree-of-Freedom Linear Systems 7.1 Introduction • Vibration: System oscillates about a certain equilibrium position. • Mathematical models: (1) Discrete-parameter systems, or lumped systems. • (2) Distributed-parameter systems, or continuous systems. • Usually a discrete system is a simplification of a continuous system through a suitable “lumping” modelling. • Importance: performance, strength, resonance, risk analysis, wide engineering applications Single-Degree-of-Freedom (Single DOF) linear system • Degree of freedom: the number of independent coordinates required to describe a system completely. • Single DOF linear system: • Two DOF linear system: System response • Defined as the behaviour of a system characterized by the motion caused by excitation. • Free Response: The response of the system to the initial displacements and velocities. • Forced Response: The response of the system to the externally applied forces.

2. 7.2 Characteristics of Discrete System Components The elements constituting a discrete mechanical system are of three types: The elements relating forces to displacements, velocities and accelerations. • Spring: relates forces to displacements X1 X2 Fs Fs Fs Slope K is the spring stiffness, its unit is N/m. X2-X1 0 Fs is an elastic force known as restoring force.

3. Damper: relates force to velocity The damper is a viscous damper or a dashpot Fd Fd c Fd Slope C is the viscous damping coefficient, its unit is N·s/m 0 Fd is a damping force that resists an increase in the relative velocity

4. Discrete Mass: relates force to acceleration m Fm Fm Slope m, its unit is Kg 0 Note: 1. Springs and dampers possess no mass unless otherwise stated 2. Masses are assumed to behave like rigid bodies

5. Spring Connected in Parallel k1 x1 x2 Fs Fs k2 • Spring Connected in Series x0 x2 x1 Fs Fs k1 k2

6. 7.3 Differential Equations of Motion for First Order and Second Order Linear Systems • A First Order System: Spring-damper system k x(t) Free body diagram: Fs(t) F(t) F(t) Fd(t) c • A Second Order System: Spring-damper-mass system k x(t) Free body diagram: Fs(t) m m F(t) F(t) Fd(t) c

7. 7.4 Harmonic Oscillator k x(t) Second order system: m F(t) Undamped case, c=0: (1) Solution: is called Phase angle With initial conditions and

8. Period(second): Natural frequency: Hertz(Hz) Example:

9. 7.5 Free Vibration of Damped Second Order Systems • A Second Order System: Spring-damper-mass system k x(t) Free body diagram: Fs(t) m m F(t) F(t) Fd(t) c • Express it in terms of nondimensional parameters: (1.7.1) Viscous damping factor: • The solution of (1.7.1) can be assumed to have the form, We can obtain the characteristic equation With solution:

10. The locus of roots plotted as a function of (1) Undamped case, the motion is pure oscillation (2) s1 ,s2 are complex conjugates. Underdamped Case (3) Critical damping (4) Overdamped case, the motion is aperiodic and decay exponentially in terms of

11. Critical damping Overdamped case

12. , where the frequency of the damped free vibration Figure 1.7.3 Underdamped Case as

13. 7.6 Logarithmic Decrement • Experimentally determine the damping of a system from the decay of the vibration amplitude during ONE complete cycle of vibration: Let , We obtain Introduce logarithmic decrement for small damping, • For any number of complete cycles:

14. 7.7 Energy Method Total energy of a spring-mass system on a horizontal plane: