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.

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 X1X1 X2X2 FsFs FsFs FsFs X 2 -X 1 Slope K is the spring stiffness, its unit is N/m. F s is an elastic force known as restoring force. 0

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

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

Spring Connected in Parallel k1k1 k2k2 x1x1 x2x2 FsFs FsFs Spring Connected in Series x1x1 x2x2 x0x0 k1k1 k2k2 FsFs FsFs

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

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

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

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

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

Overdamped case Critical damping

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

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:

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