ME 425 Mechanical Vibrations

ME 425 Mechanical Vibrations

Newton’s Laws : The equations of motion of a mechanical system is determined from Newton’s laws of motion :

where G is the center of mass of the body.

Energy methods: KE+PE = constant for conservative systems. Also : (KE)max=(PE)max .

Mechanical System Elements: Elastic elements store potential energy, do not dissipate energy F=k(x2-x1)

Viscous damping: Dissipates energy, forces are always opposite to the velocity of the body

SINGLE DEGREE OF FREEDOM SYSTEMS

Harmonic Motion : Mass spring system .

Solution :

: Natural frequency : Period of motion

Damped system :

Underdamped solution :

where : damped natural frequency

and : Damping ratio.

Forced Vibration

Or :

Harmonic excitation :

Steady state response :

where and

Base excitation :

, response:

: Transmissibility.

Force transmitted to the base :

Rotating Unbalance:

Response :

,

Measuring devices :

Let : Motion of mass relative to the base.

If , equation becomes:

solution :

,

The device becomes an accelerometer for low frequencies (r<0.2) and a seismometer (r>3) for high frequencies.

Periodic Excitation : A periodic function F(t) with period T can be expressed in Fourier series as :

Where

Single DOF system:

where:

Non-periodic Forcing Single DOF system :

Impulse response function : Convolution integral :

Energy loss in damping in one period:

In viscous damping:

In Coulomb damping:

In structural damping:

MULTI DEGREE OF FREEDOM SYSTEMS

Lagrange Equations

where L = T-V, qi : generalized coordinates, Qi : generalized forces.

Equation of Motion : where M, C and K are nxn symmetric mass, damping and stiffnes matrices, x is the displacement vector, F is the forcing vector.

Undamped Free Vibration

Assume : Eigenvalue problem :

Solution gives natural frequencies w, and the mode shapes u.

Eigenvalue Problem: has a nontrivial solution if: . Solution gives gives ui.

Procedure for modal analysis

1.  Calculate .

2.  Calculate

3.  Solve the symmetric Eigenvalue Problem with , to get wi and vi. Form .

4.  Calculate and .

5.  Calculate the modal initial conditions : and Solve .

6.  Obtain the solution in physical coordinates by:

Eigenvalue Problem: has a nontrivial solution if: . Solution gives gives vi.

Mode Expansion Method : Let . Then, the equations are tranformed to. Solution is :

Where: and . Physical solution is: .

Forced Response: The same procedure for modal analysis applies.

·  Make the coordinate transformation x(t)=Sr(t).

·  Modal equations become : . where .

·  Solve the decoupled equations in modal coordinates r(t) and then back transform to physical coordinates x(t).

Damped Systems: In general modal analysis does not apply to damped systems. It applies only if the system is proportionally damped. In this case: . In this case the modal equations are: . where .

Impedance Method for Harmonic Forcing: Assume , where Z is the system impedance matrix.

Rayleigh Quotient: Is used to estimate the first natural frequency if an approximate first mode shape is known.

Vibration Absorbers: Steady state vibration amplitude of the main mass becomes zero when . Define and . Frequency equation is : where .