1. Theoretical Mechanics DYNAMICS
Technical University of Sofia
Branch Plovdiv
Theoretical Mechanics DYNAMICS
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2. Lecture 4 Mechanical Vibrations
A mechanical vibration is the motion of a particle or a body which oscillates about a position of equilibrium.
Free vibrations - occurs naturally with no energy being added to the vibrating system; the motion is maintained by the gravitational or elastic restoring forces only.
Forced vibrations - a periodic force is applied to the system.
The motion is excited by a force,
The motion is excited by the support displacement.
Undamped vibrations - can continue indefinitely as the effects of friction are neglected.
Damped vibrations – in real systems
resistance force is applied.

3. Lecture 4 Mechanical Vibrations
Vibrations without Damping. Free vibrations
W – gravity force;
T – spring (elastic) force.
circular natural frequency
equation of simple harmonic motion

4. Lecture 4
Linear ordinary differential equations (LODE) with constant coefficients
(*)
where n is the order of the equation. If f(t) = 0, the equation is homogeneous, otherwise – non-homogeneous:
(**)
Method of Euler: search the solution of (**) as e λt , where λ is an unknown constant. Substitute x(t) with e λt in (**) to obtain the characteristic equation for λ:
(***)
which has n roots. The general solution is determined by these characteristic roots. Several cases are possible:

5. Lecture 4
1. The roots are all real and different. Denote them by λ1, λ2, ..., λn. Then, the general solution of the original equation (**) is
(1)
where C1, C2, ..., Cn
2. There are m ≤ n equal real roots, λ1= λ2 = …= λm, while the other roots are real and different. In this case, the general solution is
(2)
3. There are m equal pairs (2m ≤ n) of complex conjugate roots,
λ = α ± iβ, while the other roots are real and different. Then, the general solution has the form
(3)
where A1, A2, ..., Am, В1, В2, ..., Bm are arbitrary constants connected with C1, C2, ..., Cm.

6. The roots of the characteristic equation are imaginary and different
Lecture 4
Mechanical Vibrations
Vibrations without Damping. Solution of the equation.
Equation is a homogeneous, second-order, linear, differential equation with constant coefficients.
Substituting into the equation and dividing through by , we write the characteristic equation
and obtain the roots
The roots of the characteristic equation are imaginary and different

7. Using the simple relation
It can be shown, that
and since c1 and c2 are unknown constants, it can be rewritten as
where C1 , C2 and A are unknown constants.

8. Lecture 4 Mechanical Vibrations
Vibrations without Damping. Graphical representation of solution.
A – amplitude of vibration;  - phase angle; one cycle in time - period

9. Lecture 4 Mechanical Vibrations
Vibrations without Damping. Graphical representation of solution.
The maximum displacement of the block from its equilibrium position is defined as the amplitude of vibration – A
A full cycle is described as the angle increases by 2p rad. The corresponding value of t, denoted by t, is called the period of the free vibration and is measured in seconds.
The number of cycles described per unit of time is denoted by fn and is known as the natural frequency of the vibration.
The frequency is expressed in cycles/s. This ratio of units is called a hertz (Hz).
1 Hz = 1 cycle/s = 2 rad/s.

10. Lecture 4 Mechanical Vibrations Sample Problem - 1
A 32-kg block attached to a spring of constant k=12 kN/m can move without friction in a slot as shown. The block is given an initial 300-mm displacement downward from its equilibrium position and released. Determine 1.5 s after the block has been released
the total distance traveled by the block,
the acceleration of the block.

11. Lecture 4 Mechanical Vibrations Sample Problem - 2
The two fixed counter rotating pulleys are driven at the same angular speed w0. A round bar is placed off center on the pulleys as shown. Determine the natural frequency of the resulting bar motion. The coefficient of kinetic friction between the bar and pulleys is m.

12. Lecture 4 Mechanical Vibrations Damped Free Vibrations.
c - coefficient of viscous damping
[c] = N • s/m
Mechanical Vibrations
Damped Free Vibrations.
A type of damping of special interest is the viscous damping caused by fluid friction at low and moderate speeds.
- viscous damping force
equation of simple harmonic motion with damping
n – damping coefficient
dashpot

13. Lecture 4
Mechanical Vibrations
Damped Free Vibrations. Solution of the equation.
Substituting into the equation and dividing through by , we write the characteristic equation
and obtain the roots
We can distinguish three different cases of damping, depending upon the value of the
Heavy damping (overdamped system):
Critical damping (critically damped system):
Light damping (underdamped system):

14. Lecture 4 Mechanical Vibrations
Damped Free Vibrations. Solution of the equation.
Heavy damping –
The roots of the characteristic equation are real and different
This solution corresponds to a nonvibrating motion. The effect of
damping is so strong that when the block is displaced and released, it
simply creeps back to its original position without oscillating.
Critical damping –
The characteristic equation has a double root and the general solution is
Critically damped systems are of special interest in engineering applications since they regain their equilibrium position in the shortest possible time without oscillation.

15. Lecture 4 Mechanical Vibrations
Damped Free Vibrations. Graphical representation of solution.
Heavy and critical damping

16. Lecture 4 Mechanical Vibrations
Damped Free Vibrations. Solution of the equation.
Light damping –
The roots of the equation are complex and conjugate, and the general solution of equation is of the form
where is the circular frequency of the damped vibration (damped natural frequency of the system).
The general solution of equation is:
The motion is vibratory with diminishing amplitude and the time interval
is known as the period of the damped vibration.
Since , then

17. Lecture 4 Mechanical Vibrations
Damped Free Vibrations. Graphical representation of solution.
Light damping
The initial amplitude, A, diminishes with each cycle of vibration.
The motion is
confined within the bounds of the exponential curve:
Thus
and the vibrations fade out in time!

18. Lecture 4 Mechanical Vibrations Sample Problem - 3
The 8-kg body is moved 0.2 m to the right of the equilibrium position and released from rest at time t = 0. Determine its displacement at time t = 2 s. The viscous damping coefficient c is 20 N.s/m, and the spring stiffness is 32 N/m.