All Flashcards
What will solving [M]-1[K]{X} = w^2{x} find?
Need to solve iteratively
Finds information on highest mode of vibration
What will solving [A][M]{X} = 1/w^2{x} find?
Need to solve iteratively
Finds information on lowest mode of vibration
What is virtual displacement
An instantaneous imaginary infinitesimal variation of the coordinate. It must be compatible with the restraints of the system
What is virtual work
The work done b all the active forces
Give D’Alembert’s Principle (virtual work)
𝛿W = Σ (Fi - mr(dotdot))𝛿r = 0
Lagrange:
T =
U =
T = 1/2 m x(dot)^2 U = 1/2 k(xi-xi-1)^2
Give Lagrange equation
d/dt(dT/dq(dot)i) - dT/dqi + dU/dqi = 0
What is assumption in Rayleigh method
- Assumes 1 effective mass for entire system
- Approximate method that treats a continuous system as a discrete system
(advantage is that doesn’t deal with partial differential equations)
(deals with large number of ordinary differential equations which is great for fast computing machines)
Write Rayleigh equation
.
Assumption for Rayleigh-Ritzs
Based on premise that a closer approximation to exact natural mode can be obtained by:
- Superposing a number of assumed functions than by using a single assumed function
- Also provides values of the higher natural frequencies and the mode shapes
- Number of frequencies = number of functions
Describe viscous damping
- commonly used
- proportional to velocity
Equation for critical damping
Cc = 2Sqrt(km)
Equation for damping ratio
l = C / Cc
Describe coulomb damping
- steady friction force that occurs in structures
- forces independent of amplitude and frequency
Describe hysteretic damping
- internal dissipation of energy observed when a material is under cyclic stresses
- independent of frequency
Describe complex stiffness
- most real structure not possible to distinguish stiffness and damping effects - therefore often considered together
When would you use damping for troubleshooting
- Issues with excessive vibration in a machine or structure
- Can re-sit or change machinery causing vibration but vibration caused by external forces need to be controlled by damping
- Vibration isolation
- Vibration absorbers attached to machinery to alleviate vibration at resonance
What happens to natural frequency of system after vibration absorbes added
- System now possesses two naturla frequencies that oscillate less at frequencies alongside original natural frequency
What is the expansion theorem
- for free vibration, the vibration of a system is a superposition of the normal modes
What is the importance of finding modal properties
- Avoid resonance through knowledge of natural frequency
- Provide info for calculating fatigue life of components
- Understand system response to shock and random or periodic excitation
- Predict where in the system vibration and noise might be an issue
If response is (1 / 2 / 3), [H(w)] is made up of _____ FRFs
- Displacement
- Velocity
- Acceleration
- Receptance
- Mobility
- Accelerance
Define no-linearity
A system where superposition does not hold true
Define superposition
- consider system with inputs xi and outputs yi with initial conditions yo = y(dot)(o) = 0
- superposition holds if the system response to an input axi + Bxj is ayi + byj for all values of a, B, i and j
Give duffing equation
my(dotdot) + cy(dot) + k1y + k3y^3 = Xsin(wt)
What is reciprocity
For linear system when the frequency response is the same when excited at x1, measure at x2 and when excited at x2 measured at x1
Give 5 types of common non-linearity
- Cubic stiffness
- Bilinear stiffness and damping
- Piecewise linear stiffness
- Non-linear damping - quadratic
- Coulomb friction
What does a phase portrait provide
- information about equilibrium and stability in a system without solving the porblem
Difference of stable and unstable on a phase portrait
Stable: if the motion follows a closed loop after being displaced from equilibrium
Unstable: if the motion followed is large after displaced from equilibrium
Define chaos
when a system, given initial conditions, responds in what seems to be a completely random and unpredictable manner
Understand jumping phenomena
Different from low –> high frequency and
high —> low frequency
When Θ is very small;
sinΘ =
cosΘ =
sinΘ = Θ = 0 cosΘ = 1-1/2Θ^2
Describe theoretical modal analysis
- Description of structure (spatial model)
- Vibration modes (modal model)
- Response levels (response model)
Describe experimental modal analysis
- Response properties
- Vibration modes
- Structural model
