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Course Outline 1. MATLAB tutorial 2. Motion of systems that can be idealized as particles • • • Description of motion; Newton’s laws; Calculating forces required to induce prescribed motion; Deriving and solving equations of motion 3. Conservation laws for systems of particles • • • Work, power and energy; Linear impulse and momentum Angular momentum Exam topics 4. Vibrations • • • Characteristics of vibrations; vibration of free 1 DOF systems Vibration of damped 1 DOF systems Forced Vibrations 5. Motion of systems that can be idealized as rigid bodies • Description of rotational motion; Euler’s laws; deriving and solving equations of motion; motion of machines Particle Dynamics: Concept Checklist • • • • • • • • • • • • Understand the concept of an ‘inertial frame’ Be able to idealize an engineering design as a set of particles, and know when this idealization will give accurate results Describe the motion of a system of particles (eg components in a fixed coordinate system; components in a polar coordinate system, etc) Be able to differentiate position vectors (with proper use of the chain rule!) to determine velocity and acceleration; and be able to integrate acceleration or velocity to determine position vector. Be able to describe motion in normal-tangential and polar coordinates (eg be able to write down vector components of velocity and acceleration in terms of speed, radius of curvature of path, or coordinates in the cylindrical-polar system). Be able to convert between Cartesian to normal-tangential or polar coordinate descriptions of motion Be able to draw a correct free body diagram showing forces acting on system idealized as particles Be able to write down Newton’s laws of motion in rectangular, normal-tangential, and polar coordinate systems Be able to obtain an additional moment balance equation for a rigid body moving without rotation or rotating about a fixed axis at constant rate. Be able to use Newton’s laws of motion to solve for unknown accelerations or forces in a system of particles Use Newton’s laws of motion to derive differential equations governing the motion of a system of particles Be able to re-write second order differential equations as a pair of first-order differential equations in a form that MATLAB can solve Particle Kinematics Inertial frame – non accelerating, non rotating reference frame Particle – point mass at some position in space Position Vector Velocity Vector j r(t ) x(t )i y(t ) j z(t )k v(t) dt r(t) v(t ) vx (t )i v y (t ) j vz (t )k d dx dy dz O xi yj zk i j k k dt dt dt dt dx dy dz vx (t ) v y (t ) vz (t ) dt dt dt path of particle r(t+dt) • Direction of velocity vector is parallel to path • Magnitude of velocity vector is distance traveled / time Acceleration Vector dv y dvx dv d a(t ) ax (t )i a y (t ) j a z (t )k vx i v y j vz k i j z k dt dt dt dt dv y dvx dvz d 2x d2y d 2z ax (t ) a y (t ) az (t ) 2 2 dt dt dt dt dt dt 2 i Particle Kinematics • Straight line motion with constant acceleration 1 r X 0 V0t at 2 2 i v V0 at i a ai • Time/velocity/position dependent acceleration – use calculus t r X 0 v(t )dt i 0 v (t ) t v V0 a(t )dt i 0 x (t ) vdv V0 dv g (t ) a dt f (v) dx g (t ) v dt f (v) a( x)dx 0 v f (v)dv g (t )dt V0 x (t ) X0 t 0 t f ( x)dv v(t )dt 0 Particle Kinematics • Circular Motion at const speed sin t cos t s R V R r R cos i sin j R j v R sin i cos j Vt i V2 a R(cos i sin j) Rn n R 2 2 • General circular motion d / dt d / dt d 2 / dt 2 s R V ds / dt R r R cos i sin j v R sin i cos j Vt a R ( sin i cos j) R 2 (cos i sin j) dV V2 Rt Rn t n dt R 2 Rcos n Rsin Particle Kinematics • Arbitrary path v Vt dV V2 a t n dt R t R r x( )i y( ) j n dx d 2 y dy d 2 x d d2 d d2 t 1 R 2 3/2 2 dx dy d d • Polar Coordinates dr d v er r e dt dt 2 d 2r d 2 dr d d a r e er r 2 2 dt 2 dt dt dt dt e r j i er Newton’s laws • For a particle F ma • For a rigid body in motion without rotation, or a particle on a massless frame Mc 0 j You MUST take moments about center of mass i NA W TB NB Calculating forces required to cause prescribed motion of a particle • • • • • • Idealize system Free body diagram Kinematics F=ma for each particle. M c 0 (for rigid bodies or frames only) Solve for unknown forces or accelerations Deriving Equations of Motion for particles 1. Idealize system 2. Introduce variables to describe motion (often x,y coords, but we will see other examples) 3. Write down r, differentiate to get a 4. Draw FBD 5. F ma 6. If necessary, eliminate reaction forces 7. Result will be differential equations for coords defined in (2), e.g. m d 2 x dx kx kY sin t dt 2 dt 0 8. Identify initial conditions, and solve ODE Motion of a projectile r X 0i Y0 j Z 0k t 0 dr Vx 0i V y 0 j Vz 0k dt V0 k X0 j i 1 2 r X 0 Vx0t i Y0 V y 0t j Z 0 Vz 0t gt k 2 v Vx 0 i V y 0 j Vz 0 gt k a gk Rearranging differential equations for MATLAB • Example d2y dt 2 2n dy n2 y 0 dt • Introduce v dy / dt • Then v d y v dt 2n v n2 y • This has form dw f (t , w ) dt y w v Conservation laws for particles: Concept Checklist • • • • • • • • • • • • • • • • • • • • Know the definitions of power (or rate of work) of a force, and work done by a force Know the definition of kinetic energy of a particle Understand power-work-kinetic energy relations for a particle Be able to use work/power/kinetic energy to solve problems involving particle motion Be able to distinguish between conservative and non-conservative forces Be able to calculate the potential energy of a conservative force Be able to calculate the force associated with a potential energy function Know the work-energy relation for a system of particles; (energy conservation for a closed system) Use energy conservation to analyze motion of conservative systems of particles Know the definition of the linear impulse of a force Know the definition of linear momentum of a particle Understand the impulse-momentum (and force-momentum) relations for a particle Understand impulse-momentum relations for a system of particles (momentum conservation for a closed system) Be able to use impulse-momentum to analyze motion of particles and systems of particles Know the definition of restitution coefficient for a collision Predict changes in velocity of two colliding particles in 2D and 3D using momentum and the restitution formula Know the definition of angular impulse of a force Know the definition of angular momentum of a particle Understand the angular impulse-momentum relation Be able to use angular momentum to solve central force problems Work and Energy relations for particles k Rate of work done by a force (power developed by force) Total work done by a force P F v W F vdt W 0 Kinetic energy F dr r0 v j F(t) r1 1 2 1 T m v m vx2 v2y vz2 2 2 P O i t1 F P r0 k j O i r1 v Power-kinetic energy relation r1 Work-kinetic energy relation r0 k dT P dt W F dr T T0 r0 i P O j r1 Potential energy Type of force Potential energy m Potential energy of a conservative force (pair) Gravity acting on a particle near earths surface F j V mgy y i r V (r ) F dr constant r0 F grad(V ) P r0 k i Gravitational force exerted on mass m by mass M at the origin Force exerted by a spring with stiffness k and unstretched length 1 2 V k r L0 2 j r F i j r1 m r GMm V r r L0 F O F 2 Force acting between two charged particles V Q1Q2 4 r j +Q1 i r 1 Force exerted by one molecule of a noble gas (e.g. He, Ar, etc) on another (Lennard Jones potential). a is the equilibrium spacing between molecules, and E is the energy of the bond. 6 a 12 a E 2 r r F +Q2 2 F j i 1 r Energy relations for conservative systems subjected to external forces F1ext Internal Forces: (forces exerted by one part of the system on another) R ij m4 F3ext m1 R13 R31 R12 m R32 3 R23 R21 m2 External Forces: (any other forces) Fiext System is conservative if all internal forces are conservative forces (or constraint forces) F2ext Energy relation for a conservative system t0 t t0 t Kinetic and potential energy at time t0 T0 V0 Kinetic and potential energy at time t T V Work done by external forces Wext t0 t forces Fiext (t ) v(t )dt t0 Wext T V T0 V0 Impulse-momentum relations Impulse-momentum for a single particle t1 Linear Impulse of a force F(t) F(t )dt k t0 Linear momentum of a particle p mv Impulse-momentum relations F dp dt v m j O i p1 p 0 Impulse-momentum for a system of particles Total external impulse tot particles Total linear momentum F1ext t0 t ptot ext i F (t )dt t0 mi vi Fiext (t ) particles Special case (closed system) dptot dt F3ext m1 R13 R31 R12 m R32 3 R23 R21 m2 F2ext particles Impulse-momentum m4 tot ptot tot 0 ptot 0 (momentum conserved) Collisions vB0 x A0 vx A mAvxA1 mB vxB1 mAvxA0 mB vxB 0 Momentum B Restitution formula * vA1 x vB1 x B A A v A0 B Momentum B0 v v A1 v B1 v B 0 mA (1 e) v B 0 v A0 mA mB v A1 v A0 mB (1 e) v B 0 v A0 mA mB mB v B1 mA v A1 mB v B 0 mA v A0 B1 A1 B0 A0 B0 A0 Restitution formula v v v v (1 e) v v n n n A v B1 v A1 e v B 0 v A0 v A1 v A0 B B1 v v B1 v B 0 mB (1 e) v B 0 v A0 n n mB mA mA (1 e) v B 0 v A0 n n mB mA Angular Impulse-Momentum Equations for a Particle k Angular Impulse t0 t r (t ) F(t )dt Angular Momentum h r p r mv Impulse-Momentum relations Special Case r(t) O t0 dh M dt 0 h1 h0 x i F(t) j z y h1 h0 Angular momentum conserved Useful for central force problems Free Vibrations – concept checklist You should be able to: 1. Understand simple harmonic motion (amplitude, period, frequency, phase) 2. Identify # DOF (and hence # vibration modes) for a system 3. Understand (qualitatively) meaning of ‘natural frequency’ and ‘Vibration mode’ of a system 4. Calculate natural frequency of a 1DOF system (linear and nonlinear) 5. Write the EOM for simple spring-mass-damper systems by inspection 6. Understand natural frequency, damped natural frequency, and ‘Damping factor’ for a dissipative 1DOF vibrating system 7. Know formulas for nat freq, damped nat freq and ‘damping factor’ for spring-mass system in terms of k,m,c 8. Understand underdamped, critically damped, and overdamped motion of a dissipative 1DOF vibrating system 9. Be able to determine damping factor and natural frequency from a measured free vibration response 10. Be able to predict motion of a freely vibrating 1DOF system given its initial velocity and position, and apply this to design-type problems Vibrations and simple harmonic motion Typical vibration response • Period, frequency, angular frequency amplitude Displacement or Acceleration Period, T y(t) Peak to Peak Amplitude A time Simple Harmonic Motion x(t ) X 0 X sin t v(t ) V cos(t ) a (t ) A sin t ) V X A V Vibration of 1DOF conservative systems Harmonic Oscillator Derive EOM (F=ma) m d 2s s L0 k dt 2 Compare with ‘standard’ differential equation xs 1 n2 Solution s(t ) L0 (s0 L0 )2 v02 / n2 sin(nt ) Natural Frequency k n m C L0 x0 s0 m k Vibration modes and natural frequencies •Vibration modes: special initial deflections that cause entire system to vibrate harmonically •Natural Frequencies are the corresponding vibration frequencies x1 x2 k k m k m Number of DOF (and vibration modes) In 2D: # DOF = 2*# particles + 3*# rigid bodies - # constraints In 3D: # DOF = 3*# particles + 6*# rigid bodies - # constraints Expected # vibration modes = # DOF - # rigid body modes A ‘rigid body mode’ is steady rotation or translation of the entire system at constant speed. The maximum number of ‘rigid body’ modes (in 3D) is 6; in 2D it is 3. Usually only things like a vehicle or a molecule, which can move around freely, have rigid body modes. x1 x2 k k m k m Calculating nat freqs for 1DOF systems – the basics m y k,L0 EOM for small vibration of any 1DOF undamped system has form 1 d2y y C 2 2 n dt n is the natural frequency 1. Get EOM (F=ma or energy) 2. Linearize (sometimes) 3. Arrange in standard form 4. Read off nat freq. Tricks for calculating natural frequencies of 1DOF undamped systems • Using energy conservation to find EOM s 2 1 ds 1 KE PE m k ( s L0 ) 2 const 2 dt 2 k, d m 2 d ds ds d s ( KE PE ) m 2 k ( s L0 ) 0 dt dt dt dt d 2s m 2 ks kL0 dt • Nat freq is related to static deflection n g k,L0 m L0+ Linearizing EOM d2y f ( y) C 2 dt Sometimes EOM has form We cant solve this in general… Instead, assume y is small d2y df m 2 f (0) dt dy d 2 y df m 2 dt dy y ... C y 0 y C f (0) y 0 There are short-cuts to doing the Taylor expansion Writing down EOM for spring-mass-damper systems Commit this to memory! (or be able to derive it…) s=L0+x k, L0 F ma m c d2x dt 2 d2x dt 2 2n c dx k x 0 m dt m dx n2 x 0 dt n k m c 2 km x(t) is the ‘dynamic variable’ (deflection from static equilibrium) k1 k1 k2 Parallel: stiffness k k1 k2 c1 c2 Parallel: coefficient c c1 c2 k2 1 1 1 Series: stiffness k k1 k2 c1 c2 1 1 1 Parallel: coefficient c c c 1 2 Canonical damped vibration problem 2 EOM m d s dt c 2 ds ks kL0 dt Standard Form sx n k m with s s0 ds v0 dt 2 dx xC x x0 2 2 dt n dt n c x0 s0 C L0 2 km 1 d 2x s=L0+x k, L0 t 0 dx v0 dt m t 0 c d n 1 2 Overdamped 1 Critically Damped 1 Underdamped 1 Overdamped 1 v (n d )( x0 C ) v (n d )( x0 C ) x(t ) C exp(nt ) 0 exp(d t ) 0 exp(d t ) 2d 2d Critically Damped 1 Underdamped 1 x(t ) C ( x0 C) v0 n ( x0 C)t exp(nt ) v n ( x0 C ) x(t ) C exp(nt ) ( x0 C )cos d t 0 sin d t d Calculating natural frequency and damping factor from a measured vibration response Displacement x(t0) x(t1) x(t2) x(t3) time t0 t1 t2 t3 t4 T Measure log decrement: Measure period: Then 1 n x(t0 ) x ( t ) n log T 4 2 2 4 2 2 n T Forced Vibrations – concept checklist You should be able to: 1. Be able to derive equations of motion for spring-mass systems subjected to external forcing (several types) and solve EOM using complex vars, or by comparing to solution tables 2. Understand (qualitatively) meaning of ‘transient’ and ‘steady-state’ response of a forced vibration system (see Java simulation on web) 3. Understand the meaning of ‘Amplitude’ and ‘phase’ of steady-state response of a forced vibration system 4. Understand amplitude-v-frequency formulas (or graphs), resonance, high and low frequency response for 3 systems 5. Determine the amplitude of steady-state vibration of forced spring-mass systems. 6. Deduce damping coefficient and natural frequency from measured forced response of a vibrating system 7. Use forced vibration concepts to design engineering systems EOM for forced vibrating systems L0 k, L0 x(t) 1 d 2x F(t)=F0 sin t n2 dt 2 m 2 km , K 1 k x(t) L0 k, L0 1 d 2x m n2 dt 2 Base Excitation n y(t)=Y0 sint L0 x(t) k, L0 k , m n External forcing 2 dx x KF0 sin t n dt 2 dx 2 dy x Ky n dt n dt k , m , K 1 2 km y(t)=Y0sint m m0 Y0 2 2 dx K d2y Rotor Excitation 2 dt 2 dt x 2 dt 2 K 2 sin t n n n n 1 d 2x n k M 2 kM m K 0 M m m0 M Steady-state and Transient solution to EOM Equation 1 d 2x n2 Full Solution dt 2 2 dx x C KF0 sin(t ) n dt Initial Conditions x x0 dx v0 dt x(t ) C xh (t ) x p (t ) Steady state part (particular integral) x p (t ) X 0 sin t X0 KF0 1/2 2 2 2 2 1 / n 2 / n tan 1 Transient part (complementary integral) Overdamped 1 Critically Damped 1 Underdamped 1 2 / n 1 2 / n2 h h v h (n d ) x0h v (n d ) x0 xh (t ) C exp(nt ) 0 exp(d t ) 0 exp(d t ) 2d 2d xh (t ) C x0h v0h n x0h t exp(nt ) v h n x0h xh (t ) C exp(nt ) x0h cos d t 0 sin d t d d n 1 2 x0h x0 C x p (0) x0 C X 0 sin v0h v0 dx p dt t 0 v0 X 0 cos t 0 Canonical externally forced system (steady state solution) Steady state solution to 1 d 2x n2 dt 2 M 2 dx x C KF0 sin(t ) n dt 1 1/2 2 2 2 2 1 / n 2 / n 2 / T x p (t ) X 0 sin t k c 1 n K m k 2 km X 0 KF0 M ( / n , ) tan 1 2 / n 1 2 / n2 s=L0+xF (t ) F0 sin t k, L0 m c Canonical base excited system (steady state solution) 1 d 2x Steady state solution to n n2 dt 2 k c K 1 m 2 km X 0 KY0 M ( , n , ) M 2 dx 2 dy xCKy n dt n dt x p (t ) X 0 sin t 1 2 / n 2 1 2 / n2 2 1/2 1/2 2 2 / n 1 (1 4 2 ) 2 / n2 = 0.01 0 0 Magnification X /KY tan 2 3 / n3 1 10 = 0.05 1 = 0.1 = 0.2 = 0.4 10 0 = 0.6 10 = 1.0 -1 0 0.5 1 1.5 2 Frequency Ratio /n 2.5 3 Canonical rotor excited system (steady state solution) c 2 dx K d2y xC n2 dt 2 n dt n2 dt 2 1 d 2x Steady state solution to n m0 k c K m m0 m m0 2 k (m m0 ) X 0 KY0 M ( , n , ) M x p (t ) X 0 sin t 2 / n2 1/2 2 2 2 2 1 / n 2 / n tan 1 2 / n 1 2 / n2 y Y0 sin t